On the Value Functions of the Discrete-Time Switched LQR Problem

• Published in: IEEE transactions on automatic control Vol. 54; no. 11; pp. 2669 - 2674
• Main Authors: Wei Zhang, Jianghai Hu, Abate, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Applied sciences; Automatic control; Computer science; control theory; systems; Control theory. Systems; Convergence; Cost function; Discrete-time switched LQR (DSLQR); Exact sciences and technology; Linear systems; Mapping; Mathematical analysis; Matrices; Matrix methods; Optimal control; Piecewise linear techniques; Riccati equations; State feedback; State-space methods; Switched systems; Discrete function; Multimodel control; Finite horizon; LQ control; Discrete-time switched LQR (DSLQR); Discrete time; Infinite horizon; Riccati equation; Numerical convergence; Quadratic function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2009.2031574

In this paper, we derive some important properties for the finite-horizon and the infinite-horizon value functions associated with the discrete-time switched LQR (DSLQR) problem. It is proved that any finite-horizon value function of the DSLQR problem is the pointwise minimum of a finite number of quadratic functions that can be obtained recursively using the so-called switched Riccati mapping . It is also shown that under some mild conditions, the family of the finite-horizon value functions is homogeneous (of degree 2), is uniformly bounded over the unit ball, and converges exponentially fast to the infinite-horizon value function. The exponential convergence rate of the value iterations is characterized analytically in terms of the subsystem matrices.