**4. Robust decentralized PID controller design in the frequency domain**

This section deals with an original frequency domain robust decentralized controller design methodology applicable for uncertain systems described by a set of transfer function matrices. The design methodology is based on the Equivalent Subsystems Method (ESM) - a frequency domain decentralized controller design technique to guarantee stability and specified performance of multivariable systems and is applicable for both continuous- and discrete-time controller designs (Kozáková et al., 2009). In contrast to the two stage robust decentralized controller design method based on the M-structure stability conditions (Kozáková & Veselý, 2009), the recent innovation (Kozáková et al., 2011) consists in that robust stability conditions are directly integrated into the ESM, thus providing a one-step (direct) robust decentralized controller design for robust stability and plant-wide performance.

#### **4.1 Preliminaries and problem formulation**

Consider a MIMO system described by a transfer function matrix ( ) *m m Gs R* and a controller ( ) *m m Rs R* in the standard feedback configuration according to Fig. 5,

Fig. 5. Standard feedback configuration

where w, u, y, e, d are respectively vectors of reference, control, output, control error and disturbance of compatible dimensions. Necessary and sufficient conditions for closed-loop stability are given by the Generalized Nyquist Stability Theorem applied to the closed-loop characteristic polynomial

$$\det F(\mathbf{s}) = \det[I + Q(\mathbf{s})] \tag{40}$$

where *Qs GsRs* () () () *m m R* is the open-loop transfer function matrix.

Characteristic functions of *Q s*( ) are the set of m algebraic functions ( ), 1,..., *<sup>i</sup> qs i m* defined as follows:

$$\det[q\_i(\mathbf{s})I\_m - \mathbb{Q}(\mathbf{s})] = 0 \quad \text{i} = 1, \ldots, m \tag{41}$$

Characteristic loci (CL) are the set of loci in the complex plane traced out by the characteristic functions of Q(s), *s j* .

*Theorem 4.1* (Generalized Nyquist Stability Theorem)

The closed-loop system in Fig. 1 is stable if and only if

Robust Decentralized PID Controller Design 151

*u*

Fig. 6. Standard feedback configuration with additive perturbation (left) recast into the

According to the general robust stability condition (Skogestad & Postlethwaite, 2009), if both

For individual uncertainty forms ( ) ( ), , *M k k s M s k ao* the corresponding matrices ( ) *Mk s* are given by (49) and (50), respectively (disregarding negative signs which do not affect resulting robustness condition). The nominal model 0 *G s*( ) is usually obtained as a

Consider an uncertain system that consists of m subsystems and is given as a set of N transfer function matrices obtained in N working points of plant operation. Let the uncertain system be described by a nominal model 0 *G s*( ) and any unstructured uncertainty

is to be designed to guarantee stability over the whole operating range of the plant specified by (46) or (47) (robust stability) and a specified plant-wide performance (nominal

 [ ( )] 1 , *M j*

( )*s* : max 

 

 

1 <sup>0</sup> ( ) ( ) ( )[ ( ) ( )] ( ) ( ) *<sup>M</sup> <sup>a</sup> a a s sRs I G sRs sM s* (49)

1 0 0 ( ) ( ) ( ) ( )[ ( ) ( )] ( ) ( ) *<sup>M</sup> <sup>o</sup> o o s sG sRs I G sRs sM s* (50)

0() () () *Gs Gs G s d m* (51)

<sup>0</sup> *G s Gs Gs m d* () () () (53)

( ) { ( )} , det ( ) 0 *G s diag G s G s <sup>d</sup> i mm <sup>d</sup>* (52)

( ) { ( )} *R s diag R s i mm* det ( ) 0 *R s* (54)

() 1 if and only if

*u*

(48)

( )*s* are stable, the *M*

*M(s)* 

*(s)* 

*y*

*w e y* 

a

*R(s) G0(s)* 

system in Fig. 2 is stable for all perturbations

*y*

the nominal closed-loop system M(s) and the perturbations

form (46), (47). Consider the following splitting of 0 *G s*( ) **:**

max 

*(s)*


structure (right)

model of mean parameter values.

**4.1.1 Problem formulation** 

A decentralized controller

performance).

where

*M* 

$$\text{a. } \det F(s) \neq 0 \qquad \forall s$$

$$\text{b. } N[0, \det F(\mathbf{s})] = \sum\_{i=1}^{m} N\{0, [1 + q\_i(\mathbf{s})] \} = n\_q$$

where *Fs I Qs* ( ) ( ( )) and nq is the number of unstable poles of Q(s). Let the uncertain plant be given as a set of N transfer function matrices

$$\text{If } \Pi = \{\mathbf{G}^k(\mathbf{s})\}, k = 1, 2, \dots, N \quad \text{where} \quad \mathbf{G}^k(\mathbf{s}) = \left\langle \mathbf{G}^k\_{ij}(\mathbf{s}) \right\rangle\_{m \times m} \tag{43}$$

The simplest uncertainty model is the unstructured perturbation. A set of unstructured perturbations DU is defined as

$$D\_{\rm II} \coloneqq \{ E(jo) \colon \sigma\_{\rm max}[E(jo)] \le \ell(o) \}, \quad \ell(o) = \max\_{k} \ \sigma\_{\rm max}[E(jo)] \} \tag{44}$$

where ( ) is a scalar weight on the norm-bounded perturbation *m m s R* , max [ ( )] 1 *j* over given frequency range, max ( ) is the maximum singular value of (.), hence

$$E(j\alpha) = \ell(\alpha)A(j\alpha) \tag{45}$$

Using unstructured perturbation, the set can be generated by either additive (Ea), multiplicative input (Ei) or multiplicative output (Eo) uncertainties, or their inverse counterparts (Skogestad & Postlethwaite, 2009) thus specifying pertinent uncertainty regions. In the sequel, just additive (a) and multiplicative output (o) perturbations will be considered; results for other uncertainty types can be obtained by analogy.

Denote *G s*( ) any member of a set of possible plants , , *<sup>k</sup> k ai* ; 0 *G s*( ) the nominal model used to design the controller, and ( ) *<sup>k</sup>* the scalar weight on a normalized perturbation. The sets *<sup>k</sup>* generated by the two considered uncertainty forms are: *Additive uncertainty:* 

$$\begin{aligned} \Pi\_a &:= \{ \mathbf{G}(\mathbf{s}) : \mathbf{G}(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) + E\_a(\mathbf{s}), E\_a(joo) \le \ell\_a(oo)A(joo) \} \\ \ell\_a(oo) &= \max\_k \sigma\_{\max} \{ \mathbf{G}^k(joo) - \mathbf{G}\_0(joo) \}, \ k = 1, 2, \dots, N \end{aligned} \tag{46}$$

*Multiplicative output uncertainty:* 

$$\begin{aligned} \Pi\_o &:= \{ \mathbf{G}(s) : \mathbf{G}(s) = [I + \mathbf{E}\_o(s)] \mathbf{G}\_0(s), \ A(joo) \le \ell\_o(joo)A(joo) \} \\ \mathcal{L}\_o(oo) &= \max\_k \sigma\_{\max} \{ [\mathbf{G}^k(joo) - \mathbf{G}\_0(joo)] \mathbf{G}\_0^{-1}(joo) \}, \ k = 1, 2, \dots, N \end{aligned} \tag{47}$$

Standard feedback configuration with uncertain plant modelled using any unstructured uncertainty form can be recast into the *M* structure (for additive perturbation see Fig. 6) where M(s) is the nominal model and *s* is the norm-bounded complex perturbation.

a. det ( ) 0 *Fs s*

1 [0,det ( )] {0,[1 ( )]} *m*

*Gs k N* where () () *k k*

 

The simplest uncertainty model is the unstructured perturbation. A set of unstructured

max max *<sup>U</sup>* : { ( ) : [ ( )] ( ), ( ) max [ ( )]} *<sup>k</sup>*

*Ej j* ( ) ( )( )

multiplicative input (Ei) or multiplicative output (Eo) uncertainties, or their inverse counterparts (Skogestad & Postlethwaite, 2009) thus specifying pertinent uncertainty regions. In the sequel, just additive (a) and multiplicative output (o) perturbations will be

: { ( ) : ( ) ( ) ( ), ( ) ( ) ( )}

0 <sup>1</sup> max 0 0 : { ( ) : ( ) [ ( )] ( ), ( ) ( ) ( )} ( ) max {[ ( ) ( )] ( )}, 1,2, ,

*Gs Gs I E s G s j j j*

Standard feedback configuration with uncertain plant modelled using any unstructured

*Gs Gs G s E s E j j*

( ) max [ ( ) ( )], 1,2, , *a aa a*

 

*Gj Gj k N*

 

(46)

*Gj Gj G j k N*

(47)

considered; results for other uncertainty types can be obtained by analogy.

