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)
Analysis and design for discrete-time linear systems subject to actuator saturation. (English) Zbl 0987.93027
Summary: We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback that makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.

93B50Synthesis problems
93C55Discrete-time control systems
93D20Asymptotic stability of control systems
93B51Design techniques in systems theory
93C10Nonlinear control systems
93C73Perturbations in control systems
15A39Linear inequalities of matrices
Full Text: DOI
[1] Blanchini, F.: Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov function. IEEE trans. Automat. control 39, 428-433 (1994) · Zbl 0800.93754
[2] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in systems and control theory. (1994) · Zbl 0816.93004
[3] Davison, E. J.; Kurak, E. M.: A computational method for determining quadratic Lyapunov functions for non-linear systems. Automatica 7, 627-636 (1971) · Zbl 0225.34027
[4] Gilbert, E. G.; Tan, K. T.: Linear systems with state and control constraints: the theory and application of maximal output admissible sets. IEEE trans. Automat. control 36, 1008-1020 (1991) · Zbl 0754.93030
[5] H. Hindi, S. Boyd, Analysis of linear systems with saturation using convex optimization, Proceedings of the 37th IEEE CDC, Florida, 1998, pp. 903--908.
[6] Hu, T.; Lin, Z.: Control systems with actuator saturation: analysis and design. (2001) · Zbl 1061.93003
[7] T. Hu, Z. Lin, Exact characterization of invariant ellipsoids for linear systems with saturating actuators, IEEE Trans. Automat. Control, to appear.
[8] T. Hu, Z. Lin, B.M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance, Proceedings of the American Control Conferences, 2000, pp. 725--729; also in Automatica, to appear.
[9] Khalil, H.: Nonlinear systems. (1996) · Zbl 0842.93033
[10] Kosut, R. L.: Design of linear systems with saturating linear control and bounded states. IEEE trans. Automat. control 28, No. 1, 121-124 (1983) · Zbl 0501.93053
[11] Loparo, K. A.; Blankenship, G. L.: Estimating the domain of attraction of nonlinear feedback systems. IEEE trans. Automat. control 23, No. 4, 602-607 (1978) · Zbl 0385.93023
[12] C. Pittet, S. Tarbouriech, C. Burgat, Stability regions for linear systems with saturating controls via circle and Popov criteria, Proceedings of the 36th IEEE CDC, San Diego, 1997, pp. 4518--4523.
[13] Romanchuk, B. G.: Computing regions for attraction with polytopes: planar case. Automatica 32, No. 12, 1727-1732 (1996) · Zbl 0869.93036
[14] Vanelli, A.; Vidyasagar, M.: Maximal Lyapunov functions and domain of attraction for autonomous nonlinear systems. Automatica 21, No. 1, 69-80 (1985) · Zbl 0559.34052