**Comparison of Two Approaches to Count Derivations for Continuous-Time Adaptive Control**

Karel Perutka *Tomas Bata University in Zlin, Faculty of Applied Informatics Czech Republic* 

#### **1. Introduction**

162 Cutting Edge Research in New Technologies

[Toth & Vigo, 2002] Toth, P. & Vigo, D. (Eds.) (2002). *The Vehicle Routing Problem*, SIAM,

[Wang et al., 2006] Wang, Q., Yang, J., Ren, M. and Zheng, Y. (2006). *Driver fatigue detection: a* 

Zhan & Noon, 1996 Zhan, F. B., & Noon, C.E. (1996). *Shortest Path Algorithms: An Evaluation* 

[Zhou & Geng, 2004] Zhou, Z.H. and Geng, X. (2004). *Projection functions for eye detection*,

*Using Real Road Networks*, Transportation Science.

Pattern Recognit., vol. 37, no. 5, (2004), pp. 1049-1056.

*survey*, Proceedings of the World Congress on Intelligent Control and Automation,

Philadelphia.

(2006), pp. 8587-8591.

The control of continuous-time systems can be realized by adaptive controllers. Self-tuning controllers are adaptive controllers which call on-line identification and controllers parameters tuning in one step of computation. Supervision enlarges the area of usage of controllers. It is necessary to count derivations of action and output signals during control, which is usually realized by filters. Settings of filters are directly connected with the model of the system. Another approach allows us to use the regression polynomials instead of filters, because the general form of derivations is known before the control. Without filters, this approach keeps the signal unchanged, but the choice of inappropriate length of time interval for polynomial regression increases the amplitude of noise. The chapter shows two examples of control and suggests the appropriate length of time interval for polynomial regression.

Many processes can be viewed in the point of control as continuous-time systems. The implementation of pseudo-continuous model on continuous-time system is called as hybrid system (De Santis et al., 2009). Mostly these systems are nonlinear and specific method of control is needed (Gregorčič and Lightbody, 2010). This chapter uses the method adaptive control, because adaptive control is often used and gives adequate results (Pasik-Duncan, 2001). At adaptive control, the usage of the appropriate identification method is very important. This paper uses recursive instrumental variable method, but there are several other good methods and papers dealing with identification and parameters tuning (Flores and Pastor, 2005, Tzes and Le, 1996, Coello, 2000). The controlled process in this chapter has multi-inputs multi-outputs and the decentralized controller was used. It is common approach in practice (Martínez-Rosas et al., 2006). Decentralized control can be realized by PID controllers. These controllers are very popular due to their advantages, such as simplicity (Vrančić et al., 2010).

The ideas and results obtained in control can be useful in many different areas, for example in robotics or in production systems. Nice paper about spatial ontology for human-robot interaction was written by Belouaer at al. (Belouaer at al., 2010). A special framework to generate configurations in production systems was written by Kanso et al. (Kanso et al., 2010).

Comparison of Two Approaches to Count Derivations for Continuous-Time Adaptive Control 165

<sup>2</sup> <sup>222</sup> ' ' *x bc x ac x ab <sup>f</sup> x Px f a f b f c abac b abc c ac b*

<sup>222</sup> '' '' *fa fb fc fx P x abac b abc c acb*

Instrumental variable method is a modification of the least squares method. The least squares method uses the quadratic criterion and the existence of one global minimum. The instrumental variable method does not allow us to obtain the properties of noise, but it has inferior presumptions than the least square method. It is possible to formulate it recursively

<sup>Θ</sup> 0 1 deg 0 1 deg <sup>ˆ</sup> ˆˆ ˆ ˆˆ ˆ , ,..., , , ,..., , *<sup>T</sup>*

1 1 ,..., , ,..., ,1 *<sup>T</sup> n m*

The used suboptimal method was introduced by Dostal (Dostal, 1997). Let us minimize

2 2

 

are penalty constants. Stable polynomials *g* and *n* are counted as results

*J e t u t dt*

 **Cz C**

*k k kk k k*

1

0

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

 (2)

