4. Conclusions and future works

From these figures, the proposed decentralized adaptive gain robust controller stabilizes the uncertain large-scale interconnected system with system parameters of Eq. (79). Furthermore, one can see that each subsystem achieves good transient behavior close to nominal subsystems by the proposed decentralized robust controller. Thus, the effectiveness of the proposed robust

Figure 6. Time histories of x3(t), x3ð Þt , u(t) and u tð Þ. (a) Time histories of x3(t) and x3ð Þt , (b) Time histories of u(t) and u tð Þ.


(t) and xið Þt (i = 1, 2). (a) The time histories of x1(t) and x1ð Þt , (b) Time histories of x2(t) and x2ð Þt .


Control Input

(a) (b)

(a) (b)

0 0.5 1 1.5 2 2.5 3

Time *t*

0 0.5 1 1.5 2 2.5 3

Time *t*

*u*1(*t*) *u*2(*t*) *u*3(*t*) *u*1(*t*) *u*2(*t*)

*u*3(*t*)

<sup>1</sup>) *<sup>x</sup>* ( <sup>2</sup> (*t*) <sup>2</sup>) *<sup>x</sup>* ( <sup>2</sup> (*t*) <sup>1</sup>) *<sup>x</sup>* ( <sup>2</sup> (*t*) <sup>2</sup>) *<sup>x</sup>* ( <sup>2</sup> (*t*)

 0.5 1 1.5 2

State

In this section, on the basis of the result of Section 2, we have suggested the decentralized adaptive gain robust controller for the large-scale interconnected system with uncertainties.

control strategy is shown.



210 Adaptive Robust Control Systems

Figure 5. Time histories of xi

State

State

0 0.5 1 1.5 2 2.5 3

Time *t*

0 0.5 1 1.5 2 2.5 3

Time *t*

<sup>1</sup>) *<sup>x</sup>* ( <sup>3</sup> (*t*) <sup>2</sup>) *<sup>x</sup>* ( <sup>3</sup> (*t*) <sup>1</sup>) *<sup>x</sup>* ( <sup>3</sup> (*t*) <sup>2</sup>) *<sup>x</sup>* ( 3

(*t*)

<sup>1</sup>) *<sup>x</sup>* ( <sup>1</sup> (*t*) <sup>2</sup>) *<sup>x</sup>* ( <sup>1</sup> (*t*) *x* (1) <sup>1</sup> (*t*) *x* (2) <sup>1</sup> (*t*)

3.4. Summary

In this chapter, firstly the centralized adaptive gain robust controller for a class of uncertain linear systems has been proposed, and through a simple numerical example, we have shown the effectiveness/usefulness for the proposed adaptive gain robust control strategy. Next, for a class of uncertain large-scale interconnected systems, we have presented an LMI-based design method of decentralized adaptive gain robust controllers. In the proposed controller robust synthesis, advantages are as follows: the proposed adaptive gain robust controller can achieve satisfactory transient behavior and/or avoid the excessive control input, that is, the proposed robust controller with adjustable time-varying parameters is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the unknown parameter variations. Moreover, in this chapter we have derived the allowable perturbation region of unknown parameters, and the proposed robust controller can be obtained by solving constrained convex optimization problems. Although the solution of the some matrix inequalities can be applied to the resultant robust controller in the general controller design strategies for the conventional fixed gain robust control, the solutions of the constrained convex optimization problem derived in this chapter cannot be reflected to the resultant robust controller. Note that the proposed controller design strategy includes this fascinating fact.

In Section 2 for a class of uncertain linear systems, we have dealt with a design problem of centralized adaptive gain robust state feedback controllers. Although the standard LQ regulator theory for the purpose of generating the desired response is adopted in the existing result [32], the nominal control input is designed by using pole placement constraints. By using the controller gain for the nominal system, the proposed robust control with adjustable timevarying parameter has been designed by solving LMIs. Additionally, based on the derived LMI-based conditions, the constrained convex optimization problem has been obtained for the purpose of the maximization of the allowable perturbation region of uncertainties included in the controlled system. Section 3 extends the result for the centralized adaptive gain robust state feedback controller given in Section 2 to decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems. In this section, an LMI-based controller synthesis of decentralized adaptive gain robust state feedback control has also been presented. Furthermore, in order to maximize the allowable region of uncertainties, the design problem of the decentralized adaptive gain robust controller for the uncertain large-scale interconnected system has been reduced to the constrained convex optimization problem.