0 max 0

*o oo*

 

 

Denote *G s*( ) any member of a set of possible plants , , *<sup>k</sup>*

*<sup>k</sup> <sup>a</sup> <sup>k</sup>*

*<sup>k</sup> <sup>o</sup> <sup>k</sup>*

   

*<sup>k</sup>* generated by the two considered uncertainty forms are:

is a scalar weight on the norm-bounded perturbation *m m*

 

*i N Fs N q s n* 

where *Fs I Qs* ( ) ( ( )) and nq is the number of unstable poles of Q(s). Let the uncertain plant be given as a set of N transfer function matrices

{ ( )}, 1,2,..., *<sup>k</sup>*

*D Ej Ej* 

*j* over given frequency range, max

*i q*

 

*ij m m Gs Gs* (43)

 *E j* (44)

( ) is the maximum singular value of (.),

can be generated by either additive (Ea),

the scalar weight on a normalized perturbation. The

 

 

structure (for additive perturbation see

*s* is the norm-bounded complex

(45)

*k ai* ; 0 *G s*( ) the nominal model

*s R* ,

(42)

b.

Using unstructured perturbation, the set

used to design the controller, and ( ) *<sup>k</sup>*

*Multiplicative output uncertainty:* 

uncertainty form can be recast into the *M*

Fig. 6) where M(s) is the nominal model and

perturbations DU is defined as

where ( )

 [ ( )] 1 

max 

hence

sets 

*Additive uncertainty:* 

perturbation.

Fig. 6. Standard feedback configuration with additive perturbation (left) recast into the *M* structure (right)

According to the general robust stability condition (Skogestad & Postlethwaite, 2009), if both the nominal closed-loop system M(s) and the perturbations ( )*s* are stable, the *M* system in Fig. 2 is stable for all perturbations ( )*s* : max () 1 if and only if

$$
\sigma\_{\text{max}}[M(j\rho)] < 1 \quad \forall \, \rho \tag{48}
$$

For individual uncertainty forms ( ) ( ), , *M k k s M s k ao* the corresponding matrices ( ) *Mk s* are given by (49) and (50), respectively (disregarding negative signs which do not affect resulting robustness condition). The nominal model 0 *G s*( ) is usually obtained as a model of mean parameter values.

$$M(\mathbf{s}) = \ell\_a(\mathbf{s})R(\mathbf{s})[I + \mathbf{G}\_0(\mathbf{s})R(\mathbf{s})]^{-1} = \ell\_a(\mathbf{s})M\_a(\mathbf{s})\tag{49}$$

$$M\text{(s)} = \ell\_o \text{(s)} G\_0 \text{(s)} R \text{(s)} \text{[}I + G\_0 \text{(s)} R \text{(s)}\text{]}^{-1} = \ell\_o \text{(s)} M\_o \text{(s)}\tag{50}$$

#### **4.1.1 Problem formulation**

Consider an uncertain system that consists of m subsystems and is given as a set of N transfer function matrices obtained in N working points of plant operation. Let the uncertain system be described by a nominal model 0 *G s*( ) and any unstructured uncertainty form (46), (47). Consider the following splitting of 0 *G s*( ) **:**

$$\mathbf{G}\_0(\mathbf{s}) = \mathbf{G}\_d(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s}) \tag{51}$$

where

$$G\_d(\mathbf{s}) = \operatorname{diag} \{ G\_i(\mathbf{s}) \}\_{m \times m'} \qquad \det G\_d(\mathbf{s}) \neq 0 \tag{52}$$

$$\mathbf{G}\_m(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) - \mathbf{G}\_d(\mathbf{s}) \tag{53}$$

A decentralized controller

$$R(s) = \operatorname{diag} \{ R\_i(s) \}\_{m \times m} \quad \text{det} \, R(s) \neq 0 \tag{54}$$

is to be designed to guarantee stability over the whole operating range of the plant specified by (46) or (47) (robust stability) and a specified plant-wide performance (nominal performance).

Robust Decentralized PID Controller Design 153

characteristic functions of [ ( )] *G s <sup>m</sup>* (the set of characteristic functions are denoted

*m km k i*

det[ ( ) ( )] det{ [ ( ) ( )] ( )}

is a diagonal matrix of m equivalent subsystems generated as follows

*I GsRs I G s G s Rs*

1 det[ ( ) ( )] det[ ] [ ( ) ( )] 0, 1,2,... *m*

(57)

( ) 1 ( ) ( ) 1,2,..., *eq eq CLCP s R s G s i m i i <sup>i</sup>* (61)

(58)

(62)

*i Ps G s pI G g s g s k m* 

With respect to stability, the interactions matrix *G s <sup>m</sup>*( ) can thus be replaced by [-P(s)]

det[ ( ) ( ) ( )]det ( ) det[ ( ) ( )]

( ) { ( )} *eq eq G s diag G s <sup>i</sup> m m* (59)

( ) ( ) ( ), 1,2, , *eq G s Gs g s i m ik i k* (60)

As all matrices are diagonal, on subsystems level (58) breaks down into m equivalent closed-

Considering (58)-(61), stability conditions stated in the Generalized Nyquist Stability

The closed-loop in Fig. 3 comprising the system (51) and the decentralized controller (54) is

1. det[ ( ) ] 0, *p sI G k m* for a fixed *k m* {1,..., }

2. all equivalent characteristic polynomials (61) have roots with Re{ } 0 *s* ;

3. [0,det ( )] *N Fs n <sup>q</sup>*

where N[0,g(s)] is number of anticlockwise encirclements of the complex plane origin by the

The decentralized controller design technique for nominal stability resulting from *Corollary 4.1* enables to independently design stabilizing local controllers for individual single inputsingle output equivalent subsystems using any standard frequency-domain design method, e.g. (Bucz et al., 2010; Drahos, 2000). In the originally developed ESM version (Kozáková et al., 2009) it was proved that local controllers tuned for a specified feasible degree-of-stability of equivalent subsystems constitute the decentralized controller guaranteeing the same

stable if and only if there exists a diagonal matrix () ()() *Ps p sIs <sup>k</sup>* such that

Nyquist plot of g(s); *nq* is number of open loop poles with Re{ } 0 *s* .

*R s G s Ps Rs I G sRs*

*d m eq <sup>d</sup>*

( ), 1,2,..., *<sup>i</sup> gsi m* ); thus

where

yielding the important relationship

loop characteristic polynomials (CLCP)

Theorem modify as follows:

*Corollary 4.1* 

1

To solve this problem, a frequency domain robust decentralized controller design technique has been developed (Kozáková and Veselý, 2009; Kozáková et. al., 2011); the core of it is the Equivalent Subsystems Method (ESM).

