Hankel singular value functions from Schmidt pairs for nonlinear input–output systems

This paper presents three results in singular value analysis of Hankel operators for nonlinear input–output systems. First, the notion of a Schmidt pair is defined for a nonlinear Hankel operator. This makes it possible to define a Hankel singular value function from a purely input–output point of view and without introducing a state space setting. However, if a state space realization is known to exist then a set of sufficient conditions is given for the existence of a Schmidt pair, and the state space provides a convenient representation of the corresponding singular value function. Finally, it is shown that in a specific coordinate frame it is possible to relate this new singular value function definition to the original state space notion used to describe nonlinear balanced realizations.     

On strong stabilization of asymptotically time-invariant linear time-varying systems

This paper considers the strong stabilization problem: given a linear time-varying system which is stabilizable by dynamic feedback, when can the stabilizer be chosen to be itself stable? We consider here the case of algebras of discrete time, time-varying systems which are asymptotically time-invariant, in the sense that as time evolves the time-varying transfer operator converges to a time-invariant transfer operator. Convergence here is in the sense of uniform or strong convergence of sequences of operators on an appropriate Hilbert space of input–output signals.     

On the computation of the gap metric for LTV systems

In this paper, we consider the robust stabilization problem for linear discrete time-varying (LTV) systems using the gap metric. In particular, we show that the time-varying (TV) directed gap reduces to an operator with a TV Hankel plus Toeplitz structure. Computation of the norm of such an operator can be carried out using an iterative scheme involving a TV Hankel operator defined on a space of Hilbert–Schmidt causal operators. The “infimization” in the TV directed gap formula is shown to be, in fact, a minimum by using duality theory. The latter holds as well in the time-invariant case.     

Fliess operators on Lp spaces: convergence and continuity

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, L_p function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in L_p[t₀,t₀+T], where T is bounded and p1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on L_p[t₀,t₀+T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition.     

On the distributed stable full information H∞ minimax problem

We study the distributed parameter suboptimal full information H^∞ problem for a stable well-posed linear system with control u, disturbance w, state x, and output y. Here u, w, and y are L²-signals on (0,∞) with values in the Hilbert spaces U, W, and Y, and the state x is a continuous function of time with values in the Hilbert space H. The problem is to determine if there exists a (dynamic) γ-suboptimal feedforward compensator, i.e., a compensator 𝒰 such that the choice u = 𝒰w makes the norm of the input/output map from w to y less than a given constant γ. A sufficient condition for the existence of a γ-suboptimal compensator is that an appropriately extended input/output map of the system has a (J,S)-inner–outer factorization of a special type, and if the control and disturbance spaces are finite-dimensional and the system has an L¹ impulse response, then this condition is also necessary. Moreover, in this case there exists a central state feedback/feedforward controller, which can be used to give a simple parameterization of the set of all γ-suboptimal compensators. Our proof use a game theory approach. © 1998 John Wiley & Sons, Ltd.     

Nonlinear feedback control of multivariable non-minimum-phase processes

A new method of controlling nonlinear processes with a non-minimum-phase delay-free part is presented. Two control laws are derived for stable, multiple-input multiple-output processes. They are obtained by requesting an approximately linear, input–output response and exploiting the connections between model-predictive control and input–output linearization. Conditions under which the closed-loop system is asymptotically stable are given. The application and performance of the control laws are illustrated using numerical simulation of two chemical reactor examples that exhibit non-minimum-phase behavior.     

WeightedH 2 approximation of transfer functions

The aim of this work is to generalize to the weighted case some results and algorithms concerningL ² approximation by analytic and rational functions which are useful when performing the identification of unknow transfer functions of a class of stable (linear causal time-invariant) systems from incomplete frequency data.     

Input–output approach to stability and -gain analysis of systems with time-varying delays

Stability and L₂ (l₂)-gain of linear (continuous-time and discrete-time) systems with uncertain bounded time-varying delays are analyzed under the assumption that the nominal delay values are not equal to zero. The delay derivatives (in the continuous-time) are not assumed to be less than q<1. An input–output approach is applied by introducing a new input–output model, which leads to effective frequency domain and time domain criteria. The new method significantly improves the existing results for delays with derivatives not greater than 1, which were treated in the past as fast-varying delays (without any constraints on the delay derivatives). New bounded real lemmas (BRLs) are derived for systems with state and objective vector delays and norm-bounded uncertainties. Numerical examples illustrate the efficiency of the new method.     

Approximation of unstable infinite-dimensional systems using coprime factors

The effectiveness of comprime factor techniques in L₂ and L_∞ model reduction of unstable linear systems is analysed. Asymptotic estimates are given of the achievable error in the stable and unstable parts of the approximate system, measured in a number of different norms, some involving the associated Hankel operators. The chordal metric is introduced as a measure of approximation and is shown to yield the graph topology. Finally it is deduced that asymptotically optimal L₂ and L_∞ convergence rates can be obtained for a large class of unstable systems.     

