Model reduction on Markovian jump systems with partially unknown transition probabilities: balanced truncation approach

Abstract

In this study, the problem of model reduction based on balancing is investigated for both discrete- and continuoustime Markovian jump linear systems with partially unknown transition probabilities. By balancing transformation, the reduced-order model with the same structure as that of the original one is obtained by truncating the balanced model. For the obtain reduced order model, stability property is preserved under simultaneous balanced truncation. An upper bound of the model reduction error is guaranteed in the sense of a perturbation operator norm. Finally, two illustrative examples are provided to show the feasibility and effectiveness of the method presented in this study. 1 Introduction Markovian jump linear systems (MJLSs) have been attracting increasing attention because of their widely practical applications in manufacturing systems, power systems, economics systems, communication systems, network-based control systems and so on [1, 2]. In a practical application, a wide class of dynamic systems in engineering experience random abrupt changes because of the component failures or repairs, abrupt environmental disturbances and changing subsystem interconnections. MJLSs are widely investigated to deal with these systems with variable structures whose structures change randomly at discrete time instances governed by a Markovian process. As a dominant factor, the transition probabilities in the jumping process determine the system behaviour to a great degree and so far, a great deal of attention has been devoted to the analysis and synthesis of MJLSs, assuming the complete knowledge of the transition probabilities, including stability analysis [3, 4], dissipation analysis, filter design [5–7], model reduction [8–10], composite control [11, 12] and so on. The ideal knowledge on the transition probabilities is definitely expected to simplify the system analysis and design. In fact, whether in theory or in practice, the possibilities of obtaining the completely knowledge of the transition probabilities are questionable and the cost is probably high [13, 14]. Thus, rather than the complexity of measuring or estimating all the elements of the transition probabilities matrix, it is significant and necessary from control perspectives to study further more general MJLSs with partially unknown transition probabilities. Zhang and Boukas [15] considered the stability and stabilisation of discrete- and continuous-time MJLSs with partially unknown transition probabilities. In the paper, when the terms which contained unknown transition probabilities were separated from the others, the fixed connection weighting matrices ATi Pi + PiAi were selected for continuous-time MJLSs and Pi were introduced for discrete-time MJLSs. The finite-time stochastic control problem has been considered for discrete-time Markovian jump non-linear quadratic systems subject to exogenous disturbance and partial information on transition probabilities in [16]. Currently, the authors [17, 18] obtained a less conservative stability criterion of continuous-time MJLSs with partially unknown transition