Linear State-space Systems Research Articles

A Flexible Identification Strategy for DABNet Models

Abstract The focus of this paper is to introduce a flexible identification strategy for a general class of models. We use DABNet models, which are composed of a linear state space system whose states, de- coupled by input, are mapped by a neural network. The linear state space matrices can be represented as loosely coupled first and second order sections. A set of conditions will be analyzed for this kind of model that allows an identification process to be as flexible as possible. By flexible we mean a set of features that would be desirable to find in an identification approach: constraint handling and optimal-input design.

Min–max predictive control techniques for a linear state-space system with a bounded set of input matrices

Min–max predictive control of a linear state-space system with a bounded set of input matrices is studied based on a quadratic performance criterion. Systems with stable and integrating dynamics as well as time-varying and time-invariant uncertainties are treated. Algorithms that are computationally manageable and ensure closed-loop stability are developed and tested.

A necessary condition for quantitative exponential stability of linear state-space systems

For an arbitrary n×n constant matrix A the two following facts are well known: • (1/n) Re(trace A) − max j=1,…,n Re λ j(A)⩽0 ; • If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U=e A . In this note we present the extension of these two facts to the context of linear time-varying dynamical systems x ̇ =A(t)x, t⩾0. As a by-product, this result suggests that, the notion of “slowly varying state-space systems”, commonly used in literature, is mathematically not natural to the problem of exponential stability.

The output regulation problem of linear singular systems

This paper discusses the output regulation problem of the disturbed singular systems. The relation between this problem and the output regulation problem of standard state-space linear systems is given based on the Weierstrass canonical form of singular systems. For the case of that the disturbance is described by differential equation, the sufficient and necessary condition of which the output regulation problem is solvable and a way to design corresponding controller are given.

Singular optimal control problem of linear singular systems with linear-quadratic cost *

This paper investigates the singular linear-quadratic optimal control problem of a kind of linear time-in variant singular systems containing impulse mode. The relation between this problem and the linear-quadratic optimal control problem of standard state-space linear systems is established by using an equivalent transformation based on the Weierstrass canonical form of singular systems. The existence and uniqueness of the solution, the method to compute the optimal control and corresponding state trajectory, and the feedback synthesis of the optimal control of this problem is also given

Balanced canonical forms for system identification

Parameterizations of linear state-space systems by balanced canonical forms are considered. The structure, in particular the topological properties, of these parameterizations are investigated. Based on these results an algorithm for identification obtained by optimizing a likelihood-type criterion function is proposed and analyzed. In the algorithm both the estimation of real-valued parameters and of integer parameters (defining dynamic specification) is incorporated. The analysis of the algorithm emphasizes consistency results.

Robust stabilization of linear systems in the presence of Gaussian perturbation of parameters

Stabilization of linear systems in state space in the presence of parametric perturbations is considered. The perturbed system is represented by a matrix differential equation with the elements of the matrices given by Gaussian processes with known mean and covariance. Using methods from stochastic control theory, certain pole-placement-like results are derived which hold in the mean square sense. In the absence of any perturbation, these results reduce to the well-known results of pole placement for deterministic linear systems. Minimizing the real part of the largest eigenvalue of the expected closed-loop matrix, we obtain the optimal feedback gain that stabilizes the system at the fastest possible rate. The question of existence of a guaranteed stabilizing feedback is also investigated. As a consequence of the main result we obtain a method of designing fault-tolerant systems that will survive in the events of catastrophic controller failure. An extension of the Luenberger observer for uncertain systems is also presented. © 1998 John Wiley & Sons, Ltd.

A Quasi-Decoupling Approach for Nonclassical Linear Systems in State Space

By adopting the orthogonal transformations provided by the generalized real Schur decomposition, it is shown that every nonclassical linear system in state space can be transformed into block upper triangular form, to which the quasi-decoupling solution can be progressively carried out by solving the either first or second-order component equations with the “back substitution.” The distinct characteristics of generalized eigenvalue problems from those of standard ones are discussed. Favorable properties of the proposed method include: no inverting of any system matrix, indiscriminate applicability to both defective and nondefective systems, the simultaneous decoupling of the adjoint problem, and numerical stability. Illustrative examples are also presented.

State-space/descriptor models and asymptotic behavior of continuous-time positive control systems

This paper deals with new aspects of the theory of nonnegative matrices and their applications to some problems arising in the studies of continuous-time control systems. The criterion for asymptotic stability of linear state-space control systems with positive state matrix A and respective initial conditions is established and proven. It states that the system is asymptotically stable if and only if the state variables are linearly dependent. Moreover, the case of dynamical systems in descriptor forms with positive leading and state matrices is studied. The asymptotic stability of descriptor systems characterized by positive leading and state matrices is expressed in terms of the geometric relations between matrices of a proper part of the corresponding regular matrix pencil inverse and the initial conditions. Several illustrating examples are also included.

