**Quantitative Feedback Theory and Its Application in UAV's Flight Control**

Xiaojun Xing and Dongli Yuan *Northwestern Polytechnical University, Xi'an, China* 

## **1. Introduction**

34 Will-be-set-by-IN-TECH

36 Automatic Flight Control Systems – Latest Developments

Kuipers, J. (1999). *Quaternions and Rotation Sequences*, Princeton University Press, Princeton. NIMA (2000). *Department of Defense World Geodetic System 1984*, National Imagery and

Pio, R.L. (1966). Euler Angle Transformations. *IEEE Transactions on Automatic Control*, AC-11,

Ristic, B.; Arulampalam, S. & Gordon, N. (2004). *Beyond the Kalman Filter: Particle Filters for*

Rogers, R. (2003). *Applied Mathematics in Integrated Navigation Systems, Second Edition*,

Schwarz, K.P. & Wei, M. (1990). Efficient Numerical Formulas for the Computation of Normal

Titterton, D. & Weston, J. (2004). *Strapdown Inertial Navigation Technology, 2nd Edition*, The

Mapping Agency, NIMA Stock Number DMATR83502WGS84.

American Institute of Aeronautics and Astronautics, Reston.

Torge, W. (2001). *Geodesy, 3rd Edition*, Walter de Gruyter, Berlin and New York.

Gravity in a Cartesian Frame. *Manuscripta Geodetica*, 4, 15, 228-234.

4, 1966.

Papoulis, A. *Signal Analysis*, McGraw-Hill, New York.

*Tracking Applications*, Artech House, Boston.

Institution of Electrical Engineers, Herts.

Quantitative feedback theory (hereafter referred as QFT), developed by Isaac Horowitz (Horowitz, 1963; Horowitz and Sidi, 1972), is a frequency domain technique utilizing the Nichols chart in order to achieve a desired robust design over a specified region of plant uncertainty. Desired time-domain responses are transformed into frequency domain tolerances, which lead to bounds (or constraints) on the loop transmission function. The design process is highly transparent, allowing a designer to see what trade-offs are necessary to achieve a desired performance level.

QFT is also a unified theory that emphasizes the use of feedback for achieving the desired system performance tolerances despite plant uncertainty and plant disturbances. QFT quantitatively formulates these two factors in the form of (a) the set { } *R R T* of acceptable command or tracking input-output relationships and the set { } *D D T* of acceptable disturbance input-output relationships, and (b) a set { } *P* of possible plants which include the uncertainties. The objective is to guarantee that the control ratio *T YR <sup>R</sup>* / is a member of *R* and *T YD <sup>D</sup>* / is a member of *<sup>D</sup>* , for all plants *P* which are contained in . QFT has been developed for control systems which are both linear and nonlinear, timeinvariant and time-varying, continuous and sampled-data, uncertain multiple-input singleoutput (MISO) and multiple-input multiple-output (MIMO) plants, and for both output and internal variable feedback.

The QFT synthesis technique for highly uncertain linear time-invariant MIMO plants has the following features:


This design technique is applicable to the following problem classes:


Quantitative Feedback Theory and Its Application in UAV's Flight Control 39

Assume that the control system has negligible sensor noise and sufficient control effort authority, then for a stable LTI minimum-phase plant, a LTI compensator may be designed

Representing the characteristics of the plant and the desired system performance

Representing the nonlinear plant characteristics by a set of LTI transfer functions that

Representing the system performance specifications (see Fig.1) by LTI transfer functions

 Reducing the effect of parameter uncertainty by shaping the open-loop frequency responses so that the Bode plots of the *J* closed-loop systems fall between the boundaries *BU* and *BL* , while simultaneously satisfying all performance specifications. Obtaining the stability, tracking, disturbance, and cross-coupling (for MIMO systems) boundaries on the Nichols chart in order to satisfy the performance specifications.

Consider the control system of Fig.2, where *G s*( ) is a compensator, *F s*( ) is a prefilter, and

is the nonlinear plant with structured parametric uncertainty. To carry out a QFT design:

The nonlinear plant is described by a set of *J* minimum-phase LTI plants, i.e.,

plotted on the Nichols chart. A contour is drawn through the data points that described the boundary of the region that contains all *J* points. This contour is referred to as a template. It represents the region of structured plant parametric uncertainty on the

bandwidth (BW) of concern. Six data points (log magnitude and phase angle) for each

 The system performance specifications are represented by LTI transfer functions, and their corresponding Bode plots are shown in Fig. 3 by the upper and lower bounds *BU*

*<sup>i</sup>* are obtained, as shown in Fig. 4a, for a certain example to plot the templates,

*Ps t J <sup>t</sup>* which define the structured plant parameter uncertainty.

the Bode plots of the LTI plants as shown in Fig. 3 which is for a certain plant. *J* data points (log magnitude and phase angle), for each value of frequency,

The magnitude variation due to the plant parameter uncertainty, ( ) *P i*

Nichols chart and are obtained for specified values of frequency,

*<sup>i</sup>* , as shown in Fig. 4b.

, is depicted by

 *<sup>i</sup>* , are

*<sup>i</sup>* , within the

 *j*

> 

to achieve the desired control system performance specifications.

Using these representations to design a compensator (controller).

that form the upper *BU* and lower *BL* boundaries for the design.

cover the range of structured parametric uncertainty.

**2.3 Implementation of QFT design objective** 

specifications in the frequency domain.

The QFT design objective is achieved by:

**2.4 QFT basics** 

value of

and *BL* , respectively.

Fig. 2. Compensated nonlinear system

for each value of

{ ( )}( 1,2, , )


Problem classes 3 and 4 are converted into equivalent sets of MISO systems to which the QFT design technique is applied. The objective is to solve the MISO problems, i.e., to find compensation functions which guarantee that the performance tolerances for each MISO problem are satisfied for all *P* in.

This chapter is essentially divided into two parts. The first part, consisting of Sections 2 through 4, presents the fundamentals of the QFT robust control system design technique for the tracking and regulator control problems. The second part consists of Seciton 5 which focuses on the application of QFT techinique to the flight control design for a certain Unmaned Aerial Vehicle (UAV). This is accomplished by decomposing the UAV's MIMO plant to 2 MISO plants whose controllers are both synthisized using QFT techique for MISO systems. And the effectiveness of both controllers is verified according the digital simulation results. Besides, Sections 6 through 8 are about summary of whole chapter, references and symbols used in the chapter.

## **2. Overview of QFT**

## **2.1 Design objective of QFT**

Objective of QFT is to design and implement robust control for a system with structured parametric uncertainty that satisfies the desired performance specifications.

## **2.2 Performance specifications for control system**

In many control systems the output *y*( )*t* must lie between specified upper and lower bounds, ( )*<sup>U</sup> y t* and ( )*<sup>L</sup> y t* , respectively, as shown in Fig.1a. The conventional time-domain figures of merit, based upon a step input signal *r t*( ) are shown in Fig.1a. They are: *MP* , peak overshoot; *<sup>r</sup> t* , rise time; *<sup>p</sup> t* , peak time; and *<sup>s</sup> t* , settling time. Corresponding system performance specifications in the frequency domain are, *BU* and *BL* , the upper and lower bounds respectively, peak overshoot *Lm Mm* , and the frequency bandwidth *<sup>h</sup>* which are shown in Fig.1b.

(a) time domain response specifications (b) frequency domain response specifications Fig. 1. Desired system performance specifications

Assume that the control system has negligible sensor noise and sufficient control effort authority, then for a stable LTI minimum-phase plant, a LTI compensator may be designed to achieve the desired control system performance specifications.

## **2.3 Implementation of QFT design objective**

The QFT design objective is achieved by:


## **2.4 QFT basics**

38 Automatic Flight Control Systems – Latest Developments

Problem classes 3 and 4 are converted into equivalent sets of MISO systems to which the QFT design technique is applied. The objective is to solve the MISO problems, i.e., to find compensation functions which guarantee that the performance tolerances for each MISO

This chapter is essentially divided into two parts. The first part, consisting of Sections 2 through 4, presents the fundamentals of the QFT robust control system design technique for the tracking and regulator control problems. The second part consists of Seciton 5 which focuses on the application of QFT techinique to the flight control design for a certain Unmaned Aerial Vehicle (UAV). This is accomplished by decomposing the UAV's MIMO plant to 2 MISO plants whose controllers are both synthisized using QFT techique for MISO systems. And the effectiveness of both controllers is verified according the digital simulation results. Besides, Sections 6 through 8 are about summary of whole chapter, references and

Objective of QFT is to design and implement robust control for a system with structured

In many control systems the output *y*( )*t* must lie between specified upper and lower bounds, ( )*<sup>U</sup> y t* and ( )*<sup>L</sup> y t* , respectively, as shown in Fig.1a. The conventional time-domain figures of merit, based upon a step input signal *r t*( ) are shown in Fig.1a. They are: *MP* , peak overshoot; *<sup>r</sup> t* , rise time; *<sup>p</sup> t* , peak time; and *<sup>s</sup> t* , settling time. Corresponding system performance specifications in the frequency domain are, *BU* and *BL* , the upper and lower

*<sup>h</sup>* which are

parametric uncertainty that satisfies the desired performance specifications.

bounds respectively, peak overshoot *Lm Mm* , and the frequency bandwidth

(a) time domain response specifications (b) frequency domain response specifications

**2.2 Performance specifications for control system** 

Fig. 1. Desired system performance specifications

6. Sampled-data systems as well as continuous systems for all of the preceding.

.

5. Distributed systems.

problem are satisfied for all *P* in

symbols used in the chapter.

**2.1 Design objective of QFT** 

**2. Overview of QFT** 

shown in Fig.1b.

Consider the control system of Fig.2, where *G s*( ) is a compensator, *F s*( ) is a prefilter, and is the nonlinear plant with structured parametric uncertainty. To carry out a QFT design:


Fig. 2. Compensated nonlinear system

Quantitative Feedback Theory and Its Application in UAV's Flight Control 41

Therefore, the QFT robust design technique assures that the desired performance specifications are satisfied over the prescribed region of structured plant parametric

> ( ) ( )( ) *a*

where *K Ka* and *i J* 1,2,..., . The log magnitude changes in a prescribed range due to the

*ss a ss a*

(1)

() () () *L s GsP s t t* (2)

*K K P s*

Consider a certain position control system whose plant transfer function is given by

*t*

plant parameter uncertainty. The loop transmission *L s*( ) is defined as

The control ratio *TL* of the unity-feedback system of Fig. 2 is

Fig. 5. Closed-loop responses: LTI plants with G(s)

Fig. 6. Closed-loop responses: LTI plants with G(s) and F(s)

uncertainty.

**3.1 Open-loop plant** 

**3. Insight to the QFT technique** 

**3.2 Closed-loop formulation** 

Fig. 3. LTI plants

Fig. 4. (a) Bode plots of 6 LTI plants; (b) template construction for =3 rad/sec; (c) construction of the Nichols chart plant templates

## **2.5 QFT design**

The tracking design objective is to

	- results in satisfying the desired performance specifications of Fig. 1
	- results in the closed-loop frequency responses *TLi* shown in Fig. 5
	- results in the ( ) *L i j* of Fig. 5 of the compensated system, being equal to or smaller than ( ) *P i j* of Fig. 3 for the uncompensated system and that it is equal or less than ( ) *R i j* , for each value of *<sup>i</sup>* of interest; that is: () () () *Li Ri Pi jjj*

Fig. 5. Closed-loop responses: LTI plants with G(s)

Fig. 6. Closed-loop responses: LTI plants with G(s) and F(s)

Therefore, the QFT robust design technique assures that the desired performance specifications are satisfied over the prescribed region of structured plant parametric uncertainty.

## **3. Insight to the QFT technique**

#### **3.1 Open-loop plant**

40 Automatic Flight Control Systems – Latest Developments

(a) (b) (c)

 results in satisfying the desired performance specifications of Fig. 1 results in the closed-loop frequency responses *TLi* shown in Fig. 5

b. Synthesize a prefilter *F s*( ) of Fig. 2 that results in shifting and reshaping the *TLi* responses in order that they lie within the *BU* and *BL* boundaries in Fig. 5 as shown in

of Fig. 5 of the compensated system, being equal to or smaller

*<sup>i</sup>* of interest; that is: () () () *Li Ri Pi* 

 

*jjj*

 

of Fig. 3 for the uncompensated system and that it is equal or less

=3 rad/sec;

Fig. 4. (a) Bode plots of 6 LTI plants; (b) template construction for

(c) construction of the Nichols chart plant templates

a. Synthesize a compensator *G s*( ) of Fig. 2 that

 *j*

, for each value of

Fig. 3. LTI plants

**2.5 QFT design** 

Fig. 6.

The tracking design objective is to

results in the ( ) *L i*

than ( ) *P i j*

than ( ) *R i j*

Consider a certain position control system whose plant transfer function is given by

$$P\_s(\mathbf{s}) = \frac{K\_s}{\mathbf{s}(\mathbf{s}+\mathbf{a})} = \frac{K'}{\mathbf{s}(\mathbf{s}+\mathbf{a})} \tag{1}$$

where *K Ka* and *i J* 1,2,..., . The log magnitude changes in a prescribed range due to the plant parameter uncertainty. The loop transmission *L s*( ) is defined as

$$L\_r(\mathbf{s}) = G(\mathbf{s})P\_r(\mathbf{s})\tag{2}$$

#### **3.2 Closed-loop formulation**

The control ratio *TL* of the unity-feedback system of Fig. 2 is

Quantitative Feedback Theory and Its Application in UAV's Flight Control 43

of transfer functions which describe the region of plant parameter uncertainty, *G* is the cascade compensator, and *F* is an input prefilter transfer function. The output *y t*( ) is required to track the command input *r t*( ) and to reject the external disturbances 1 *d t*( ) and <sup>2</sup> *d t*( ) . The compensator *G* in Fig. 7 is to be designed so that the variation of *y t*( ) to the uncertainty in the plant *P* is within allowable tolerances and the effects of the disturbances <sup>1</sup> *d t*( ) and 2 *d t*( ) on *y*( )*t* are acceptably small. Also, the prefilter properties of *F s*( ) must be designed to the desired tracking by the output *y*( )*t* of the input *r t*( ) . Since the control system in Fig. 7 has two measurable quantities, *r t*( ) and *y*( )*t* , it is referred to as a two degree-of-freedom (DOF) feedback structure. If the two disturbance inputs are measurable, then it represents a four DOF structure. The actual design is closely related to the extent of the uncertainty and to the narrowness of the performance tolerances. The uncertainty of the

> ( ) ( ) *<sup>K</sup> P s*

where the value of *K* is in the range [1, 10] and *a* is in the range [-2, 2]. The design objective is to guarantee that ( ) ( )/ ( ) *T s Ys Rs <sup>R</sup>* and ( ) ( )/ ( ) *T s Ys Ds <sup>D</sup>* are members of the sets of acceptable *R* and *D* for changes of *K* and *a* . In a feedback control system, the principal challenge in the control system design is to relate the system performance specifications to the requirements on the loop transmission function *Ls GsPs* () () () in order to achieve the desired benefits of feedback, i.e., the desired reduction in sensitivity to plant uncertainty and desired disturbance attenuation. The advantage of the frequency domain is that *Ls GsPs* () () () is simply the multiplication of complex numbers. In the frequency domain it

*P where t J <sup>t</sup>* (5)

*i* separately, and thus, at each

*ss a* (6)

*<sup>i</sup>* , the optimal

represents the set

**4.2 The QFT method (single-loop MISO system)** 

plant transfer function is denoted by the set

Given that the plant transfer function is

and is illustrated as follows.

is possible to evaluate *L j* ( )

Fig. 7. A feedback structure

**4.3 QFT design procedure** 

bounds on *L j* ( )

 { } 1,2,..., 

design procedure to accomplish this objective is as follows:

can be determined.

at every

The objective is to design the prefilter *F s*( ) and the compensator *G s*( ) of Fig.7 so that the specified robust design is achieved for the given region of plant parameter uncertainty. The

Basic structure of a feedback control system is given in Fig.7 , in which

$$T\_{L\_t} = \frac{Y}{R\_t} = \frac{L\_t}{1 + L\_t} \tag{3}$$

The overall system control ratio *TR*

is given by:

$$T\_{\mathbb{X}\_{\mathbb{f}}}(\mathbf{s}) = \frac{F(\mathbf{s})L\_{\text{\tiny s}}(\mathbf{s})}{1 + L\_{\text{\tiny s}}(\mathbf{s})} \tag{4}$$

## **3.3 Results of applying the QFT design technique**

The proper application of the robust QFT design technique requires the utilization of the prescribed performance specifications from the onset of the design process, and the selection of a nominal plant *Po* from the *J* LTI plants. Once the proper loop shaping of () () () *L s GsP s o o* is accomplished, a synthesized *G s*( ) is achieved that satisfies the desired performance specifications. The last step of this design process is the synthesis of the prefilter that ensures that the Bode plots of *TRi* all lie between the upper and lower bounds *BU* and *BL* .