### **4.2 Decentralized controller design for performance: Equivalent Subsystems Method**

The Equivalent Subsystems Method (ESM) is a Nyquist-based technique to design decentralized controller for stability and specified plant-wide performance. According to it, local controllers ( ), 1,..., *Rsi m <sup>i</sup>* are designed independently for so-called equivalent subsystems obtained from frequency responses of decoupled subsystems by shaping each of them using one of m characteristic loci of the interactions matrix Gm(s). If local controllers are independently tuned for specified degree-of-stability of equivalent subsystems, the resulting decentralized controller guarantees the same degree-of-stability plant-wide (Kozáková et al., 2009). Unlike standard robust approaches, the proposed technique considers full nominal model of mean parameter values, thus reducing conservatism of resulting robust stability conditions. In the context of robust decentralized controller design, the Equivalent Subsystems Method is directly applicable to design DC for the nominal model (Fig. 3).

Fig. 7. Standard feedback loop under decentralized controller

The key idea behind the method is factorization of the closed-loop characteristic polynomial (40) in terms of the nominal system (51) under the decentralized controller (54). Then

$$\det F(\mathbf{s}) = \det[\mathbf{R}^{-1}(\mathbf{s}) + \mathbf{G}\_d(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s})] \det \mathbf{R}(\mathbf{s}) \tag{55}$$

Denote the sum of the diagonal matrices in (55) as

$$R^{-1}(\mathbf{s}) + G\_d(\mathbf{s}) = P(\mathbf{s})\tag{56}$$

where ( ) { ( )} *P s diag p s i mm* .

In order to "counterbalance" interactions *G s <sup>m</sup>*( ) , consider the closed-loop being at the limit of instability and choose the diagonal matrix () () *Ps p sI <sup>k</sup>* to have identical entries pk(s); then by similarity with (41) the bracketed term in (55) defines the k-th of the m characteristic functions of [ ( )] *G s <sup>m</sup>* (the set of characteristic functions are denoted ( ), 1,2,..., *<sup>i</sup> gsi m* ); thus

$$\det[P(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s})] = \det[p\_k I + \mathbf{G}\_m] = \prod\_{i=1}^m [-\mathbf{g}\_k(\mathbf{s}) + \mathbf{g}\_i(\mathbf{s})] = \mathbf{0}, \ k = 1, 2, \dots \\ m \tag{57}$$

With respect to stability, the interactions matrix *G s <sup>m</sup>*( ) can thus be replaced by [-P(s)] yielding the important relationship

$$\begin{aligned} \det[I + G(s)R(s)] &= \det[I + [G\_d(s) + G\_m(s)]R(s)] = \\ &= \det[R^{-1}(s) + G\_d(s) - P(s)]\det R(s) = \det[I + G^{\circ \eta}(s)R(s)] \end{aligned} \tag{58}$$

where

152 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

To solve this problem, a frequency domain robust decentralized controller design technique has been developed (Kozáková and Veselý, 2009; Kozáková et. al., 2011); the core of it is the

**4.2 Decentralized controller design for performance: Equivalent Subsystems Method**  The Equivalent Subsystems Method (ESM) is a Nyquist-based technique to design decentralized controller for stability and specified plant-wide performance. According to it, local controllers ( ), 1,..., *Rsi m <sup>i</sup>* are designed independently for so-called equivalent subsystems obtained from frequency responses of decoupled subsystems by shaping each of them using one of m characteristic loci of the interactions matrix Gm(s). If local controllers are independently tuned for specified degree-of-stability of equivalent subsystems, the resulting decentralized controller guarantees the same degree-of-stability plant-wide (Kozáková et al., 2009). Unlike standard robust approaches, the proposed technique considers full nominal model of mean parameter values, thus reducing conservatism of resulting robust stability conditions. In the context of robust decentralized controller design, the Equivalent Subsystems Method is directly applicable to design DC for the nominal

*<sup>w</sup>***e** *u* **+ +** *<sup>y</sup>*

The key idea behind the method is factorization of the closed-loop characteristic polynomial

In order to "counterbalance" interactions *G s <sup>m</sup>*( ) , consider the closed-loop being at the limit of instability and choose the diagonal matrix () () *Ps p sI <sup>k</sup>* to have identical entries pk(s); then by similarity with (41) the bracketed term in (55) defines the k-th of the m

(40) in terms of the nominal system (51) under the decentralized controller (54). Then

1

*Gd(s)* 

**G11 0 … 0 0 G22 … 0 ………………... 0 0 … Gmm**

<sup>1</sup> det ( ) det[ ( ) ( ) ( )]det ( ) *Fs R s G s G s Rs d m* (55)

() () () *R s G s Ps <sup>d</sup>* (56)

*Gm(s)* 

**0 G12 … G1m G21 0 … G2m ……………….… Gm1 Gm2 … 0**

*G0(s)* 

Equivalent Subsystems Method (ESM).

**-** 

*R(s)* 

Fig. 7. Standard feedback loop under decentralized controller

Denote the sum of the diagonal matrices in (55) as

where ( ) { ( )} *P s diag p s i mm* .

**R1 0 … 0 0 R2 … 0 ……………….. 0 0 … Rm** 

model (Fig. 3).

$$\mathbf{G}^{\alpha \dagger}(\mathbf{s}) = \operatorname{diag} \{ \mathbf{G}\_i^{\alpha \dagger}(\mathbf{s}) \}\_{m \times m} \tag{59}$$

is a diagonal matrix of m equivalent subsystems generated as follows

$$\mathbf{G}\_{lk}^{\prime \neq l}(\mathbf{s}) = \mathbf{G}\_l(\mathbf{s}) + \mathbf{g}\_k(\mathbf{s}), \qquad i = 1, 2, \dots, m \tag{60}$$

As all matrices are diagonal, on subsystems level (58) breaks down into m equivalent closedloop characteristic polynomials (CLCP)

$$\text{CLCP}\_{i}^{\text{eq}}\text{(s)} = 1 + R\_{i}(\text{s})\text{G}\_{i}^{\text{eq}}\text{(s)}\qquad i = 1, 2, \dots, m \tag{61}$$

Considering (58)-(61), stability conditions stated in the Generalized Nyquist Stability Theorem modify as follows:

#### *Corollary 4.1*

The closed-loop in Fig. 3 comprising the system (51) and the decentralized controller (54) is stable if and only if there exists a diagonal matrix () ()() *Ps p sIs <sup>k</sup>* such that

$$\mathbf{1.}\ \det[p\_k(\mathbf{s})\mathbf{I} + \mathbf{G}\_m] = \mathbf{0}, \quad \text{for a fixed} \quad k \in \{\mathbf{1}, \dots, m\}$$

2. all equivalent characteristic polynomials (61) have roots with Re{ } 0 *s* ; (62)

$$\text{3. } N[0, \det F(s)] = n\_q$$

where N[0,g(s)] is number of anticlockwise encirclements of the complex plane origin by the Nyquist plot of g(s); *nq* is number of open loop poles with Re{ } 0 *s* .