Minimal parametrizations of single-input linear dynamical systems

The parametrization of the set of all controllable single-input time-invariant linear dynamical systems is defined as a map π from a parameter set M_π onto a given set of canonical forms 𝒸_π. A parametrization is called ‘minimal’ if it induces a continuous canonical map and the number of parameters is minimal. Some of its general properties, necessary and sufficient conditions, and several new concrete forms suitable for identification are also given.     

Optimal deterministic observer design for linear continuous-time systems with unknown output disturbances in a ball in L2

The problem of the existence and design of an optimal observer for linear systems with unknown disturbances additive to the output is considered. The disturbances are assumed to be arbitrary L² functions constrained to a ball in L². The problem of reconstructing the best estimate of the state with respect to the worst disturbances is reformulated in terms of finding an operator L²-Rⁿ Satisfying some linear equalities and of minimal norm. The necessary and sufficient condition for the existence of such an operator is proved to be the well-known observability condition. The optimal observer is in the form of an integral operator. An explicit formula for its integrand is given. In addition, differential equations fulfilled by the optimal state estimate for the cases of a fixed or time-variable observation interval are derived.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

Extended Ho–Kalman algorithm for systems represented in generalized orthonormal bases

This paper considers the construction of minimal state space models of linear time-invariant systems on the basis of system representations in terms of generalized orthogonal basis function expansions. Starting from the classical Ho–Kalman algorithm that solves the problem using Markov parameter expansions, a generalization is obtained by analysing the matrix representations of the Hankel operators in generalized orthonormal bases. Using the so-called Hambo-domain techniques an efficient algorithm is given to implement the proposed method.     

Robust control of uncertain differential linear repetitive processes

For two-dimensional (2-D) systems, information propagates in two independent directions. 2-D systems are known to have both system-theoretical and applications interest, and the so-called linear repetitive processes (LRPs) are a distinct class of 2-D discrete linear systems. This paper is concerned with the problem of L₂–L_∞ (energy to peak) control for uncertain differential LRPs, where the parameter uncertainties are assumed to be norm-bounded. For an unstable LRP, our attention is focused on the design of an L₂–L_∞ static state feedback controller and an L₂–L_∞ dynamic output feedback controller, both of which guarantee the corresponding closed-loop LRPs to be stable along the pass and have a prescribed L₂–L_∞ performance. Sufficient conditions for the existence of such L₂–L_∞ controllers are proposed in terms of linear matrix inequalities (LMIs). The desired L₂–L_∞ dynamic output feedback controller can be found by solving a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures.     

Connectivity and the measurement of operational risk: an input–output approach

In this article I propose a framework for the analysis of the interdependencies within a financial institution that is based on the input–output framework originally developed by Leontiev (1941). After discussing the state of the art of operational risk measurement, I briefly review the foundations of input–output analysis and explain how to build an input–output model at the business unit level for a financial institution. I also discuss the suitability of an input–output model in capturing the impact on operational risk losses of the interdependencies within a financial institution and then present, through some numerical examples, how to implement the model within a quantitative framework for the measurement of operational risk. In Ersilia, to establish the relationships that sustain the city's life, the inhabitants stretch strings from the corners of the houses, white or black or gray or black-and-white according to whether they mark a relationship of blood, of trade, authority, agency. When the strings become so numerous that you can no longer pass among them, the inhabitants leave: the houses are dismantled; only the strings and their supports remain. From a mountainside, camping with their household goods, Ersilia's refugees look at the labyrinth of taut strings and poles that rise in the plain. That is the city of Ersilia still, and they are nothing. Italo Calvino, Invisible Cities (1972)     

Bounded real lemma and robust control of 2-D singular Roesser models

This paper discusses the problem of robust H_∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.     

On the robust stabilizability of uncertain linear time-invariant plants using nonlinear time-varying controllers

We consider general families of linear time-invariant plants described by a nominal plant model with additive or multiplicative unstructured modeling uncertainty. We show that such families of plants are robustly (uniformly) stabilizable by nonlinear time-varying controllers, if and only if they are robustly stabilizable by linear time-invariant ones. Some auxiliary results on Hankel and L_∞-norm approximation are also obtained.     

Deadbeat control in multivariable non-linear time-varying systems with constraints of control inputs

A new approach to deadbeat control of a class of multivariable non-linear time-varying systems is presented. The approach enables us to find a non-linear control law, minimal positive integer N and a piecewise-constant control input vector subjected to constraints such that the output of closed-loop system coincides with given reference vector r( l) for ≥ NT, where r(<) represents the class of vector functions which can be generated in linear time-invariant systems, T is a given sampling period.     

Well-posedness, regularity and exact controllability of the SCOLE model

The SCOLE model is a coupled system consisting of a flexible beam (modelled as an Euler–Bernoulli equation) with one end clamped and the other end linked to a rigid body. Its inputs are the force and the torque acting on the rigid body. It is well-known that the SCOLE model is not exactly controllable with L ² input signals in the natural energy state space H ^c , because the control operator is bounded from the input space \mathbbC²{\mathbb{C}^2} to H ^c , and hence compact. We regard the velocity and the angular velocity of the rigid body as the output signals of this system. Using the theory of coupled linear systems (one infinite-dimensional and one finite-dimensional) developed by us recently in another paper, we show that the SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space X ì H^c{\mathcal{X}\subset H^c}.