## **3.4 Benefits of QFT**

The benefits of the QFT technique may be summarized as follows:


## **4. QFT design for the MISO analog control system**

## **4.1 Introduction**

The MIMO synthesis problem is converted into a number of single-loop feedback problems in which parameter uncertainty, cross-coupling effects, and system performance tolerances are derived from the original MIMO problem. The solutions to these single-loop problems represent a solution to the MIMO plant. It is not necessary to consider the complete system characteristic equation. The design is tuned to the extent of the uncertainty and the performance tolerances.

Here, we will present an in-depth understanding and appreciation of the power of the QFT technique through apply QFT to a robust single-loop MISO system, which has two inputs, a tracking and an external disturbance input, respectively, and a single output control system.

## **4.2 The QFT method (single-loop MISO system)**

Basic structure of a feedback control system is given in Fig.7 , in which represents the set of transfer functions which describe the region of plant parameter uncertainty, *G* is the cascade compensator, and *F* is an input prefilter transfer function. The output *y t*( ) is required to track the command input *r t*( ) and to reject the external disturbances 1 *d t*( ) and <sup>2</sup> *d t*( ) . The compensator *G* in Fig. 7 is to be designed so that the variation of *y t*( ) to the uncertainty in the plant *P* is within allowable tolerances and the effects of the disturbances <sup>1</sup> *d t*( ) and 2 *d t*( ) on *y*( )*t* are acceptably small. Also, the prefilter properties of *F s*( ) must be designed to the desired tracking by the output *y*( )*t* of the input *r t*( ) . Since the control system in Fig. 7 has two measurable quantities, *r t*( ) and *y*( )*t* , it is referred to as a two degree-of-freedom (DOF) feedback structure. If the two disturbance inputs are measurable, then it represents a four DOF structure. The actual design is closely related to the extent of the uncertainty and to the narrowness of the performance tolerances. The uncertainty of the plant transfer function is denoted by the set

$$\mathfrak{sp} = \{P\_i\} \quad \text{where } t = 1, 2, \dots, I \tag{5}$$

and is illustrated as follows.

42 Automatic Flight Control Systems – Latest Developments

*Y L <sup>T</sup>*

*Lt*

*R*

*t*

The overall system control ratio *TR*

**3.3 Results of applying the QFT design technique** 

The benefits of the QFT technique may be summarized as follows:

There can be one robust design for the full, operating envelope.

Design limitations are apparent up front and during the design process.

The structure of the compensator (controller) is determined up front.

There is less development time for a full envelope design.

**4. QFT design for the MISO analog control system** 

is given by:

*BU* and *BL* .

**3.4 Benefits of QFT** 

**4.1 Introduction** 

performance tolerances.

control system.

the QFT CAD package.

1 *t*

*R L* (3)

(4)

*L t*

() () ( ) 1 () *t*

The proper application of the robust QFT design technique requires the utilization of the prescribed performance specifications from the onset of the design process, and the selection of a nominal plant *Po* from the *J* LTI plants. Once the proper loop shaping of () () () *L s GsP s o o* is accomplished, a synthesized *G s*( ) is achieved that satisfies the desired performance specifications. The last step of this design process is the synthesis of the prefilter that ensures that the Bode plots of *TRi* all lie between the upper and lower bounds

It results in a robust design which is insensitive to structured plant parameter variation.

 The achievable performance specifications can be determined in the early design stage. If necessary, one can redesign for changes in the specifications quickly with the aid of

The MIMO synthesis problem is converted into a number of single-loop feedback problems in which parameter uncertainty, cross-coupling effects, and system performance tolerances are derived from the original MIMO problem. The solutions to these single-loop problems represent a solution to the MIMO plant. It is not necessary to consider the complete system characteristic equation. The design is tuned to the extent of the uncertainty and the

Here, we will present an in-depth understanding and appreciation of the power of the QFT technique through apply QFT to a robust single-loop MISO system, which has two inputs, a tracking and an external disturbance input, respectively, and a single output

*FsL s T s L s*

*t*

Given that the plant transfer function is

$$P(\mathbf{s}) = \frac{\mathbf{K}}{\mathbf{s}(\mathbf{s} + \mathbf{a})} \tag{6}$$

where the value of *K* is in the range [1, 10] and *a* is in the range [-2, 2]. The design objective is to guarantee that ( ) ( )/ ( ) *T s Ys Rs <sup>R</sup>* and ( ) ( )/ ( ) *T s Ys Ds <sup>D</sup>* are members of the sets of acceptable *R* and *D* for changes of *K* and *a* . In a feedback control system, the principal challenge in the control system design is to relate the system performance specifications to the requirements on the loop transmission function *Ls GsPs* () () () in order to achieve the desired benefits of feedback, i.e., the desired reduction in sensitivity to plant uncertainty and desired disturbance attenuation. The advantage of the frequency domain is that *Ls GsPs* () () () is simply the multiplication of complex numbers. In the frequency domain it is possible to evaluate *L j* ( ) at every *i* separately, and thus, at each *<sup>i</sup>* , the optimal bounds on *L j* ( ) can be determined.

Fig. 7. A feedback structure

#### **4.3 QFT design procedure**

The objective is to design the prefilter *F s*( ) and the compensator *G s*( ) of Fig.7 so that the specified robust design is achieved for the given region of plant parameter uncertainty. The design procedure to accomplish this objective is as follows:

Quantitative Feedback Theory and Its Application in UAV's Flight Control 45

It is desirable to synthesize the control ratios corresponding to the upper and lower bounds

increases as

synthesizing the loop transmission () () () *L s GsP s o o* as discussed in Sec. 4.13 of this chapter.

. This characteristic of ( ) *R i*

*<sup>i</sup>* increases.

 *j*

> *j*

*<sup>i</sup>* increases above the 0-dB crossing

simplifies the process of

(see Sec. 4.9)

(10)

ensures that the

 *j*

 *j*

Fig. 8. System time domain tracking performance specifications

(a) Ideal simple second-order models (b) The augmented models

*R*

*p p* and *t T ss n D* 4/ 4/

*<sup>U</sup> T s*

*R*

*U*

*T s*

An approach to the modeling process is to start with a simple second-order model of the

( ) <sup>2</sup> ( )( ) *n n*

( ) ( ) ( ) () 1 ()

*Y s G s*

*eq*

*eq*

*s s spsp*

*n n*

 

2 2

 

( ) *RU T s* of Eq. (10) can be represented by an equivalent unity-feedback system so that

2 2

1 2

(the desired settling time). The control ratio

*Rs G s* (11)

 

*cf* (see Fig. 9b) of *RU T* . This characteristic of ( ) *R i*

To synthesize ( ) *L s <sup>o</sup>* , it is necessary to determine the tracking bounds ( ) *B j R i*

decrease in magnitude as

*RU T* and *RL T* , respectively, so that ( ) *R i*

which are obtained based upon ( ) *R i*

frequency

tracking bounds ( ) *B j R i*

Fig. 9. Bode plots of *TR*

*n* 1 2 

where 2

where

desired control ratio *RU T* having the form


The following sections will illustrate the design procedure step by step.

#### **4.4 Minimum-phase system performance specifications**

In order to apply the QFT technique, it is necessary to synthesize the desired model control ratio based upon the system's desired performance specifications in the time domain. For the minimum-phase LTI MISO system of Fig. 7, the control ratios for tracking and for disturbance rejection are, respectively,

$$T\_{\pi}(\mathbf{s}) = \frac{F(\mathbf{s})G(\mathbf{s})P(\mathbf{s})}{1 + G(\mathbf{s})P(\mathbf{s})} = \frac{F(\mathbf{s})L(\mathbf{s})}{1 + L(\mathbf{s})} = F(\mathbf{s})T(\mathbf{s}) \quad \text{with} \quad d\_{\mathbf{i}}(\mathbf{t}) = d\_{\mathbf{i}}(\mathbf{t}) = \mathbf{0} \tag{7}$$

$$T\_{D1} = \frac{P(\text{s})}{1 + G(\text{s})P(\text{s})} = \frac{P}{1 + L} \quad \text{with} \quad r(t) = d\_z(t) = 0 \tag{8}$$

$$T\_{D2} = \frac{1}{1 + G(s)P(s)} = \frac{1}{1 + L} \quad \text{with} \quad r(t) = d\_i(t) = 0 \tag{9}$$

#### **4.4.1 Tracking models**

The QFT technique requires that the desired tracking control ratios be modeled in the frequency domain to satisfy the required gain *Km* and the desired time domain performance specifications for a step input. Thus, the system's tracking performance specifications for a simple second-order system are based upon satisfying some or all of the step forcing function figures of merit (FOM) for under-damped ( , , , , ) *Mppsr m tttK* and over-damped (,, ) *sr m ttK* responses, respectively. These are graphically depicted in Fig. 8. The time responses ( )*<sup>U</sup> y t* and ( )*<sup>L</sup> y t* in this figure represent the upper and lower bounds, respectively, of the tracking performance specifications; that is, an acceptable response *y t*( ) must lie between these bounds. The Bode plots of the upper bound *BU* and lower bound *BL* for ( ) *Lm T j <sup>R</sup>* vs. are shown in Fig. 9.

It is desirable to synthesize the control ratios corresponding to the upper and lower bounds *RU T* and *RL T* , respectively, so that ( ) *R i j* increases as *<sup>i</sup>* increases above the 0-dB crossing frequency *cf* (see Fig. 9b) of *RU T* . This characteristic of ( ) *R i j* simplifies the process of synthesizing the loop transmission () () () *L s GsP s o o* as discussed in Sec. 4.13 of this chapter. To synthesize ( ) *L s <sup>o</sup>* , it is necessary to determine the tracking bounds ( ) *B j R i* (see Sec. 4.9) which are obtained based upon ( ) *R i j* . This characteristic of ( ) *R i j* ensures that the tracking bounds ( ) *B j R i* decrease in magnitude as *<sup>i</sup>* increases.

Fig. 8. System time domain tracking performance specifications

(a) Ideal simple second-order models (b) The augmented models

Fig. 9. Bode plots of *TR*

An approach to the modeling process is to start with a simple second-order model of the desired control ratio *RU T* having the form

$$T\_{\text{eq}I} \text{ (s)} = \frac{\text{o}\_{\text{\tiny u}}^{2}}{\text{s}^{2} + 2\text{\tiny}\varphi\text{o}\_{\text{\tiny u}}\text{s} + \text{o}\_{\text{\tiny u}}^{2}} = \frac{\text{o}\_{\text{\tiny u}}^{2}}{\text{(s-p\_{1})(s-p\_{2})}} \tag{10}$$

where 2 *n* 1 2 *p p* and *t T ss n D* 4/ 4/ (the desired settling time). The control ratio ( ) *RU T s* of Eq. (10) can be represented by an equivalent unity-feedback system so that

$$T\_{\%I} \text{(s)} = \frac{Y \text{(s)}}{R \text{(s)}} = \frac{G\_{eq} \text{(s)}}{1 + G\_{eq} \text{(s)}} \tag{11}$$

where

44 Automatic Flight Control Systems – Latest Developments

**Step 3.** Specify the *J* LTI plant models that define the boundary of the region of plant

**Step 4.** Obtain plant templates at specified frequencies that pictorially describe the region

**Step 7-9.** Determine the disturbance, tracking, and optimal bounds on the Nichols chart. **Step 10.** Synthesize the nominal loop transmission function () () () *L s GsP s o o* that satisfies all

**Step 12.** Simulate the system in order to obtain the time response data for each of the *J*

In order to apply the QFT technique, it is necessary to synthesize the desired model control ratio based upon the system's desired performance specifications in the time domain. For the minimum-phase LTI MISO system of Fig. 7, the control ratios for tracking and for

> () () () ()() ( ) ( ) ( ) with ( ) ( ) 0 1 () () 1 () *<sup>R</sup> FsGsPs FsLs T s FsTs d t d t GsPs Ls*

> > 1 2 ( ) with ( ) ( ) 0 1 () () 1 *<sup>D</sup> Ps P <sup>T</sup> rt d t*

2 1 1 1 with ( ) ( ) 0 1 () () 1

The QFT technique requires that the desired tracking control ratios be modeled in the frequency domain to satisfy the required gain *Km* and the desired time domain performance specifications for a step input. Thus, the system's tracking performance specifications for a simple second-order system are based upon satisfying some or all of the step forcing function figures of merit (FOM) for under-damped ( , , , , ) *Mppsr m tttK* and over-damped (,, ) *sr m ttK* responses, respectively. These are graphically depicted in Fig. 8. The time responses ( )*<sup>U</sup> y t* and ( )*<sup>L</sup> y t* in this figure represent the upper and lower bounds, respectively, of the tracking performance specifications; that is, an acceptable response *y t*( ) must lie between these bounds. The Bode plots of the upper bound *BU* and lower bound *BL* for

*T r <sup>D</sup> t d t*

1 2

*GsPs L* (8)

*GsPs L* (9)

(7)

of plant parameter uncertainty on the Nichols chart.

**Step 11.** Based upon Steps 1 through 10, synthesize the prefilter *F s*( ) .

The following sections will illustrate the design procedure step by step.

**Step 6.** Determine the stability contour (*U* -contour) on the Nichols chart.

**Step 5.** Select the nominal plant transfer function ( ) *P s <sup>o</sup>* .

the bounds and the stability contour.