The decentralized controller design technique for nominal stability resulting from *Corollary 4.1* enables to independently design stabilizing local controllers for individual single inputsingle output equivalent subsystems using any standard frequency-domain design method, e.g. (Bucz et al., 2010; Drahos, 2000). In the originally developed ESM version (Kozáková et al., 2009) it was proved that local controllers tuned for a specified feasible degree-of-stability of equivalent subsystems constitute the decentralized controller guaranteeing the same

Robust Decentralized PID Controller Design 155

3. Specification of a minimum phase margin PM for equivalent subsystems using (63). 4. Local controller design for specified PM in equivalent subsystems using appropriate

**4.3 Decentralized controller design for robust stability using the Equivalent** 

5. Verification of achieved performance by evaluating frequency domain performance

In the context of robust control approach, the ESM method in its original version is inherently appropriate to design decentralized controller guaranteeing stability and specified performance of the nominal model (nominal stability, nominal performance). If, in addition, the decentralized controller has to guarantee closed-loop stability over the whole operating range of the plant specified by the chosen uncertainty description (robust stability), the ESM can be used either within a two-stage design procedure or a direct design

1. Two stage robust decentralized controller design for robust stability and nominal

In the first stage, the decentralized controller for the nominal system is designed using ESM, afterwards, fulfilment of the M-stability condition (48) is examined; if satisfied, the design procedure stops, otherwise in the second stage the controller parameters are modified to satisfy robust stability conditions in the tightest possible way, or local controllers are

<sup>1</sup>

Expressions on the r.h.s. of (69) and (70) do not depend on a particular controller and can be

 

multiplicative output uncertainty (70)

*additive uncertainty* (69)

can be obtained using the singular value properties in manipulations of the M-condition (48) considering (49) or (50). The following bounds for the nominal complementary

00 0 *T s G sRs I G sRs* ( ) ( ) ( )[ ( ) ( )] (68)

redesigned using modified performance requirements (Kozáková & Veselý, 2009). 2. Direct decentralized controller design for robust stability and nominal performance By direct integration of robust stability condition (48) in the ESM, a "one-shot" design of local controllers for both nominal performance and robust stability can be carried out. In case of decentralized controller design for guaranteed maximum overshoot and specified settling time, the upper bound for the maximum peak of the nominal complementary

max , ts and MT using (66), (67).

1. Generating frequency responses of equivalent subsystems. 2. Specification of performance requirements in terms of

procedure for robust stability and nominal performance.

sensitivity over the given frequency range

sensitivity have been derived:

max 0

*T j*

max 0

<sup>1</sup> [ ( )] ( ) ( ) *<sup>O</sup> o*

evaluated prior to designing the controller. In this way, if

*MT max{ [T ( j )]} max* <sup>0</sup>

min 0

[ ( )] [ ( )] ( ) ( ) *<sup>A</sup> a G j T j <sup>L</sup>* 

 

*L*

*Design procedure:* 

**Subsystems Method** 

performance

frequency domain method.

measure and via simulation.

degree-of-stability plant-wide. To design local controllers of equivalent subsystems, the general conditions in *Corollary 4.1* allow using any frequency domain performance measure that can appropriately be interpreted for the full system. In the next subsection, the plant wide performance is specified in terms of maximum overshoot which is closely related to phase margins of equivalent subsystems.

#### **4.2.1 Decentralized controller design for guaranteed maximum overshoot and specified settling time**

The ESM can be applied to design decentralized controller to guarantee specified maximum overshoot of output variables of the multivariable system. The design procedure evolves from the known relationship between the phase margin (PM) and the maximum peak of the complementary sensitivity (Skogestad & Postlethwaite, 2009)

$$PM \ge 2 \arcsin\left(\frac{1}{2M\_T}\right) \ge \frac{1}{M\_T} [rad] \tag{63}$$

where

$$M\_T = \sigma\_{\text{max}}[T(j\rho)]\tag{64}$$

is the maximum peak of the complementary sensitivity T(s) defined as

$$T(s) = G(s)R(s)[I + G(s)R(s)]^{-1} \tag{65}$$

Relation between the maximum overshoot max and MT is given by (Bucz et al., 2010)

$$
\eta\_{\text{max}} \le \frac{1.18M\_T - \left| T(0) \right|}{\left| T(0) \right|} 100 \left[ \% \right] \tag{66}
$$

According to the ESM philosophy, local controllers are designed using frequency domain methods; if PID controller is considered, the most appropriate ones are e.g. the Bode diagram design or the Neymark D-partition method. If using the Bode diagram design, in addition to max it is also possible to specify the required settling time ts related with the closed-loop bandwidth frequency<sup>0</sup> defined as the gain crossover frequency. The following relations between ts and 0 are useful (Reinisch, 1974).

$$t\_s \approx \frac{3}{o\_0} \text{ for } M\_T \in \{1.3; 1.5\}$$

$$\frac{\pi}{t\_s} < o\_0 < \frac{4\pi}{t\_s} \tag{67}$$

In general, a larger bandwidth corresponds to a smaller rise time, since high frequency signals are more easily passed on to the outputs. If the bandwidth is small, the time response will generally be slow and the system will usually be more robust.

#### *Design procedure:*

154 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

degree-of-stability plant-wide. To design local controllers of equivalent subsystems, the general conditions in *Corollary 4.1* allow using any frequency domain performance measure that can appropriately be interpreted for the full system. In the next subsection, the plant wide performance is specified in terms of maximum overshoot which is closely related to

The ESM can be applied to design decentralized controller to guarantee specified maximum overshoot of output variables of the multivariable system. The design procedure evolves from the known relationship between the phase margin (PM) and the maximum peak of the

**4.2.1 Decentralized controller design for guaranteed maximum overshoot and** 

1 1 2arcsin [ ] <sup>2</sup> *T T PM rad*

is the maximum peak of the complementary sensitivity T(s) defined as

max

0 are useful (Reinisch, 1974).

0 <sup>3</sup> *st* 

response will generally be slow and the system will usually be more robust.

*M M*

max[ ( )] *MT T j*

1.18 (0) 100[%] (0) *M T <sup>T</sup> T*

max it is also possible to specify the required settling time ts related with the

According to the ESM philosophy, local controllers are designed using frequency domain methods; if PID controller is considered, the most appropriate ones are e.g. the Bode diagram design or the Neymark D-partition method. If using the Bode diagram design, in

for (1.3; 1.5) *MT*

0 4

In general, a larger bandwidth corresponds to a smaller rise time, since high frequency signals are more easily passed on to the outputs. If the bandwidth is small, the time

*s s t t*

 

(63)

<sup>1</sup> *Ts GsRs I GsRs* ( ) ( ) ( )[ ( ) ( )] (65)

(66)

<sup>0</sup> defined as the gain crossover frequency. The following

(67)

max and MT is given by (Bucz et al., 2010)

(64)

complementary sensitivity (Skogestad & Postlethwaite, 2009)

phase margins of equivalent subsystems.

Relation between the maximum overshoot

**specified settling time** 

where

addition to

relations between ts and

closed-loop bandwidth frequency


#### **4.3 Decentralized controller design for robust stability using the Equivalent Subsystems Method**

In the context of robust control approach, the ESM method in its original version is inherently appropriate to design decentralized controller guaranteeing stability and specified performance of the nominal model (nominal stability, nominal performance). If, in addition, the decentralized controller has to guarantee closed-loop stability over the whole operating range of the plant specified by the chosen uncertainty description (robust stability), the ESM can be used either within a two-stage design procedure or a direct design procedure for robust stability and nominal performance.

1. Two stage robust decentralized controller design for robust stability and nominal performance

In the first stage, the decentralized controller for the nominal system is designed using ESM, afterwards, fulfilment of the M-stability condition (48) is examined; if satisfied, the design procedure stops, otherwise in the second stage the controller parameters are modified to satisfy robust stability conditions in the tightest possible way, or local controllers are redesigned using modified performance requirements (Kozáková & Veselý, 2009).

