# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Robust $H\sb{\infty}$ control for linear time-invariant systems with norm-bounded uncertainty in the input matrix. (English) Zbl 0698.93054
Summary: This paper focuses on the problem of robust $H\sb{\infty}$ control design for a class of linear time-invariant systems with uncertainty in the state space model. We consider uncertain systems with norm-bounded parameter uncertainty in the input matrix. The paper presents a state feedback control design which stabilizes the plant for all admissible uncertainties and also guarantees an $H\sb{\infty}$-norm bound constraint on disturbance attenuation. Paralleling to the theory of robust control, the robust $H\sb{\infty}$ control problem is solved via the notion of `quadratic stabilization with an $H\sb{\infty}$-norm bound constraint’. Necessary and sufficient conditions for quadratic stabilization with an $H\sb{\infty}$-norm bound are derived. It is shown that the solution to this problem involves solving a parameter-dependent algebraic Riccati equation. The results can be regarded as extensions of existing results on robust stabilization of linear uncertain systems and $H\sb{\infty}$ optimal control.

##### MSC:
 93D15 Stabilization of systems by feedback 93B50 Synthesis problems 93B35 Sensitivity (robustness) of control systems 93C05 Linear control systems 93C15 Control systems governed by ODE
Full Text:
##### References:
 [1] De Souza, C. E.; Xie, L.: On the properties of indefinite algebraic Riccati equations. Technical report EE8952 (August 1989) [2] Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A.: State-space solutions to standard H2 and H$\infty$control problems. IEEE trans. Auto. control 34, 831-847 (1989) · Zbl 0698.93031 [3] P.P. Khargonekar, I.R. Petersen and K. Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and H\infty co control theory, IEEE Trans. Automat. Control, to appear. · Zbl 0707.93060 [4] Madiwale, A. N.; Hadddad, W. M.; Bernstein, D. S.: Proc. 27th IEEE conf. Decision and control. 965-972 (Dec. 1988) [5] Petersen, I. R.; Hollot, C. V.: A Riccati equation approach to the stabilization of uncertain linear systems. Automatica 22, 397-411 (1986) · Zbl 0602.93055 [6] Petersen, I. R.: Disturbance attenuation and H$\infty$optimization: A design method based on the algebraic Riccati equation. IEEE trans. Automat. control 32, 427-429 (1987) · Zbl 0626.93063 [7] Petersen, I. R.: A stabilization algorithm for a class of uncertain linear systems. Systems control lett. 8, 351-357 (1987) · Zbl 0618.93056 [8] Petersen, I. R.: Stabilization of an uncertain linear system in which uncertain parameters enter into the input matrix. SIAM J. Control optim. 26, 1257-1264 (1988) · Zbl 0667.93087 [9] Petersen, I. R.: Some new results on algebraic Riccati equations arising in linear quadratic differential games and the stabilization of uncertain linear systems. Systems control lett. 10, 341-348 (1988) · Zbl 0663.93022 [10] Willems, J. C.: Least squares, stationary optimal control and algebraic Riccati equation. IEEE trans. Automat. control 16, 621-634 (1971) [11] Xie, L.; De Souza, C. E.: State feedback H$\infty$optimal control problems for non-detectable systems. Systems control lett. 13, 315-319 (1989) · Zbl 0684.93031 [12] Zhou, K.; Khargonekar, P. P.: Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Systems control lett. 10, 17-20 (1988) · Zbl 0634.93066 [13] Zhou, K.; Khargonekar, P. P.: An algebraic Riccati equation approach to H$\infty$optimization. Systems control lett. 11, 85-91 (1988) · Zbl 0666.93025