**4.4 Minimum-phase system performance specifications** 

are shown in Fig. 9.

**Step 1.** Synthesize the desired tracking model. **Step 2.** Synthesize the desired disturbance model.

parameter uncertainty.

disturbance rejection are, respectively,

plants.

**4.4.1 Tracking models** 

( ) *Lm T j <sup>R</sup>* vs. 

Quantitative Feedback Theory and Its Application in UAV's Flight Control 47

In order to minimize the iteration process in achieving acceptable models for ( ) *RU T s* and

process: (a) first synthesize the second-order model of Eq. (15) containing the zero at

necessary, one or more of these poles are moved right and/or left until the desired specifications are satisfied. As illustrated by the slopes of the straight-line Bode plots in Fig. 9b, selecting the value of all three poles in the range specified above insures an

*RL <sup>T</sup>* may be such that a pair of complex poles and a real pole need to be chosen for the model response. For this situation, the real pole must be more dominant than the complex poles, (d) depending on the performance specifications, ( ) *RU T s* may require two real poles and a zero "close" to the origin, i.e., select 1 *z* very much less than 1 *p* and 2 *p* in order to

For the case where *y*( )*t* , corresponding to *RU T* , is to have an allowable "large" overshoot followed by a small tolerable undershoot, a dominant complex pole pair is not suitable for *RU T* . An acceptable overshoot with no undershoot for *RU T* can be achieved by *RU T* having two real dominant poles 1 2 *p p* , a dominant real zero ( 1 1 *z p* ) "close"' to <sup>1</sup> *p* , and a far off pole 3 2 *p p* . The closeness of the zero dictates the value of *MP* . Thus, a designer selects a

The simplest disturbance control ratio model specification is ( ) ( / ( )) *TD P j*

constant, [the desired maximum magnitude of the output based upon a unit-step disturbance input]; i.e., for <sup>1</sup> *d t*( ) : ( )*p p y t a* , and for: 2 () () *<sup>p</sup> d t y t a* for *<sup>x</sup> t t* . Thus, the

pole-zero combination to yield the form of the desired time-domain response.

frequency domain disturbance specification is ( ) *<sup>D</sup> <sup>p</sup> Lm T j*

Fig. 10. Bode plots of disturbance models for ( ) *TD j*

that meets the desired FOM; and (b) then, as a first trial, select all three real

 *a aa* 

. Other possibilities are as follows: (c) the specified values of *<sup>p</sup> t* and *<sup>s</sup> t* for

(see Fig. 9b) must be larger than the actual variation in the plant, *<sup>P</sup>*

, the following procedure may expedite the design

 

 *Y j D j a* , a

*Lm a* over the desired specified

. For succeeding trials, if

.

 *j*

poles of Eq. (16) to have the value of 3 3 *n D* <sup>2</sup> <sup>1</sup>

( ) *RL T s* which have an increasing ( ) *<sup>R</sup>*

effectively have an under-damped response.

**4.4.2 Disturbance rejection models** 

<sup>1</sup> *<sup>n</sup> z a* 

increasing *<sup>R</sup>*

At high frequencies *hf*

BW (see Fig. 10).

$$G\_{\circ \eta}(\mathbf{s}) = \frac{\alpha\_{\mathbf{s}}^{\natural}}{s(s + 2\xi \alpha\_{\mathbf{s}})} \tag{12}$$

The gain constant of this equivalent Type1 transfer function ( ) *G s eq* is 1 <sup>0</sup> lim[ ( )] *eq <sup>s</sup> K sG s* / 2 *<sup>n</sup>* .

The simplest over-damped model for ( ) *RL T s* is of the form

$$T\_{\aleph\_L}(\mathbf{s}) = \frac{Y(\mathbf{s})}{R(\mathbf{s})} = \frac{K}{(\mathbf{s} - \sigma\_1)(\mathbf{s} - \sigma\_2)} = \frac{G\_{eq}(\mathbf{s})}{1 + G\_{eq}(\mathbf{s})} \tag{13}$$

where

$$G\_{\circ\_{\mathfrak{q}}}(\mathbf{s}) = \frac{\sigma\_{\mathfrak{i}}\sigma\_{\mathfrak{i}}}{\mathbf{s}\left[\mathbf{s} - \left(\sigma\_{\mathfrak{i}} + \sigma\_{\mathfrak{i}}\right)\right]}$$

and *K*1 12 1 2 /( ) . Selection of the parameters <sup>1</sup> and <sup>2</sup> is used to meet the specifications for *<sup>s</sup> t* and *K*<sup>1</sup> .

Once the ideal models ( ) *RU <sup>T</sup> <sup>j</sup>* and ( ) *RL <sup>T</sup> <sup>j</sup>* are determined, the time and frequency response plots of Figs. 8 and 9a, respectively, can then be drawn. The high-frequency range in Fig. 9a is defined as *<sup>b</sup>* , where *<sup>b</sup>* is the model BW frequency of *BU* . In order to achieve the desired characteristic of an increasing magnitude of *<sup>R</sup>* of *BU* for*i c <sup>f</sup>* , an increasing spread between *BU* and *BL* is required in the high-frequency range (see Fig. 9b), that is,

$$
\delta\_{\circ\_{\circ}} = B\_{\cup} - B\_{\cup} \tag{14}
$$

must increase with increasing frequency. This desired increase in *<sup>R</sup>* is achieved by changing *BU* and *BL* by augmenting *RU T* with a zero [see Eq. (15)] as close to the origin as possible without significantly affecting the time response. This additional zero raises the curve *BU* for the frequency range above*cf* . The spread can be further increased by augmenting *RL <sup>T</sup>* with a negative real pole [see Eq. (16)] which is as close to the origin as possible but far enough away not to significantly affect the time response. Note that the straight-line Bode plot is shown only for *RL <sup>T</sup>* . This additional pole lowers *BL* for this frequency range.

$$T\_{\%I} \left( \text{s} \right) = \frac{\left( \text{o} \right)\_{\text{n}}^{2} / \text{a} \left( \text{s} + \text{a} \right)}{\text{s}^{2} + 2 \text{\text{\textdegree}} \text{o} \text{o} \text{\textdegree s} + \text{o} \text{o}^{2}} = \frac{\left( \text{o} \right)\_{\text{n}}^{2} / \text{a} \left( \text{s} - \text{z}\_{1} \right)}{\left( \text{s} - \text{\textdegree c}\_{1} \right) \left( \text{s} - \text{\textdegree c}\_{2} \right)} \tag{15}$$

$$T\_{\mathbb{K}\_L}(\mathbf{s}) = \frac{\mathbf{K}}{(\mathbf{s} + a\_1)(\mathbf{s} + a\_2)(\mathbf{s} + a\_3)} = \frac{\mathbf{K}}{(\mathbf{s} - \sigma\_1)(\mathbf{s} - \sigma\_2)(\mathbf{s} - \sigma\_3)}\tag{16}$$

Thus, the magnitude of ( ) *R i j* increases as*<sup>i</sup>* , increases above*cf* . 46 Automatic Flight Control Systems – Latest Developments

*s s* 

*eq*

*G s eq*

/( ) . Selection of the parameters

 and ( ) *RL <sup>T</sup> <sup>j</sup>*

The simplest over-damped model for ( ) *RL T s* is of the form

*R*

*L*

the desired characteristic of an increasing magnitude of *<sup>R</sup>*

*<sup>b</sup>* , where

curve *BU* for the frequency range above

*R*

 *j*

*U*

*T s*

/ 2 *<sup>n</sup>*

where

.

and *K*1 12 1 2 

is defined as

frequency range.

Thus, the magnitude of ( ) *R i*

that is,

specifications for *<sup>s</sup> t* and *K*<sup>1</sup> .

Once the ideal models ( ) *RU <sup>T</sup> <sup>j</sup>*

 

The gain constant of this equivalent Type1 transfer function ( ) *G s eq* is 1 <sup>0</sup>

*G s*

2 ( ) (2 ) *n*

1 2 ( ) ( ) ( ) ( ) ( )( ) 1 ( )

 

 1 2 1 2

 

( ) ( )

 

*s s*

plots of Figs. 8 and 9a, respectively, can then be drawn. The high-frequency range in Fig. 9a

increasing spread between *BU* and *BL* is required in the high-frequency range (see Fig. 9b),

*hf B B U L*

changing *BU* and *BL* by augmenting *RU T* with a zero [see Eq. (15)] as close to the origin as possible without significantly affecting the time response. This additional zero raises the

augmenting *RL <sup>T</sup>* with a negative real pole [see Eq. (16)] which is as close to the origin as possible but far enough away not to significantly affect the time response. Note that the straight-line Bode plot is shown only for *RL <sup>T</sup>* . This additional pole lowers *BL* for this

2 2

*n n*

( ) ( )( )( ) ( )( )( ) *RL*

*sasasa s s s*

*as a as z T s s s ss*

( / )( ) ( / )( ) ( ) <sup>2</sup> ( )( ) *n n*

 

123 1 2 3

 

*<sup>i</sup>* , increases above

2 2

*K K T s*

must increase with increasing frequency. This desired increase in *<sup>R</sup>*

increases as

*Rs s s G s* 

*Ys K G s*

*n*

*eq*

<sup>1</sup> and

*<sup>b</sup>* is the model BW frequency of *BU* . In order to achieve

(14)

*cf* . The spread can be further increased by

1

(15)

1 2

(16)

 

*cf* .  

*eq*

are determined, the time and frequency response

of *BU* for

<sup>2</sup> is used to meet the

*i c <sup>f</sup>* , an

is achieved by

(13)

(12)

lim[ ( )] *eq <sup>s</sup> K sG s* 

In order to minimize the iteration process in achieving acceptable models for ( ) *RU T s* and ( ) *RL T s* which have an increasing ( ) *<sup>R</sup> j* , the following procedure may expedite the design process: (a) first synthesize the second-order model of Eq. (15) containing the zero at <sup>1</sup> *<sup>n</sup> z a* that meets the desired FOM; and (b) then, as a first trial, select all three real poles of Eq. (16) to have the value of 3 3 *n D* <sup>2</sup> <sup>1</sup> *a aa* . For succeeding trials, if necessary, one or more of these poles are moved right and/or left until the desired specifications are satisfied. As illustrated by the slopes of the straight-line Bode plots in Fig. 9b, selecting the value of all three poles in the range specified above insures an increasing *<sup>R</sup>* . Other possibilities are as follows: (c) the specified values of *<sup>p</sup> t* and *<sup>s</sup> t* for *RL <sup>T</sup>* may be such that a pair of complex poles and a real pole need to be chosen for the model response. For this situation, the real pole must be more dominant than the complex poles, (d) depending on the performance specifications, ( ) *RU T s* may require two real poles and a zero "close" to the origin, i.e., select 1 *z* very much less than 1 *p* and 2 *p* in order to effectively have an under-damped response.

At high frequencies *hf* (see Fig. 9b) must be larger than the actual variation in the plant, *<sup>P</sup>* . For the case where *y*( )*t* , corresponding to *RU T* , is to have an allowable "large" overshoot followed by a small tolerable undershoot, a dominant complex pole pair is not suitable for *RU T* . An acceptable overshoot with no undershoot for *RU T* can be achieved by *RU T* having two real dominant poles 1 2 *p p* , a dominant real zero ( 1 1 *z p* ) "close"' to <sup>1</sup> *p* , and a far off pole 3 2 *p p* . The closeness of the zero dictates the value of *MP* . Thus, a designer selects a pole-zero combination to yield the form of the desired time-domain response.

## **4.4.2 Disturbance rejection models**

The simplest disturbance control ratio model specification is ( ) ( / ( )) *TD P j Y j D j a* , a constant, [the desired maximum magnitude of the output based upon a unit-step disturbance input]; i.e., for <sup>1</sup> *d t*( ) : ( )*p p y t a* , and for: 2 () () *<sup>p</sup> d t y t a* for *<sup>x</sup> t t* . Thus, the frequency domain disturbance specification is ( ) *<sup>D</sup> <sup>p</sup> Lm T j Lm a* over the desired specified BW (see Fig. 10).

Fig. 10. Bode plots of disturbance models for ( ) *TD j*

Quantitative Feedback Theory and Its Application in UAV's Flight Control 49

Thus, it is necessary to synthesize an ( ) *<sup>o</sup> L s* so that the disturbances are properly attenuated. For the present, only one aspect of this disturbance-response problem is considered, namely


 ( ) ( ) 1 () *L j T j <sup>L</sup> <sup>j</sup>*

1 *<sup>L</sup> L T M L* 

and over the whole range of

'

therefore be transformed into a constraint on the maximum value *T*max of Eq. (20). This results in limiting the peak of the disturbance response. A value of *ML* can be selected to correspond to the maximum value of *TR* . Therefore, the top portion, efa as shown in Fig.11, of the M-contour on the NC, which corresponds to the value of the selected value of *ML* ,

> lim[ ( )] *<sup>K</sup> P j*

of the dominant complex-pole pair of *TD*

(20)

(21)

parameter values. This

of the dominant complex-pole pair of *TD* . This constraint can

, the limiting value of the plant transfer function

Fig. 11. U-contour construction (stability contour)

a constraint is placed on the damping ratio

Therefore, it is reasonable to add the requirement

represents the excess of poles over zeros of *P s*( ) .

nearest the

approaches

where  where *ML* is a constant for all

becomes part of the U-contour.

For a large class of problems, as

results in a constraint on

## **4.5** *J* **LTI plant models**

The simple plant of Eq. (17)

$$P\_r(s) = \frac{Ka}{s(s+a)}\tag{17}$$

where K {1,10} and a {1,10}, is used to illustrate the MISO QFT design procedure. The region of plant parameter uncertainty may be described by *J* LTI plants, where *i J* 1,2,..., which lie on its boundary.

#### **4.6 Plant templates of** ( ), ( ) *Ps Pj t i*

With *L GP* , Eq. (7) yields

$$\text{L.m.}\ T\_n = \text{L.m.}\ F + \text{L.m.} \left[\frac{L}{1+L}\right] = \text{L.m.}\ F + \text{L.m.}\ T \tag{18}$$

The change in *TR* due to the uncertainty in P, since F is LTI, is

$$\Delta\text{(L.m.T.}\_{k}) = \text{L.m.T}\_{k} - \text{L.m.F} = \text{L.m.}\left[\frac{L}{1+L}\right] \tag{19}$$

The proper design of *<sup>o</sup> L L* and *F* , must restrict this change in *TR* so that the actual value of *<sup>R</sup> Lm T* always lies between *BU* and *BL* of Fig. 9b. The first step in synthesizing an *Lo* is to make NC templates which characterize the variation of the plant uncertainty for various values of*<sup>i</sup>* , over a frequency range *x i hR* , where*x c <sup>f</sup>* . For the plant of Eq. (17) the values K = a = 1 represent the lowest point of each of the templates ( ) *P <sup>i</sup> j* and may be selected as the nominal plant *Po* for all frequencies.

#### **4.7 Nominal plant**

While any plant case can be chosen, it is a common practice to select, whenever possible, a plant whose NC point is always at the lower left corner of the template for all frequencies for which the templates are obtained.