<sup>2</sup>

 

(3)

*a b k aa a bb b d* (4)

**C z** (6)

(7)

*k kk k L L k yt y t ut u t* (5)

 1 1

*k ut ut ut k k* , ,..., <sup>1</sup> *knm* **z** (8)

Θ ˆ ˆ 1 *<sup>T</sup> ek k k k* **y** (9)

Θ Θ ˆ ˆ *k k kek* 1 **L** ˆ (10)

(11)

**C z** 

*kk k*

 1 1 11 *<sup>T</sup>*

1 1

*kk k* **C z**

The first derivation is

and second derivation is

(Zhu & Backx, 1993).

**2.3 Recursive instrumental variable** 

**L**

**C C**

**2.4 Suboptimal linear quadratic controller** 

quadratic functional

where

 0, 0 

of spectral factorizations

#### **2. Theoretical background**

#### **2.1 Self-tuning control**

Self-tuning controllers (STC) are based on on-line identification and on tuning the controller parameters with respect to identified changes in controlled systems. The self-tuning controllers can be further divided to the STC with explicit identification and the STC with implicit identification, the STC with implicit identification directly identifies the controller parameters. On the other hand, the STC with explicit identification computes the controller parameters using the parameters of the system model (Bobal et al., 2005).

#### **2.2 On-line identification**

When self-tuning controller was used, the scheme of input and output signal modification depicted in figure 1 is applied, because the continuous-time system parameters *ai* and *bj* are estimated using recursive instrumental variable method. The action (input) signal *u*(*t*) is continuously approximated by Lagrange regression polynomial on an interval of given length during entire control. The structure of Lagrange regression polynomial (1) together with its derivation (2), (3) is generally known before the start of identification, only the numerical values of their parameters are needed and counted. It is the alternative way to obtain values of derivations needed for identification. After the polynomial approximation, the approximating polynomial derivation *u(i)L*(*t*) is counted. It is sampled in purpose to count the values of subsystem parameters using recursive identification algorithm.

Lagrange polynomial of second order was used in the paper in the form

$$P\_2\left(\mathbf{x}\right) = \frac{(\mathbf{x}-b)(\mathbf{x}-c)}{(a-b)(a-c)}f\left(a\right) + \frac{(\mathbf{x}-a)(\mathbf{x}-c)}{(b-a)(b-c)}f\left(b\right) + \frac{(\mathbf{x}-a)(\mathbf{x}-b)}{(c-a)(c-b)}f\left(c\right) \tag{1}$$

The first derivation is

164 Cutting Edge Research in New Technologies

Self-tuning controllers (STC) are based on on-line identification and on tuning the controller parameters with respect to identified changes in controlled systems. The self-tuning controllers can be further divided to the STC with explicit identification and the STC with implicit identification, the STC with implicit identification directly identifies the controller parameters. On the other hand, the STC with explicit identification computes the controller

When self-tuning controller was used, the scheme of input and output signal modification depicted in figure 1 is applied, because the continuous-time system parameters *ai* and *bj* are estimated using recursive instrumental variable method. The action (input) signal *u*(*t*) is continuously approximated by Lagrange regression polynomial on an interval of given length during entire control. The structure of Lagrange regression polynomial (1) together with its derivation (2), (3) is generally known before the start of identification, only the numerical values of their parameters are needed and counted. It is the alternative way to obtain values of derivations needed for identification. After the polynomial approximation, the approximating polynomial derivation *u(i)L*(*t*) is counted. It is sampled in purpose to

count the values of subsystem parameters using recursive identification algorithm.

Fig. 1. Scheme of I/O signals modification for STC.

Lagrange polynomial of second order was used in the paper in the form

 <sup>2</sup> *xbxc xaxc xaxb P x f a f b f c abac b abc c acb*

(1)

parameters using the parameters of the system model (Bobal et al., 2005).

