Equivalent characterizations of detectability and stabilizability for a class of linear time-varying systems

This paper addresses four characterizations of detectability and stabilizability for a broad class of linear continuous-time time-varying systems. It is shown that for the class of constant-rank systems all these notions of detectability and stabilizability are actually equivalent to each other.     

Discrete-time linear periodic systems: A note on the reachability and controllability interval length

Discrete-time linear systems with periodic coefficients of period T are considered in this paper. The reachability and controllability indices at time t for periodic systems, μ_rt and μ_ct, are defined. It is shown that the reachability [controllability] subspace at time t does coincide with the reachability [controllability] subspace over the interval (t − μ_rt,T, t) [(t, t + μ_ct.,T)] and properly includes the reachability [controllability] subspace over the interval (t −(μ_rt−1) T, t) [(t, t +(μ_cr−1)T)].     

Stabilizability of systems with exponential dichotomy

In this paper we introduce the concept of controllability into a closed subspace for time-varying linear systems. Various characterizations are given and the dual relation is discussed. This concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy.     

Disturbance localization for left invertible linear periodic discrete-time systems

The disturbance localization problem for left invertible linear periodic discrete-time systems is solved using periodic state feedback controllers. The proposed technique is of algebraic nature and has the following two main characteristics: (i) It yields simple algebraic criteria for testing the solvability of the problem, as compared to known geometric criteria, which may not be so easy to check. (ii) It derives analytically the general expressions of all periodic controllers admissible for disturbance localization, as compared to known techniques, which lead to nonanalytic parametrizations of the admissible controllers via constructive procedures. Moreover, for the aforementioned class of periodic systems, the state feedback simultaneous disturbance localization and stabilization or pole placement problem is treated, and conditions for its solvability are established, on the basis of a decentralized control approach, that makes use of the equivalence between the above problem and the stabilization or pole placement problem of a general proper multichannel system by decentralized static output feedback.     

On the detectability and observability of discrete-time Markov jump linear systems

This paper presents a new detectability concept for discrete-time Markov jump linear systems with finite Markov state, which generalizes the MS-detectability concept found in the literature. The new sense of detectability can similarly assure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is a stabilizing solution. In addition, the paper introduces a related observability concept that also generalizes previous concepts. A test for detectability based on a coupled matrix equation is derived from the definition, and a test for observability is presented, which can be performed in a finite number of steps. The results are illustrated by examples, including one that shows that a system may be detectable in the new sense but not in the MS sense.     

Compensating periodic descriptor systems

The compensating problem for forward–backward periodic systems in discrete-time has been studied. A normal compensator for this kind of systems is proposed using a suitable observer. In addition, some properties to construct the periodic observer and compensator can be obtained from the associated invariant systems.     

A structural approach to detectability for a class of hybrid systems

We address detectability of linear switching systems. We show that detectability of a linear switching system reduces to asymptotic stability of a suitable switching system with guards extracted from it. A condition for checking asymptotic stability of linear switching systems with guards is also derived.     

Stabilizability,representations and factorizations for time-varying linear systems

We study the system representations, factorizations and stabilizability of a discrete-time time-varying linear system in the framework of nest algebra. We shown that every time-varying linear system admits a normalized left factorization, while it may possibly not have a right factorization. The relationship between the system representation and factorization is studied, and some new stabilizability criteria are established.     

A note on asymptotic stabilization of linear systems by periodic, piecewise constant, output feedback

This note studies the asymptotic stabilization problem for controllable and observable, single-input single-output, linear, time-invariant, continuous-time systems by means of memoryless output feedback of the form u(t)=k(t)y(t), with k(t) periodic and piecewise constant. A necessary and sufficient condition, together with a simpler to test sufficient condition, given in terms of a bilinear matrix inequality, is presented. A few illustrative examples complete the paper.     

On detectability of stochastic systems

We discuss notions of detectability for stochastic linear control systems of Itô type. A natural concept of detectability requires a non-zero output, if the state process is unstable. We show that this property can equivalently be characterized by a generalized version of the Hautus-test. The proof is based on spectral theory for positive operators.     