## **4.8** *U***-contour (stability bound)**

The specifications on system performance in the time domain (see Fig. 8) and in the frequency domain (see Fig. 9) identify a minimum damping ratio for the dominant roots of the closed-loop system which corresponds to a bound on the value of *Mp Mm* . On the NC this bound on *Mp ML* (see Fig. 11) establishes a region which must not be penetrated by the templates and the loop transmission functions ( ) *L j <sup>t</sup>* for all . The boundary of this region is referred to as the stability bound, the U-contour, because this becomes the dominating constraint on *L*( ) *j* . Therefore, the top portion, indicated by the coordinates efa, of the *ML* contour becomes part of the U-contour. The formation of the U -contour is discussed in this section. For the two cases of disturbance rejection depicted in Fig. 7 the control ratios are, respectively, as given in Eqs. (8) and (9).

48 Automatic Flight Control Systems – Latest Developments

( ) ( ) *<sup>t</sup> Ka P s ss a*

where K {1,10} and a {1,10}, is used to illustrate the MISO QFT design procedure. The region of plant parameter uncertainty may be described by *J* LTI plants, where *i J* 1,2,...,

> *L Lm T Lm F Lm Lm F Lm T L*

> > *L*

 

*L*

( ) <sup>1</sup> *R R*

The proper design of *<sup>o</sup> L L* and *F* , must restrict this change in *TR* so that the actual value of *<sup>R</sup> Lm T* always lies between *BU* and *BL* of Fig. 9b. The first step in synthesizing an *Lo* is to make NC templates which characterize the variation of the plant uncertainty for various values

While any plant case can be chosen, it is a common practice to select, whenever possible, a plant whose NC point is always at the lower left corner of the template for all frequencies

The specifications on system performance in the time domain (see Fig. 8) and in the

of the closed-loop system which corresponds to a bound on the value of *Mp Mm* . On the NC this bound on *Mp ML* (see Fig. 11) establishes a region which must not be penetrated

region is referred to as the stability bound, the U-contour, because this becomes the

of the *ML* contour becomes part of the U-contour. The formation of the U -contour is discussed in this section. For the two cases of disturbance rejection depicted in Fig. 7 the

*x c* 

*Lm T Lm T Lm F Lm*

values K = a = 1 represent the lowest point of each of the templates ( ) *P <sup>i</sup> j*

The change in *TR* due to the uncertainty in P, since F is LTI, is

 , where

frequency domain (see Fig. 9) identify a minimum damping ratio

by the templates and the loop transmission functions ( ) *L j <sup>t</sup>*

control ratios are, respectively, as given in Eqs. (8) and (9).

*<sup>i</sup>* , over a frequency range *x i hR*

for which the templates are obtained.

**4.8** *U***-contour (stability bound)** 

dominating constraint on *L*( ) *j*

selected as the nominal plant *Po* for all frequencies.

1 *<sup>R</sup>*

(17)

(18)

(19)

*<sup>f</sup>* . For the plant of Eq. (17) the

. Therefore, the top portion, indicated by the coordinates efa,

for all

for the dominant roots

. The boundary of this

and may be

**4.5** *J* **LTI plant models**  The simple plant of Eq. (17)

which lie on its boundary.

With *L GP* , Eq. (7) yields

of

**4.7 Nominal plant** 

**4.6 Plant templates of** ( ), ( ) *Ps Pj t i*

Fig. 11. U-contour construction (stability contour)

Thus, it is necessary to synthesize an ( ) *<sup>o</sup> L s* so that the disturbances are properly attenuated. For the present, only one aspect of this disturbance-response problem is considered, namely a constraint is placed on the damping ratio of the dominant complex-pole pair of *TD* nearest the -axis. This damping ratio is related to the peak value of

$$\left| T(j\omega o) \right| = \left| \frac{L(j\omega o)}{1 + L(j\omega o)} \right| \tag{20}$$

Therefore, it is reasonable to add the requirement

$$\left|T\right| = \left|\frac{L}{1+L}\right| \le M\_\perp \tag{21}$$

where *ML* is a constant for all and over the whole range of parameter values. This results in a constraint on of the dominant complex-pole pair of *TD* . This constraint can therefore be transformed into a constraint on the maximum value *T*max of Eq. (20). This results in limiting the peak of the disturbance response. A value of *ML* can be selected to correspond to the maximum value of *TR* . Therefore, the top portion, efa as shown in Fig.11, of the M-contour on the NC, which corresponds to the value of the selected value of *ML* , becomes part of the U-contour.

For a large class of problems, as , the limiting value of the plant transfer function approaches

$$\lim\_{\longleftrightarrow} \left[ P(j\alpha o) \right] = \frac{K^\*}{o\nu^\*}$$

where represents the excess of poles over zeros of *P s*( ) .

Quantitative Feedback Theory and Its Application in UAV's Flight Control 51

<sup>1</sup> ( ) ( 1)

*s s* (22)

= 2 [-13.0 dB, -153.4°}. Note, once a nominal

*<sup>L</sup>* (23)

(24)

, i.e.,

.

*P s <sup>o</sup>*

plant is chosen, it must be used for determining all the bounds ( ) *BR i j*

 on ( ) *<sup>o</sup> L j* for= 2

**CONTROL RATIO**. From Fig. 7 , the disturbance control ratio for input 2 *d t*( ) is

Two disturbance inputs are shown in Fig. 7. It is assumed that only one disturbance input

<sup>1</sup> ( ) <sup>1</sup> *T s <sup>D</sup>*

( ) <sup>1</sup> *T s <sup>D</sup>*

this equation has the mathematical format required to use the NC. Over the specified BW it

 

, which results in the requirement, from Eq.(24), that *L j* () 1

**: CASE 1** 

and is represented in Fig. 13 by point *A* for

Fig. 13. Derivation of bounds ( ) *BR i j*

**4.10 Disturbance bounds** ( ) *BD i j*

**CASE 1** [ 2 01 1 *d t Du t d t* ( ) ( ), ( ) 0 ]

exists at a time. Both cases are analyzed.

Substituting 1 / *L* into Eq. (23) yields

is desired that () 1 *T j <sup>D</sup>*

#### **4.9 Tracking bounds** ( ) *B j R i*

Consider the plot of *Lm P*( ) *j* vs. *P*( ) *j* for a plant shown in Fig. 12 (the solid curve). With *Gs A* () 1 and *F s*() 1 in Fig. 7, *L P* . The plot of *Lm L*( ) *j* vs. *L*( ) *j* is tangent to the M = 1dB curve with a resonant frequency 1.1 *<sup>m</sup>* . If 2 *Lm Mm* dB is specified for *Lm TR* , the gain A is increased, raising *Lm L*( ) *j* , until it is tangent to the 2-dB M-curve. For this example the curve is raised by *Lm A dB G A* 4.5 ( 1.679) and the resonant frequency is *<sup>m</sup>* = 2.09.

Now consider that the plant uncertainty involves only the variation in gain A between the values of 1 and 1.679. It is desired to find a cascade compensator *G s*( ) , in Fig. 7, such that the specification 1 2 *<sup>m</sup> dB Lm M dB* is always maintained for this plant gain variation while the resonant frequency *<sup>m</sup>* remains constant. This requires that the loop transmission *L*( ) ( )( ) *j G j P j* be synthesized so that it is tangent to an M-contour in the range of 1 2 *dB Lm M dB* for the entire range of 1 <A <1.679 and the resultant resonant frequency satisfies the requirement 2.09 *m m* .

Fig. 12. Log magnitude-angle diagram

It is assumed for Eq. (19) that the compensators *F* and *G* are fixed (LTI), that is, they have negligible uncertainty. Thus, only the uncertainty in *P* contributes to the change in *TR* given by Eq. (19). The solution requires that the actual () () *LmT j j Ri Ri* dB in Fig. 9b. Thus, it is necessary to determine the resulting constraint, or bound ( ) *BR i j* , on ( )*<sup>i</sup> L j* . The procedure is to select a nominal plant ( ) *P s <sup>o</sup>* and to derive the bounds on the resulting nominal loop transfer function () () () *L s GsP s o o* .

As an illustration, consider the plot of *Lm P*( 2) . ( 2) *j vs P j* for the plant of Eq. (17). As shown in Fig. 13, the plant's region of uncertainty *P*( 2) *j* is given by the contour *ABCD*, i.e., *Lm P*( 2) *j* lies on or within the boundary of this contour. The nominal plant transfer function, with 1 *Ko* and 1 *<sup>o</sup> a* , is

50 Automatic Flight Control Systems – Latest Developments

example the curve is raised by *Lm A dB G A* 4.5 ( 1.679) and the resonant frequency is

Now consider that the plant uncertainty involves only the variation in gain A between the values of 1 and 1.679. It is desired to find a cascade compensator *G s*( ) , in Fig. 7, such that the specification 1 2 *<sup>m</sup> dB Lm M dB* is always maintained for this plant gain variation while the

1 2 *dB Lm M dB* for the entire range of 1 <A <1.679 and the resultant resonant frequency

It is assumed for Eq. (19) that the compensators *F* and *G* are fixed (LTI), that is, they have negligible uncertainty. Thus, only the uncertainty in *P* contributes to the change in *TR* given

is to select a nominal plant ( ) *P s <sup>o</sup>* and to derive the bounds on the resulting nominal loop

As an illustration, consider the plot of *Lm P*( 2) . ( 2) *j vs P j* for the plant of Eq. (17). As shown in Fig. 13, the plant's region of uncertainty *P*( 2) *j* is given by the contour *ABCD*, i.e., *Lm P*( 2) *j* lies on or within the boundary of this contour. The nominal plant transfer function,

 

 , on ( )*<sup>i</sup> L j*

dB in Fig. 9b. Thus, it

. The procedure

by Eq. (19). The solution requires that the actual () () *LmT j j Ri Ri*

is necessary to determine the resulting constraint, or bound ( ) *BR i j*

for a plant shown in Fig. 12 (the solid curve). With

, until it is tangent to the 2-dB M-curve. For this

 vs. *L*( ) *j*

*<sup>m</sup>* . If 2 *Lm Mm* dB is specified for *Lm TR* ,

is tangent to the

*<sup>m</sup>* remains constant. This requires that the loop transmission

*P j* be synthesized so that it is tangent to an M-contour in the range of

**4.9 Tracking bounds** ( ) *B j R i*

Consider the plot of *Lm P*( ) *j*

resonant frequency

*L*( ) ( )( ) *j*

 *G j* 

*<sup>m</sup>* = 2.09. M = 1dB curve with a resonant frequency 1.1

the gain A is increased, raising *Lm L*( ) *j*

*m m*

.

satisfies the requirement 2.09

Fig. 12. Log magnitude-angle diagram

transfer function () () () *L s GsP s o o* .

with 1 *Ko* and 1 *<sup>o</sup> a* , is

vs. *P*( ) *j*

*Gs A* () 1 and *F s*() 1 in Fig. 7, *L P* . The plot of *Lm L*( ) *j*

$$P\_\circ(s) = \frac{1}{s(s+1)}\tag{22}$$

and is represented in Fig. 13 by point *A* for = 2 [-13.0 dB, -153.4°}. Note, once a nominal plant is chosen, it must be used for determining all the bounds ( ) *BR i j*.

Fig. 13. Derivation of bounds ( ) *BR i j* on ( ) *<sup>o</sup> L j* for= 2

#### **4.10 Disturbance bounds** ( ) *BD i j***: CASE 1**

Two disturbance inputs are shown in Fig. 7. It is assumed that only one disturbance input exists at a time. Both cases are analyzed.

**CASE 1** [ 2 01 1 *d t Du t d t* ( ) ( ), ( ) 0 ]

**CONTROL RATIO**. From Fig. 7 , the disturbance control ratio for input 2 *d t*( ) is

$$T\_o(\mathbf{s}) = \frac{1}{\mathbf{1} + L} \tag{23}$$

Substituting 1 / *L* into Eq. (23) yields

$$T\_{\rm{D}}(\mathbf{s}) = \frac{\ell}{\mathbf{1} + \ell} \tag{24}$$

this equation has the mathematical format required to use the NC. Over the specified BW it is desired that () 1 *T j <sup>D</sup>* , which results in the requirement, from Eq.(24), that *L j* () 1 , i.e.,

Quantitative Feedback Theory and Its Application in UAV's Flight Control 53

**DISTURBANCE RESPONSE CHARACTERISTICS.** Based on Eq. (25), the time and frequency-domain response characteristics, for a unit-step disturbance forcing function, are

> ( ) ( ) ( ) ( ) *p D p p y t M t y t d t*

( ) () () ( ) *D D <sup>m</sup> <sup>p</sup>*

transfer function ( ) *<sup>o</sup> L s* is obtained in the manner shown in Fig. 14. The composite

the largest values. For the situation of Fig. 14b, the outermost of the two boundaries ( ) *BR i j*

bounds have one or more intersections. If there are no intersections, then the bound with the

A realistic definition of optimum in an LTI system is the minimization of the high-frequency loop gain *K* while satisfying the performance bounds. This gain affects the high-frequency

where

 largest value or with the outermost boundary dominates. The synthesized ( ) *o i L j*

must not lie in the interior of the ( ) *Bo i j*

*Y j Mj Tj D j*

is composed of those portions of each respective bound ( ) *BR i j*

becomes the perimeter of ( ) *Bo i j*

situation of Fig. 14a, must be on or just above the bound ( ) *Bo i j*

 

that are the most restrictive. For the case shown in Fig. 14a the bound

(33)

that is used to synthesize the desired loop transmission

*<sup>i</sup>* , is composed of those portions of each respective bound

. The situations of Fig. 14 occur when the two

contour.

is the excess of poles over zeros assigned

and ( ) *BD i j*

. For the situation of Fig. 14b

that have

, for the

(32)

given, respectively, by

where *<sup>p</sup> t* is the peak time.

