Padé Discretization for Linear Systems With Polyhedral Lyapunov Functions

In this paper we study the robustness analysis problem for linear continuous-time systems subject to parametric time-varying uncertainties making use of piecewise linear (polyhedral) Lyapunov functions. A given class of Lyapunov functions is said to be “universal” for the uncertain system under consideration if the robust stability of the system is equivalent to the existence of a Lyapunov function belonging to the class. In the literature it has been shown that the class of polyhedral functions is universal, while, for instance, the class of quadratic functions is not. This fact justifies the effort of developing efficient algorithms for the construction of polyhedral Lyapunov functions. In this context, we provide a low computational cost procedure, based on a novel convex condition, for the construction of a polyhedral Lyapunov function. In the section on the numerical examples, we consider some benchmark problems for the robust stability analysis and we show that the proposed low computational cost approach, though only sufficient, is less conservative than all the other approaches presented so far in the literature.

Recently, we showed that certain types of polyhedral
\nLyapunov functions for linear time-invariant systems, are preserved by diagonal Pad ́ e approximations, under the assumption
\nthat the continuous-time system matrix Ac has distinct eigenvalues. In this paper we show that this result also holds true in the
\ncase that Ac has non-trivial Jordan blocks

This paper addresses the issue of constructing piecewise linear Lyapunov functions (PWLLF) for nonlinear systems satisfying the generalized sector condition, which is a generalization of the polytopic uncertainty. The main result of this paper is to propose a fast method to solving linear programming problems (LPs) which appear repeatedly in constructing PWLLF. The method uses a special structure of LPs which appear in constructing PWLLF.

This chapter studies null controllability of continuous-time linear systems with bounded controls. Null controllability of a system refers to the possibility of steering its state to the origin in a finite time by an appropriate choice of the admissible control input. If the system is linear and is controllable, then any state can be steered to any other location, for example, the origin, in the state space in a finite time by a control input. This implies that any controllable linear system is null controllable from any point in the state space. The control inputs that are used to drive certain states to the origin, however, might have to be large in magnitude. As a result, when the control input is limited by the actuator saturation, a linear controllable system might not be globally null controllable. In this situation, it is important to identify the set of all the states that can be steered to the origin with the bounded controls delivered by the actuators. This set is referred to as the null controllable region and is denoted as C

Polyhedral lyapunov functions computation for robust and gain scheduled design

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

New tests for checking asymptotic stability of positive 1D continuous-time and discrete-time linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models are proposed. Checking of the asymptotic stability of positive 2D linear systems is reduced to checking of suitable corresponding 1D positive linear systems. It is shown that the stability tests can be also applied to checking the asymptotic stability of fractional discrete-time linear systems with delays. Effectiveness of the tests is shown on numerical examples.

This paper deals with the stability of continuous-time 2D nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are extended to 2D nonlinear systems. To check the stability a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what proposed in this paper are the first results of this kind on stability of continuous-time 2D nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems is proposed. The proposed method is based on the sum of squares decomposition, therefore it formulates the Lyapunov function search algorithmically. In this way polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

This paper considers the problem of constrained stabilization of linear continuous-time systems when a quadratic cost criterion is imposed in the design of the state feedback control law. The linear quadratic regulator (LQR) is solved under positivity constraint, which means that the resulting closed-loop systems are not only optimally stable, but also positive. We focus on the class of linear continuous-time positive systems (Metzlerian systems) and use the interesting properties of Metzler matrices to provide the necessary ingredients for the main results of the paper. A two-step procedure is proposed to solve the problem. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the Metzlerian constraint, and the constrained stabilization is solved using linear programming (LP) or linear matrix inequality (LMI). Second, a sufficient condition is outlined for the existence of a solution to maintain the positivity of the first step while achieving the optimality of LQR. Finally, the robustness of the design is analyzed and possible extension of the design is proposed for an uncertain interval systems. A numerical example is included to illustrate the procedure.

This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov’s inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.

Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.

A procedure for identifying linear continuous-time multivariable systems is presented. The scheme is based on a discrete identification method which requires no structural identification. This is accomplished by the use of pseudo-observable forms whose structure is defined by their corresponding pseudo-observability indices. A First-Order Hold transformation of this discrete-time model to an equivalent continuous-time model is then performed. First, the state transition matrix is transformed to an equivalent continuous-time system matrix using a procedure which is insensitive to the size of the spectral radii of the system. The remaining system matrices are then transformed using a generalized First-Order Hold transformation technique which places no restrictions on the singularity of the continuous-time system.

A new digital redesign method is proposed for feedback control systems, by which the states in the continuous-time system are completely reserved in the redesigned sampled-data system. The features of the proposed method are: 1) the N-delay control is employed, and the lth plant input is changed N/sub l/ times during one sampling period; 2) the states of the redesigned sampled-data system completely match those of the original continuous-time closed-loop system at every sampling period; and 3) the proposed redesign method can be applicable for a static state-feedback and/or a dynamic controller. An illustrative example of position control using a DC servo motor is presented. The advantages of the proposed redesign approach are demonstrated both in the time and frequency domains.

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

A sufficient condition for the existence of a common quadratic Lyapunov function (CQLF) for the linear systems x/spl dot/=A/sub i/x, A/sub i//spl isin/{A/sub 1/,A/sub 2/,...,A/sub m/}, A/sub i//spl isin/IR/sup n/spl times/n/ is that the matrices can be simultaneously triangularized using a non-singular transformation T. In this paper, we show that this result follows trivially from the structure of the matrices in the set A, and that the switching system, constructed by switching between the matrices in this set, is benign from a stability viewpoint. Finally, we then discuss several conditions under which a transformation T exists.