References

pany; 1960

1995;40(4):704-707

2006;193:383-396

Matemática Mexicana. 1960;(5):102-199

Journal of Basic Engineering. 1964;86(1):51-60

Inc.; 2001

[1] Bermish BR, Corless M, Leitmann G. A new class of stabilizing controllers for uncertain dynamical systems. SIAM Journal on Control and Optimization. 1983;21(2):246-255

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

http://dx.doi.org/10.5772/intechopen.71733

213

[2] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in system and

[3] Chen YH. Decentralized robust control system Design for Large-Scale Uncertain Systems.

[4] Choi YK, Chung MJ, Bien Z. An adaptive control scheme for robot manipulators. Inter-

[5] Davison EJ, Tripathi NK. The optimal decentralized control of a large power system: Load and frequency control. IEEE Transactions on Automatic Control. 1978;23(2):312-325

[6] Doyle JC, Glover K, Khargonekar PP, Francis BA. State-space solutions to standard H<sup>2</sup> and H<sup>∞</sup> control problems. IEEE Transactions on Automatic Control. 1989;34(8):831-847

[7] Faub PL, Wolovich WA. Decoupling in the design and synthesis of multivariable control

[8] Chilali M, Gahinet P, Apkarian P. Robust pole placement in LMI regions. IEEE Trans-

[9] Gantmacher FR. The Theory of Matrices. Vol. 1. New York: Chelsea Publishing Com-

[10] Gilbert EG, Kolmanovsky I. Nonlinear tracking control in the presence of state and control constraints: A generalized reference governor. Automatica. 2002;38(12):2071-2077

[11] Gong Z. Decentralized robust control of uncertain interconnected systems with prescribed degree of exponential convergence. IEEE Transactions on Automatic Control.

[12] Goodwin GC, Graebe SF, Saogado ME. Control System Design. NJ, USA: Prentice-Hall

[13] Hua C, Guan X, Shi P. Decentralized robust model reference adaptive control for interconnected time-delay systems. Journal of Computational and Applied Mathematics.

[15] Kalman RE. Contributions to the theory of optimal control. Boletín de la Sociedad

[16] Kalman RE. When is a linear control system optimal? Transactions of ASME Series D,

[14] Khalil HK. Nonlinear Systems. 3rd ed. New Jersey: Prentice-Hall Inc.; 2002

systems. IEEE Transactions on Automatic Control. 1967;12(6):651-659

control theory. SIAM Studies in Applied Mathmatics. 1994

International Journal of Control. 1988;47(5):1195-1205

national Journal of Control. 1986;44(4):1185-1191

actions on Automatic Control. 1999;44(12):2257-2270

In the future research, an extension of the proposed adaptive gain robust state feedback controller to output feedback control systems or observer-based control ones is considered. Moreover, the problem for the extension to such a broad class of systems as uncertain timedelay systems, uncertain discrete-time systems, and so on should be tackled. Furthermore, we will examine the conservativeness of the proposed adaptive gain robust control strategy and online adjustment way of the design parameter which plays important roles such as avoiding the excessive control input.

On the other hand, it is well known that the design of control systems is often complicated by the presence of physical constraints: temperatures, pressures, saturating actuators, within safety margins, and so on. If such constraints are violated, serious consequences may ensue. For example, physical components will suffer damage from violating some constraints, or saturations for state/input constraints may cause a loss of closed-loop stability. In particular, input saturation is a common feature of control systems, and the stabilization problems of linear systems with control input saturation have been studied (e.g., [33, 40]). Additionally, some researchers have investigated analysis of constrained systems and reference managing for linear systems subject to input and state constraints (e.g., [10, 19]). Therefore, the future research subjects include the constrained robust controller design reducing the effect of unknown parameters.