The composite bound ( ) *Bo i j*

 and ( ) *BD i j*

the synthesized ( ) *o i L j*

Fig. 14. Composite ( ) *B j o i*

**4.13 Shaping of** ( ) *L j <sup>o</sup>*

response since lim[ ( )] ( ) *Lj Kj*

bound ( ) *Bo i j*

and ( ) *BD i j*

( ) *BR i j*

( ) *Bo i j*

**4.12 The composite boundary** ( ) *B j o i*

, for each value of

*i*

and

$$T\_{\circ}(joo) \ast \frac{1}{\left| L(joo) \right|} = \left| \ell(joo) \right|.$$

**DISTURBANCE RESPONSE CHARACTERISTIC.** A time-domain tracking response characteristic based upon 1 *rt u t* () () often specifies a maximum allowable peak overshoot *Mp* . In the frequency domain this specification may be approximated by

$$\left| M\_{\kappa}(j\alpha) \right| = \left| T\_{\kappa}(j\alpha) \right| = \left| \frac{Y(j\alpha)}{R(j\alpha)} \right| \le M\_{\kappa} \approx M\_{\nu} \tag{25}$$

The corresponding time- and frequency-domain response characteristics, based upon the step disturbance forcing function 2 1 *dt u t* () () , are, respectively,

$$\left| M\_{\nu}(t) \right| = \left| \frac{Y(t)}{d(t)} \right| \le \alpha\_{\nu} \qquad \text{for } t \ge t\_{\ast} \tag{26}$$

and

$$\left| M\_{\alpha}(jo) \right| = \left| T\_{\alpha}(jo) \right| = \left| \frac{Y(jo)}{D(jo)} \right| \le \alpha\_{\ast} \approx \alpha\_{\ast} \tag{27}$$

#### **4.11 Disturbance bounds** ( ) *BD i j***: CASE 2**

**CASE 2** [ 1 01 2 *d t Du t d t* ( ) ( ), ( ) 0 ]

**CONTROL RATIO.** From Fig. 7, the disturbance control ratio for the input 1 *d t*( ) is

$$T\_o(jo) = \frac{P(jo)}{1 + G(jo)P(jo)}\tag{28}$$

Assuming point *A* of the template represents the nominal plant *Po* . Eq. (28) is multiplied by *P P o o* / and rearranged as follows:

$$T\_o = \frac{P\_s}{P\_s} \left| \frac{\mathbf{1}}{\frac{\mathbf{1}}{P} + \mathbf{G}} \right| = \frac{P\_s}{\frac{P\_s}{P} + \mathbf{G} P\_s} = \frac{P\_s}{\frac{P\_s}{P} + L\_s} = \frac{P\_s}{W} \tag{29}$$

where

$$\mathcal{W} = \begin{pmatrix} P\_\circ \end{pmatrix} \begin{pmatrix} P \end{pmatrix} + L\_\circ \tag{30}$$

Thus Eq.(29) with *D D Lm T* yields

$$\text{L.m.}\,\text{V}\,\text{V} = \text{L.m.}\,P\_o - \mathcal{S}\_D \tag{31}$$

**DISTURBANCE RESPONSE CHARACTERISTICS.** Based on Eq. (25), the time and frequency-domain response characteristics, for a unit-step disturbance forcing function, are given, respectively, by

$$\left| M\_{\cup}(t) \right| = \left| \frac{y(t\_{\circ})}{d(t)} \right| = \left| y(t\_{\circ}) \right| \le \alpha\_{\circ} \tag{32}$$

and

52 Automatic Flight Control Systems – Latest Developments

1 () () ( ) *<sup>D</sup> T j <sup>j</sup> L j*

**DISTURBANCE RESPONSE CHARACTERISTIC.** A time-domain tracking response characteristic based upon 1 *rt u t* () () often specifies a maximum allowable peak

> ( ) () () ( ) *R R m P Y j <sup>M</sup> <sup>j</sup> <sup>T</sup> <sup>j</sup> M M R j*

The corresponding time- and frequency-domain response characteristics, based upon the

( ) ( ) ( ) *D Px Y t M t for t t d t* 

( ) () () ( ) *D D m P*

( ) ( ) 1 ( )( ) *<sup>D</sup> P j T j <sup>G</sup> <sup>j</sup> <sup>P</sup> <sup>j</sup>*

Assuming point *A* of the template represents the nominal plant *Po* . Eq. (28) is multiplied by

*o o o o*

*P P PP*

*P W P P G GP L PP P*

*o o o*

 

*Lm W Lm Po D*

*o o*

  

*Y j Mj Tj D j*

**CONTROL RATIO.** From Fig. 7, the disturbance control ratio for the input 1 *d t*( ) is

1 1

 

*D*

*T*

yields

**: CASE 2** 

(25)

(27)

(28)

( /) *W PP L o o* (30)

(26)

(29)

(31)

overshoot *Mp* . In the frequency domain this specification may be approximated by

 

step disturbance forcing function 2 1 *dt u t* () () , are, respectively,

and

where

**4.11 Disturbance bounds** ( ) *BD i j*

**CASE 2** [ 1 01 2 *d t Du t d t* ( ) ( ), ( ) 0 ]

*P P o o* / and rearranged as follows:

Thus Eq.(29) with *D D Lm T*

$$\left| M\_{\Box}(j\phi) \right| = \left| T\_{\Box}(j\phi) \right| = \left| \frac{Y(j\phi)}{D(j\phi)} \right| \le \alpha\_{\Box} = \alpha\_{\Box} \tag{33}$$

where *<sup>p</sup> t* is the peak time.

#### **4.12 The composite boundary** ( ) *B j o i*

The composite bound ( ) *Bo i j* that is used to synthesize the desired loop transmission transfer function ( ) *<sup>o</sup> L s* is obtained in the manner shown in Fig. 14. The composite bound ( ) *Bo i j* , for each value of*<sup>i</sup>* , is composed of those portions of each respective bound ( ) *BR i j* and ( ) *BD i j* that are the most restrictive. For the case shown in Fig. 14a the bound ( ) *Bo i j* is composed of those portions of each respective bound ( ) *BR i j* and ( ) *BD i j* that have the largest values. For the situation of Fig. 14b, the outermost of the two boundaries ( ) *BR i j* and ( ) *BD i j* becomes the perimeter of ( ) *Bo i j* . The situations of Fig. 14 occur when the two bounds have one or more intersections. If there are no intersections, then the bound with the largest value or with the outermost boundary dominates. The synthesized ( ) *o i L j* , for the situation of Fig. 14a, must be on or just above the bound ( ) *Bo i j* . For the situation of Fig. 14b the synthesized ( ) *o i L j* must not lie in the interior of the ( ) *Bo i j*contour.

Fig. 14. Composite ( ) *B j o i* 

#### **4.13 Shaping of** ( ) *L j <sup>o</sup> i*

A realistic definition of optimum in an LTI system is the minimization of the high-frequency loop gain *K* while satisfying the performance bounds. This gain affects the high-frequency response since lim[ ( )] ( ) *Lj Kj* where is the excess of poles over zeros assigned

Quantitative Feedback Theory and Its Application in UAV's Flight Control 55

frequency domain specifications. For the example of this chapter the magnitude of the frequency response must be within the bounds *BU* and *BL* shown in Fig. 9b, which are redrawn in Fig. 16. A method for determining the bounds on *F s*( ) is as follows: Place the

(see Fig. 17). Traversing the template, determine the maximum max *Lm T* and

( ) ( ) 1 ()

*i*

*L j Lm T j <sup>L</sup> <sup>j</sup>*

obtained from the M-contours. These values are plotted as shown in Fig. 16. The tracking

() () () *R i <sup>i</sup> <sup>i</sup> Lm T j*

max min ( ) *L i RUL*

*j Lm T Lm T B B*

 *Lm F j*

*i*

*i*

 

point of the ( ) *<sup>o</sup> L j*

*Lm T j* (36)

(37)

(35)

, i.e., *TR* , is less than or

within the

curve on the NC

Design of a proper ( ) *<sup>o</sup> L s* guarantees only that the variation in ( ) *TR j*

equal to that allowed. The purpose of the prefilter is to position *Lm T*( ) *j*

*<sup>i</sup>* plant template on the ( ) *o i L j*

The variations in Eqs. (35) and (36) are both due to the variation in P; thus

 

**4.14 Design of the prefilter** *F s*( )

nominal point *A* of the

minimum min *Lm T* , values of

control ratio is /[1 ] *T FL L <sup>R</sup>* and

Fig. 16. Requirements on *F s*( )

Fig. 17. Prefilter determination

to *L*( ) *j* . Thus, only the gain *K* has a significant effect on the high-frequency response, and the effect of the other parameter uncertainty is negligible. Also, the importance of minimizing the high-frequency loop gain is to minimize the effect of sensor noise whose spectrum, in general, lies in the high-frequency range.

For the plant of Eq. (17), the shaping of ( ) *<sup>o</sup> L j* is shown by the dashed curve in Fig. 15. A point such as ( 2) *<sup>o</sup> Lm L j* must be on or above the curve labeled ( 2) *Bo j* . Further, in order to satisfy the specifications, ( ) *<sup>o</sup> L j* cannot violate the U-contour. In this example a reasonable ( ) *<sup>o</sup> L j* closely follows the U-contour up to 40 rad/sec and stays below it above 40 as shown in Fig 15. Additional specifications are = 4, i.e., there are 4 poles in excess of zeros, and that it also must be Type 1 (one pole at the origin).A representative procedure for choosing a rational function ( ) *L s <sup>o</sup>* which satisfies the above specifications is now described. It involves building up the function

$$L\_s(joo) = L\_{sk}(joo) = P\_s(joo) \prod\_{l=0}^{v} [K\_l G\_l(joo)] \tag{34}$$

where for k = 0, 1 0 *Go* , and 0 *w k k K K* 

In order to minimize the order of the compensator, a good starting point for "building up" the loop transmission function is to initially assume that 0 ( ) *<sup>o</sup> L j* = ( ) *Po j* as indicated in Eq. (34). ( ) *<sup>o</sup> L j* is built up term-by-term in order to stay just outside the U-contour in the NC of Fig. 15. The first step is to find the ( ) *Bo i j* which dominates ( ) *<sup>o</sup> L j*.

Fig. 15. Shaping of ( ) *<sup>o</sup> L j*on the Nichols chart for the plant of Eq. (17)

## **4.14 Design of the prefilter** *F s*( )

54 Automatic Flight Control Systems – Latest Developments

point such as ( 2) *<sup>o</sup> Lm L j* must be on or above the curve labeled ( 2) *Bo j* . Further, in order to

zeros, and that it also must be Type 1 (one pole at the origin).A representative procedure for choosing a rational function ( ) *L s <sup>o</sup>* which satisfies the above specifications is now described.

( ) ( ) ( ) [ ( )]

*o ok o k k*

*L j L j P j KG j*

In order to minimize the order of the compensator, a good starting point for "building up"

*w*

*k*

is built up term-by-term in order to stay just outside the U-contour in the NC of

which dominates ( ) *<sup>o</sup> L j*

on the Nichols chart for the plant of Eq. (17)

spectrum, in general, lies in the high-frequency range.

closely follows the U-contour up to 40

0

, and

the loop transmission function is to initially assume that 0 ( ) *<sup>o</sup> L j*

0 *w k k K K* 

as shown in Fig 15. Additional specifications are

For the plant of Eq. (17), the shaping of ( ) *<sup>o</sup> L j*

satisfy the specifications, ( ) *<sup>o</sup> L j*

It involves building up the function

Fig. 15. The first step is to find the ( ) *Bo i j*

where for k = 0, 1 0 *Go*

Fig. 15. Shaping of ( ) *<sup>o</sup> L j*

(34). ( ) *<sup>o</sup> L j*

 . Thus, only the gain *K* has a significant effect on the high-frequency response, and the effect of the other parameter uncertainty is negligible. Also, the importance of minimizing the high-frequency loop gain is to minimize the effect of sensor noise whose

is shown by the dashed curve in Fig. 15. A

= 4, i.e., there are 4 poles in excess of

as indicated in Eq.

rad/sec and stays below it above 40

cannot violate the U-contour. In this example a reasonable

 

(34)

 = ( ) *Po j*

.

to *L*( ) *j*

( ) *<sup>o</sup> L j*

Design of a proper ( ) *<sup>o</sup> L s* guarantees only that the variation in ( ) *TR j* , i.e., *TR* , is less than or equal to that allowed. The purpose of the prefilter is to position *Lm T*( ) *j* within the frequency domain specifications. For the example of this chapter the magnitude of the frequency response must be within the bounds *BU* and *BL* shown in Fig. 9b, which are redrawn in Fig. 16. A method for determining the bounds on *F s*( ) is as follows: Place the nominal point *A* of the *<sup>i</sup>* plant template on the ( ) *o i L j* point of the ( ) *<sup>o</sup> L j* curve on the NC (see Fig. 17). Traversing the template, determine the maximum max *Lm T* and minimum min *Lm T* , values of

$$Lm\ T(j\phi\_{\cdot}) = \frac{L(j\phi\_{\cdot})}{1 + L(j\phi\_{\cdot})} \tag{35}$$

obtained from the M-contours. These values are plotted as shown in Fig. 16. The tracking control ratio is /[1 ] *T FL L <sup>R</sup>* and

$$\text{L.m.T}\_{\text{x}}(jao\_{\cdot}) = \text{L.m.F}(jao\_{\cdot}) + \text{L.m.T}(jao\_{\cdot}) \tag{36}$$

The variations in Eqs. (35) and (36) are both due to the variation in P; thus

$$\mathcal{S}\_{\perp}(j\alpha\_{i}) = \text{L.m.}\ T\_{u\alpha\iota} - \text{L.m.}\ T\_{u\alpha\iota} \le \mathcal{S}\_{\kappa} = B\_{\omega} - B\_{\iota} \tag{37}$$

Fig. 16. Requirements on *F s*( )

Fig. 17. Prefilter determination

Quantitative Feedback Theory and Its Application in UAV's Flight Control 57

external step disturbance inputs 1 *d t*( ) and <sup>2</sup> *d t*( ) . An outline of the basic design procedure for

1. Synthesize the tracking model control ratio ( ) *T s <sup>R</sup>* in the way described in Sec. 4.4, based

2. Synthesize the disturbance-rejection model control ratios ( ) *T s <sup>D</sup>* in the manner described

disturbance rejection, and *V* for the universal high frequency boundary (UHFB) in

phase systems this requires that the synthesized loop transmission ( ) *D i Lm L j*

 versus ( ) *R Ri B j*

on the Nichols diagram.

**5. Robust QFT flight control design for a certain UAV** 

 *j*

(see Fig. 9b), and *ML* [see Eq.(21)]. For minimum-phase systems this

on the Nichols diagram.

and ( ) *D i Lm B j*

*<sup>i</sup>* at various values of the angle

that lies between *BU* and *BL* of Fig. 9b.

, whichever is the largest value (termed the "worst" or "most

. Repeat this procedure for sufficient values of

on the loop transmission ( ) ( )( ) *D i io i L j*

*P j* , using the tracking model (step 1), the templates ( ) *P <sup>i</sup> j*

requires that the synthesized loop transmission satisfy the requirement that ( ) *o i Lm L j*

severe" boundary). Draw a curve through these points. The resulting plot defines the

boundaries and U-contour so that ( ) *o i Lm L j*

to be as close as possible to the boundary value ( ) *Bo i j*

10. Based upon the information available from steps 1 and 9, synthesize an *F s*( ) those

11. Obtain the time-response data for *y*( )*t* : (a) with 1 *dt u t* () () and *r t*() 0 and (b) with <sup>1</sup> *rt u t* () () and *d t*() 0 for sufficient points around the parameter space describing the

Unmanned Aerial Vehicles (hereafter referred as UAVs) play a very important role in modern war. Whereas flight stability of UAVs is easily affected by airflow, model perturbation and other uncertainty. To enhance flight stability and robustness of UAVs,

that pictorially describe the plant uncertainty on the Nichols

 *G j*

 *j* for tracking, *ML* for

*P j* . For minimum-

must be

(step 3), the

on the

(see Fig. 10) to determine the

*d Di B j*

> *G j P j*

, select the value of

*i* .

is on or above the

by selecting an

 

on the nominal transmission

 versus ( ) 

. Synthesize an ( ) ( )( ) *o o L j*

on the Nichols diagram (see Fig. 15

the QFT technique, as applied to a minimum-phase plant, is as follows:

upon the desired tracking specifications (see Figs. 8 and 9b).

chart for the desired pass-band frequency range.

