Carl N. Nett

Control oriented system identification: a worst-case/deterministic approach in H/sub infinity /

The authors formulate and solve two related control-oriented system identification problems for stable linear shift-invariant distributed parameter plants. In each of these problems the assumed a priori information is minimal, consisting only of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The first of these problems involves identification of a point sample of the plant frequency response from a noisy, finite, output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest. This problem leads naturally to the second problem, which involves identification of the plant transfer function in H/sub infinity / from a finite number of noisy point samples of the plant frequency response. Concrete plans for identification algorithms are provided for each of these two problems. >

Algebraic aspects of linear control system stability

In this paper the problem of linear control system stability is formulated in very general terms and then resolved by (i) establishing a necessary and sufficient condition for stability, (ii) exploring in some detail the nature of this condition, and (iii) developing a parametric characterization of all controllers which stabilize a given plant. The problem formulation developed herein is based on a ring theoretic setting for linear systems, and is distinguished by the use of a novel control system configuration. Any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems. Moreover, no other configuration has this property. Benefits and extensions accrued as a result of this property are thoroughly discussed, as are some promising directions for future research.

Linear state-space systems in infinite-dimensional space: the role and characterization of joint stabilizability/detectability

Several fundamental results from the theory of linear state-space systems in finite-dimensional space are extended to encompass a class of linear state-space systems in infinite-dimensional space. The results treated are those pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and the concept of joint stabilizability/detectability. A complete structural characterization of jointly stabilizable/detectable systems is obtained. The generalized theory applies to a large class of linear state-space systems, assuming only that: (i) the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators; (ii) the state space is Hilbert space; (iii) the input and output spaces are finite-dimensional; and (iv) the sensing and control operators are bounded. General conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space can be drawn from these results. >

A new method for computing delay margins for stability of linear delay systems

This note is concerned with stability properties of linear time-invariant delay systems. We consider retarded delay systems modeled both as a high order scalar differential difference equation and as a set of first order differential-difference equations expressed in state space form. We provide a computational method that can be used to compute a delay interval such that the delay system under consideration is stable for all delay values that lie in the computed interval. This method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently. Based on this method, we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models.<<ETX>>

Nonlinear control of surge in axial compression systems

Abstract This paper discusses the nonlinear control of surge in axial compression systems. A nonlinear model-based Luenberger-type observer is first developed from a simple incompressible flow model of an axial compressor rig; then the input-output feedback linearizing control technique is employed to derive an appropriate control law. The use of the output observer as the basis for the control design enables the reconstruction of approximate system states from a single output measurement, it enhances transient tracking of measured output, and it reduces the destabilizing effect of measurement noise in closed loop. The nonlinear controller described in this work achieves an implicit linearization of the dynamics between the measured output variable, the dynamic pressure at the compressor inlet and the system input variable (the throttle area parameter). The designed controller is validated by both simulation and experimental demonstration on an axial compressor experimental rig. The controlled system responses show adequate stabilization of the surge dynamics in the presence of impulsive and persistent disturbances in system conditions. Further evaluation of the overall controlled system performance is enhanced by comparisons with a linear surge controller. It is shown that gain scheduling along nominal conditions is not required to maintain uniform closed-loop performance, during tracking of setpoint changes, with the nonlinear controller.

Worst case system identification in l 1 ne-equation> 1 : optimal algorithms and error bounds

Abstract This paper analyses worst case identification of linear shift invariant systems with an l 1 error criterion. The assumed apriori information on the plant are bounds on the system gain and decay rate of the impulse response of the unknown system and the experimental data is assumed to be a finite set of corrupted impulse or step response samples of the system.

Euclidean condition and block relative gain: Connections, conjectures, and clarifications

Given an arbitrary nonsingular complex matrix, certain quantitative relationships between its Euclidean condition number and its associated block relative gains are established. In addition, an as yet unproven relation of this type is conjectured. The control theoretic implications of the established relations are briefly discussed, clarifying the role of the block relative gain concept in control theory.

Paper: Sensitivity integrals for multivariable discrete-time systems

The purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discrete-time systems. We present extensions to both the classical Poisson integral and Bodes sensitivity integral formulae. The main utility of these results is that they can be used to quantify design limitations that arise in multivariable discrete-time systems, due to such system characteristics as open-loop unstable poles and nonminimum-phase zeros, and to such design requirements as stability and bandwidth constraints. These formulae are similar to those for multivariable continuous-time systems obtained elsewhere, and they reveal that feedback design limitations depend on directionality properties of the sensitivity function, and on those of unstable poles and nonminimum-phase zeros in the open loop transfer function.

Structured singular values and stability analysis of uncertain polynomials, part 2: a missing link

Abstract In Part 1 of this paper, a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one. In Part 2 of this series, we demonstrate that the framework developed in Part 1 is particularly suitable for a class of stability problems and its unifies several frequency-based stability conditions obtained via alternative approaches. Generally, these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion. Several stability conditions are derived readily from solution of the structured singular value presented in Part 1, which either extend or coincide with previously known results. In particular, for family of interval and diamond polynomials, we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of Kharitonov theorem lead to one another, thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series. A related robust stabilization problem is also investigated in this framework.

Structured singular values and stability analysis of uncertain polynomials, part 1: the generalized m

Abstract One of the primary goals in this two-part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials. Another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis. To achieve these goal, we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework. Part 1 of this series is devoted entirely to the generalized structured singular values, specifically for the case when a certain matrix representing the ‘nominal system’ is of rank one. An analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable. In particular, when the general framework is specialized to that of the standard structured singular value, the expression is solved explicitly. Also for several additional important cases, explicit solutions are obtained. The framework as well as results will then be used in Part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties. We demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches. For several problems of interest, we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of Kharitonov theorem can be derive from one another, and hence establish a link between two drastically different type of results.