×

zbMATH — the first resource for mathematics

Explicit controller formulas for LMI-based \(H_ \infty\) synthesis. (English) Zbl 0855.93025
For a given linear time-invariant plant \(P(s)\) with state-space realization, we have to find an \(H_\infty\)-suboptimal, internally stabilizing controller \(u = K(s)y\) that makes the closed-loop \(L_2\) gain from the disturbance to the error signal less than a prescribed value. The design is based on linear matrix inequalities (LMI). The synthesis is performed in two steps: First, solve a system of three LMIs, where the unknowns are two symmetric matrices \(R\), \(S\) of size equal to the plant order. Second, given \(R\), \(S\) and the plant, compute the controller by solving another LMI called the controller LMI. For the second part the paper finds explicit formulas. These were implemented in the LMI Control Toolbox for use with MATLAB.

MSC:
93B36 \(H^\infty\)-control
93B40 Computational methods in systems theory (MSC2010)
15A39 Linear inequalities of matrices
Software:
Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, B.D.O.; Vongpanitlerd, S., ()
[2] Apkarian, P.; Gahinet, P.; Apkarian, P.; Gahinet, P., A convex characterization of parameter dependent H∞ controllers, IEEE trans. autom. control, IEEE trans. autom. control, AC-40, 1681-864, (1995), see also · Zbl 0825.93174
[3] Apkarian, P.; Gahinet, P.; Becker, G., Selfscheduled H∞ control of linear parameter-varying systems, Automatica, 31, 1251-1261, (1995) · Zbl 0825.93169
[4] Boyd, S.P.; El Ghaoui, L., Method of centers for minimizing generalized eigenvalues, Lin. alg. applic., 188, 63-111, (1993) · Zbl 0781.65051
[5] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., ()
[6] Chilali, M.; Gahinet, P., H∞ design with pole placement constraints: an LMI approach, IEEE trans. autom. control, AC-41, 358-367, (1996) · Zbl 0857.93048
[7] David, J.; De Moor, B., The opposite of analytic centering for solving minimum rank problems in control and identification, ()
[8] Doyle, J.C.; Glover, K.; Khargonekar, P.; Francis, B., State-space solutions to standard H2 and H∞ control problems, IEEE trans. autom. control, AC-34, 831-847, (1989) · Zbl 0698.93031
[9] El Ghaoui, L.; Gahinet, P., Rank-minimization under LMI constraints: a framework for output feedback problems, (), 1176-1179
[10] Fan, M.K.H., A second-order interior point method for solving linear matrix inequality problems, SIAM J. control optim., (1993), Submitted to
[11] Francis, B., A course in H∞ control theory, ()
[12] Gahinet, P., A convex parametrization of H∞ suboptimal controllers, (), 937-942
[13] Gahinet, P.; Apkarian, P., A linear matrix inequality approach to H∞ control, Int. J. robust nonlinear control, 4, 421-448, (1994) · Zbl 0808.93024
[14] Gahinet, P.; Ignat, A., Low-order H∞ synthesis via lmis, (), 1499-1500
[15] Gahinet, P.; Laub, A.J.; Gahinet, P.; Laub, A.J., Reliable computation of γopt in singular H∞ control, (), 1527-1532, (1995), Also in · Zbl 0728.93022
[16] Gahinet, P.; Nemirovski, A.; Laub, A.J.; Chilali, M.; Gahinet, P.; Nemirovski, A., (), 2038-2041, Also in
[17] Glover, K.; Doyle, J.C., State-space formulas for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensitivity, Syst. control lett., 11, 167-172, (1988) · Zbl 0671.93029
[18] Grigoriadis, K.M.; Skelton, R.E., Fixed-order control design for LMI control design problem using alternating projection methods, (), 2003-2008 · Zbl 0854.93146
[19] Iwasaki, T.; Skelton, R.E., All controllers for the general H∞ control problem: LMI existence conditions and state-space formulas, Automatica, 30, 1307-1317, (1994) · Zbl 0806.93017
[20] Limebeer, D.J.N.; Green, M.; Walker, D., Discrete-time H∞ control, ()
[21] Nemirovski, A.; Gahinet, P.; Nemirovski, A.; Gahinet, P., The projective method for solving linear matrix inequalities, (), 840-844, (1994), Also in
[22] Nesterov, Yu.; Nemirovski, A., ()
[23] Packard, A., Gain scheduling via linear fractional transformations, Syst. control lett., 22, 79-92, (1994) · Zbl 0792.93043
[24] Packard, A.; Zhou, K.; Pandey, P.; Becker, G., A collection of robust control problems leading to LMI’s, (), 1245-1250
[25] Peterson, I.R.; Anderson, B.D.O.; Jonckheere, E.A., A first principles solution to the non-singular H∞ control problem, Int. J. robust nonlinear. control, 1, 171-185, (1991) · Zbl 0759.93027
[26] Safonov, M.G.; Limebeer, D.J.; Chiang, R.Y., Simplifying the H∞ theory via loop-shifting, matrix-pencil and descriptor concepts, Int. J. control, 50, 2467-2488, (1989) · Zbl 0695.93034
[27] Scherer, C., The Riccati inequality and state-space H∞-optimal control, ()
[28] Scherer, C., H∞ optimization without assumptions on finite or infinite zeros, SIAM J. control optim., 30, 143-166, (1992) · Zbl 0748.93017
[29] Stoorvogel, A.A., The discrete-time H∞ control problem with measurement feedback, SIAM J. control optim., 30, 182-202, (1992) · Zbl 0748.93042
[30] Stoorvogel, A.A.; Saberi, A.; Chen, B.M., The discrete-time H∞ control problem with measurement feedback, Int. J. robust nonlin. control, 4, 457-479, (1994) · Zbl 0812.93028
[31] Vandenberghe, L.; Boyd, S., Primal-dual potential reduction method for problems involving matrix inequalities, Math. programming, ser. B, 69, 205-236, (1995) · Zbl 0857.90104
[32] Wu, F.; Yang, X.; Packard, A.; Becker, G., Induced \(L\)_{2} norm control for LPV systems with bounded parameter variation rates, ()
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.