LTI modelling of NFIR systems: near-linearity and control, LS estimation and linearization

Linear time-invariant (LTI) modelling of nonlinear finite impulse response (NFIR) systems is studied from a control point of view. Nearly linear NFIR systems and their control-relevant properties are analysed in detail. The main modelling interest is in the analysis of least squares (LS) LTI identification when the true system is an NFIR system, which is possibly nearly linear. Linearization is used for comparison purposes as the second LTI modelling technique. Nearly linear systems provide a natural generalization of LTI systems to include nonlinearities that allow globally good LTI approximations, while at the same time, such nonlinearities can have a very dramatic effect on the local characteristics of the system. Several control-oriented examples illustrate the possible weaknesses and strengths of the studied LTI modelling techniques. Linearization is found to be especially vulnerable to the presence of even very small, only locally significant, nonlinearities. LS estimation can largely avoid such difficulties, but input design becomes a more critical issue than in standard linear estimation theory. Certain counter-intuitive properties of commonly used input-output stability notions, such as ?₂ stability, are discussed via the concept of near-linearity.     

Least-squares LTI approximation of nonlinear systems and quasistationarity analysis

Least-squares linear time-invariant (LTI) approximation of discrete-time nonlinear systems is studied in a generalized harmonic analysis setting extending an earlier result based on quasistationary signals. The least-squares optimal LTI model is such that the crosscorrelation between the input and the LTI model output equals the crosscorrelation between the input and the output of the nonlinear system. New results for limits of sample averages of signals are derived via Riemann-Stieltjes integration theory. These results are applied to crosscorrelation and quasistationarity analysis of input-output signals for several important classes of nonlinear systems, including stable finite memory, Wiener and Hammerstein systems. This analysis demonstrates that the assumptions used in the least-squares LTI approximation setup are fairly mild. Finally, an illustrative example is provided.     

LTI approximation of nonlinear systems via signal distribution theory

L₂ and L₁ optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L₂ (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L₁ (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.     

Linear approximations of nonlinear FIR systems for separable input processes

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgang's classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener-Hammerstein systems.     

Equivalence of Convolution Systems in a Behavioral Framework

In this paper the problem of system equivalence is tackled for a rather general class of linear time-invariant systems. We consider AR-systems described by linear continuous shift-invariant operators with finite memory, acting on Fréchet-signal spaces, containing the space {\cal E} ({\open R}) of infinitely differentiable functions on {\open R}. This class is in one–one correspondence with matrices of suitable sizes over the convolution algebra {\cal E} ({\open R}) of all compactly supported distributions. Using some deep results from the theory of Fréchet spaces, various necessary and sufficient conditions for system equivalence and system inclusion are formulated. It is shown that a surjectivity demand on the system defining convolution operator matrix is necessary and sufficient for being able to translate the problem of system equivalence into division properties over the convolution algebra {\cal E}({\open R}). This surjectivity condition is guaranteed if the system defining matrix over {\cal E}({\open R}) has a right-inverse over {\cal D}({\open R}), the space of all Schwartz distributions. Date received: February 9, 2000. Date revised: April 11, 2003.     

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.     

On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces

In this paper, we investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. The results are obtained by using the α-resolvent family (S_α(t))_t≥0 on a complex Banach space X combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.     

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.     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

FIR-type robust <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H ₂ and H _∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.     

A distributed controller approach for delay-independent stability of networked control systems

This article introduces a novel distributed controller approach for networked control systems (NCS) to achieve finite gain L₂ stability independent of constant time delay. The proposed approach represents a generalization of the well-known scattering transformation which applies for passive systems only. The main results of this article are (a) a sufficient stability condition for general multi-input-multi-output (MIMO) input-feedforward-output-feedback-passive (IF-OFP) nonlinear systems and (b) a necessary and sufficient stability condition for linear time-invariant (LTI) single-input-single-output (SISO) systems. The performance advantages of the proposed approach are reduced sensitivity to time delay and improved steady state error compared to alternative known delay-independent small gain type approaches. Simulations validate the proposed approach.     

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input-output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.     

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.     

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.     

Fault detection filter design for stochastic time-delay systems with sensor faults

This article considers the fault detection (FD) problem for a class of Itô-type stochastic time-delay systems subject to external disturbances and sensor faults. The main objective is to design a fault detection filter (FDF) such that it has prescribed levels of disturbance attenuation and fault sensitivity. Sufficient conditions for guaranteeing these levels are formulated in terms of linear matrix inequalities (LMIs), and the corresponding fault detection filter design is cast into a convex optimisation problem which can be efficiently handled by using standard numerical algorithms. In order to reduce the conservatism of filter design with mixed objectives, multi-Lyapunov functions approach is used via Projection Lemma. In addition, it is shown that our results not only include some previous conditions characterising H _∞ performance and H _? performance defined for linear time-invariant (LTI) systems as special cases but also improve these conditions. Finally, two examples are employed to illustrate the effectiveness of the proposed design scheme.     

Linear periodic controller design for decentralized control systems

In the decentralized control of linear time-invariant (LTI) systems, a decentralized fixed mode (DFM) is a system mode which is immoveable using an LTI controller, while a quotient DFM (QDFM) is one which is immoveable using any form of nonlinear time-varying compensation. If a system has no unstable DFMs, there are well-known procedures for designing an LTI stabilizing controller; for systems which have unstable DFMs but no unstable QDFMs, we provide a simple design algorithm which yields a linear periodic sampled-data stabilizing controller.     

On ultimate boundedness around non-assignable equilibria of linear time-invariant systems

In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y_?. Assume there is no assignable equilibrium point corresponding to y_?. How “close” to y_? can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y_? be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability.     

An efficient LMI approach for the quadratic stabilisation of a class of linear,uncertain, time-varying systems

The purpose of this article is to provide a numerically efficient method for the quadratic stabilisation of a class of linear, discrete-time, uncertain, time-varying systems. The considered class of systems is characterised by an interval time-varying (ITV) matrix and constant sensor and actuator matrices. It is required to find a linear time-invariant (LTI) static output feedback controller yielding a quadratically stable closed-loop system independently of the parameter variation rate. The solvability conditions are stated in terms of linear matrix inequalities (LMIs). The set of LMIs includes the stability conditions for the feedback connection of a unique suitably defined extreme plant with an LTI output controller and the positivity of a closed-loop extremal matrix. A consequent noticeable feature of the article is that the total number of LMIs is independent of the number of uncertain parameters. This greatly enhances the numerical efficiency of the design procedure.