A Design of Gain Switching Control Systems with Quadratic Performance Index

Abstract

The VSS design problem for a linear system has been investigated using a quadratic performance index, the value of which can be more lowered by logically switching admissible gains {K₀, K₁, …, K_p} than by the single constant gain K₀. A unique guaranteed cost control (GCC) method is introduced to derive the gain switching law proposed here.
The problem considered in this paper can be stated as follows: Given a controllable linear system,
x(t)=Ax(t)+Bu(t), x∈Rⁿ, u∈R^m,
with a set of admissible gains {K₀, K₁, …, K_p} and the quadratic performance index,
J=∫∞_t0e^2α(t-t₀)x(t)'Qx(t)dt, Q>0,
find a variable gain state feedback control law which gives a lower cost than the control law with the constant gain K₀.
The main results we have obtained are as follows:
(i) A solution to the problem has the form,
u(t)=K_j(t)x(t), j(t)={j:min_iV_i=V_j}, V_i=2x(t)'PB(K_i-K₀)x(t),
where the matrix P is the solution of the Lyapunov equation,
(A+BK₀)'P+P(A+BK₀)+2αP+Q=0.
(ii) The designed VSS is stable if there hold the conditions that both the matrices A+BK₀ and A+BK₀+αI are stable and the matrix Q+2αP is positive definite.
(iii) Even the case where the system matrices A and B include some bounded modeling errors can be treated successfully by introducing a parameter σ>0 into the switching law as
The parameter σ represents the size of the modeling errors; the trade-off between robustness and control performance can be performed by selecting the value of the parameter σ.
The results of some simulation experiments are also presented to demonstrate the effectiveness of the proposed design technique and to illustrate the corresponding relationship between the parameter σ and the robustness property.