2. Direct decentralized controller design for robust stability and nominal performance

By direct integration of robust stability condition (48) in the ESM, a "one-shot" design of local controllers for both nominal performance and robust stability can be carried out. In case of decentralized controller design for guaranteed maximum overshoot and specified settling time, the upper bound for the maximum peak of the nominal complementary sensitivity over the given frequency range

$$M\_T = \max\_{\mathbf{o}} \{ \sigma\_{\max} \{ T\_0(j \mathbf{o}) \} \} \qquad T\_0(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) \mathbf{R}(\mathbf{s}) \mathbf{I} \mathbf{I} + \mathbf{G}\_0(\mathbf{s}) \mathbf{R}(\mathbf{s}) \}^{-1} \tag{68}$$

can be obtained using the singular value properties in manipulations of the M-condition (48) considering (49) or (50). The following bounds for the nominal complementary sensitivity have been derived:

$$\sigma\_{\text{max}}[T\_0(j\alpha)] < \frac{\sigma\_{\text{min}}[\mathcal{G}\_0(j\alpha)]}{\left|\ell\_d(o\rho)\right|} = L\_A(o) \qquad \forall \, o \quad additive \, uncertainty \tag{69}$$

$$\sigma\_{\text{max}}[T\_0(joo)] < \frac{1}{\left| \ell\_o(o) \right|} = \mathcal{L}\_0(o) \qquad \forall o \quad \text{multiplicative output uncertainty} \tag{70}$$

Expressions on the r.h.s. of (69) and (70) do not depend on a particular controller and can be evaluated prior to designing the controller. In this way, if

$$M\_T = \max\_{o \boldsymbol{o}} \{ \sigma\_{\max} [T\_0(jo)] \} \tag{71}$$

Robust Decentralized PID Controller Design 157

0 20 40 *s* 

where s is sampling frequency, and 0 is control system bandwidth, i.e. the maximum frequency at which the system output still tracks and input sinusoid in a satisfactory manner (Lian et al., 2002). A proper choice of sampling period is crucial for achievable bandwidth and feasibility of the required phase margin. Given a discrete-time transfer function *G z*( ) , the frequency response can be studied by plotting Nyquist or Bode plots of

. The discrete-time robust controller design for maximum overshoot and settling

**4.5 Decentralized discrete-time PID Controller design for the Quadruple tank process**  In the frequency domain, the direct robust decentralized PID design procedure has been applied for the transfer function matrix (3) identified in three working points within the minimum and nonminimum phase regions (7) and (8), respectively. In both cases the

From three plant models (3) evaluated in working points taken from the minimum phase

62 1 (23 1)(62 1) ( ) 1.5667 3.1333

(30 1)(90 1) 90 1

*ss s*

All three transfer function matrices were discretized using the sampling period

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>500</sup> <sup>0</sup>

time(s)

30 *T s <sup>S</sup>* chosen as approx. 1/10 of the settling time of plant step responses in Fig. 8.

2.4667 1.2333

*s ss*

uncertainty region as specified in (7), the resulting continuous-time nominal model is

( ) *<sup>j</sup> <sup>T</sup> z e G z* 

time is described in the next Section.

*Minimum phase configuration* 

nominal model is a mean value parameter model.

0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fig. 8. Step response of the quadruple tank process

Discrete-time transfer function matrix of the nominal plant is

y1,y2

*G s*

(72)

(73)

is used in the Design procedure, the resulting decentralized controller will simultaneously guarantee achieving the required maximum overshoot of all output variables (nominal performance) and stability over the whole operating range of the plant specified by selected working points (robust stability).

### **4.4 Discrete-time robust decentralized controller design using the Equivalent Subsystems Method**

Controllers for continuous-time plants are mostly implemented as discrete-time controllers. A common approach to discrete-time controller design is the continuous controller redesign i.e. conversion of the already designed continuous controller into its discrete counterpart. This approach, however, is only an approximate scheme; performance under these controllers deteriorates with increasing sampling period. This drawback may be improved by modifying the continuous controller design before it is discretized which can often allow significantly larger sampling periods (Lewis, 1992). Then, the ESM design methodology can be applied in a similar way as in the continuous-time case using discrete characteristic loci, discrete Nyquist plots and discrete Bode diagrams of equivalent subsystems. Local controllers designed as continuous-time ones are subsequently converted into their discretetime counterparts. Closed-loop performance under a discrete-time controller is verified using simulations and the discrete-time maximum singular value of the sensitivity *<sup>M</sup>*[ ( )] *S z* where

$$S(z) = \left[I + G(z)R(z)\right]^{-1}, \ z = e^{j\alpha T\_s} \tag{72}$$

The maximum singular value *j T max [S(e )] max* plotted as function of frequency should

be small at low frequencies where feedback is effective, and approach 1 at high frequencies, as the system is strictly proper, having a peak larger than 1 around the crossover frequency. The peak is unavoidable for real systems. Bandwidth frequency is

defined as frequency where [ ( )] *<sup>s</sup> j T <sup>M</sup> S e* crosses 0.7 from below (Skogestad & Postlethwaite, 2009). Similarly, a discretized version of robust stability conditions (69), (70) based on (46) and (47) is applied.

## **4.4.1 Design of continuous controllers for discretization**

The crucial step for the discrete controller design is proper choice of the sampling time T. Then, frequency response of the discretized system matches the one of the continuous time system up to a certain frequency / 2 *<sup>S</sup>* , and the discrete controller can be obtained by converting the continuous–time controller designed from the discrete frequency responses to its discrete-time counterpart.

The sampling period T is to be selected according to the Shannon-Kotelnikov sampling theorem, or using common rules of thumb, e.g. as ~ 1/10 of the settling time of the plant step response, or from control system bandwidth according to the relation

*MT* max{ [ ( )]} max 0 *T j* 

is used in the Design procedure, the resulting decentralized controller will simultaneously guarantee achieving the required maximum overshoot of all output variables (nominal performance) and stability over the whole operating range of the plant specified by selected

Controllers for continuous-time plants are mostly implemented as discrete-time controllers. A common approach to discrete-time controller design is the continuous controller redesign i.e. conversion of the already designed continuous controller into its discrete counterpart. This approach, however, is only an approximate scheme; performance under these controllers deteriorates with increasing sampling period. This drawback may be improved by modifying the continuous controller design before it is discretized which can often allow significantly larger sampling periods (Lewis, 1992). Then, the ESM design methodology can be applied in a similar way as in the continuous-time case using discrete characteristic loci, discrete Nyquist plots and discrete Bode diagrams of equivalent subsystems. Local controllers designed as continuous-time ones are subsequently converted into their discretetime counterparts. Closed-loop performance under a discrete-time controller is verified

  (71)

*<sup>M</sup>*[ ( )] *S z*

**4.4 Discrete-time robust decentralized controller design using the Equivalent** 

using simulations and the discrete-time maximum singular value of the sensitivity

<sup>1</sup> ( ) [ ( ) ( )] , *<sup>s</sup> <sup>j</sup> <sup>T</sup> Sz I GzRz z e*

be small at low frequencies where feedback is effective, and approach 1 at high frequencies, as the system is strictly proper, having a peak larger than 1 around the crossover frequency. The peak is unavoidable for real systems. Bandwidth frequency is

Postlethwaite, 2009). Similarly, a discretized version of robust stability conditions (69),

The crucial step for the discrete controller design is proper choice of the sampling time T. Then, frequency response of the discretized system matches the one of the continuous time

converting the continuous–time controller designed from the discrete frequency responses

The sampling period T is to be selected according to the Shannon-Kotelnikov sampling theorem, or using common rules of thumb, e.g. as ~ 1/10 of the settling time of the plant

 (72)

plotted as function of frequency should

crosses 0.7 from below (Skogestad &

*<sup>S</sup>* , and the discrete controller can be obtained by

working points (robust stability).

