Orthogonal Embeddings of Linear Time-Varying Systems

 Given a contractive linear system T which is represented as a causal linear contractive operator on a Hilbert space H, we study the question: When does there exist an isometric or unitary causal extension of T to H⊕H? Here causality is formulated as a lower triangularity condition and we assume the same causality structure for the extension. This question is often referred to as the generalized Darlington synthesis problem for linear time-varying systems. Date received: November 26, 2001. Date revised: June 19, 2002.     

Computation of maximal pre-controllability submodules over a Noetherian ring

For dynamical systems with coefficient in a ring it is not always possible to compute by a finite procedure the maximal (A,B)-invariant submodule contained in the kernel of the output map C, namely . This difficulty prevents one from checking necessary and sufficient geometric conditions for the solvability of noninteracting control problems. In this paper we will prove that, for dynamical systems with coefficients in a Noetherian ring, it is always possible to compute by a finite procedure the maximal pre-controllability submodule contained in the kernel of the output map C, namely . This result is useful in dealing with the block decoupling problem or the disturbance rejection problem.     

Induced Convolution Operator Norms of Linear Dynamical Systems

In this paper we develop explicit formulas for induced convolution operator norms and their bounds. These results generalize established induced operator norms for linear dynamical systems with various classes of input–output signal pairs. Date received: April 1, 1999. Date revised: February 21, 2000.     

On the fuzzy difference equation xn+1 = A+B/xn

 In this paper we study the existence the oscillatory behavior the boundedness and the asymptotic behavior of the positive solutions of the fuzzy equation x _n+1 =A+B/x _n ,n=0,1,… where x _n is a sequence of fuzzy numbers, A, B are fuzzy numbers.     

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.     

On the problem of general structural assignments of linear systems through sensor/actuator selection

A systematic method is developed for determining an output matrix C for a given matrix pair (A,B) such that the resulting linear system characterized by the matrix triple (A,B,C) has the pre-specified system structural properties, such as the finite and infinite zero structure and the invertibility structures. Since the matrix C describes the locations of the sensors, the procedure of choosing C is often referred to as sensor selection. The method developed in this paper for sensor selection can be applied to the dual problem of actuator selection, where, for a given matrix pair (A,C), a matrix B is to be determined such that the resulting matrix triple (A,B,C) has the pre-specified structural properties.     

Separation principle for quasi‐one‐sided Lipschitz nonlinear systems with time delay

In this article, we address the problem of output stabilization for a class of nonlinear time‐delay systems. First, an observer is designed for estimating the state of nonlinear time‐delay systems by means of quasi‐one‐sided Lipschitz condition, which is less conservative than the one‐sided Lipschitz condition. Then, a state feedback controller is designed to stabilize the nonlinear systems in terms of weak quasi‐one‐sided Lipschitz condition. Furthermore, it is shown that the separation principle holds for stabilization of the systems based on the observer‐based controller. Under the quasi‐one‐sided Lipschitz condition, state observer and feedback controller can be designed separately even though the parameter (A,C) of nonlinear time‐delay systems is not detectable and parameter (A,B) is not stabilizable. Finally, a numerical example is provided to verify the efficiency of the main results.     

Some generalizations of comparison results for fractional differential equations

In this paper, by using the technique of upper and lower solutions together with the theory of strict and nonstrict fractional differential inequalities involving Riemann–Liouville differential operator of order q, 0<q<1, some necessary comparison results for further generalizations of several dynamical concepts are obtained. Furthermore, these results are extended to the finite systems of fractional differential equations.     

Controllability for Discrete Systems with a Finite Control Set

In this paper we consider the problem of controllability for a discrete linear control system x _k+1=Ax _k+Bu _k, u _k∈U, where (A,B) is controllable and U is a finite set. We prove the existence of a finite set U ensuring density for the reachable set from the origin under the necessary assumptions that the pair (A,B) is controllable and A has eigenvalues with modulus greater than or equal to 1. In the case of A only invertible we obtain density on compact sets. We also provide uniformity results with respect to the matrix A and the initial condition. In the one-dimensional case the matrix A reduces to a scalar λ and for λ>1 the reachable set R(0,U) from the origin is?

