Márcio F. Braga

Robust state feedback control for discrete-time linear systems via LMIs with a scalar parameter

This paper proposes an improved approach to ℋ2 and ℋ∞ robust state feedback control design for discrete-time polytopic time-invariant linear systems based on Linear Matrix Inequalities (LMIs) with a scalar parameter. The synthesis conditions, that depend on a real parameter lying in the interval (-1,1), become LMIs for fixed values of the scalar, reducing to standard conditions in the literature when the scalar is equal to zero. At the price of line searches combined with LMIs, less conservative results for robust state feedback control are obtained. The closed-loop stability and the ℋ2 and ℋ∞ guaranteed costs are certified by means of an affine parameter-dependent Lyapunov function. The validity and the efficiency of the method are illustrated by means of examples and exhaustive numerical comparisons.

ℋ∞ state feedback control for MJLS with uncertain probabilities

This paper addresses the problem of ? ∞ state feedback control design for discrete-time Markov jump linear systems (MJLS) with uncertain transition probability matrix. The main novelty is that, differently from the existing approaches in the literature, the proposed conditions allow the use of polynomially parameter-dependent Lyapunov matrices to certify the closed-loop stability of the MJLS. Therefore, the method is able to provide ? ∞ controllers in cases where the other techniques fail. The synthesis conditions are given in terms of linear matrix inequality relaxations. Examples illustrate the main advantages of the proposed control design method when compared to other approaches from the literature.

A new procedure for discretization and state feedback control of uncertain linear systems

This paper addresses the problem of constant sampling discretization of uncertain time-invariant continuous-time linear systems in polytopic domains. To circumvent the difficulty of dealing with the exponential of uncertain matrices, a new discretization method, based on Taylor series expansion, is proposed. The resulting discrete-time uncertain system is described in terms of homogeneous polynomial matrices with parameters lying in the unit simplex and an additive norm-bounded uncertainty which represents the discretization residual error. As a second contribution, linear matrix inequality (LMI) based conditions for the synthesis of a stabilizing state feedback control for discrete-time linear systems with polynomial dependence on the uncertain parameters and an additive norm-bounded uncertainty are proposed. Numerical experiments illustrate the discretization technique advantages of using higher orders in the Taylor series expansion to obtain more precise approximations. The examples also show that, at the price of simple line searches in a scalar parameter, and using Lyapunov functions of higher degrees, less conservative results for robust state feedback control design of discretized uncertain systems can be obtained.

Discretisation and control of polytopic systems with uncertain sampling rates and network-induced delays

This paper investigates the problems of uncertain sampling rate discretisation and the networked control of uncertain time-invariant continuous-time linear systems in polytopic domains. The sampling period is assumed to be unknown but belonging to a given interval. To avoid the difficulty of dealing with the exponential of uncertain matrices, a discrete-time model is obtained by applying a Taylor series expansion of degree ℓ to the original system. The resulting discrete-time model is composed of homogeneous polynomial matrices with parameters lying in the Cartesian product of simplexes, called a multi-simplex, plus an additive norm-bounded term representing the discretisation residual error. The original continuous-time system is controlled through a communication network that introduces a time delay in the process. Linear matrix inequality relaxations that include a scalar parameter search are proposed for the design of a digital robust state feedback controller that guarantees the closed-loop stability of the networked control system. Numerical experiments are presented to illustrate the versatility of the proposed method, which can be applied to a more general class of networked control problems than the existing approaches in the literature.

Discretization and event triggered digital output feedback control of LPV systems

Abstract This paper investigates the problem of discretization and digital output feedback control design for continuous-time linear parameter-varying (LPV) systems subject to a time-varying networked-induced delay. The proposed discretization procedure converts a continuous-time LPV system into an equivalent discrete-time LPV system based on an extension of the Taylor series expansion and using an event-based sampling. The scheduling parameters are continuously measured and modeled as piecewise constant. A new transmission of the measured output to the controller is triggered by significant changes in the parameters, yielding time-varying transmission intervals. The obtained discretized model has matrices with polynomial dependence on the time-varying parameters and an additive norm-bounded term representing the discretization residual error. A two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the LPV model. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through numerical examples.

