Approximation and Monotonicity of the Maximal Invariant Ellipsoid for Discrete-Time Systems by Bounded Controls

• Published in: IEEE transactions on automatic control Vol. 55; no. 2; pp. 440 - 446
• Main Authors: BIN ZHOU, DUAN, Guang-Ren, ZONGLI LIN
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuator saturation; Algorithms; Applied sciences; Approximation; Computer science; control theory; systems; Control equipment; Control system analysis; Control systems; Control theory. Systems; Convergence; Cyclic redundancy check; Delay; Differential equations; Ellipsoids; Exact sciences and technology; Feedback; invariant set; Invariants; Linear systems; Mathematical analysis; maximal invariant ellipsoid; minimal energy control with guaranteed convergence rate (MECGCR); Polynomials; Read only memory; Size control; Studies; Tradeoffs; Bounded control; maximal invariant ellipsoid; Invariant; minimal energy control with guaranteed convergence rate (MECGCR); Necessary and sufficient condition; Saturation; Nonlinear problems; Invariant set; Fork join problem; Actuator saturation; Closed feedback; Positive definite matrix; Discrete time; Algebraic equation; Linear matrix inequality; Convergence rate; Ellipsoid; Numerical convergence; Monotonicity; Actuator; Mathematical programming
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2009.2036324

An analytic approximation of the maximal invariant ellipsoid for a discrete-time linear system with bounded controls is derived. The approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid. It is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an LMI-based optimization problem or a nonlinear algebraic equation. Furthermore, the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established. On the other hand, the monotonicity of the maximal invariant ellipsoid resulting from the ¿minimal energy control with guaranteed convergence rate¿ problem is established that shows a trade-off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed-loop system under a bounded control. Two illustrative examples demonstrate of the effectiveness of the results.