zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The generalized $H\sb 2$ control problem. (English) Zbl 0772.93027
Summary: We consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary (but bounded) energy exogenous inputs. More specifically, we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from $L\sb 2[0,\infty)$ to $L\sb \infty[0,\infty)$ is below a specified level. We consider both state-feedback and output feedback problems. In the state-feedback case it is shown that if this synthesis problem is solvable, then a solution can be chosen to be a constant state-feedback gain. Necessary and sufficient conditions for the existence of solutions as well as a formula for a state-feedback gain that solves this control problem are obtained in terms of a finite-dimensional convex feasibility program. After showing the separation properties of this synthesis problem, the output feedback case is reduced to a state-feedback problem. It is shown that, in the output feedback case, generalized $H\sb 2$ controllers can be chosen to be observer based controllers. The theory is demonstrated with a numerical example.

93B50Synthesis problems
93C05Linear control systems
Full Text: DOI
[1] Bernussou, J.; Peres, P. L. D.; Geromel, J. C.: An LP oriented procedure for quadratic stabilization of uncertain systems. Syst. contr. Lett. 13, 65-72 (1989) · Zbl 0678.93042
[2] Boyd, S. P.; Barratt, C. H.: Linear controller design: limits of performance. (1990) · Zbl 0748.93003
[3] Boyd, S. P.; Yang, Q.: Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Control 49, 2215-2240 (1989) · Zbl 0683.93057
[4] Corless, M.; Zhu, G.; Skelton, R. E.: Robustness properties of covariance controllers. Proc. of the 28th conf. On decision and control (1989)
[5] Doyle, J. C.: ONR/honeywell workshop on advances in multivariable control. (1984)
[6] Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A.: State-space solutions to standard H2 and H$\infty $and control problems. IEEE trans. Aut. control 34, 831-847 (1989) · Zbl 0698.93031
[7] Grimble, M. J.: Relationship between the trace and the maximum eigenvalue norms for linear quadratic control design. IEEE trans. Aut. control 35, 1176-1181 (1990) · Zbl 0724.93034
[8] Kalman, R. E.: Contributions to the theory of optimal control. Bul. soc. Mat. Mexico 5, 102 (1960) · Zbl 0112.06303
[9] Kaminer, I.; Khargonekar, P. P.; Rotea, M. A.: Mixed H2H$\infty $control for discrete-time systems via convex optimization. Automatica 29, 57-70 (1993) · Zbl 0772.93055
[10] Khargonekar, P. P.; Rotea, M. A.: Multiple objective control of linear systems: the quadratic norm case. IEEE trans. Aut. control 36, 14-24 (1991) · Zbl 0723.93021
[11] Khargonekar, P. P.; Rotea, M. A.: Mixed H2H$\infty $control: A convex optimization approach. IEEE trans. Aut. control 36, 824-837 (1991) · Zbl 0748.93031
[12] Khargonekar, P. P.; Petersen, I. R.; Rotea, M. A.: H\infty-optimal control with state-feedback. IEEE trans. Aut. control 33, 786-788 (1988) · Zbl 0655.93026
[13] Marshall, A. W.; Olkin, I.: Inequalities: theory of majorization and its applications. (1979) · Zbl 0437.26007
[14] Packard, A.; Becker, G.: State-feedback solutions to quadratic stabilization. Proc. of the 28th annual allerton conf. On communication, control, and computing, 768-769 (1990)
[15] Rotea, M. A.: The generalized H2 control problem. Proc. of 1st IFAC symposium on design methods of control systems, Zürich 1, 112-117 (1991)
[16] Rotea, M. A.; Khargonekar, P. P.: Generalized H2H$\infty $control via convex optimization. Proc. of the 30th conference on decision and control, brighton, UK 3, 2719-2720 (1991)
[17] Wilson, D. A.: Convolution and Hankel operator norms for linear systems. IEEE trans. Aut. control 34, 94-97 (1989) · Zbl 0661.93022
[18] Wilson, D. A.: Extended optimality properties of the linear quadratic regulator and stationary Kalman filter. IEEE trans. Aut. control 35, 583-585 (1990) · Zbl 0708.93091
[19] Zhu, G.; Skelton, R. E.: Mixed L2 and L$\infty $problems by weight selection in quadratic optimal control. Int. J. Control 53, 1161-1176 (1991) · Zbl 0725.93041