SUMMARYRobust λ-contractive sets have been proposed in previous literature for uncertain polytopic linear systems. It is well known that, if initial state is inside such sets, it is guaranteed to converge to the origin. This work presents the generalization of such concepts to systems whose behaviour changes among different linear models with probability given by a Markov chain. We propose sequence-dependent sets and associated controllers which can ensure a reliability bound when initial conditions are outside the maximal λ-contractive set. Such reliability bound will be understood as the probability of actually reaching the origin from a given initial condition without violating constraints. As initial condition are further away from the origin, the likelihood of reaching the origin decreases.

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This paper presents a methodology to obtain a guaranteed-reliability controller for constrained linear systems, which switch between different modes according to a Markov chain (Markov jump linear systems). Inside the classical maximal robust controllable set, there is 100% guarantee of never violating constraints at future time. However, outside such set, some sequences might make hitting constraints unavoidable for some disturbance realisations. A guaranteed-reliability controller based on a greedy heuristic approach was proposed in an earlier work for disturbance-free, robustly stabilisable Markov jump linear systems. Here, extensions are presented by, first, considering bounded disturbances and, second, presenting an iterative algorithm based on dynamic programming. In non-stabilisable systems, reliability is zero; therefore, prior results cannot be applied; in this case, optimisation of a mean-time-to-failure bound is proposed, via minor algorithm modifications. Optimality can be proved in the disturbance-free, finitely generated case. Copyright understood as the average time trajectories take to hit constraints, should be considered. Obviously, reliability and MTTF will depend on how close initial state is from constraint boundaries.The aforementioned issues arise in practise in, for instance, networked control systems with random packet losses [11], control of processes subject to randomly occurring faults (fault-tolerant control [12]), predictive control under scenarios discarding some possible outcomes [13], etc.Specifically, this paper extends the classical set-invariance ideas to provide some robust reliability and MTTF bounds (and associated controllers), in processes where disturbances are bounded and parameter changes can be modelled as a set of discrete-time linear systems (operation modes), with mode transitions governed by a Markov chain (Markov jump linear systems (MJLS) [14,15]). In this MJLS case, the maximal sets for which a mode-dependent control action exists making them robustly invariant can, too, be computed by adapting the aforementioned concepts [16, Lemma 16], [6,17]. Of course, linear matrix inequalities may be used to handle ellipsoidal invariant sets for MJLS [18], but this approach will not be pursued in this work.As discussed in the aforementioned text, the work [8] defined reliability as the probability of avoiding future constraint violations, but only in a disturbance-free setting. Basically, the main idea proposed there is the fact that, if the initial state is outside the maximal robust controllable set, there might exist mode sequences such that constraint violation is unavoidable ('failure'): if we are 'close' to the maximal invariant set, such failure can be avoided with high probability, but it will not be the case if we are 'far away' from it. The cited work proposed a solution, based on so-called 'sequencedependent' sets; a suboptimal 'greedy' action was chosen to be the one associated to the most likely sequence with which the invariant set was reachable.Results in th...

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This work considers fault-tolerant control (FTC) for discrete-time Markov Jump Linear System (MJLS) subject to constraints on the state an control variables. The objective is to design a control law which depends on the jump variable, minimizing an average quadratic function. Improving over previous MJLS literature, input and state constraints are enforced. The initial condition of the system and the transition probability of the Markov chain are available to the controller at each instant of time. Concepts arising from the receding horizon framework and invariant set theory are incorporated in a constrained fault-tolerant predictive control approach.