OBSERVABILITY OF DISCRETE TIME LINEAR SWITCHING SYSTEMS1

This paper considers a subclass of hybrid systems, which is the discrete-time linear switching systems whose changes in the dynamics are determined by exogenous, uncontrollable events, and resets are present. In the presence of uncertain parameter and non-symmetrical constrained control, based on the results of nominal linear switching systems, a necessary and sufficient condition to test the robustness of a state feedback controller is given. The candidate state feedback controller for this test is that stabilizing nominal discrete-time linear switching systems. Also, the method in this paper can be extended to the case which has state constraints. At last, a numerical example is presented to illustrate the technique.

Feedback Control of Discrete Linear Switching Systems with Bounded State and Control Input in the Presence of Disturbances

In this paper we analyse the stability robustness of linear continuous-time and discrete-time systems with structured time-varying uncertainties. The system to be controlled is described by the usual state-space formulation but includes time-varying uncertainties. The first new robust stability sufficient condition we have derived in this paper may become the necessary and sufficient condition of system stability in some situations. The second new condition is always less conservative than the sufficient condition recently proposed by Sobel et al., (1989). The third new condition, using the similarity transformation, provides the possibility of getting even less conservative results. In deriving these new conditions we also present some new lemmas which are valuable in related research. The results are extended to linear discrete-time systems and some illustrative examples are given.

As with a continuous-time linear system, the predictor state feedback law for a discrete-time linear system with input delay is the product of a feedback gain matrix with the predicted state at a future time instant ahead of the current time instant by the amount of the delay, which is the sum of the zero input solution and the zero state solution of the system. The zero state solution for a discrete-time system is a finite summation that involves past input. The zero input solution is the product of the transition matrix and the initial state. The truncated predictor feedback (TPF), which results from discarding the finite summation part of the predictor feedback law, reduces implementation complexity. The delay independent TPF law further discards the delay-dependent transition matrix in the TPF law and is thus robust to unknown delays. It was pointed out in Lin (Commun. Inf. Syst. 7(3), 227–264 (2007)) and Zhou and Lin (IEEE Trans. Circuits Syst. I Regular Papers 58(11), 2741–2754 (2011)) that such a delay independent TPF law stabilizes a discrete-time linear system with all its poles at z = 1 or inside the unit circle no matter how large the delay is.

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

Inverse systems of linear systemsThe concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

SUMMARY The LQ+ problem, i.e. the finite-horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ+-optimal control to be given by the standard LQ-optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ-optimal closed-loop system is derived for positive discrete-time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright q 2010 John Wiley & Sons, Ltd.

SUMMARY This paper deals with the problem of finite frequency H∞ full-order filter design for discrete-time and continuous-time linear systems, with polytopic uncertainties. Based on the generalized Kalman–Yakubovich–Popov lemma and a parameter-dependent Lyapunov function, a set of sufficient conditions are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level, from disturbance to the estimation error, is smaller than a given value over a prescribed finite frequency domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we introduce a large number of slack variables by applying Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance. This leads to performance improvement and reduction of conservatism in the solution. It is shown later that the robust filter gains can be obtained by solving a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.

A common quadratic Lyapunov function (CQLF) guarantees the asymptotic stability of a set of systems. A complete characterization of the set of systems with such a property have been unsuccessful (except for second-order systems). Thus, for both the continuous-time and discrete-time cases, several subsets of linear systems which have a CQLF are known. Some results indicate that there is a parallelism between the continuous-time case and the discrete-time case. In this paper, we show a new subclass for continuous-time systems which have a CQLF by using a property of M-matrices. We also show the discrete-time counterpart of the above new subclass. Next, it is shown that the whole class of continuous-time linear systems having a CQLF is connected directly with its discrete-time counterpart by using a bilinear transformation. For some known subclasses of systems having a CQLF, the transformation gives a one-to-one correspondence between the continuous-time and discrete-time cases. We further show relationships among the obtained results and other, known results.

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.

Bisimilar linear systems

In this paper, the problem of designing Robust Gain-Scheduling observers for continuous-time Linear Parameter-Varying systems via parameter-dependent Lyapunov function is addressed. The scheduling parameters are assumed to be imprecisely measured, i.e., corrupted with additive noise. Multi-simplex modeling approach is utilized to model the time-varying parameters and associated uncertainties. Sufficient conditions are given for the synthesis problem in terms of Parameterized Linear Matrix Inequalities. The designed observer not only guarantees asymptotic stability of estimation error dynamics but also minimizes the estimation errors under exogenous disturbances. Both, ℋ 2 and ℋ ∞ performances have been considered. An illustrative numerical example is presented to demonstrate the benefit of the developed approach.

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

This communication is concerned with stabilization via output static feedback of discrete-time linear time-invariant finite-dimensional systems. A new concept is introduced: output static stabilization in the relaxed sense. This (conservative) stabilization concept is defined so that it is characterized in terms of the discrete-time LQR problem. Properties, and alternative equivalent definitions of this new concept are analyzed. It is shown that for a particular class of plants, the stabilization problem in the above new sense can be cast as a convex programming problem (which moreover can be used as a tool for synthesis). A full characterization for the class of plants that are stabilizable in the above sense is presented. Important consequences of this characterization are somehow surprising. It is shown that the introduced new (conservative) stabilization concept is indeed much more important than it seems. The general problem of output static stabilization can be transformed into the stabilization problem in this new sense. A relationship between output static stabilizability and the discrete-time LQR problem is proved. Convex necessary and sufficient conditions, for joint stabilization (of multiple plants) in the above sense and for decentralized stabilization in the above sense, are presented.