6. Use the data of steps 2 and 3 and the values of ( ) *D i*

on or above the curve for ( ) *D i Lm B j*

7. Determine the tracking bound ( ) *BR i j*

 *vs* 

appropriate compensator transfer function *G*( ) *j*

conjunction with steps 6 through 8.

 

is on or above the curve for ( ) *R i Lm B j*

same NC. For a given value of

 or ( ) *R i Lm B j*

overall boundary ( ). *o i Lm B j*

results in a *<sup>R</sup> Lm T* [Eq. (7)] vs.

using the ( ) *o i Lm B j*

curve for ( ) *o i Lm B j*

plant uncertainty.

**5.1 Introduction** 

disturbance bound ( ) *BD i j*

assuming *B B D o* ).

( ) ( )( ) *o i io i L j*

8. Plot curves of ( ) *R i Lm B j*

values of ( ) *R i j*

 *G j*

( ) *D i Lm B j*

9. Design ( ) *o i L j*

3. Obtain templates of ( ) *P <sup>i</sup> j*

in Sec. 4.10 based upon the disturbance-rejection specifications.

4. Select a nominal plant from the set of Eq. (5) and denote it as ( ) *P s <sup>o</sup>* . 5. Determine the U-contour based upon the specified values of ( ) *R i*

If values of ( ) *<sup>o</sup> L j* , for each value*<sup>i</sup>* , lie exactly on the tracking bounds ( ) *BR i j* , then *L R* . Therefore, based upon Eq. (36), it is necessary to determine the range in dB by which ( )*<sup>i</sup> Lm T j* must be raised or lowered to fit within the bounds of the specifications by use of the prefilter ( )*<sup>i</sup> F j* . The process is repeated for each frequency corresponding to the templates used in the design of ( ) *<sup>o</sup> L j* . Therefore, in Fig. 18 the difference between the *RU* max *Lm T Lm T* and the *RL* min *Lm T Lm T* curves yields the requirement for *Lm F*( ) *<sup>j</sup>* , i.e., from Eq. (36).

$$\text{L.m.F}(jao) = \text{L.m.T}\_{s}(jao) - \text{L.m.T}(jao) \tag{38}$$

Fig. 18. Frequency bounds on the prefilter *F s*( )

The procedure for designing *F s*( ) is summarized as follows:


$$\lim\_{s \to 0} [F(s)] = 1\tag{39}$$

#### **4.15 Basic design procedure for a MISO system**

The basic concepts of the QFT technique are explained by means of a design example. The system configuration shown in Fig. 7 contains three inputs. The first objectives are to track a step input 1 *rt u t* () () with no steady-state error and to satisfy the performance specifications of Fig. 8. An additional objective is to attenuate the system response caused by 56 Automatic Flight Control Systems – Latest Developments

Therefore, based upon Eq. (36), it is necessary to determine the range in dB by which

() () () *<sup>R</sup> Lm F j*

. Then use the M-contours to determine max ( ) *T <sup>i</sup> j*

3. From the values obtained in steps 1 and 2, plot *RU* max *Lm T Lm T* and *RL* min *Lm T Lm T*

<sup>0</sup> lim[ ( )] 1

The basic concepts of the QFT technique are explained by means of a design example. The system configuration shown in Fig. 7 contains three inputs. The first objectives are to track a step input 1 *rt u t* () () with no steady-state error and to satisfy the performance specifications of Fig. 8. An additional objective is to attenuate the system response caused by

2. Obtain the values of *RU Lm T* and *RL Lm T* for various values of a, from Fig. 9b.

4. Use straight-line approximations to synthesize an *F s*( ) so that ( )*<sup>i</sup> Lm F j*

plots of step 3. For step forcing functions the resulting *F s*( ) must satisfy

 *Lm T j*

must be raised or lowered to fit within the bounds of the specifications by use of

. The process is repeated for each frequency corresponding to the

*<sup>i</sup>* , lie exactly on the tracking bounds ( ) *BR i j*

 

. Therefore, in Fig. 18 the difference between the

*Lm T j* (38)

plot on the NC to determine *T*max and *T*min

*<sup>s</sup> F s* (39)

with its nominal point on the

 and min ( ) *T <sup>i</sup> j*

(see Fig.

lies within the

 , then *L R* .

, i.e., from

If values of ( ) *<sup>o</sup> L j*

the prefilter ( )*<sup>i</sup> F j*

for each

17).

vs. 

point ( ) *<sup>o</sup> Lm L j*

as shown in Fig. 18.

**4.15 Basic design procedure for a MISO system** 

( )*<sup>i</sup> Lm T j*

Eq. (36).

templates used in the design of ( ) *<sup>o</sup> L j*

Fig. 18. Frequency bounds on the prefilter *F s*( )

1. Use templates in conjunction with the ( ) *<sup>o</sup> L j*

The procedure for designing *F s*( ) is summarized as follows:

*<sup>i</sup>* . This is done by placing ( ) *P <sup>i</sup> j*

, for each value

*RU* max *Lm T Lm T* and the *RL* min *Lm T Lm T* curves yields the requirement for *Lm F*( ) *<sup>j</sup>*

external step disturbance inputs 1 *d t*( ) and <sup>2</sup> *d t*( ) . An outline of the basic design procedure for the QFT technique, as applied to a minimum-phase plant, is as follows:


## **5. Robust QFT flight control design for a certain UAV**

#### **5.1 Introduction**

Unmanned Aerial Vehicles (hereafter referred as UAVs) play a very important role in modern war. Whereas flight stability of UAVs is easily affected by airflow, model perturbation and other uncertainty. To enhance flight stability and robustness of UAVs,

Quantitative Feedback Theory and Its Application in UAV's Flight Control 59

[] 0 *P PW <sup>n</sup> ijn ijn p where p f or i j*

.

. In some design problems it may be

(40)

is sideslip angle, *p* is roll

is rudder

resulting in a diagonal *Pn* matrix for *P* representing the nominal plant case in the set

comparison to *P*, for the nonnominal plant cases in

State equation of the UAV is generally expressed as:

 

**5.3.1 Mathematical model of the UAV** 

Fig. 19. QFT control structure of loop I

Fig. 20. QFT control structure of loop II

is yaw angle rate,

where *<sup>T</sup> X p a r*

**5.3.2 System decomposition** 

deflection angle, *ac*

angle rate,

**5.3 QFT design and simulation for a certain UAV's lateral motion** 

subsystem. QFT control structures of both loops are given in Fig.19 and Fig.20.

 

 ; *<sup>T</sup> u rc ac* 

*y t Cx t*

is roll angle, *<sup>a</sup>*

coupling effects.

section.

With plant uncertainty the off-diagonal terms of *Pn* will not be zero but "very small" in

necessary or desired to determine a *Pn* upon which the QFT design is accomplished. Doing this minimizes the effort required to achieve the desired BW and minimizes the cross-

QFT approach for MIMO system will be applied to a certain UAV's lateral motion in this

() () () ( ) *x t Ax t Bu t*

> ; *<sup>T</sup> Y p*

 ; 

is rudder deflection angle command input, *<sup>A</sup>*, , *B C* are system matrix,

is aileron deflection angle, *rc*

input matrix and input-output matrix respectively. By way of wind tunnel test and mathematic method, matrices *A, B* and *C* in eqs.(40) for the small UAV can be derived.

The UAV's lateral state equation described in Eq.(40) has two inputs and four outputs. According to QFT approach for MIMO system, we decompose Eq.(40) into two MISO subsystems using BNIA, one is yaw loop (loop I) subsystem, the other is roll loop (loop II)

*<sup>H</sup>* control, QFT technique, linear quadratic Gaussian (LQG) have been applied to UAVs' flight control system at present. Comparatively, QFT can take uncertainty's scopes and performance requirements into account, analyze and design robust controller on Nichols chart quantitatively in order to make the open-loop frequency curve comply with boundary conditions and have robust stability and performance robustness.

QFT has been widely used in aerospace field and is mature for robust controller design of LTI/SISO system. But QFT design for MIMO system still faces many difficulties. In view of the characteristics of a certain small UAV which used in tracking and surveillance, a novel QFT controller design method for the UAV's lateral motion is introduced in this section.

## **5.2 QFT design for MIMO systems**

## **5.2.1 Overview**

The QFT design for MIMO systems is based upon the mathematical means which results in the representation of a MIMO control system by 2 *m* MISO equivalent control systems. The highly structured uncertain LTT MIMO plant has the following features:


The design process for these individual loops is the same as the design of a MISO system described in previous sections.

Pure mathematical transformation method used in QFT design for MIMO systems tends to cause a larger super-margin design and is very complicated when system is of higher order. Comparatively, Basically Non-interacting (hereafter referred as BNIA) is commonly used in practical applications. Note that principle of BNIA, which will be negligible here, can be found in references of this chapter.

## **5.2.2 Non-interacting (BNIA) loops**

A BNIA loop is one in which the output ( ) *<sup>K</sup> y s* due to the input ( ) *<sup>j</sup> r s* is ideally zero. Plant uncertainty and loop interaction (cross-coupling) makes the ideal response unachievable. Thus, the system performance specifications describe a range of acceptable responses for the commanded output and a maximum tolerable response for the uncommanded outputs. The uncommanded outputs are treated as cross-coupling effects.

For an LTI plant having no parameter uncertainty, it is possible to essentially achieve zero cross-coupling effects, i.e., the output 0 *<sup>K</sup> y* . This desired result can be achieved by post multiplying *P* by a matrix *W* to yield:

[] 0 *P PW <sup>n</sup> ijn ijn p where p f or i j*

resulting in a diagonal *Pn* matrix for *P* representing the nominal plant case in the set . With plant uncertainty the off-diagonal terms of *Pn* will not be zero but "very small" in comparison to *P*, for the nonnominal plant cases in . In some design problems it may be necessary or desired to determine a *Pn* upon which the QFT design is accomplished. Doing this minimizes the effort required to achieve the desired BW and minimizes the crosscoupling effects.

## **5.3 QFT design and simulation for a certain UAV's lateral motion**

QFT approach for MIMO system will be applied to a certain UAV's lateral motion in this section.

## **5.3.1 Mathematical model of the UAV**

58 Automatic Flight Control Systems – Latest Developments

*<sup>H</sup>* control, QFT technique, linear quadratic Gaussian (LQG) have been applied to UAVs' flight control system at present. Comparatively, QFT can take uncertainty's scopes and performance requirements into account, analyze and design robust controller on Nichols chart quantitatively in order to make the open-loop frequency curve comply with boundary

QFT has been widely used in aerospace field and is mature for robust controller design of LTI/SISO system. But QFT design for MIMO system still faces many difficulties. In view of the characteristics of a certain small UAV which used in tracking and surveillance, a novel QFT controller design method for the UAV's lateral motion is introduced in this

The QFT design for MIMO systems is based upon the mathematical means which results in

1. The synthesis problem is converted into a number of single-loop problems, in which structured parameter uncertainty, external disturbance, and performance tolerances are derived from the original MIMO problem. The solutions to these single-loop problems are guaranteed to work for the MIMO plant. It is not necessary to consider the system

2. The design is tuned to the extent of the uncertainty and the performance tolerances. The design for a MIMO system, as stated previously, involves the design of an equivalent

The design process for these individual loops is the same as the design of a MISO system

Pure mathematical transformation method used in QFT design for MIMO systems tends to cause a larger super-margin design and is very complicated when system is of higher order. Comparatively, Basically Non-interacting (hereafter referred as BNIA) is commonly used in practical applications. Note that principle of BNIA, which will be negligible here, can be

A BNIA loop is one in which the output ( ) *<sup>K</sup> y s* due to the input ( ) *<sup>j</sup> r s* is ideally zero. Plant uncertainty and loop interaction (cross-coupling) makes the ideal response unachievable. Thus, the system performance specifications describe a range of acceptable responses for the commanded output and a maximum tolerable response for the uncommanded outputs. The

For an LTI plant having no parameter uncertainty, it is possible to essentially achieve zero cross-coupling effects, i.e., the output 0 *<sup>K</sup> y* . This desired result can be achieved by post

*m* MISO equivalent control systems. The

conditions and have robust stability and performance robustness.

section.

**5.2.1 Overview** 

**5.2 QFT design for MIMO systems** 

characteristic equation.

described in previous sections.

found in references of this chapter.

**5.2.2 Non-interacting (BNIA) loops** 

multiplying *P* by a matrix *W* to yield:

uncommanded outputs are treated as cross-coupling effects.

set of MISO system feedback loops.

the representation of a MIMO control system by 2

highly structured uncertain LTT MIMO plant has the following features:

State equation of the UAV is generally expressed as:

$$\begin{cases} \dot{\mathbf{x}}(t) = A\mathbf{x}(t) + Bu(t) \\ y(t) = \mathbf{C}\mathbf{x}(t) \end{cases} \tag{40}$$

where *<sup>T</sup> X p a r* ; *<sup>T</sup> u rc ac* ; *<sup>T</sup> Y p* ; is sideslip angle, *p* is roll angle rate, is yaw angle rate, is roll angle, *<sup>a</sup>* is aileron deflection angle, *rc* is rudder deflection angle, *ac* is rudder deflection angle command input, *<sup>A</sup>*, , *B C* are system matrix, input matrix and input-output matrix respectively. By way of wind tunnel test and mathematic method, matrices *A, B* and *C* in eqs.(40) for the small UAV can be derived.

#### **5.3.2 System decomposition**

The UAV's lateral state equation described in Eq.(40) has two inputs and four outputs. According to QFT approach for MIMO system, we decompose Eq.(40) into two MISO subsystems using BNIA, one is yaw loop (loop I) subsystem, the other is roll loop (loop II) subsystem. QFT control structures of both loops are given in Fig.19 and Fig.20.

Fig. 19. QFT control structure of loop I

Fig. 20. QFT control structure of loop II

Quantitative Feedback Theory and Its Application in UAV's Flight Control 61

upper tracking boundary when upper tracking boundary cross over 0 dB line, then the final

3.6 ( ) ( 0.9) 1 4 *RL T j s ss*

> 11 1 11 1

> > 11 11 1

<sup>1</sup> *Km u* 1 1.9091=5.5155dB ╱

1 2 180 , cos (0.5 / 1) 54.062 *<sup>m</sup>*

2. **Plant Template and Border Calculation for Loop I.** According to the requirements of performance index, generate the tracking response boundary, robust stability boundary

3. **Controller and Prefilter Design for Loop I.** In Fig. 21(a), the open-loop frequency characteristics curve (noted by black solid line) of the nominal plant (corresponding to G(s) =1) and the compound boundary (the region embraced by green and red solid line) are drawn up in Nichols chart. Apparently, the open-loop frequency curve locates under tracking performance boundary curve, open-loop frequency characteristics curve cross over the instability boundary (red solid ring line in Fig. 21(a)) which make the MISO system of loop I instable or unsatisfactory for corresponding performance requirements. So, it is necessary to enlarge the controller gain and introduce into dynamic compensation element to shape the open loop frequency characteristic curve to ensure shaped open-loop frequency characteristic meet the requirtments of stability

> <sup>1</sup>

> > /0.6 1 *f s <sup>s</sup>*

The open-loop frequency characteristics curve with G(s) is shown in Fig.21 (b). Clearly, the shaped curve does not cross over the instability region (red solid ring line),i.e. the shaped

*s s* 

8.855 / 2.045 1 /8.68 1 /113.5 1 /907.9 1 *s s*

*q s q sg s*

*q sg s q sg s*

Stability performance index and robust performance index are respectively

1

1

 

and dynamic performance indics. Using MATLAB QFT toolbox, we get

system is stable. Besides, the characteristic of tracking boundary is met.

and inference rejection boundary in Nichols chart.