Robust stability and stabilization of discrete-time Markov jump linear systems with partly unknown transition probability matrix

An improved linear matrix inequality (LMI) approach is proposed to deal with the problems of stability, mode-dependent and mode-independent stabilization of discrete-time Markov jump linear systems (MJLS) with partly unknown transition probability matrix. As a first contribution, the uncertain parameters of the transition probability matrix are modeled in terms of the Cartesian product of simplexes, called multi-simplex. Then, convergent LMI relaxations with improved trade-off between precision and computational effort are proposed for the stability analysis of this class of MJLS. Finally, new design conditions based on LMIs with a scalar parameter are proposed for state feedback control, in both mode independent and mode dependent scenarios, providing less conservative results when compared to other conditions available in the literature, as illustrated by numerical examples.

Reduced-order dynamic output feedback control of uncertain discrete-time Markov jump linear systems

ABSTRACT This paper deals with the problem of designing reduced-order robust dynamic output feedback controllers for discrete-time Markov jump linear systems (MJLS) with polytopic state space matrices and uncertain transition probabilities. Starting from a full order, mode-dependent and polynomially parameter-dependent dynamic output feedback controller, sufficient linear matrix inequality based conditions are provided for the existence of a robust reduced-order dynamic output feedback stabilising controller with complete, partial or none mode dependency assuring an upper bound to the or the norm of the closed-loop system. The main advantage of the proposed method when compared to the existing approaches is the fact that the dynamic controllers are exclusively expressed in terms of the decision variables of the problem. In other words, the matrices that define the controller realisation do not depend explicitly on the state space matrices associated with the modes of the MJLS. As a consequence, the method is specially suitable to handle order reduction or cluster availability constraints in the context of or dynamic output feedback control of discrete-time MJLS. Additionally, as illustrated by means of numerical examples, the proposed approach can provide less conservative results than other conditions in the literature.

ℋ 2 filter design through multi-simplex modeling for discrete-time Markov jump linear systems with partly unknown transition probability matrix

This paper is concerned with the ℋ2 robust filtering problem for discrete-time Markov jump linear systems (MJLS) with transition probability matrix affected by uncertainties. Differently from previous approaches in the literature, the proposed strategy presents a systematic way to handle, simultaneously, different types of uncertainties commonly appearing in the transition probability matrix of MJLS. Full-order filters with partial, complete or null Markov mode observation are synthesized via a linear matrix inequality (LMI) based formulation. The main novelty of the proposed filter design procedure is the use of parameter-dependent Lyapunov matrices of arbitrary degree to certify the stochastic stability and to guarantee an upper bound to the ℋ2 norm of the filtering error system. Moreover, the proposed conditions also include slack variables and scalars. For fixed values of the scalar parameters, the conditions become LMIs. Numerical examples borrowed from the literature illustrate that the proposed filter can provide better ℋ2 guaranteed costs when compared to other existing methods.

ℋ ∞ static output feedback control of discrete-time Markov jump linear systems with uncertain transition probability matrix

This paper investigates the problem of ℋ∞ static output feedback control design for discrete-time Markov jump linear systems (MJLS), assuming that the transition probability matrix is not precisely known, but affected by different classes of uncertainties: polytopic, bounded or completely unknown elements. All types of uncertainties are modeled through one single representation, expressed in terms of the Cartesian product of simplexes, called multi-simplex. The main novelty of the proposed design procedure is that, differently from previous approaches in the literature, parameter-dependent Lyapunov matrices are used to certify the closed-loop stability with an ℋ∞ bound for the discrete-time MJLS. The proposed conditions are based on linear matrix inequality relaxations performed in two steps: the first step generates a parameter-dependent state feedback controller that is employed as an input for the second stage, which synthesizes a robust static output feedback gain assuring an ℋ∞ guaranteed cost. The proposed strategy can also cope with ℋ∞ state feedback control for discrete-time MJLS. Numerical examples illustrate the advantages of the proposed methodology when compared to other methods from the literature.

Discretization and discrete-time output feedback control of linear parameter varying continuous-time systems

This paper is concerned with the problem of discretization and discrete-time output feedback control design for polytopic continuous-time linear parameter-varying (LPV) systems with network-induced delay. Using a constant sampling period and an extension of the Taylor series expansion applied to the exponential of a parameter-dependent matrix, a discretized model whose matrices depend polynomially on the time-varying parameters that are used to schedule the control gain is obtained. The discrete-time model also has additive norm bounded terms, representing the discretization errors, and a network-induced delay in the control signal. A two-step strategy based on linear matrix inequality (LMI) conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the original continuous-time LPV system. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through a numerical example.