A Parametric Lyapunov Equation Approach to the Design of Low Gain Feedback

• Published in: IEEE transactions on automatic control Vol. 53; no. 6; pp. 1548 - 1554
• Main Authors: Zhou, Bin, Duan, Guangren, Lin, Zongli
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuator saturation; Applied sciences; Computer science; control theory; systems; Control system synthesis; Control systems; Control theory; Control theory. Systems; Design engineering; Exact sciences and technology; Feedback; Gain; global stabilization; Hydraulic actuators; Linear systems; low gain feedback; Mathematical analysis; Mathematical models; Nonlinear control systems; Optimal control; parametric Lyapunov equation; Performance gain; Poles and zeros; Riccati equations; Robust stability; Scalars; semiglobal stabilization; set invariance; State feedback; Studies; State feedback; Non linear control; Feedback regulation; Control synthesis; Invariant set; Actuator saturation; Pole assignment; Control constraint; Eigenvalue problem; low gain feedback; Actuator; Modal control; set invariance; Polynomial matrix; Saturation; Lyapunov equation; Constrained optimization; parametric Lyapunov equation; Robust control; semiglobal stabilization; Global stabilization; Small gain theorem; Open loop; Algebraic equation; Matrix polynomial; Riccati equation
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2008.921036

Low gain feedback has found several applications in constrained control systems, robust control and nonlinear control. Low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero. Such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic Riccati equation (ARE). The eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter, while the ARE approach requires the solution of an ARE for each value of the parameter. This note proposes an alternative approach to low gain feedback design based on the solution of a parametric Lyapunov equation. Such an approach possesses the advantages of both the eigenstructure assignment approach and the ARE-based approach. It also avoids the possible numerical stiffness in solving a parametric ARE and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach.