Dynamic Output Feedback Control of Discrete-Time Markov Jump Linear Systems through Linear Matrix Inequalities

Abstract

This paper addresses the H 2 and H∞ dynamic output feedback control design problems of discrete-time Markov jump linear systems. Under the mode-dependent assumption, which means that the Markov parameters are available for feedback, the main contribution is the complete characterization of all full order proper Markov jump linear controllers such that the H 2 or H∞ norm of the closed loop system remains bounded by a given prespecified level, yielding the global solution to the corresponding mode-dependent optimal control design problem, expressed in terms of pure linear matrix inequalities. Some academic examples are solved for illustration and comparison. As a more consequent practical application, the networked control of a vehicle platoon using measurement signals transmitted in a Markov channel, as initially proposed in [P. Seiler and R. Sengupta, IEEE Trans. Automat. Control, 50 (2005), pp. 356-364], is considered.

H∞ robust and networked control of discrete-time MJLS through LMIs

“…Also notice that the inequality (9) is indexed in both i 2 K and j 2 K and it is needed to linearize the term P À1 pi in inequality (4). Those constraints are similar to the ones developed for output feedback at [14].…”

Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities

“…It is not possible to use Theorem 1 for this design, since the LMIs (16) are unfeasible for a plant that is not robustly stable. Using Theorem 2, however, we get the minimum upper bound = 1.8778 corresponding to the worst vector of initial probabilities ∈ R 2 treated as an additional optimization variable, see [16]. The associated filter gains are In this case, we have to stress that once the internal model of the plant is available, an observerbased filter can be designed in order to make the estimation error stable even though the system under consideration is itself unstable.…”

“…If it is not possible to determine that value, one can consider the vector of initial probabilities ∈ R N as an additional optimization variable and minimize the worst case norm as indicated in [16]. From the H ∞ -norm definition [17], it is also possible to calculate G 2 ∞ as the optimal solution of a convex programming problem expressed by LMIs, which can be obtained from the Bounded Real Lemma presented in [18] …”

Stability analysis of two-dimensional Markovian jump state-delayed systems in the Roesser model with uncertain transition probabilities

“…Although the system performance may be influenced by various factors, transition probabilities (TPs) of the jumping process play an essential role in determining the system behavior and performance. In the existing literature, many analysis and synthesis issues on MJSs have been exploited with the assumption that the TPs can be completely accessible [2,4,9,23,27,31]. However, incomplete TPs of the jumping process are often encountered in practice, especially when adequate samples of the transitions are costly or time-consuming to obtain.…”

“…Recently, Markovian jump systems (MJSs), an important class of stochastic hybrid systems in which the switching between subsystems are governed by a finite-state Markov chain, have received an extensive attention due to their flexibility in modeling real-world problems [9,11,21,34]. For example, MJSs can be used to model control systems where the plant and controllers are connected via a wireless communications network subject to random abrupt changes in the inputs or Markov sensors assignment [12].…”