Unknown input observability for discrete-time linear multivariable systems and its application

A new concept of the unknown input observability, namely (r, s ; K) observability, for discrete-time linear multivariable systems is introduced. This concept is an extension of the conventional unknown input observability and reconstructibility. Necessary and sufficient conditions for the (r, s ; K) observability of the given system are derived by introducing a new unobservable subspace and presenting algorithms for obtaining the subspace. It is also shown that these new concepts are closely related to the existence condition of a dead-beat observer for a decentralized control system.     

Discrete-time observability for distributed parameter systems

The aim of this paper is to study the observability for systems described by first-order evolution equations and for those described by second-order evolution equations in the case of discrete-time observations. For the systems with a finite number of sensors we present necessary and sufficient conditions for observability. We show that these distributed parameter systems are never finite-step observable. We give the restricted sets of the initial-state spaces whose elements are N-step observable. We also investigate the relations between the systems with discrete-time observations and the systems with continuous-time observations from the viewpoint of observability Moreover, we see the essential difference between the parabolic case and the hyperbolic case.     

How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

On the design of linear time-invariant systems by-periodic output feedback Part II. Output feedback controllability†

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for the complete controllability of linear time-invariant systems by discrete output feedback. In the case where the desired final state is the origin, the resultant, output feedback gain is periodic and the transfer is accomplished in at most vd periods, where vd is the discrete-time index of observability. In the more general case where the desired final state is non-zero, the transfer is accomplished using at most vd output measurements.     

Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations

Discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control and H _∞-control of Markovian jump linear systems are considered. First, the equations that arise from the quadratic optimal control problem are studied. The matrix cost is only assumed to be hermitian. Conditions for the existence of the maximal hermitian solution are derived in terms of the concept of mean square stabilizability and a convex set not being empty. A connection with convex optimization is established, leading to a numerical algorithm. A necessary and sufficient condition for the existence of a stabilizing solution (in the mean square sense) is derived. Sufficient conditions in terms of the usual observability and detectability tests for linear systems are also obtained. Finally, the coupled algebraic Riccati equations that arise from the H _∞-control of discrete-time Markovian jump linear systems are analyzed. An algorithm for deriving a stabilizing solution, if it exists, is obtained. These results generalize and unify several previous ones presented in the literature of discrete-time coupled Riccati equations of Markovian jump linear systems. Date received: November 14, 1996. Date revised: January 12, 1999.     

Worst case optimal observer design for linear discrete-time systems with unknown bounded square summable disturbances

The problem of the existence and design of an optimal observer is considered for linear discrete-time systems with unknown disturbances additive to the output. The optimal observer reconstructs the best estimate of the state at a given time with respect to the worst disturbances constrained to a ball in l ². It is proved that an observability condition is necessary and sufficient for the existence of such an observer. An explicit formula for the optimal observer is derived (it includes the degenerate case when some of the outputs are disturbance free).     

Robust passivity‐based H∞ control for uncertain switched nonlinear systems

In this paper, we investigate the H_∞ control problem for uncertain switched nonlinear systems with passive and non‐passive subsystems. For any given average dwell time, any given passivity rate and any given disturbance attenuation level, we design feedback controllers of subsystems, which may depend on the pre‐given constants, to solve the H_∞ control problem for the uncertain switched nonlinear systems for all admissible uncertainties. For linear systems, the exponential small‐time norm‐observability is shown to be preserved under disturbance. Two examples are provided to demonstrate the effectiveness of the proposed design method. Copyright © 2016 John Wiley & Sons, Ltd.     

Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems

A continuous-time system cannot be recovered solely from its uniformly sampled discrete-time model through the zero-order hold discretization or step-invariant transformation, but our studies indicate that it can be recovered uniquely from its non-uniformly sampled discrete-time model. In this paper, we discuss some related issues of non-uniformly sampled systems, including model derivation, controllability and observability, computation of single-rate models with different sampling periods, reconstruction of continuous-time systems, and parameter identification of non-uniformly sampled discrete-time systems. A numerical example is also given for illustration.     

Exponential stability and robust H∞ control of a class of discrete-time switched non-linear systems with time-varying delays via T-S fuzzy model

This paper deals with stability and robust H_∞ control of discrete-time switched non-linear systems with time-varying delays. The T-S fuzzy models are utilised to represent each sub-non-linear system. Thus, with two level functions, namely, crisp switching functions and local fuzzy weighting functions, we introduce a discrete-time switched fuzzy systems, which inherently contain the features of the switched hybrid systems and T-S fuzzy systems. Piecewise fuzzy weighting-dependent Lyapunov–Krasovskii functionals (PFLKFs) and average dwell-time approach are utilised in this paper for the exponentially stability analysis and controller design, and with free fuzzy weighting matrix scheme, switching control laws are obtained such that H_∞ performance is satisfied. The conditions of stability and the control laws are given in the form of linear matrix inequalities (LMIs) that are numerically feasible. The state decay estimate is explicitly given. A numerical example and the control of delayed single link robot arm with uncertain part are given to demonstrate the efficiency of the proposed method.     