The maximum singular value *j T max [S(e )] max*

defined as frequency where [ ( )] *<sup>s</sup> j T <sup>M</sup> S e*

(70) based on (46) and (47) is applied.

system up to a certain frequency / 2

to its discrete-time counterpart.

**4.4.1 Design of continuous controllers for discretization** 

 

step response, or from control system bandwidth according to the relation

**Subsystems Method** 

where

$$20 < \frac{\alpha\_s}{\alpha\_0} < 40\tag{72}$$

where s is sampling frequency, and 0 is control system bandwidth, i.e. the maximum frequency at which the system output still tracks and input sinusoid in a satisfactory manner (Lian et al., 2002). A proper choice of sampling period is crucial for achievable bandwidth and feasibility of the required phase margin. Given a discrete-time transfer function *G z*( ) , the frequency response can be studied by plotting Nyquist or Bode plots of ( ) *<sup>j</sup> <sup>T</sup> z e G z* . The discrete-time robust controller design for maximum overshoot and settling time is described in the next Section.

#### **4.5 Decentralized discrete-time PID Controller design for the Quadruple tank process**

In the frequency domain, the direct robust decentralized PID design procedure has been applied for the transfer function matrix (3) identified in three working points within the minimum and nonminimum phase regions (7) and (8), respectively. In both cases the nominal model is a mean value parameter model.

#### *Minimum phase configuration*

From three plant models (3) evaluated in working points taken from the minimum phase uncertainty region as specified in (7), the resulting continuous-time nominal model is

$$\mathbf{G}\_{0}(\mathbf{s}) = \begin{bmatrix} \frac{2.4667}{62s+1} & \frac{1.2333}{(23s+1)(62s+1)}\\ \frac{1.5667}{(30s+1)(90s+1)} & \frac{3.1333}{90s+1} \end{bmatrix} \tag{73}$$

All three transfer function matrices were discretized using the sampling period 30 *T s <sup>S</sup>* chosen as approx. 1/10 of the settling time of plant step responses in Fig. 8.

Fig. 8. Step response of the quadruple tank process

Discrete-time transfer function matrix of the nominal plant is

Robust Decentralized PID Controller Design 159

Relevant parameters read form discrete Bode plots of uncompensated equivalent

subsystem PM Crossover

<sup>12</sup>( ) *eq G z* 53.90 0.048 rad/ s-1 <sup>22</sup>( ) *eq G z* 58.350 0.0448 rad/s-1

For both equivalent subsystems the required settling time and maximum overshoot were chosen with respect to plant dynamics: 600 , 1.05 *s T t sM* corresponding to max

Related values of other design parameters obtained from (63) and (67) respectively are:

*PM PM req* min was chosen <sup>0</sup> <sup>65</sup> *PMreq* . To design local controllers, Bode design procedure (Kuo, 2003) has been applied independently for each equivalent subsystem to achieve the

is designed. If 0 ( ) *PM PM*

designed. The resulting PID controller is obtained in the series form

. Achieved design results are summarized in Tab. 4.

subsyst. Ri(s) Ri(z) PMachieved achieved

Table 4. Design results and achieved frequency domain performance measures (minimum

1 1 0.199 0.082 ( ) <sup>1</sup> *<sup>z</sup> R z*

2 1 0.221 0.119 ( ) <sup>1</sup> *<sup>z</sup> R z*

*z*

*z*

is found on the magnitude Bode plot; if 0 ( ) *PM PM*

1

1

58.350 0.0122

65.70 0.0121

frequency

*req* , a PD controller

, and subsequently a PI controller is

0.0131 . The required phase margin

5% .

*req* , a

rad/s-1

rad/s-1

subsystems in Fig. 10 are summarized in Tab. 3.

0

required phase margin: 0 *PM*( )

PI controller ( ) *<sup>I</sup>*

( ) ( )(1 ) *<sup>I</sup> PID P D <sup>K</sup> G s K Ks s*

Eq.

<sup>12</sup>( ) *eq G z* <sup>1</sup>

<sup>22</sup>( ) *eq G z* <sup>2</sup>

phase configuration)

*PI P <sup>K</sup> Gs K*

0.0039 *R s*( ) 0.1988

0.0034 *R s*( ) 0.2212

min *PM* 56.87 and required crossover frequency <sup>0</sup>

*s*

*s*

*s*

() 1 *G s Ks PD <sup>D</sup>* is designed first, to provide 0 ( ) *PMreq*

Equivalent

Table 3. Relevant parameters of equivalent subsystems generated by g2(z)

$$G(z) = \begin{bmatrix} \frac{0.9462z^{-1}}{1 - 0.6164z^{-1}} & \frac{0.2221z^{-1} + 0.1226z^{-2}}{1 - 0.8877z^{-1} + 0.1673z^{-2}}\\ \frac{0.1710z^{-1} + 0.1097z^{-2}}{1 - 1.0840z^{-1} + 0.2636z^{-2}} & \frac{0.8882z^{-1}}{1 - 0.7165z^{-1}} \end{bmatrix} \tag{74}$$

From the discretized transfer function matrices and the nominal model (74), upper bounds for max 0 [ ( )] *T j*were evaluated according to (69) and (70).

Fig. 9. Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70)

Inspection of Fig. 9 reveals, that \_ min 0.77 1 *M L TA A* is not feasible for the local controller design (closed-loop design magnitude less than 1 does not guarantee proper setpoint tracking, even at =0); hence \_ min 1.22 *MM L T TO O* has been considered in the sequel.

Characteristic loci g1(z), g2(z) of Gm(z) were calculated; 2 *g z*( ) was selected to generate the equivalent subsystems according to (60). Bode plots of resulting equivalent subsystems are shown in Fig. 10.

Fig. 10. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left), <sup>22</sup>( ) *eq G z* (right) (min. phase case)

1 0.6164 1 0.8877 0.1673 ( ) 0.1710 0.1097 0.8882 1 1.0840 0.2636 1 0.7165

*z zz G z*

1 2 1 1 2 1

10-2

has been considered in the sequel.

0 10 20


Phase [deg]

Fig. 10. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left),

magnitude [dB]

evaluated according to (69) and (70)

omega [rad/s]

design (closed-loop design magnitude less than 1 does not guarantee proper setpoint tracking,

Characteristic loci g1(z), g2(z) of Gm(z) were calculated; 2 *g z*( ) was selected to generate the equivalent subsystems according to (60). Bode plots of resulting equivalent subsystems are

10-1

is not feasible for the local controller

10-4 10-3 10-2 10-1 <sup>100</sup> -10

10-4 10-3 10-2 10-1 <sup>100</sup> -200

omega [rad/s]

omega [rad/s]

 

*zz z zz z*

From the discretized transfer function matrices and the nominal model (74), upper bounds

LA LO

were evaluated according to (69) and (70).

10-3

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

even at =0); hence \_ min 1.22 *MM L T TO O*

10-4 10-3 10-2 10-1 <sup>100</sup> -5

omega [rad/s]

10-4 10-3 10-2 10-1 <sup>100</sup> -200

<sup>22</sup>( ) *eq G z* (right) (min. phase case)

omega [rad/s]

Inspection of Fig. 9 reveals, that \_ min 0.77 1 *M L TA A*

 [ ( )] *T j*

Fig. 9. Upper bounds for max 0

shown in Fig. 10.


Phase [deg]

magnitude [dB]

for max 0 

 [ ( )] *T j*

0.9462 0.2221 0.1226

1 1 2 1 1 2

(74)

 

*z zz*

Relevant parameters read form discrete Bode plots of uncompensated equivalent subsystems in Fig. 10 are summarized in Tab. 3.


