Mixed H ₂ / H _∞ control of time‐varying stochastic discrete‐time systems under uniform detectability

Abstract

This study is concerned with the infinite horizon H 2 /H ∞ control of time-varying stochastic discrete-time systems with state and disturbance dependent noise. The notion of uniform detectability for time-varying stochastic discrete-time systems is introduced, and some good properties are derived. Under uniform detectability, a necessary and sufficient condition for the existence of time-varying stochastic H 2 /H ∞ control is presented, which transforms the design of H 2 /H ∞ control into the solvability of coupled time-varying matrix-valued equations. Finally, an iterative algorithm is proposed to solve the coupled time-varying matrix-valued equations, and a numerical example is provided to show the effectiveness of the obtained results.

Corrections to ‘Mixed control of time‐varying stochastic discrete‐time systems under uniform detectability’

“…appeared in the proof of Lemma 1-(ii) in [1]? In general, the rightness of Lemma 1 [1] needs a complete proof.…”

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This note first points out some errors of the above mentioned paper [1], and then presents a complete proof for Lemma 1-(i) in [1].

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“…In the proof of Theorem 1 [1], P k = ∞ i=k φ i,k D i D i φ i,k is wrong, because the column number of D i is n but the row number of φ i,k is 2 l−k n, φ i,k D i D i φ i,k is meaningless. Hence, the necessity proof of Theorem 1 [1] is wrong. A correct proof can also be found in [2].…”

Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations

“…The robust stabilisation and robust control were considered for discrete‐time stochastic systems with time‐varying delays, and a linear matrix inequality approach was provided to solve these two problems in [2]. In [3], the infinite horizon mixed control was solved for time‐varying stochastic discrete‐time systems under uniform detectability. A linear matrix inequality approach was developed for almost sure stabilisation of linear discrete‐time stochastic systems in [4].…”

“…Similarly in [8], with these two notations the discrete‐time stochastic Lyapunov matrix equations were also investigated. In [3], the concept of uniform detectability was introduced for discrete time‐varying stochastic linear systems, and then a criterion was established in terms of a sequence of stochastic Lyapunov matrix equations.…”