Stability analysis of continuous-time periodic systems via theharmonic analysis

Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDLCP model whose transition matrix can be determined explicitly. By extending the Gerschgorin theorem to linear operators on the linear space l₂, yet another disc-group criterion is derived, which is only sufficient. Stability of the lossy Mathieu equation is analyzed as a numerical example to illustrate the results     

A complete stability analysis for planar delta operator systems subject to state saturation

In this paper, stability analysis is investigated for planar delta operator systems subject to state saturation. General properties of limit trajectories are characterized for the planar delta operator systems subject to state saturation. A nontrivial limit trajectory only intersects with two pairs of opposite sides but not neighboring sides of a unit square. Hence, a relation is established between the present intersection of a trajectory with lines x₂ = ±1 and the next intersection. Necessary and sufficient conditions on global asymptotic stability are obtained for the planar delta operator systems with full and partial saturated state. Simulation results are given to show the effectiveness of the proposed methods.     

Control of an unstable reaction–diffusion PDE with long input delay

A Smith Predictor-like design for compensation of arbitrarily long input delays is available for general, controllable, possibly unstable LTI finite-dimensional systems. Such a design has not been proposed previously for problems where the plant is a PDE. We present a design and stability analysis for a prototype problem, where the plant is a reaction–diffusion (parabolic) PDE, with boundary control. The plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay, with an unbounded input operator. The predictor-based feedback design extends fairly routinely, within the framework of infinite-dimensional backstepping. However, the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ODE. The unbounded character of the input operator requires that the stability be characterized in terms of the H₁ (rather than the usual L₂) norm of the actuator state. The analysis involves an interesting structure of interconnected PDEs, of parabolic and first-order hyperbolic types, where the feedback gain kernel for the undelayed problem becomes an initial condition in a PDE arising in the compensator design for the problem with input delay. Space and time variables swap their roles in an interesting manner throughout the analysis.     

Zeros and Poles of Linear Continuous-Time Periodic Systems: Definitions and Properties

The paper deals with definitions of zeros and poles and their features in finite-dimensional linear continuous-time periodic (FDLCP) systems under a harmonic framework. More precisely, system and transfer zeros and poles in the harmonic wave-to-wave sense are defined on what we call the regularized harmonic system operators and the harmonic transfer operators of FDLCP systems by means of regularized determinants; then their composition and properties related to system structures are examined via the Floquet theory and controllability/observability decompositions of FDLCP systems. The study shows that under mild assumptions, the harmonic transfer operators of FDLCP systems are analytic and meromorphic, on which zeros and poles are well-defined. Basic zero/pole relationships are established, which are similar to their linear time-invariant counterparts and in particular explicate some interesting harmonic wave-to-wave behaviors of FDLCP systems. The results are significant in analysis and synthesis of FDLCP systems when the harmonic approach is adopted.     

Stabilization of linear multivariable constant delay-differential systems

The article presents the iterative method of stabilizing control signal selection in the form of the state-feedback of a multivariable linear time-invariant plant with delays. The proposed method is based on the consecutive assignment of the single system eigenvalues λ_j so that its stabilizability will be preserved for a given iteration step and the assigned eigenvalues will not be changed, even though a dynamical element identified by the matrix K_j(z) (where z is an elementary delay operator) is placed into the state feedback. To illustrate the abovementioned method, a simplified example is given.     

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

Asymptotic stability of a parallel repairable system with warm standby

This paper presents the asymptotic stability of a repairable system associated with two identical units parallel and one unit warm standby. It is shown that the system operator generates a positive C ₀-semigroup in the state Banach space. The steady-state solution is shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. By analysing the spectrum of the system, it is found that 0 is the unique spectral point of the system on the imaginary axis. As a result, the asymptotic stability of the system is obtained.     

H2 and H∞ norm computations of linear continuous-time periodic systems via the skew analysis of frequency response operators

The H₂ and H_∞ norm computations of finite-dimensional linear continuous-time periodic (FDLCP) systems through the frequency response operators defined by steady-state analysis are discussed. By the skew truncation, the H₂ norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time-invariant (LTI) continuous-time system. The H_∞ norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent LTI continuous-time system over a certain frequency range via the modified skew truncation. By the latter result, a Hamiltonian test is proved for FDLCP systems in an LTI fashion, based on which a modified bisection algorithm is developed.     

On the stability of the information state system

The purpose of this paper is to present some preliminary results on the stability of the information state system. The information state system underlies the (infinite dimensional) dynamics of an H_∞ controller for a nonlinear system. Thus it is important to understand its stability and the structure of its equilibrium points. We analyse the important case corresponding to the mixed sensitivity problem. We prove the existence of an equilibrium information state, convergence under very general conditions to such an equilibrium state p_e and uniqueness of this state (up to an irrelevant constant). In this case the equilibrium p_e is usually singular in the sense that it takes on the value − ∞ except on a low dimensional subset of its domain.This meshes with the article [9] which analysed the effect of using p_e to initialize the information state controller and gave explicit formulas which in many cases produce a dramatic reduction in the amount of computation required to implement the controller. What this article suggests is that indeed p_e is the only equilibrium initialization possible.     

Solvability conditions and design for synchronization of discrete‐time multiagent systems

This paper provides solvability conditions for state synchronization with homogeneous discrete‐time multiagent systems with a directed and weighted communication network under partial‐ or full‐state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H₂ discrete‐time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H₂ discrete‐time algebraic Riccati equation.     