*g s*

<sup>11</sup> 1.275

Corresponding minimum amplitude margin and phase margin are respectively

1.1

0.1

(44)

(45)

(46)

lower boundary transfer function is

and

and

where , *<sup>c</sup> <sup>c</sup>* are sideslip angle input and roll angle input respectively; 1 2 *g g*, are QFT controllers; 11 22 *f* , *f* are QFT prefilters; 11 22 *c c*, are disturbance inputs; 11 22 *q q*, is controlled plants.

Decomposed state equation has relationship with that of the original system as follows:

$$A\_{\ c} = A\_{\prime} \ B\_{\ c} = B\_{\prime} \ \mathbf{C}\_{\ c} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \mathbf{C}\_{\epsilon}$$

Transfer function matrices *P* of decomposed plant can be easily derived as

$$P = \mathbf{C}\_c (\mathbf{s}I - A\_c)^{-1} B\_c = \begin{bmatrix} p\_{11} & p\_{12} \\ p\_{21} & p\_{22} \end{bmatrix}$$

where <sup>11</sup> *p* is the transfer function from *rc* to ; <sup>22</sup> *p* represents the transfer function from *ac* to ; <sup>12</sup> *p* is the transfer function from *rc* to , 21 *p* represents the transfer function from *ac* to .

Next, we adopt 5 flight states to develop the QFT controllers of both loops.

#### **5.3.3 QFT design for loop I**

For loop I, we ensure 1 *g* ( )*<sup>s</sup>* and <sup>11</sup> *f* ( )*<sup>s</sup>* meet requirements of robust stability when *<sup>c</sup>* acts as command input and 11 *c* as disturbance input. Besides, both subsystems should own ideal tracking performance and preferable noise restraint capability.

1. **Selection of Performance Indices.** Tracking performances indices of sideslip angle are overshoot % 2% , settling time *ts* 6% . Given the original model of upper tracking boundary is

2 2 2 ( ) <sup>2</sup> *n R n n <sup>U</sup> T j s s* (41)

According to % and *<sup>s</sup> t* , damping ratio and natural oscillation frequency *<sup>n</sup>* is adopted as 0.78 and 0.8978. Add a zero (z=-1) as close to the origin as possible without significantly affecting the time response(see Sec.4.4.1). This additional zero raises tracking boundary curve above*cf* , the final transfer function of tracking curve's upper boundary is

$$T\_{s\&I} \text{(jo)} = \frac{0.806 \text{(s+1)}}{\text{s}^2 + 1.4 \text{s} + 0.806} \tag{42}$$

the lower boundary original model of tracking curve as

$$T\_{\%} \, (jo) = \frac{0.9}{\text{s} + 0.9} \tag{43}$$

Adding two poles (p1=-1, p2=-4), which locate in left half s-plane to ensure stability of *RL <sup>T</sup>* and are as close to the origin as possible but far enough away not to significantly affect the time response (see Sec.4.4.1), to eq. (43) to make lower tracking boundary separate from upper tracking boundary when upper tracking boundary cross over 0 dB line, then the final lower boundary transfer function is

$$T\_{\%} \text{(jo)} = \frac{3.6}{(s+0.9)(s+1)(s+4)} \tag{44}$$

Stability performance index and robust performance index are respectively

$$\left| \frac{q\_{\text{i}1}(\mathbf{s}) \mathbf{g}\_{\text{i}}(\mathbf{s})}{1 + q\_{\text{i}1}(\mathbf{s}) \mathbf{g}\_{\text{i}}(\mathbf{s})} \right| \le 1.1$$

and

60 Automatic Flight Control Systems – Latest Developments

controllers; 11 22 *f* , *f* are QFT prefilters; 11 22 *c c*, are disturbance inputs; 11 22 *q q*, is controlled

<sup>1000</sup> , , <sup>0001</sup> *A AB BC c cc <sup>C</sup>* 

1 11 12

% 2% , settling time *ts* 6% . Given the original model of upper tracking

*n n*

 

and natural oscillation frequency

2

21 22

; <sup>22</sup> *p* represents the transfer function from *ac*

, 21 *p* represents the transfer function from *ac*

to

(41)

(43)

*<sup>n</sup>* is adopted

 to .

acts as

(42)

*p p* 

Decomposed state equation has relationship with that of the original system as follows:

( ) *c cc <sup>p</sup> <sup>p</sup> P C sI A B*

For loop I, we ensure 1 *g* ( )*<sup>s</sup>* and <sup>11</sup> *f* ( )*<sup>s</sup>* meet requirements of robust stability when *<sup>c</sup>*

command input and 11 *c* as disturbance input. Besides, both subsystems should own ideal

1. **Selection of Performance Indices.** Tracking performances indices of sideslip angle are

2 2 ( ) <sup>2</sup> *n*

as 0.78 and 0.8978. Add a zero (z=-1) as close to the origin as possible without significantly affecting the time response(see Sec.4.4.1). This additional zero raises tracking boundary

*cf* , the final transfer function of tracking curve's upper boundary is

*s s*

2 0.806( 1) ( ) 1.4 0.806 *RU <sup>s</sup> T j*

> 0.9 ( ) 0.9 *RL T j s*

Adding two poles (p1=-1, p2=-4), which locate in left half s-plane to ensure stability of *RL <sup>T</sup>* and are as close to the origin as possible but far enough away not to significantly affect the time response (see Sec.4.4.1), to eq. (43) to make lower tracking boundary separate from

*s s* 

Transfer function matrices *P* of decomposed plant can be easily derived as

 to 

 to 

tracking performance and preferable noise restraint capability.

and *<sup>s</sup> t* , damping ratio

the lower boundary original model of tracking curve as

*R*

*<sup>U</sup> T j*

Next, we adopt 5 flight states to develop the QFT controllers of both loops.

where <sup>11</sup> *p* is the transfer function from *rc*

; <sup>12</sup> *p* is the transfer function from *rc*

**5.3.3 QFT design for loop I** 

overshoot

According to %

curve above

boundary is

are sideslip angle input and roll angle input respectively; 1 2 *g g*, are QFT

where , *<sup>c</sup> <sup>c</sup>* 

plants.

$$\left| \frac{q\_{\text{in}}(s)}{1 + q\_{\text{in}}(s)g\_{\text{i}}(s)} \right| \le 0.1$$

Corresponding minimum amplitude margin and phase margin are respectively

$$K\_{\rm u} = 1 + \text{\textquotedblleft } \underline{\text{\textquotedblleft}}\_{\text{\textquotedblright}} = 1.9091 \text{\textquotedblright} \text{\textquotedblright} \text{\textquotedblleft} \text{\textquotedblleft}$$

and

$$\Phi\_{\rm \tiny \tiny \textrm \}} = 180^{\circ} - \theta \,\prime \theta = \cos^{-1}(0.5 \,/\, \mu^2 - 1) = 54.062^{\circ}$$


$$g\_{\rm s}(s) = \frac{8.855 \left(s \,/\, 2.045 + 1\right) \left(s \,/\, 8.68 + 1\right)}{\left(s \,/\, 113.5 + 1\right) \left(s \,/\, 907.9 + 1\right)}\tag{45}$$

$$f\_{\rm in}(s) = \frac{1.275}{s / 0.6 + 1} \tag{46}$$

The open-loop frequency characteristics curve with G(s) is shown in Fig.21 (b). Clearly, the shaped curve does not cross over the instability region (red solid ring line),i.e. the shaped system is stable. Besides, the characteristic of tracking boundary is met.

Quantitative Feedback Theory and Its Application in UAV's Flight Control 63

Robust performance index curve Interfence rejection curve

10-2 10-1 100 101

Frequency /rad/sec

10-2 10-1 100

Upper boundary Lower boundary Closed-loop output1 Closed-loop output1

Frequency /rad/sec

The time-domain simulation results of closed-loop system under 5 design envelopes are shown in Fig.25 and Fig.26. The unit step-response of sideslip angle lies between the upper and lower boundary response curve; the unit step-response of disturbance input are located under the given boundary. Apparently, the closed-loop system satisfies the requirements of robust stability and tracking boundary requirements, and owns strong disturbance rejection capability.



Fig. 24. Tracking boundary




Magnitude /dB


0

Fig. 23. Disturbance rejection boundary



Magnitude /dB




(a) Open-loop frequency response when G(s) =1 (b) Open-loop frequency response with controller

Fig. 21. Open loop frequency characteristics in Nichols Chart

4. **Verification and Simulation for Loop I.** Closed-loop system stability margin analysis curve, inference rejection boundary analysis curves and tracking boundary analysis curves in loop I are given in Fig.22 ,Fig.23 and Fig.24. Clearly, the stability margin curve, inference rejection boundary curve and tracking boundary curve are all under the stability performance index curve, the performance index curve and between the upper and lower boundaries of tracking curves. Obviously, Closed-loop control system satisfies the performance requirements in loop I.

Fig. 22. Stability margin

Fig. 23. Disturbance rejection boundary

62 Automatic Flight Control Systems – Latest Developments

(a) Open-loop frequency response when G(s) =1 (b) Open-loop frequency response with

4. **Verification and Simulation for Loop I.** Closed-loop system stability margin analysis curve, inference rejection boundary analysis curves and tracking boundary analysis curves in loop I are given in Fig.22 ,Fig.23 and Fig.24. Clearly, the stability margin curve, inference rejection boundary curve and tracking boundary curve are all under the stability performance index curve, the performance index curve and between the upper and lower boundaries of tracking curves. Obviously, Closed-loop control system

10-2 10-1 100 101 102 103

Stability performance index curve

Stability margin curve

Frequency /rad/sec

Fig. 21. Open loop frequency characteristics in Nichols Chart

satisfies the performance requirements in loop I.


Fig. 22. Stability margin




Magnitude /dB


0

controller

Fig. 24. Tracking boundary

The time-domain simulation results of closed-loop system under 5 design envelopes are shown in Fig.25 and Fig.26. The unit step-response of sideslip angle lies between the upper and lower boundary response curve; the unit step-response of disturbance input are located under the given boundary. Apparently, the closed-loop system satisfies the requirements of robust stability and tracking boundary requirements, and owns strong disturbance rejection capability.

Quantitative Feedback Theory and Its Application in UAV's Flight Control 65

3.6 ( ) ( 0.9) 1 4 *RL T j s ss*

Stability performance index and robust performance index are defined as

( ) 1.1

Minimum amplitude margin and phase margin are 5.5155B and 54.062

 

 

22 2 22 2

Similar to loop I, using MATLAB QFT toolbox, we can get

*g s*

1 () *q sg s q sg s*

2. **Controller and Prefilter Design for Loop II.** 

rejection capability.

is 5

results are shown in Fig.28 and Fig.29. The overshoot of

performance indices, own better flight stability and robustness.

Fig. 27. QFT control structure for the UAV's lateral motion

0, the initial value of

and <sup>22</sup> 22 2

 <sup>2</sup> 11.8 27.94 1 1.18 1 ( ) 1280 1 1926 1 *s s*

> <sup>22</sup> 1.01 ( ) 0.7 1 *f s*

3. **Verification and Simulation for loop II.** Closed-loop system satisfies requirements of robust stability and tracking boundary requirements and owns strong disturbance

state equation, 1 11 222 *g* ( ), ( ), ( ), ( ) *s f sgs f s* , models of rudder and ailerons into Fig.27. The simulation

about 1 second. The settling time of yaw angle rate, roll angle rate and roll angle are all about 0.1 second. Besides, the initial value of sideslip angle almost have no influence in roll angle response, the settling time of yaw angle rate, roll angle rate is no more than 1 second. Clearly, QFT controller for the UAV's lateral motion satisfies the requirements of

**5.3.5 Performance analysis of QFT controller for the UAV's lateral motion**  QFT control structure for the UAV's lateral motion is shown in Fig.27 .Given

, the initial sideslip angle

*s s*

1 () *q s q sg s*

> is 1

is about 0.064

(48)

( ) 0.1

(49)

*<sup>s</sup>* (50)

respectively.

*c* and *<sup>c</sup>* are

and settling time is

, substitute the UAV's lateral

Fig. 25. The unit step response of 

Fig. 26. The unit step response of with disturbance

#### **5.3.4 QFT design for loop II**

QFT design for loop II is similar to that for loop I.

1. **Selection of Performance Indices.** Tracking performance indices of roll angle is overshoot % 5% and settling time <sup>12</sup> *<sup>s</sup> t <sup>s</sup>* , the upper and lower boundary tracking curve are respectively

$$T\_{\%I} \text{(jo)} = \frac{0.25(1.7s + 1)}{s^2 + 0.78s + 0.25} \tag{47}$$

$$T\_{\%} \, (jo) = \frac{3.6}{(s+0.9)(s+1)(s+4)} \tag{48}$$

Stability performance index and robust performance index are defined as

$$\left| \frac{q\_{\text{22}}(\text{s}) \mathcal{g}\_{\text{2}}(\text{s})}{1 + q\_{\text{22}}(\text{s}) \mathcal{g}\_{\text{2}}(\text{s})} \right| \le \mu = 1.1 \text{ and } \left| \frac{q\_{\text{22}}(\text{s})}{1 + q\_{\text{22}}(\text{s}) \mathcal{g}\_{\text{2}}(\text{s})} \right| \le 0.1$$

Minimum amplitude margin and phase margin are 5.5155B and 54.062 respectively.

#### 2. **Controller and Prefilter Design for Loop II.**

64 Automatic Flight Control Systems – Latest Developments

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>0</sup>

Upper boundary Lower boundary fliight state 1 fliight state 2 fliight state 3 fliight state 4 fliight state 5

> flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

Time/second

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

with disturbance

1. **Selection of Performance Indices.** Tracking performance indices of roll angle is

2 0.25(1.7 1) ( ) 0.78 0.25 *RU <sup>s</sup> T j*

*s s*

Time/second

% 5% and settling time <sup>12</sup> *<sup>s</sup> t <sup>s</sup>* , the upper and lower boundary tracking

(47)

0.2

Fig. 25. The unit step response of

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Fig. 26. The unit step response of

curve are respectively

QFT design for loop II is similar to that for loop I.

**5.3.4 QFT design for loop II** 

overshoot

Sideslip angle/degree

0.4

0.6

0.8

Sideslip angle/degree

1

1.2

1.4

Similar to loop I, using MATLAB QFT toolbox, we can get

$$\lg\_2(s) = \frac{11.8 \, (s/27.94 + 1) \, (s/1.18 + 1)}{(s/1280 + 1) \, (s/1926 + 1)} \tag{49}$$

$$f\_{z2}(\mathbf{s}) = \frac{1.01}{\left(s/0.7 + 1\right)}\tag{50}$$

3. **Verification and Simulation for loop II.** Closed-loop system satisfies requirements of robust stability and tracking boundary requirements and owns strong disturbance rejection capability.