Table 3. Relevant parameters of equivalent subsystems generated by g2(z)

For both equivalent subsystems the required settling time and maximum overshoot were chosen with respect to plant dynamics: 600 , 1.05 *s T t sM* corresponding to max 5% . Related values of other design parameters obtained from (63) and (67) respectively are: 0 min *PM* 56.87 and required crossover frequency <sup>0</sup> 0.0131 . The required phase margin *PM PM req* min was chosen <sup>0</sup> <sup>65</sup> *PMreq* . To design local controllers, Bode design procedure (Kuo, 2003) has been applied independently for each equivalent subsystem to achieve the required phase margin: 0 *PM*( ) is found on the magnitude Bode plot; if 0 ( ) *PM PM req* , a

PI controller ( ) *<sup>I</sup> PI P <sup>K</sup> Gs K s* is designed. If 0 ( ) *PM PM req* , a PD controller () 1 *G s Ks PD <sup>D</sup>* is designed first, to provide 0 ( ) *PMreq* , and subsequently a PI controller is designed. The resulting PID controller is obtained in the series form ( ) ( )(1 ) *<sup>I</sup> PID P D <sup>K</sup> G s K Ks s* . Achieved design results are summarized in Tab. 4.


Table 4. Design results and achieved frequency domain performance measures (minimum phase configuration)

Robust Decentralized PID Controller Design 161

yq g

5 0 dB


Achieved nominal performance was verified via plotting sensitivity magnitude plot in Fig.

10-4 10-3 10-2 10-1 <sup>100</sup> <sup>0</sup>

w [rad/sec]

Fulfilment of robust stability condition (70) is examined in Fig. 15. The closed-loop system is

Real Axis


around the crossover frequency proves good

6 dB 4 dB

2 dB



14. Sensitivity peak max{ [ ( )]} 2 *<sup>M</sup> S j* 

0.2

stable over the whole minimum phase region (7).


0.4

0.6

sigmaMax(S)

0.8

1

1.2

1.4

closed-loop performance.

Fig. 14. [ ( )] *<sup>M</sup> j T z e S z*

Fig. 13. Stability test using the Nyquist plot of det[ ( ) ( )] *I GzRz*

Imaginary Axis

Design results in Tab. 4 along with Bode plots of compensated equivalent subsystems in Fig.11 prove achieving required design parameters. Closed-loop step responses are in Fig. 12.

Fig. 11. Discrete Bode plots of equivalent subsystems under designed PI controllers: <sup>12</sup>( ) *eq G z* (left), 22( ) *eq G z* (right)

Fig. 12. Nominal closed-loop step responses of the quadruple tank process (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in both cases.

Nominal closed-loop stability was verified both by calculating closed-loop poles and using the Generalized Nyquist encirclement criterion (Fig. 13).

Roots\_of\_CLCP { 0.7019 0.2572i,0.8313, 0.7167, 0.7165, 0.6164, 0.3720, 0.2637

Design results in Tab. 4 along with Bode plots of compensated equivalent subsystems in Fig.11 prove achieving required design parameters. Closed-loop step responses are in

0


Fig. 12. Nominal closed-loop step responses of the quadruple tank process (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in

Nominal closed-loop stability was verified both by calculating closed-loop poles and using

Roots\_of\_CLCP { 0.7019 0.2572i,0.8313, 0.7167, 0.7165, 0.6164, 0.3720, 0.2637

y1,y2

Phase

Fig. 11. Discrete Bode plots of equivalent subsystems under designed PI controllers:

y1 y2 Magnitude

50

10-4 10-3 10-2 10-1 <sup>100</sup> -50

10-4 10-3 10-2 10-1 <sup>100</sup> -200

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>20</sup>

t

omega

y

y1 y2

Fig. 12.

0


both cases.

y1,y2 [cm]

Phase

Magnitude

50

10-4 10-3 10-2 10-1 <sup>100</sup> -50

10-4 10-3 10-2 10-1 <sup>100</sup> -200

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>100</sup> <sup>20</sup>

t [s]

the Generalized Nyquist encirclement criterion (Fig. 13).

<sup>12</sup>( ) *eq G z* (left), 22( ) *eq G z* (right)

omega

Fig. 13. Stability test using the Nyquist plot of det[ ( ) ( )] *I GzRz*

Achieved nominal performance was verified via plotting sensitivity magnitude plot in Fig. 14. Sensitivity peak max{ [ ( )]} 2 *<sup>M</sup> S j* around the crossover frequency proves good closed-loop performance.

Fig. 14. [ ( )] *<sup>M</sup> j T z e S z* - versus –frequency plot

Fulfilment of robust stability condition (70) is examined in Fig. 15. The closed-loop system is stable over the whole minimum phase region (7).

Robust Decentralized PID Controller Design 163

Obviously, proper setpoint tracking can be guaranteed for both uncertainty types, just on a limited frequency range. Hence, 1.05 *MT* and multiplicative output uncertainty will be

Bode plots of equivalent subsystems generated using <sup>2</sup> *g z*( ) are shown in Fig. 17, and their


Phase [deg]

Fig. 17. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left),

subsystem PM Crossover

<sup>22</sup>( ) *eq G z* 44.040 0.0344

<sup>12</sup> ( ) *eq G z* 43.810 0.040rad/s-1

For both equivalent subsystems the required settling time and maximum overshoot were chosen the same as in the minimum phase case: 600 , 1.05 *s T t sM* corresponding to

<sup>0</sup> <sup>60</sup> *PMreq* . Achieved design results are summarized in Tab. 6 and Bode plots of compensated equivalent subsystems in Fig.18 prove achieving required design

frequency

rad/s-1

0.0131 . The required phase margin *PM PM req* min was chosen

Equivalent

Table 5. Relevant parameters of equivalent subsystems generated by g2(z).

5% . Related values of other design parameters are: <sup>0</sup>

magnitude [dB]

10-4 10-3 10-2 10-1 <sup>100</sup> -20

10-4 10-3 10-2 10-1 <sup>100</sup> -200

omega [rad/s]

min *PM* 56.87 and required

omega [rad/s]

considered in the sequel.


max 

parameters.

crossover frequency <sup>0</sup>

Phase [deg]

magnitude [dB]

relevant parameters are summarized in Tab. 5.

10-4 10-3 10-2 10-1 <sup>100</sup> -20

10-4 10-3 10-2 10-1 <sup>100</sup> -200

<sup>22</sup>( ) *eq G z* (right) (non-minimum phase case)

omega [rad/s]

omega [rad/s]

Fig. 15. Verification of the robust stability condition max 0 <sup>1</sup> [ ( )] ( ) ( ) *<sup>O</sup> o T j L* 

#### *Non-minimum phase configuration*

To design robust decentralized PI controller for the non-minimum phase configuration, the continuous-time nominal model was evaluated for 1 2 , taken from the non-minimum phase uncertainty region (8) and interchanged columns of the transfer function matrix (due to opposite pairing as suggested in Section 2):

$$\mathbf{G}\_{0}(\mathbf{s}) = \begin{bmatrix} \frac{3.0830}{(23s+1)(62s+1)} & \frac{0.6167}{62s+1} \\ \frac{0.7833}{90s+1} & \frac{3.9170}{(30s+1)(90s+1)} \end{bmatrix} \tag{73}$$

Discrete-time transfer function matrix of the nominal plant obtained for 30 *T s <sup>S</sup>* is

$$\mathbf{G}(z) = \begin{bmatrix} \frac{0.5554z^{-1} + 0.3065z^{-2}}{1 - 0.8877z^{-1} + 0.1673z^{-2}} & \frac{0.2366z^{-1}}{1 - 0.6164z^{-1}}\\ \frac{0.2220z^{-1}}{1 - 0.7165z^{-1}} & \frac{0.4275z^{-1} + 0.2743z^{-2}}{1 - 1.0840z^{-1} + 0.2636z^{-2}} \end{bmatrix} \tag{74}$$

Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70) are in Fig. 16.

