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In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity.Index Terms-Switched systems, stabilizability, LMI.M. Fiacchini is with GIPSA-lab,

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov-Metzler condition for stabilizability does not hold.

We consider the problem of output feedback stabilization in linear systems when the measured outputs and control inputs are subject to event-triggered sampling and dynamic quantization. A new sampling algorithm is proposed for outputs which does not lead to accumulation of sampling times and results in asymptotic stabilization of the system. The approach for output sampling is based on defining an event function that compares the difference between the current output and the most recently transmitted output sample not only with the current value of the output, but also takes into account a certain number of previously transmitted output samples. This allows us to reconstruct the state using an observer with sample-and-hold measurements. The estimated states are used to generate a control input, which is subjected to a different event-triggered sampling routine; hence the sampling times of inputs and outputs are asynchronous. Using Lyapunov-based approach, we prove the asymptotic stabilization of the closed-loop system and show that there exists a minimum inter-sampling time for control inputs and for outputs. To show that these sampling routines are robust with respect to transmission errors, only the quantized (in space) values of outputs and inputs are transmitted to the controller and the plant, respectively. A dynamic quantizer is adopted for this purpose, and an algorithm is proposed to update the range and the centre of the quantizer that results in an asymptotically stable closed-loop system.

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This paper presents a novel formulation of a robust model predictive controller (RMPCT) to track piecewise constant references. The real plant is assumed to be modelled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed MPC can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. This allows us to reject constant disturbances compensating the effect of then, changing the setpoint. Feasibility of the proposed controller for any admissible setpoint is achieved by adding an artificial steady state as decision variable. Robust constraint satisfaction is guaranteed by tube-based approach and considering nominal predictions. Robust stability and convergence to (a neighborhood of) the desired steady state is ensured by considering a modified cost function and an extended terminal constraint. The cost function penalizes the tracking error with the artificial reference and the deviation between the artificial and desired steady state; the terminal constraint restricts the terminal state and the artificial steady state. The optimization problem to be solved is a Quadratic Programming problem, which allows explicit implementations. In order to demonstrate the benefits of the proposed controller, this has been tested on a real positioning plant consisting in a linear motor driving a cart. The fast dynamics of the systems requires an explicit calculation of the controller, which has been implemented by means of a search tree strategy. The experimental results show the robust tracking and the admissible evolution of the closed loop system.

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h i g h l i g h t s • A new MPC suitable for closed-loop re-identification is proposed. • A re-identification needs to be developed in a closed-loop fashion, since the process cannot be stopped. • The main problem is the conflict between the control and identification objectives. • A generalization, from punctual stability to (invariant) set stability, is done to avoid the conflict. • The proposal could be potentially applied to real processes.