**2. Theoretical background** 

**2.1 Self-tuning control** 

**2.2 On-line identification** 

$$f'(\mathbf{x}) \equiv P'\_2(\mathbf{x}) = \frac{2\mathbf{x} - (b+c)}{(a-b)(a-c)} f(a) + \frac{2\mathbf{x} - (a+c)}{(b-a)(b-c)} f(b) + \frac{2\mathbf{x} - (a+b)}{(c-a)(c-b)} f(c) \tag{2}$$

and second derivation is

$$f^{\prime\prime}(\mathbf{x}) \equiv P^{\prime\prime}\_{\ \ 2}(\mathbf{x}) = \frac{2f(a)}{(a-b)(a-c)} + \frac{2f(b)}{(b-a)(b-c)} + \frac{2f(c)}{(c-a)(c-b)}\tag{3}$$

#### **2.3 Recursive instrumental variable**

Instrumental variable method is a modification of the least squares method. The least squares method uses the quadratic criterion and the existence of one global minimum. The instrumental variable method does not allow us to obtain the properties of noise, but it has inferior presumptions than the least square method. It is possible to formulate it recursively (Zhu & Backx, 1993).

$$\hat{\Theta}^{T}\left(k\right) = \left(\hat{a}\_{0}, \hat{a}\_{1}, \dots, \hat{a}\_{\deg(a)}, \hat{b}\_{0}, \hat{b}\_{1}, \dots, \hat{b}\_{\deg(b)}, d\right) \tag{4}$$

$$\boldsymbol{\Phi}^{T}\left(\boldsymbol{k}\right) = \left[ -y\left(\boldsymbol{t}\_{k}\right), \dots, -y\boldsymbol{\upmu}^{\left(n-1\right)}\left(\boldsymbol{t}\_{k}\right), \boldsymbol{u}\left(\boldsymbol{t}\_{k}\right), \dots, \boldsymbol{u}\_{L}^{\left(m-1\right)}\left(\boldsymbol{t}\_{k}\right), 1\right] \tag{5}$$

$$\mathbf{L}(k) = \frac{\mathbf{C}(k-1)\mathbf{z}(k)}{1 + \boldsymbol{\Phi}^T(k)\mathbf{C}(k-1)\mathbf{z}(k-1)}\tag{6}$$

$$\mathbf{C}(k) = \mathbf{C}(k-1) - \frac{\mathbf{C}(k-1)\mathbf{z}(k)\boldsymbol{\Phi}^{\top}(k)\mathbf{C}(k-1)}{1 + \boldsymbol{\Phi}^{\top}(k)\mathbf{C}(k-1)\mathbf{z}(k)}\tag{7}$$

$$\mathbf{z}\left(k\right) = \left[\boldsymbol{\mu}\left(t\_{k}\right), \boldsymbol{\mu}\left(t\_{k-1}\right), \dots, \boldsymbol{\mu}\left(t\_{k-n-m}\right)\right] \tag{8}$$

$$
\hat{e}(k) = \mathbf{y}(k) - \boldsymbol{\Phi}^T(k)\hat{\boldsymbol{\Theta}}(k-1) \tag{9}
$$

$$
\hat{\Theta}(k) = \hat{\Theta}(k-1) + \mathbf{L}(k)\hat{e}(k) \tag{10}
$$

#### **2.4 Suboptimal linear quadratic controller**

The used suboptimal method was introduced by Dostal (Dostal, 1997). Let us minimize quadratic functional

$$J = \bigcap\_{0}^{\phi} \left\{ \mu e^2 \left( t \right) + \phi \tilde{u}^2 \left( t \right) \right\} dt \tag{11}$$

where 0, 0 are penalty constants. Stable polynomials *g* and *n* are counted as results of spectral factorizations

Comparison of Two Approaches to Count Derivations for Continuous-Time Adaptive Control 167

**Step 1.** Go through the bits field row by row. The row which gives the highest number

**Step 2.** When the subsystem is set, the last bit in the row of identified subsystem is set to 1

**Step 4.** Go through the bits field row by row. The row which gives the lowest number after

Repeat Step 1 to 6 *n*/2-times at even *n* and *n*/2-times without Step 4 to 6 at last calling at odd *n* after the change of set-point. After this tuning, run the self-tuning control without