Fig. 16. Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70)

LO sigma max (T)

<sup>1</sup> [ ( )] ( ) ( ) *<sup>O</sup> o*

 

taken from the non-minimum

 *L*

(73)

(74)

<sup>10</sup>-4 <sup>10</sup>-3 <sup>10</sup>-2 <sup>10</sup>-1 <sup>10</sup><sup>0</sup> <sup>0</sup>

w [rad/sec]

To design robust decentralized PI controller for the non-minimum phase configuration, the

phase uncertainty region (8) and interchanged columns of the transfer function matrix (due

(23 1)(62 1) 62 1 ( ) 0.7833 3.9170

0.5554 0.3065 0.2366

Discrete-time transfer function matrix of the nominal plant obtained for 30 *T s <sup>S</sup>* is

1 0.8877 0.1673 1 0.6164 ( )

*zz z G z*

3.0830 0.6167

*ss s*

90 1 (30 1)(90 1)

*s ss*

1 2 1 1 2 1

 

*zz z*

<sup>10</sup>-4 <sup>10</sup>-3 <sup>10</sup>-2 <sup>10</sup>-1 <sup>10</sup><sup>0</sup> 0.5

omega [rad/s]

0.2220 0.4275 0.2743 1 0.7165 1 1.0840 0.2636

1 1 2 1 1 2

 

evaluated according to (69) and (70) are in Fig. 16.

evaluated according to (69) and (70)

LA LO

*z zz z zz*

*T j*

> ,

0.5

*Non-minimum phase configuration* 

Upper bounds for max 0

Fig. 16. Upper bounds for max 0

 [ ( )] *T j*

L

,LO

A

1 1.5 2 2.5 3 3.5 4 4.5

 [ ( )] *T j*

Fig. 15. Verification of the robust stability condition max 0

continuous-time nominal model was evaluated for 1 2

to opposite pairing as suggested in Section 2):

0

*G s*

1

1.5

2

2.5

Obviously, proper setpoint tracking can be guaranteed for both uncertainty types, just on a limited frequency range. Hence, 1.05 *MT* and multiplicative output uncertainty will be considered in the sequel.

Bode plots of equivalent subsystems generated using <sup>2</sup> *g z*( ) are shown in Fig. 17, and their relevant parameters are summarized in Tab. 5.

Fig. 17. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left), <sup>22</sup>( ) *eq G z* (right) (non-minimum phase case)


Table 5. Relevant parameters of equivalent subsystems generated by g2(z).

For both equivalent subsystems the required settling time and maximum overshoot were chosen the same as in the minimum phase case: 600 , 1.05 *s T t sM* corresponding to max 5% . Related values of other design parameters are: <sup>0</sup> min *PM* 56.87 and required crossover frequency <sup>0</sup> 0.0131 . The required phase margin *PM PM req* min was chosen <sup>0</sup> <sup>60</sup> *PMreq* . Achieved design results are summarized in Tab. 6 and Bode plots of compensated equivalent subsystems in Fig.18 prove achieving required design parameters.


Table 6. Design results and achieved frequency domain performance measures for the nonminimum phase case

Fig. 18. Discrete Bode plots of equivalent subsystems under designed PI controllers: <sup>12</sup> ( ) *eq G z* (left), 22( ) *eq G z* (right)

y

Robust Decentralized PID Controller Design 165

10-4 10-3 10-2 10-1 <sup>100</sup> <sup>0</sup>

w [rad/sec]

Fulfilment of robust stability condition (70) is examined in Fig. 21. The closed-loop system is

<sup>10</sup>-4 <sup>10</sup>-3 <sup>10</sup>-2 <sup>10</sup>-1 <sup>10</sup><sup>0</sup> <sup>0</sup>

w [rad/sec]

The robust decentralized PID controller design procedures have been developed both in frequency and time domains. The proposed controller design schemes are based on different principles, with the same control aim: to achieve robust stability and specified performance. The comparative study of both approaches is presented on robust decentralized discrete-

*T j*

LO sigma max (T)

<sup>1</sup> [ ( )] ( ) ( ) *<sup>O</sup> o*

 

 *L*

 around the

Roots\_of\_CLCP { 0.6768 0.2761i, 0.7335 0.2262i, 0. 7165, 0.6164, 0.5876, 0.3313

The sensitivity magnitude plot in Fig. 20 with the peak max{ [ ( )]} 2 *<sup>M</sup> S j*

crossover frequency proves good closed-loop nominal performance.

0.5

stable over the whole non-minimum phase region (8).

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Fig. 21. Verification of the robust stability condition max 0


sigmaMax(S)

Fig. 20. [ ( )] *<sup>M</sup> j T z e S z*

**5. Conclusion** 

1

1.5

Fig. 19. Nominal closed-loop step responses of the quadruple tank system in non-minimum phase configuration (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in both cases.

Nominal closed-loop poles verify nominal stability.

subsyst. Ri(s) Ri(z) PMachieved achieved

Table 6. Design results and achieved frequency domain performance measures for the non-

1 1 0.2083 0.0923 ( ) <sup>1</sup> *<sup>z</sup> R z*

2 1 0.2376 0.1832 ( ) <sup>1</sup> *<sup>z</sup> R z*

0


Phase

Fig. 18. Discrete Bode plots of equivalent subsystems under designed PI controllers:

y1 y2

y1,y2 [cm]

Fig. 19. Nominal closed-loop step responses of the quadruple tank system in non-minimum phase configuration (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem,

and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum

Magnitude

50

*z*

*z*

1

1

54.170 0.0122

56.89 0.0120

10-4 10-3 10-2 10-1 <sup>100</sup> -50

10-4 10-3 10-2 10-1 <sup>100</sup> -200

omega

y

y1 y2

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>20</sup>

t[s]

rad/s-1

rad/s-1

Eq.

<sup>12</sup> ( ) *eq G z* <sup>1</sup>

<sup>22</sup>( ) *eq G z* <sup>2</sup>

0


y1,y2

Phase

Magnitude

50

minimum phase case

0.0039 *R s*( ) 0.2083

0.0030 *R s*( ) 0.2376

10-4 10-3 10-2 10-1 <sup>100</sup> -50

10-4 10-3 10-2 10-1 <sup>100</sup> -200

<sup>12</sup> ( ) *eq G z* (left), 22( ) *eq G z* (right)

omega

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>100</sup> <sup>20</sup>

t

Nominal closed-loop poles verify nominal stability.

overshoot and settling time (600s) were kept in both cases.

*s*

*s*

Roots\_of\_CLCP { 0.6768 0.2761i, 0.7335 0.2262i, 0. 7165, 0.6164, 0.5876, 0.3313

The sensitivity magnitude plot in Fig. 20 with the peak max{ [ ( )]} 2 *<sup>M</sup> S j* around the crossover frequency proves good closed-loop nominal performance.

Fig. 20. [ ( )] *<sup>M</sup> j T z e S z* - versus –frequency plot

Fulfilment of robust stability condition (70) is examined in Fig. 21. The closed-loop system is stable over the whole non-minimum phase region (8).

Fig. 21. Verification of the robust stability condition max 0 <sup>1</sup> [ ( )] ( ) ( ) *<sup>O</sup> o T j L* 

## **5. Conclusion**

The robust decentralized PID controller design procedures have been developed both in frequency and time domains. The proposed controller design schemes are based on different principles, with the same control aim: to achieve robust stability and specified performance. The comparative study of both approaches is presented on robust decentralized discrete-

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