On gramians and balanced truncation of discrete-time bilinear systems

This paper obtains a balanced truncation model reduction method for discrete-time bilinear systems. After deriving the sufficient conditions for the discrete-time bilinear systems to lead to bounded outputs for some bounded inputs, we study the properties of the reachability and observability gramians. The gramians can be computed by solving two generalized Lyapunov equations. By balancing the reachability and observability gramians, a model reduction method is obtained. Some propositions of the reduced order models are given. A numerical example is given to illustrate the effectiveness of the proposed method.     

μ-Dependent model reduction for uncertain discrete-time switched linear systems with average dwell time

In this article, the model reduction problem for a class of discrete-time polytopic uncertain switched linear systems with average dwell time switching is investigated. The stability criterion for general discrete-time switched systems is first explored, and a μ-dependent approach is then introduced for the considered systems to the model reduction solution. A reduced-order model is constructed and its corresponding existence conditions are derived via LMI formulation. The admissible switching signals and the desired reduced model matrices are accordingly obtained from such conditions such that the resulting model error system is robustly exponentially stable and has an exponential H _∞ performance. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results.     

Design of H 2 Controllers for Sampled-Data Systems with Input Time Delays

This paper presents a general framework based on lifting technique for sampled-data systems with input time delays. By analyzing the properties of operator-valued matrices of lifted systems with input time delays, an extended lifting technique is obtained. It is then shown that, with the proposed lifting technique, the complex behavior of the system can be illustrated by two simple lifted systems, which construct the extended lifted system. The extended lifted system has the same induced norm as that of the original system with an input time delay, since the proposed lifting technique is an isometric isomorphism. Through applying the proposed lifting technique to sampled-data systems with input time delays, the time-invariant discrete-time system with infinite-dimensional input and output spaces is obtained. The equivalent discrete-time system, which is derived from the extended lifted system, can satisfy the problem of H ₂ sampled-data control systems with input time delays. Simulation results are given to show that the proposed method can guarantee a more stable system response than the conventional H ₂ sampled-data controller for the sampled-data systems with the various input time delays.     

Supervisory control of timed discrete event systems under partial observation based on activity models and eligible time bounds

To avoid the state–space explosion by including tick events in timed discrete event systems (DESs) under partial observation, a notion of eligible time bounds is introduced and based on the notion, controllability and observability conditions of languages are presented. In particular, this paper shows that these controllability and observability conditions are necessary and sufficient for the existence of a supervisor to achieve the given language specification.     

Decentralised H ∞ filtering of interconnected discrete-time fuzzy systems with time delays

This article considers the decentralised H _∞ filtering of interconnected discrete-time fuzzy systems with time delays based on piecewise Lyapunov–Krasovskii functionals. The fuzzy system consists of J interconnected time-delay discrete-time Takagi–Sugeno fuzzy subsystems and the decentralised H _∞ filter is designed for each subsystem. It is shown that the stability with H _∞ performance of overall filtering error system can be established if a piecewise Lyapunov–Kroasovskii functional can be constructed, and moreover, the functional can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A simulation example is given to show the effectiveness of the presented approach.     

Filtering for a class of discrete-time systems with time-delays via delta operator approach

In this article, a robust H _∞ filtering problem for a class of norm-bounded uncertain discrete-time systems with time delays is investigated using delta operator approach. Based on Lyapunov–Krasovskii functional in delta domain, a new delay-dependent sufficient condition for the solvability of this problem is presented in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an expression of a desired delta operator H _∞ filter is given. The proposed method can unify some previous related continuous and discrete systems into a delta operator systems framework. A numerical example is given to illustrate the effectiveness of the developed techniques.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.     

H2 control of discrete-time periodic systems with Markovian jumps and multiplicative noise

This paper addresses the problem of optimal and robust H₂ control for discrete-time periodic systems with Markov jump parameters and multiplicative noise. To analyse the system performance in the presence of exogenous random disturbance, an H₂ norm is firstly established on the basis of Gramian matrices. Further, under the condition of exact observability, a necessary and sufficient condition is presented for the solvability of H₂ optimal control problem by means of a generalised Riccati equation. When the transition probabilities of jump parameter are incompletely measurable, an H₂-guaranteed cost norm is exploited and the robust H₂ controller is designed through a linear matrix inequality (LMI) optimisation approach. An example of a networked control system is supplied to illustrate the proposed results.     

H ∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities

In this article, the H _∞ model reduction problem for a class of discrete-time Markov jump linear systems (MJLS) with partially known transition probabilities is investigated. The proposed systems are more general, relaxing the traditional assumption in Markov jump systems that all the transition probabilities must be completely known. A reduced-order model is constructed and the LMI-based sufficient conditions of its existence are derived such that the corresponding model error system is internally stochastically stable and has a guaranteed H _∞ performance index. A numerical example is given to illustrate the effectiveness and potential of the developed theoretical results.