In figures 2-7, there are obtained results of control of two inputs two outputs systems by two controllers. Counting step was 0.2 s. In these figures, the meaning of the symbols is following: w1 – set-point of first subsystem, u1 – action signal of first subsystem, y1 – output signal of first subsystem, w2 – set-point of second subsystem, u2 – action signal of second subsystem, y2 – output signal of second subsystem, p11, p01, q21, q11, q01 – parameters of first sub-controller, b01, a11, a01 – parameters of the model of the first controlled subsystem.

Fig. 2. History of control – too small interval for approximation by Lagrange polynomial.

supervisor until the new change of the set-point when the supervisor is called.

the on-line identification.

on-line identification.

**Step 5.** Do Step 2. **Step 6.** Do Step 3.

**3. Experimental part** 

and in remaining rows the last bit is set to 0.

**Step 3.** One bit left rotation of all rows in bits field.

after conversion also gives the number of the subsystem in which goes one step of

conversion also gives the number of the subsystem in which goes one step of the

$$\left(as\right)^{\*}\left.\rho a s + b^{\*}\,\mu b = \text{g}^{\*}\,\text{g}, \text{n}^{\*}\,\text{n} = a^{\*}a.\tag{12}$$

Solving the following diophantic equation

$$asp + bq = \text{gn} \tag{13}$$

gives the parameters of controller. If the system transfer function has the form

$$G(s) = \frac{b\_0}{s^2 + a\_1s + a\_0} \tag{14}$$

The controller is

$$FQ = \frac{q\_2s^2 + q\_1s + q\_0}{s\left(p\_2s^2 + p\_1s + p\_0\right)}\tag{15}$$

and polynomials *g* and *n* are

$$\mathbf{g(s)} = \mathbf{g\_3s^3} + \mathbf{g\_2s^2} + \mathbf{g\_1s} + \mathbf{g\_0} \tag{16}$$

$$m(s) = s^2 + n\_1s + n\_0 \tag{17}$$

Their coefficients obtained by spectral factorization are in the form

$$g\_0 = \sqrt{\mu b\_0^2} \tag{18}$$

$$\mathcal{g}\_1 = \sqrt{2\mathcal{g}\_2\mathcal{g}\_0 + \rho a\_0^2} \tag{19}$$

$$\mathcal{g}\_2 = \sqrt{2\mathcal{g}\_3\mathcal{g}\_1 + \mathcal{o}\left(a\_1^2 - 2a\_0\right)}\tag{20}$$

$$
\mathfrak{g}\_3 = \sqrt{\varphi} \tag{21}
$$

$$m\_0 = \sqrt{a\_0^2} \tag{22}$$

$$m\_1 = \sqrt{2m\_0 - a\_1^2 - 2a\_0} \tag{23}$$

#### **2.5 Supervisor**

The used supervisor is based on the supervisor introduced by Perutka (Perutka, 2007) and it is used in this paper for the first time.

Supervisor is used for decentralized or decoupled control of multi-input multi-output systems, number of inputs and outputs are the same and denoted as *n*. Such system is controlled by *n* sub-controllers. Let us suppose the existence of bits field with *n* x *n*  dimension. The initial values of the field form the identity matrix. Each row the field corresponds to one subsystem of controlled system.