State estimation and detectability of probabilistic discrete event systems

A probabilistic discrete event system (PDES) is a nondeterministic discrete event system where the probabilities of nondeterministic transitions are specified. State estimation problems of PDES are more difficult than those of non-probabilistic discrete event systems. In our previous papers, we investigated state estimation problems for non-probabilistic discrete event systems. We defined four types of detectabilities and derived necessary and sufficient conditions for checking these detectabilities. In this paper, we extend our study to state estimation problems for PDES by considering the probabilities. The first step in our approach is to convert a given PDES into a nondeterministic discrete event system and find sufficient conditions for checking probabilistic detectabilities. Next, to find necessary and sufficient conditions for checking probabilistic detectabilities, we investigate the “convergence” of event sequences in PDES. An event sequence is convergent if along this sequence, it is more and more certain that the system is in a particular state. We derive conditions for convergence and hence for detectabilities. We focus on systems with complete event observation and no state observation. For better presentation, the theoretical development is illustrated by a simplified example of nephritis diagnosis.     

IQC-based -control of linear periodic systems

Stability analysis of linear periodically time-varying systems via integral quadratic constraints is extended to the problem of control design. A full-state feedback controller that satisfies exponential stability and -gain disturbance attenuation from an external disturbance to a controlled output is designed for linear systems with periodically time-dependent system matrices. The main result relies on dual forms of certain integral quadratic constraints. The solvability conditions for the problem are cast as a set of finite-dimensional linear matrix inequalities and thus, they are easily solvable. Moreover, the best possible disturbance attenuation level can be obtained as a convex problem.     

Peak-to-peak filtering for periodic piecewise linear polytopic systems

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques.     

Balancing related methods for minimal realization of periodic systems

We propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time-varying dimensions. The first method belongs to the family of square-root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations. An alternative balancing-free square-root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems. The key numerical computation in both methods is the solution of nonnegative periodic Lyapunov equations directly for the Cholesky factors of the solutions. For this purpose, a numerically reliable computational algorithm is proposed to solve nonnegative periodic Lyapunov equations with time-varying dimensions.     

Stabilization of linear time-invariant systems with periodic communication constraints by output sample hold control

This article considers feedback control systems wherein the control loops are closed through a real-time network, and expresses the linear time-invariant system with the constraint in an input or output as a periodic discrete time system. It is shown that this system is stabilized by using output sample hold contol. This method has the merit that the capacity of a sensor-controller communication bus is small. This work was presented, in part, at the 8th International Symposium on Artificial Life and Robotics, Oita, Japan, January 24–26, 2003     

Reconstruction of the Fourier expansion of inputs of linear time-varying systems

In this paper we propose a general method to estimate periodic unknown input signals of finite-dimensional linear time-varying systems. We present an infinite-dimensional observer that reconstructs the coefficients of the Fourier decomposition of such systems. Although the overall system is infinite dimensional, convergence of the observer can be proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. Besides its low computational complexity and global convergence, this observer has the advantage of providing a simple asymptotic formula that is useful for tuning finite-dimensional filters. Two illustrative examples are presented.     

The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.     

Feedback equivalence of constant linear systems

Two constant linear systems are said to be feedback equivalent if one can be transformed into the other via an element of the “feedback group”, which acts by state space feedback and by change of basis in the state and input spaces. Let C_n,m be the space of n-dimensional completely reachable systems with m-dimensional input (pairs of matrices, n × n and n × m). The action of the feedback group partitions the space C_n,m into finitely many orbits (equivalence classes), and the closure of each orbit is a union of orbits. If one views orbit closure as ‘deformation’, then orbit closure may be considered in terms of perturbations or system failure. In this paper we determine: (1) a classification of the orbits, and (2) the orbits contained in the closure of a given orbit. Both of these problems have been solved previously (see [1,4,6,3]); here we present simple proofs and point out a connection between this problem and the analogous problem for nilpotent matrices.     

Numerical analysis of the reduction of linear systems into orthogonal canonical form

An efficient numerically stable computational algorithm for reduction of linear systems into orthogonal canonical form is described. The algorithm is based on QR decomposition with column pivoting. Exact error bounds and operation count for the algorithm are derived.