×

Linearization algorithm for a reduced order \(H_\infty\) control design of an active suspension system. (English) Zbl 1293.93184

Summary: This paper proposes a method for the design of reduced-order controllers and the proposed method is applied to the active suspension system in the Laboratoire d’Automatique de Grenoble, France. All performance specifications are posed as a single constraint on the \(H_\infty\) norm of a certain frequencyscaled closed loop transfer function along with the requirement that a controller should be strictly proper. The number of states in the generalized plant is twenty-nine.
In order to solve the proposed discrete-time \(H_\infty\) control problem, we derived necessary and sufficient conditions for the design of strictly proper \(H_\infty\) discrete-time controllers. The resulting conditions turned out to be the necessary and sufficient conditions for standard (non strictly proper) \(H_\infty\) control plus one additional matrix inequality. As in the standard \(H_\infty\) control problem, when the controller has the same order as the plant to be controlled, these conditions become convex sets on the design variables.
When the order of the controller is specified to be less than the order of the plant to be controlled, the conditions for the design of \(H_\infty\) controllers are concave. In order to compute controllers with very low order, we propose a numerical algorithm by linearizing the concave matrix inequalities so as to generate a sequential semidefinite programming problems with monotonically decreasing cost. At each iteration of this algorithm, a linearized version of the original set of (nonconvex) matrix inequalities is solved using semi-definite programming.
Our methodology provides a controller of order nine which satisfies all performance requirements. Reduced order controllers from order two to eight are also designed. The controller of order three appears to be the one that well compromises performance with complexity. Experiments using the proposed third order controller have been performed on the actual suspension system. The proposed third order controller performs better than the best third order controller shown in [I. D. Landau et al., European J. Control 9, No. 1, 3–12 (2003; Zbl 1293.93294)].

MSC:

93B18 Linearizations
93B36 \(H^\infty\)-control
93C95 Application models in control theory
90C22 Semidefinite programming

Citations:

Zbl 1293.93294
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alazard, D., Design and optimization of restricted complexity controllers: Towards a non-parametric model based solution, European J Control, 9, 100-104 (2003) · Zbl 1293.93124
[2] Campos-Delgado, D. U.; Femat, R.; Ruiz-Velazquez, E., Design of reduced-order controllers via \(H_∞\) and parametric optimization: Comparison for an active suspension system, European J Control, 9, 48-61 (2003) · Zbl 1293.93125
[3] Cao, Y.; Yan, W., Multi-objective optimization restricted complexity controllers, European J Control, 9, 61-65 (2003) · Zbl 1293.93126
[4] Chable, S.; Mahieu, S.; Chiappa, C., Design and optimization of restriced complexity controller: A modal approach for reduced-order controllers, European J Control, 9, 39-47 (2003) · Zbl 1293.93127
[5] Constantinescu, A.; Landau, I. D., Controller order reduction by identification in closed-loop applied to a benchmark problem, European J Control, 9, 84-99 (2003) · Zbl 1293.93128
[6] de Oliveira, M. C.; Camino, J. F.; Skelton, R. E., A convexifying algorithm for the design of structured linear controllers, (Proceedings of the 39th IEEE Conference on Decision and Control. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia (2000)), 2781-2786
[7] de Oliveira, M. C.; Geromel, J. C.; Bernussou, J., Extended \(H_2\) and \(H_\infty\) norm characterizations and controller parametrizations for discrete-time systems, Int J Control, 75, 9, 666-679 (2002) · Zbl 1029.93020
[8] Geromel, J. C.; de Souza, C. C.; Skelton, R. E., Static output feedback controllers: Stability and convexity, IEEE Trans Autom Control, 43, 1, 120-125 (1998) · Zbl 0952.93106
[9] El Ghaoui, L.; Oustry, F.; Rami, M. A., A cone complementarity linearization algorithm for static outputfeedback and related problems, IEEE Trans Autom Control, 42, 8, 1171-1176 (1997) · Zbl 0887.93017
[10] Grigoriadis, K. M.; Beran, E. B., Alternating projection algorithms for linear matrix inequalities problems with rank constraints, (El Ghaoui, L.; Niculescu, S. I., Advances in linear matrix inequality methods in control in design and control (2000), SIAM, Philadelphia), 251-267 · Zbl 0956.93021
[11] Hol, C. W.J.; Scherer, C. W.; van der Meche, E. G.; Bosgra, O. H., A nonlinear sdp approach to fixed-order controller synthesis and comparison with two other methods applied to an active suspension system, European J Control, 9, 13-28 (2003) · Zbl 1293.93316
[12] Horn, R. A.; Johnson, C. R., Matrix analysis (1985), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0576.15001
[13] Iwasaki, T.; Skelton, R. E., The xy-centering algorithm for the dual lmi problem: a new approach to fixed order control design, Int J Control, 62, 6, 1257-1272 (1995) · Zbl 0839.93033
[14] Iwasaki, T., The dual iteration for fixed order control, IEEE Trans Autom Control, 44, 4, 783-788 (1999) · Zbl 0957.93029
[15] Landau, I. D.; Karimi, A.; Miskovic, L.; Prochazka, H., Control of an active suspension system as a benchmark for design and optimization of restricted-complexity controllers, European J Control, 9, 3-12 (2003) · Zbl 1293.93294
[16] Le Mauf, F.; Duc, G., Designing a low order robust controller for an active suspension system thank lmi, genetic algorithm and gradient search, European J Control, 9, 29-38 (2003) · Zbl 1293.93139
[17] Miskovic, L.; Karimi, A.; Bonvin, D., Correlationbased tuning of a restricted-complexity controller for an active suspension system, European J Control, 9, 77-83 (2003) · Zbl 1293.93692
[18] Scherer, C.; Gahinet, P.; Chilali, M., Multiobjective outputfeedback control via lmi optimization, IEEE Trans Autom Control, 42, 7, 896-911 (1997) · Zbl 0883.93024
[19] Skelton, R. E.; Iwasaki, T.; Grigoriadis, K., Unified algebraic approach to linear control design (1998), Taylor & Francis: Taylor & Francis London
[20] Available at http://lawww.epfl.ch/page5323.html; Available at http://lawww.epfl.ch/page5323.html
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.