\* \* \*\* \* *as as b b g g n n a a*

, . (12)

*asp bq gn* (13)

3 2 10 *g s gs gs gs g* (16)

<sup>2</sup> *n s s ns n* 1 0 (17)

*b* (18)

*a a* (20)

<sup>2</sup> *n a* 0 0 (22)

1 01 0 *n na a* 2 2 (23)

(21)

*a* (19)

(14)

(15)

 

 <sup>0</sup> 2

> 2 2 10 2 2 10

*qs qs q FQ s ps ps p* 

3 2

Their coefficients obtained by spectral factorization are in the form

*<sup>b</sup> G s*

1 0

*s as a* 

2 0 0 *g* 

1 20 0 *g gg* 2

2 31 1 0 *g gg* 2 2 

> <sup>3</sup> *g*

> > 2

The used supervisor is based on the supervisor introduced by Perutka (Perutka, 2007) and it

Supervisor is used for decentralized or decoupled control of multi-input multi-output systems, number of inputs and outputs are the same and denoted as *n*. Such system is controlled by *n* sub-controllers. Let us suppose the existence of bits field with *n* x *n*  dimension. The initial values of the field form the identity matrix. Each row the field

2

<sup>2</sup>

gives the parameters of controller. If the system transfer function has the form

Solving the following diophantic equation

The controller is

**2.5 Supervisor** 

is used in this paper for the first time.

corresponds to one subsystem of controlled system.

and polynomials *g* and *n* are

**Step 6.** Do Step 3.

Repeat Step 1 to 6 *n*/2-times at even *n* and *n*/2-times without Step 4 to 6 at last calling at odd *n* after the change of set-point. After this tuning, run the self-tuning control without supervisor until the new change of the set-point when the supervisor is called.

#### **3. Experimental part**

In figures 2-7, there are obtained results of control of two inputs two outputs systems by two controllers. Counting step was 0.2 s. In these figures, the meaning of the symbols is following: w1 – set-point of first subsystem, u1 – action signal of first subsystem, y1 – output signal of first subsystem, w2 – set-point of second subsystem, u2 – action signal of second subsystem, y2 – output signal of second subsystem, p11, p01, q21, q11, q01 – parameters of first sub-controller, b01, a11, a01 – parameters of the model of the first controlled subsystem.

Fig. 2. History of control – too small interval for approximation by Lagrange polynomial.

Comparison of Two Approaches to Count Derivations for Continuous-Time Adaptive Control 169

Fig. 5. History of control – adequate interval for approximation by Lagrange polynomial.

Fig. 6. History of controller parameters for 1st subsystem – adequate interval for

approximation by Lagrange polynomial.

Fig. 3. History of controller parameters for 1st subsystem – too small interval for approximation by Lagrange polynomial.

Fig. 4. History of subsystem model parameters for 1st subsystem - too small interval for approximation by Lagrange polynomial.

Fig. 3. History of controller parameters for 1st subsystem – too small interval for

Fig. 4. History of subsystem model parameters for 1st subsystem - too small interval for

approximation by Lagrange polynomial.

approximation by Lagrange polynomial.

Fig. 5. History of control – adequate interval for approximation by Lagrange polynomial.

Fig. 6. History of controller parameters for 1st subsystem – adequate interval for approximation by Lagrange polynomial.

Comparison of Two Approaches to Count Derivations for Continuous-Time Adaptive Control 171

will focus on the exact mathematical derivation of the time appropriate interval with the

The author would like to mention MSM7088352101 grant, from which the work was

Belouaer, L., Bouzid, M., Mouaddib, A.-I., 2010. A Spatial Ontology for Human-Robot

Bobal, V., Böhm, J., Fessl, J., Machacek, J., 2005. *Digital Self-tuning Controllers*. Springer-

De Santis, E., Di Benedetto, M.D., Pola, G., 2009. A structural approach to detectability for a

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Gregorčič, G., Lightbody, G., 2010. Nonlinear model-based control of nonlinear processes.

Kanso, M., Berruet, P., Philippe, J.-L., 2010. A Framework Based on a High Conception Level

Martínez-Rosas, J.C., Arteaga, M.A., Castillo-Sánchez, A.M., 2006. Decentralized control of

Pasik-Duncan, B., 2001. On stochastic adaptive control of continuous-time systems.

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combination of the dynamical filter of noise.

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2007.

**5. Acknowledgements** 

supported.

**6. References** 

00-3.

Figures 2 - 4 provide the results obtained for time interval of approximation 0.41 s.

Figures 5 - 7 provide the results obtained for time interval of approximation 4.1 s.

