# zbMATH — the first resource for mathematics

Non-smooth structured control design with application to PID loop-shaping of a process. (English) Zbl 1127.93326
Summary: Feedback controllers with specific structure arise frequently in applications because they are easily apprehended by design engineers and facilitate on-board implementations and re-tuning. This work is dedicated to $$H_{\infty}$$ synthesis with structured controllers. In this context, straightforward application of traditional synthesis techniques fails, which explains why only a few ad hoc methods have been developed over the years. In response, we propose a more systematic way to design $$H_\infty$$ optimal controllers with fixed structure using local optimization techniques. Our approach addresses in principle all those controller structures which can be built into mathematical programming constraints. We apply non-smooth optimization techniques to compute locally optimal solutions, and provide practical tests for descent and optimality. In the experimental part we apply our technique to $$H_\infty$$ loop-shaping proportional integral derivative (PID) controllers for MIMO systems and demonstrate its use for PID control of a chemical process.

##### MSC:
 93B51 Design techniques (robust design, computer-aided design, etc.) 93B36 $$H^\infty$$-control 93B50 Synthesis problems 93C99 Model systems in control theory
Full Text:
##### References:
 [1] Doyle, IEEE Transactions on Automatic Control AC-34 pp 831– (1989) [2] Tsitsiklis, SIAM Journal on Control 35 pp 2118– (1997) [3] . Global optimization in control system analysis and design. In Control and Dynamic Systems: Advances in Theory and Applications, (ed.), vol. 53. Academic Press: New York, 1992. [4] . Structured controller design with evolutionary optimization. AIA2003, Benalmadena, Spain, September 2003; 163–168. [5] , , . Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Applied Mathematics, vol. 15. SIAM: Philadelphia, 1994. · Zbl 0816.93004 [6] Grigoriadis, Automatica 32 pp 1117– (1996) [7] Ebihara, International Journal of Control 77 pp 1137– (2004) [8] . Synthesis of fixed-structure controllers via numerical optimization. Proceedings of IEEE Conference on Decision and Control, Lake Buena Vista, FL, 1994; 2678–2683. [9] Iwasaki, IEEE Transactions on Automatic Control 44 pp 783– (1999) [10] . An LMI optimization approach for structured linear controllers. Proceedings of IEEE Conference on Decision and Control, Maui, HI, 2003; 5143–5148. [11] , . A pathfollowing method for solving BMI problems in control. Proceedings of American Control Conference, San Diego, U.S.A., 1999; 1385–1389. [12] Bao, Industrial and Engineering Chemistry Research 38 pp 3407– (1999) [13] A state-space algorithm for designing H loop shaping PID controllers. Technical Report, Cambridge University, Cambridge, U.K., October 2000. [14] . Robust decentralized control design based on coprime factorization representations and LMI optimization. Proceedings of the 36th SICE Annual Conference, Tokushima, Japan, 1997; 1007–1012. [15] Tan, Asian Journal of Control 4 pp 439– (2002) [16] Saeki, Automatica 42 pp 93– (2006) [17] Rotkowitz, IEEE Transactions on Automatic Control 51 pp 274– (2006) [18] Xin, IEEE Transactions on Automatic Control 49 pp 1623– (2004) [19] . Linear Controller Design: Limits of Performance. Prentice-Hall: Englewood Cliffs, NJ, 1991. · Zbl 0748.93003 [20] Scherer, Linear Algebra and Applications 351–352C pp 639– (2002) [21] Mäkilä, IEEE Transactions on Automatic Control AC-32 pp 658– (1987) [22] Polak, Automatica 18 pp 267– (1982) [23] Apkarian, IEEE Transactions on Automatic Control 51 pp 71– (2006) [24] Apkarian, European Journal of Control 12 pp 229– (2006) [25] Apkarian, Automatica (2006) [26] McFarlane, IEEE Transactions on Automatic Control 37 pp 759– (1992) · JFM 14.0176.02 [27] Noll, Mathematical Programming Series B 104 pp 701– (2005) [28] Apkarian, Systems and Control Letters 55 pp 971– (2006) [29] , . Second-order nonsmooth optimization for H synthesis. 5th IFAC Symposium on Robust Control Design, Toulouse, France, July 2006. [30] Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series. Wiley: New York, 1983. · Zbl 0582.49001 [31] Polak, IEEE Transactions on Automatic Control AC-34 pp 268– (1989) [32] Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences. Springer: New York, 1997. · Zbl 0899.90148 [33] Boyd, Systems and Control Letters 15 pp 1– (1990) [34] . Stabilizing control of an inverted pendulum system based on H loop shaping design procedure. Proceedings of the 3rd World Congress on Intelligent Control and Automation, Hefei, China, vol. 5, June 2000; 3385–3388. [35] Fujita, IEEE Control Systems Magazine 13 pp 57– (1993) [36] , . Loop shaping controller design for a binary distillation column. Control’91, Edinburgh, U.K., vol. 2, March 1991; 1271–1276. [37] . Coordinate optimization for bi-convex matrix inequalitites. Proceedings of IEEE Conference on Decision and Control, San Diego, CA, 1997; 3609–3613. [38] , . Nonsmooth technique for stabilizing linear systems. 2006, submitted for publication. [39] Suykens, International Journal of Bifurcation and Chaos 11 pp 2133– (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.