Robust Feedback Control of a Single Server Queueing System

This paper extends previous work of Ball et al. [BDKY] to control of a model of a simple queueing server. There are n queues of customers to be served by a single server who can service only one queue at a time. Each queue is subject to an unknown arrival rate, called a “disturbance” in accord with standard usage from H _∞ theory. An H _∞-type performance criterion is formulated. The resulting control problem has several novel features distinguishing it from the standard smooth case already studied in the control literature: the presence of constraining dynamics on the boundary of the state space to ensure the physical property that queue lengths remain nonnegative, and jump discontinuities in any nonconstant state-feedback law caused by the finiteness of the admissible control set (choice of queue to be served). We arrive at the solution to the appropriate Hamilton–Jacobi equation via an analogue of the stable invariant manifold for the associated Hamiltonian flow (as was done by van der Schaft for the smooth case) and relate this solution to the (lower) value of a restricted differential game, similar to that formulated by Soravia for problems without constraining dynamics. An additional example is included which shows that the projection dynamics used to maintain nonnegativity of the state variables must be handled carefully in more general models involving interactions among the different queues. Primary motivation comes from the application to traffic signal control. Other application areas, such as manufacturing systems and computer networks, are mentioned. Date received: August 14, 1998. Date revised: May 17, 1999.     

Local Approximability of Max-Min and Min-Max Linear Programs

In a max-min LP, the objective is to maximise ω subject to A x≤1, C x≥ω 1, and x≥0. In a min-max LP, the objective is to minimise ρ subject to A x≤ρ 1, C x≥1, and x≥0. The matrices A and C are nonnegative and sparse: each row a _i of A has at most Δ _I positive elements, and each row c _k of C has at most Δ _K positive elements.     

The Kleinman Iteration for Nonstabilizable Systems

 We consider the computation of Hermitian nonnegative definite solutions of algebraic Riccati equations. These solutions are the limit, P=lim_{i →∞} P _i, of a sequence of matrices obtained by solving a sequence of Lyapunov equations. The procedure parallels the well-known Kleinman technique but the stabilizability condition on the underlying linear time-invariant system is removed. The convergence of the constructed sequence {P _i }_i≥1 is guaranteed by the minimality of P _i in the set of Hermitian nonnegative definite solutions of the Lyapunov equation in the ith iteration step. Date received: October 21, 1999. Date revised: February 14, 2002. RID="*" ID="*"This work was supported by the Acciones Integradas programme of Deutscher Akademischer Austauschdienst (Germany) and Dirección General de Infraestructura y Relaciones Internacionales (Spain).     

Further results on structural assignment of linear systems via sensor selection

The problem of assigning structural properties of a linear system through sensor selection is, for a given pair (A,B), to find an output pair (C,D) such that the resulting system (A,B,C,D) has the pre-specified structural properties, such as the finite and infinite zero structures and the invertibility properties. In this paper, by introducing the notion of infinite zero assignable sets for the pair (A,B), we establish necessary and sufficient conditions for the assignability of a given set of infinite zeros and a set of structural properties which includes the left invertibility property. In establishing these conditions, we develop a numerical algorithm for the construction of the required (C,D).     

A representation of all solutions of the control algebraic Riccati equation for infinite-dimensional systems

We obtain a representation of all self-adjoint solutions of the control algebraic Riccati equation associated to the infinite-dimensional state linear system Σ(A,B,C) under the following assumptions: A generates a C ₀-group, the system is output stabilizable, strongly detectable and the dual Riccati equation has an invertible self-adjoint non-negative solution.     

Mixed output feedback of the finite frequency H∞ control for singularly perturbed system

This paper considers the H_∞ control problem for a class of linear singularly perturbed systems in the finite frequency range. A mixed output feedback controller comprising of a static output feedback controller and a dynamic output feedback controller is developed for the system stabilisation. Based on the generalised Kalman–Yakubovich–Popov (GKYP) lemma, the frequency-domain inequalities can be converted into linear matrix inequalities which are numerically tractable. Compared with the existing full frequency approaches, better results are obtained. Moreover, the selection methods of the cut-off frequencies in both low and high frequency ranges are extensively studied with a view to reduce the conservativeness in output feedback control design. Simulation results suggest the asymptotic validity of the main results in this paper.     

Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

Detection filter design for LPV systems—a geometric approach

This paper investigates fault detection and isolation of linear parameter-varying (LPV) systems by using parameter-varying (C,A)-invariant subspace and parameter-varying unobservability subspaces. The so called “detection filter” approach, formulated as the fundamental problem of residual generation (FPRG) for linear time-invariant (LTI) systems, is extended for a class of LPV systems. The question of stability is addressed in the terms of Lyapunov quadratic stability by using linear matrix inequalities. The results are applied to the model of a generic small commercial aircraft.     

The Popov criterion for strongly stable distributed parameter systems

In this article we generalize the Popov criterion to the class of strongly stable infinite-dimensional linear systems; the semigroup is strongly stable and the input to state, state to output and input to output maps are all bounded on the infinite-time interval. One application is to show that integral control can be used to track constant reference signals for positive-real strongly stable systems in the presence of sectorial non-linearities. A second application is to show the robustness of asymptotic stability of positive-real strongly stable systems to a large class of non-linear perturbations. Systems satisfying the assumptions in this paper include dissipative systems with collocated actuators and sensors.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.