From the illustratively shown results, it is clear that it is important to correctly choose the appropriate length of time interval which is used for regression by Lagrange polynomial. By repeating several experiments with different controlled systems it was verified that it is appropriate to use 20 counting steps in the presence of noise.

Fig. 7. History of subsystem model parameters for 1st subsystem - adequate interval for approximation by Lagrange polynomial.

#### **4. Conclusions**

The chapter presented simulation results of self-tuning control with polynomial regression used for derivations counting. It was shown that inappropriate selection of regression interval makes more noise. The recommended length of the interval was given. Future work will focus on the exact mathematical derivation of the time appropriate interval with the combination of the dynamical filter of noise.

#### **5. Acknowledgements**

The author would like to mention MSM7088352101 grant, from which the work was supported.

#### **6. References**

170 Cutting Edge Research in New Technologies

From the illustratively shown results, it is clear that it is important to correctly choose the appropriate length of time interval which is used for regression by Lagrange polynomial. By repeating several experiments with different controlled systems it was verified that it is

Fig. 7. History of subsystem model parameters for 1st subsystem - adequate interval for

The chapter presented simulation results of self-tuning control with polynomial regression used for derivations counting. It was shown that inappropriate selection of regression interval makes more noise. The recommended length of the interval was given. Future work

approximation by Lagrange polynomial.

**4. Conclusions** 

Figures 2 - 4 provide the results obtained for time interval of approximation 0.41 s. Figures 5 - 7 provide the results obtained for time interval of approximation 4.1 s.

appropriate to use 20 counting steps in the presence of noise.


**9** 

*Czech Republic* 

**Implementation of Control** 

*Department of Automation and Control Engineering* 

*Faculty of Applied Informatics, Tomas Bata University in Zlín* 

Radek Matušů and Roman Prokop

**Design Methods into Matlab Environment** 

Computer-aided tools for analysis and synthesis of control systems are widely employed by many users from a range of researchers, control engineers or students. The reason is obvious. Such toolboxes represent comfortable and effective way of dealing with an array of complex control problems, sometimes even without deeper knowledge of the specific method. For example, Control System Toolbox, Robust Control Toolbox or Polynomial Toolbox (PolyX, 2011) for Matlab belong among the most popular ones in the control field. The main aim of this chapter is to present two simple and freely downloadable Matlab programs which allow user-friendly work for two selected specific control design issues by

First of the packages (Matušů, 2010; Matušů & Prokop, 2011a) is focused on algebraic design of continuous-time controllers under assumption of interval plants. The program takes advantage of Matlab + Simulink + Polynomial Toolbox (PolyX, 2011) environment and it represents an easy but effective and user-friendly way to control synthesis, robust stability

The second of the presented programs (Matušů & Prokop, 2010, 2011b, 2011c) deals with control of time-delay systems using three various modifications of Smith predictor. The software implementation includes the modification for unstable and integrating processes, PI-PD modification for systems with long dead time, and modification applying control

The described software products, which can be used both for research and educational purposes, are freely available on the Internet (Matušů & Prokop, 2011a, 2011b). Their

The chapter is organized as follows. The Section 2 focuses on algebraic design of controllers for interval plants. It is divided into three partial subsections dealing with brief outline of basic theoretical background, description of the developed program itself and demonstration of its capabilities by means of an illustrative example, respectively. Analogically, the Section 3 has the very same structure but it presents the control of time-delay systems using three

The partial versions of this work have been already presented in (Matušů & Prokop, 2010,

application potential is going to be illustrated on several control examples.

modifications of Smith predictor. Finally, Section 4 offers some conclusion remarks.

**1. Introduction** 

means of Graphical User Interface (GUI).

design by Coefficient Diagram Method (CDM).

analysis and simulation.

2011c; Matušů 2010).

Zhu, Y., Backx, T., 1993. *Identification of Multivariable Industrial Processes for Simulation, Diagnosis and Control*. Springer-Verlag Ltd., London, United Kingdom, ISBN 3-540- 19835-0.