An invert-free Arnoldi method for computing interior eigenpairs of large matrices

This paper presents an invert-free Arnoldi method for extracting a few interior eigenpairs of large sparse matrices. It is derived by implicitly applying the Arnoldi process with the shifted and inverted operator (A?τ I)^?1 in a shifted Krylov subspace (A?τ I)𝒦 _m (A, v ₁). Due to a subtle relationship between the Krylov subspace 𝒦 _m (A, v ₁) and its shifted Krylov subspace, we avoid forming the shifted and inverted operator explicitly. Comparisons are drawn between the harmonic Arnoldi method and the invert-free Arnoldi method. Finally, numerical results are reported to show the efficiency of the new method.     

THE NYQUIST ROBUST STABILITY MARGIN—A NEW METRIC FOR THE STABILITY OF UNCERTAIN SYSTEMS

The Nyquist robust stability margin k_N is proposed as a new tool for analysing the robustness of uncertain systems. The analysis is done using Nyquist arguments involving eigenvalues instead of singular values, and yields exact necessary and sufficient conditions for robust stability. The concept of a critical line on the Nyquist plane is defined and used to calculate a critical perturbation radius which in turn is used to produce k_N. The new approach gives alternatives to computing exact stability margins in some cases of highly directional uncertainty templates where other models are not applicable. © 1997 by John Wiley & Sons, Ltd.     

Trace formula of linear continuous-time periodic systems via the harmonic Lyapunov equation

In the former part of this paper, a trace formula is established for the H 2 norms of a class of finite-dimensional linear continuous-time periodic (FDLCP) systems based on the solution of the harmonic Lyapunov equations. This trace formula is quite similar in form to what we have for the H 2 norm of an LTI continuous-time system apart from the fact that infinite-dimensional matrices are involved in the FDLCP setting. Based on this formula, in the latter part of this paper some trace formulas are developed via the approximate modelling and truncating approach, which are numerically implementable in most practical FDLCP systems. There are numerical examples to illustrate the efficacy of the trace formulas suggested.     

Output feedback decentralized control of large-scale systems using weighted sensitivity functions minimization

This paper considers the problem of achieving stability and desired dynamical transient behavior for linear large-scale systems, by decentralized control. It can be done by making the effects of the interconnections between the subsystems arbitrarily small. Sufficient conditions for stability and diagonal dominance of the closed-loop system are introduced. These conditions are in terms of decentralized subsystems and directly make a constructive H_∞ control design possible. A mixed H_∞ pole region placement is suggested, such that by assigning the closed-loop eigenvalues of the isolated subsystems appropriately, the eigenvalues of the overall closed-loop system are assigned in desirable range. The designs are illustrated by an example.     

Class of globally stable saturated state feedback regulators

The aim is to find a feedback matrix F for a saturated stale feedback regulator, which guarantees its global asymptotic stability. We consider the discrete-time system with the constrained input described by x_k+1 = Ax_k + Bu_k , where u_k ? Ω R and matrix A is assumed to be critically stable, i.e. there exists some eigenvalues, lambda; _i (A), such that |λ _i (A)| = 1.     

Controllability discrepancy and irreducibility/reducibility of Floquet factorisations in linear continuous-time periodic systems

The paper reports interesting but unnoticed facts about irreducibility (resp., reducibility) of Flouqet factorisations and their harmonic implication in term of controllability in finite-dimensional linear continuous-time periodic (FDLCP) systems. Reducibility and irreducibility are attributed to matrix logarithm algorithms during computing Floquet factorisations in FDLCP systems, which are a pair of essential features but remain unnoticed in the Floquet theory so far. The study reveals that reducible Floquet factorisations may bring in harmonic waves variance into the Fourier analysis of FDLCP systems that in turn may alter our interpretation of controllability when the Floquet factors are used separately during controllability testing; namely, controllability interpretation discrepancy (or simply, controllability discrepancy) may occur and must be examined whenever reducible Floquet factorisations are involved. On the contrary, when irreducible Floquet factorisations are employed, controllability interpretation discrepancy can be avoided. Examples are included to illustrate such observations.     

Improved closed‐loop stability for fixed‐point controller realizations using the delta operator

A stable discrete‐time control system may achieve a lower than predicted performance or even become unstable when the discrete‐time control law is implemented with a fixed‐point digital control processor due to the finite word length (FWL) effects, which depends on the control law state‐space realization and the discrete‐time operator (e.g., the delta operator or the forward‐shift operator) used to represent the control law. To improve the closed‐loop stability (and as a byproduct, performance) when the control law is implemented, a state‐space approach that selects the control law realization to optimize a stability‐related objective function is developed using the delta operator. Analytical and numerical comparison of the fixed‐point performance of delta control laws with the performance of the corresponding forward‐shift control laws quantifies the improved closed‐loop stability of the delta realizations over those of the corresponding forward‐shift realizations. It is also shown that there exists a simple mapping between the optimal FWL forward‐shift control law realizations and the optimal FWL delta control law realizations. The results are illustrated by delta and forward‐shift control law realizations of a discrete‐time H_∞ control law designed for a teleoperation motion‐scaling system. Copyright © 2001 John Wiley & Sons, Ltd.     

Computing the L2-induced norm of a compression operator

A new computationally viable approach is derived for computing the induced norm of a state space compression operator; that is an integral operator defined on the finite length space L₂[0,h]. Determining this norm is a crucial component in the control analysis of both delay systems and sampled-data systems. The technique developed may have application to a wider variety of integral operators.