## **5.3.5 Performance analysis of QFT controller for the UAV's lateral motion**

QFT control structure for the UAV's lateral motion is shown in Fig.27 .Given *c* and *<sup>c</sup>* are 0, the initial value of is 5 , the initial sideslip angle is 1 , substitute the UAV's lateral state equation, 1 11 222 *g* ( ), ( ), ( ), ( ) *s f sgs f s* , models of rudder and ailerons into Fig.27. The simulation results are shown in Fig.28 and Fig.29. The overshoot of is about 0.064 and settling time is about 1 second. The settling time of yaw angle rate, roll angle rate and roll angle are all about 0.1 second. Besides, the initial value of sideslip angle almost have no influence in roll angle response, the settling time of yaw angle rate, roll angle rate is no more than 1 second. Clearly, QFT controller for the UAV's lateral motion satisfies the requirements of performance indices, own better flight stability and robustness.

Fig. 27. QFT control structure for the UAV's lateral motion

Quantitative Feedback Theory and Its Application in UAV's Flight Control 67


(c) Yaw angle rate (d) Roll angle rate

This chapter is devoted to presenting an overview and in-depth expression of QFT in order to enhance the understanding and appreciation of the power of the QFT technique. Then, A QFT design of robust controller for a certain UAV's lateral motion, which is a MIMO system, is proposed base on BNIA principle in order to show how to apply QFT in flight control system of UAVs. Meantime, the simulation results show that the QFT controller own better robust stability and superior dynamic characteristics which verify the validity of presented

MIMO Multiple-input multiple-output; more than one tracking and/or external disturbance

MISO Multiple-input single-output; a system having one tracking input, one or more

( ), ( ), ( ) *D iK iO i B jw B jw B jw* The disturbance, tracking, and optimal bounds on ( )*<sup>i</sup> L j*

The magnitude variation due to the plant parameter uncertainty

Fig. 29. Responses of sideslip angle, roll angle, yaw angle rate and roll angle rate when



Roll angle rate/degree/second


0

5 x 10-3

0 2 4 6 8 10

for the MISO

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

Time/second

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -1

Time/second

**6. Summary** 

method.

*R* A set of *TR*

*D* A set of *TD*

A set of *P*

system 

( ) *P j*

**7. Symbols & terminology** 

*P* MISO plant with uncertainty

inputs and more than one output

*<sup>h</sup>* The frequency bandwidth

*TR* Acceptable command or tracking input-output responses

*TD* Acceptable disturbance input-output responses

external disturbance inputs, and a single output

0 1

Yaw angle rate/degree/second

Fig. 28. Responses of sideslip angle, roll angle, yaw angle rate and roll angle rate when 0 5

Fig. 29. Responses of sideslip angle, roll angle, yaw angle rate and roll angle rate when 0 1

## **6. Summary**

66 Automatic Flight Control Systems – Latest Developments

0



Roll angle/degree

(a) Sideslip angle (b) Roll angle



Roll angle rate/degre/second

(c) Yaw angle rate (d) Roll angle rate

Fig. 28. Responses of sideslip angle, roll angle, yaw angle rate and roll angle rate when

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5 -100

0

100

1

2

Roll angle/degree

(a) Sideslip angle (b) Roll angle

3

4 5

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> -1

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

> flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

Time/second

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -500

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> -1.2

time/second

Time/second

flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

> flight state 1 flight state 2 flight state 3 flight state 4 flight state 5

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> -0.08

Time/second

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -1

Time/second

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> -1

Time/second


0

1

2

Roll angle/degree

3

4

5

0 5

Yaw angle rate/degree/second



Sideslip angle/degree

0

0.02

0.04

This chapter is devoted to presenting an overview and in-depth expression of QFT in order to enhance the understanding and appreciation of the power of the QFT technique. Then, A QFT design of robust controller for a certain UAV's lateral motion, which is a MIMO system, is proposed base on BNIA principle in order to show how to apply QFT in flight control system of UAVs. Meantime, the simulation results show that the QFT controller own better robust stability and superior dynamic characteristics which verify the validity of presented method.

## **7. Symbols & terminology**


MIMO Multiple-input multiple-output; more than one tracking and/or external disturbance inputs and more than one output

MISO Multiple-input single-output; a system having one tracking input, one or more external disturbance inputs, and a single output

```
( ), ( ), ( ) D iK iO i B jw B jw B jw The disturbance, tracking, and optimal bounds on ( )i L j for the MISO 
system
```
*<sup>h</sup>* The frequency bandwidth

( ) *P j*The magnitude variation due to the plant parameter uncertainty

Quantitative Feedback Theory and Its Application in UAV's Flight Control 69

*RL <sup>T</sup>* The desired MISO tracking control ratio that satisfies the specified lower bound

The work of this chapter is supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2011GQ8005) and Northwestern Polytechnical University

Chen Huaimin: An Integrated QFT/EA Controller Design Method for a UAV's

Constantine H. Houpis,Steven J. Rasmussen. Quantitative Feedback Theory: Fundamentals

Horowitz I. M, and M. Sidi, "Synthesis of Feedback Systems with Large Plant Ignorance

Horowitz, I. M. and C. Loecher, "Design 3x3 Multivariable Feedback System with Large

Ibid, "Synthesis of Feedback Systems with Non-Linear Time Uncertain Plants to

Houpis, C. H. "Quantitative Feedback Theory (QFT) for the Engineer: A Paradigm for the

Houpis, C. H. and P. R. Chandler, Editors: "Quantitative Feedback Theory Symposium

O Yaniv,Y Chait:A Simplified Multi-Input Multi-Output Formulation for Quantitative

Reynolds, O. R., M Pachter, and C. H. Houpis. "Design of a Subsonic Flight Control System

Thompson, D. F., and O. D. I. Nwokah, "Optimal Loop Synthesis in Quantitative Feedback

American Control Conference, pp. 350-354,1994.

and Applications[M], Marcel Dekker, Inc. New York, Basel

Plant Uncertainty," hit. J. Control, vol. 33, pp. 677-699,1981.

Lateral Flight Control System, Mechanical Science and Technology, Vol.27-

for Prescribed Time Domain Tolerances," Int. J. of Control, vol. 16, pp 287-309,

Satisfy Quantitative Performance Specifications ," IEEE Proc., vol. 64, pp. 123-

Design of Control Systems for Uncertain Systems," WL-TR-95-3061, AF Wright Laboratory, Wright-Patterson AFB, OH, 1995 (Available from National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22151, document

Proceedings," WL-TR-92-3063, Wright Laboratories, Wright-Patterson AFB,OH,

Feedback Theory, Journal of Dynamic Systems, Measurement, and Control,Vol.114-

for the Vista F-16 Using Quantitative Feedback Theory, "Proceedings of the

Theory," Proceedings, of the American Control Conference, San Diego, CA, pp.626-

*<sup>D</sup> T* The desired MISO disturbance control ratio which satisfies the specified FOM

Foundation for Fundamental Research (No. :NPU-FFR-JC20100216)

FOM

UAV Unmanned Aerial Vehicle BNIA Basically Non-interacting

3(2008),p. 413.

1973.

130,1976.

1992.

6(1998),p.179.

631,1990.

number AD-A297571.)

**8. Acknowledgement** 

**9. References** 

*Lm* Log magnitude

LTI Linear-time-invariant

FOM figure of merit

*<sup>b</sup>* The symbol for bandwidth frequency of the models

*<sup>m</sup>* The resonant frequency

, *<sup>i</sup>* Phase margin frequency for a MISO system and for the *th i* loop of a MIMO system, respectively

*<sup>s</sup>* Sampling frequency

, {} *RR ri* The tracking input for a MISO system and the tracking input vector for a MIMO system, respectively

*<sup>U</sup> RU B Lm T* The Lm of the desired tracking control ratio for the upper bound of the MISO system

*<sup>L</sup> RL B Lm T* The Lm of the desired tracking control ratio for the lower bound of the MISO system

*Bs* Stability bounds for the discrete design

( ) *D i j* The (upper) value of ( ) *D i Lm T j*for MISO system

( ) *hf <sup>i</sup> j* The dB difference between the augmented bounds of *BU* and *BL* in the high frequency range for a MISO system

( ) *R i j* The dB difference between *BU* and *BL* for a given frequency, for a MISO system

, {}*ij F F f* The prefilter for a MISO system and the mxm prefilter matrix for a MIMO system respectively

, {} *G G ij f* The compensator or controller for a MISO system and the mxm compensator or controller matrix for a MIMO system, respectively. For a diagonal matrix { } *G ij f*

, *i* The phase margin angle for the MISO system and for the *th i* loop of the MIMO system, respectively

*J* The number of plant transfer functions for a MISO system or plant matrix for a MIMO system that describes the region of plant parameter uncertainty where i = 1, 2.....J denotes the particular plant case in the region of plant parameter uncertainty

The excess of poles over zeros of a transfer function

, *<sup>o</sup> oi L L* The optimal loop transmission function for the MISO system and the *th i* loop of the MIMO system, respectively

, *ML Li M* The specified closed-loop frequency domain overshoots constraint for the MISO system and for the *th i* loop of a MIMO system, respectively. This overshoot constraint may be dictated by the phase margin angle for the specified loop transmission function

( ) *P <sup>i</sup> j* Script cap tee in conjunction with *P* denotes a template, i.e., ( ) *P <sup>i</sup> j* and ( ) *Q <sup>i</sup> j* frequency, for a MISO and MIMO plants respectively

*RU <sup>T</sup>* The desired MISO tracking control ratio that satisfies the specified upper bound FOM

*RL <sup>T</sup>* The desired MISO tracking control ratio that satisfies the specified lower bound FOM

*<sup>D</sup> T* The desired MISO disturbance control ratio which satisfies the specified FOM

UAV Unmanned Aerial Vehicle

BNIA Basically Non-interacting

## **8. Acknowledgement**

The work of this chapter is supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2011GQ8005) and Northwestern Polytechnical University Foundation for Fundamental Research (No. :NPU-FFR-JC20100216)

## **9. References**

68 Automatic Flight Control Systems – Latest Developments

, {} *RR ri* The tracking input for a MISO system and the tracking input vector for a MIMO

*<sup>U</sup> RU B Lm T* The Lm of the desired tracking control ratio for the upper bound of the MISO

*<sup>L</sup> RL B Lm T* The Lm of the desired tracking control ratio for the lower bound of the MISO

for MISO system

, {}*ij F F f* The prefilter for a MISO system and the mxm prefilter matrix for a MIMO system

, {} *G G ij f* The compensator or controller for a MISO system and the mxm compensator or

*J* The number of plant transfer functions for a MISO system or plant matrix for a MIMO system that describes the region of plant parameter uncertainty where i = 1, 2.....J denotes

, *ML Li M* The specified closed-loop frequency domain overshoots constraint for the MISO

*RU <sup>T</sup>* The desired MISO tracking control ratio that satisfies the specified upper bound

*i* loop of a MIMO system, respectively. This overshoot constraint may

The dB difference between the augmented bounds of *BU* and *BL* in the high

*i* loop of a MIMO system,

, for a MISO system

*i* loop of the MIMO system,

*i* loop of the

 and ( ) *Q <sup>i</sup> j*

*Lm* Log magnitude LTI Linear-time-invariant FOM figure of merit

*<sup>m</sup>* The resonant frequency

*<sup>s</sup>* Sampling frequency

*Bs* Stability bounds for the discrete design

frequency range for a MISO system

The (upper) value of ( ) *D i Lm T j*

system, respectively

*<sup>b</sup>* The symbol for bandwidth frequency of the models

*<sup>i</sup>* Phase margin frequency for a MISO system and for the *th*

The dB difference between *BU* and *BL* for a given frequency

The phase margin angle for the MISO system and for the *th*

the particular plant case in the region of plant parameter uncertainty

The excess of poles over zeros of a transfer function

frequency, for a MISO and MIMO plants respectively

controller matrix for a MIMO system, respectively. For a diagonal matrix { } *G ij f*

, *<sup>o</sup> oi L L* The optimal loop transmission function for the MISO system and the *th*

be dictated by the phase margin angle for the specified loop transmission function

Script cap tee in conjunction with *P* denotes a template, i.e., ( ) *P <sup>i</sup> j*

, 

system

system

( ) *D i j*

( ) *hf <sup>i</sup> j*

( ) *R i j*

, *i* 

( ) *P <sup>i</sup> j*

FOM

respectively

respectively

MIMO system, respectively

system and for the *th*

respectively


**3** 

*Italy* 

**Gain Tuning of Flight Control Laws for** 

Urbano Tancredi1 and Federico Corraro2

*1University of Naples Parthenope 2Italian Aerospace Research Centre* 

**Satisfying Trajectory Tracking Requirements** 

The present chapter is concerned with presenting an approach for the synthesis of a gainscheduled flight control law that assures compliance to trajectory tracking requirements. More precisely, a strategy is proposed for improving the tracking performances of a baseline controller, obtained by conventional synthesis techniques, by tuning its gains. The approach is specifically designed for atmospheric re-entry applications, in which gain scheduled flight

Gain-scheduling design approaches conventionally construct a nonlinear controller by combining the members of an appropriate family of linear time-invariant (LTI) controllers (Leith & Leithead, 2000). The time-invariant feedback laws usually share the same structure, and differ only for the values of some tunable parameters, most notably the controller's gains. These gains are generally determined taking advantage of well-assessed LTI-based design techniques, such as pole placement and gain/phase margin methods. However, once a set of LTI feedback laws is specified, the nonlinear controller must be synthesized, which requires an additional design step. This step is of considerable importance since the choice of nonlinear controller realization can greatly influence the closed loop performance (Leith & Leithead, 2000). Furthermore, actual mission requirements constraint quantitatively the time response of the augmented system (Crespo et al., 2010), e.g. by imposing tracking requirements of a reference trajectory or requiring relevant output variables to be enclosed within a limited flight envelope. As such, the final gain-scheduled controller's performances are ascertained by means of numerical simulation based methods, most notably Monte Carlo, which can highlight limitations that were not apparent in the LTI design phase. As a result, in these cases one is forced to iterate the LTI design, but using analysis results that refer to the nonlinear controller rather than to the LTI ones, further complicating the design

Several methods have been proposed in the open literature both for taking into account explicitly the complex dependency of the final controller response from its gains and for dealing with quantitative performance requirements, such as tracking errors. Most, if not all, proposed approaches formulate the design task as an optimization problem, in which the merit function evaluation requires numerical simulation of the augmented system's timeresponse. For instance, (Crespo et al., 2008) develops optimization-based strategies for

**1. Introduction** 

improvement task.

control laws are typically used.

Trosen, D. W., M, Pachter, and C. H. Houpis, "Formation Flight Control Automation," Proceedings of the American Institute of Aeronautics and Astronautics (AIAA) Conference, pp. 1379-1404, Scottsdale, AZ, 1994.