Improved L/sub 2/-sensitivity for state-space digital system

An improved L/sub 2/-sensitivity measure for linear state-space digital systems is presented. It gives a more precise evaluation of the transfer function deviation /spl Delta/H(z) between the infinite wordlength system and the finite wordlength system when the structure matrices {A,b,c} contain 0 and /spl plusmn/1 coefficients. This measure can be applied in the system optimization, resulting in a computationally efficient and very low L/sub 2/-sensitivity structure. The theoretical result is verified by a numerical example.

Information Criteria for FDI

Using an information point of view, we discuss deterministic versus stochastic tools for residual generation and evaluation for FDI in linear state-space systems. Previously proposed statistical isolation methods are revisited, using both a linear transformation formulation and the information content of the corresponding residuals. New information criteria are proposed for investigating the residual optimization problem.

Experimental Design for the Control of Linear State-Space Systems

Experimental design consists in determining experimental conditions that optimize a given criterion. Here, we consider systems with fixed structure and unknown parameters, for which the criterion corresponds to a control objective. We point out that one must first solve a dual control problem, and propose a new dual control policy for discrete-time systems. The low complexity of this dual control algorithm allows one to use it in experimental design for control, which implies a simultaneous choice of the input sequence and measurement times. Simulation results are presented.

When is a periodic discrete-time system equivalent to a time-invariant one?

We give the precise conditions under which a periodic discrete-time linear state-space system can be transformed into a time-invariant one by a change of basis. Thus our theory is the discrete-time counterpart of the classical theory of Floquet transforms developed by Floquet and Lyapunov in the 1800s for continuous-time systems. We state and prove a necessary and sufficient condition for a “discrete-time Floquet transform” to exist, and give a construction for the transform when it does exist. Our results also extend to generalized state-space, or descriptor, systems.

Guardian map approach to robust stability of linear systems with constant real parameter uncertainty

New sufficient conditions are derived for stability robustness of linear time-invariant state-space systems with constant real parameter uncertainty. These bounds are obtained by applying a guardian map to the uncertain system matrices. Since this approach is only valid for constant real parameter uncertainty, these bounds do not imply quadratic stability, which guarantees robust stability with respect to time-varying uncertainty but is often conservative with respect to constant real parameter uncertainty. Numerical results are given to compare the new bounds with bounds obtained previously by means of Lyapunov methods. >

A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations

The object of this paper is to maximize the stability radius of a linear state space system by state feedback under Lipschitzian structured stochastic perturbations. The supermanl achievable stability radius is characterized via the resolution of a parametrized Riccati equation and a matrix inequality. The dependence on the parameters is investigated and the limiting behaviour is examined. An example illustrating the results is treated at the end of the paper.

Time moment and Padé approximation methods applied to the order reduction of MIMO linear systems

The combined use of the time moment and Padé approximation methods (in time space and frequency domain description), coupled with three powerful dominant pole selection criteria, is introduced for the purpose of application to high-order MIMO linear time-invariant state-space original system models, in order to obtain corresponding adequate reduced order model(s). The proposed approach has been applied successfully to a 10th-order two-input two-output time-invariant linear model of a practical power system.

Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials

In this paper we consider robustness measures (stability radii) for system matrices which are subjected to structured real and complex perturbations of the form A↦A + BDC where B, C are given matrices. Our object is twofold:(a) to present a number of new results, mainly concerning the real stability radius and its differences from the complex one; (b) to give an overview of our approach to the robustness analysis of linear state space systems, including basic properties and characterizations of the complex stability radius. Applying the results to the special case where A is in companion form and B = [0, 0, ⃛, 0, 1]T, we are able to determine stability radii for Hurwitz and Schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector. Computable formulae are obtained and illustrated by several examples

Model validation by the distortion method: linear state space systems

The distortion method was introduced by Butterfield and others as a means of verifying a nominal system model, comprising a set of ordinary, nonlinear differential equations, by comparing the model response with a set of experimentally obtained data. The essence of the method is to make the minimum distortions to the model parameters necessary to ensure that the model output exactly matches the measured data. The distortions thus obtained can then be tested against a priori knowledge of the parameters, in particular to see if the variance of the distortions is less than the uncertainty in the expected parameter values. In the paper it is shown that, if the nominal model is linear, then, under certain conditions, explicit formulas can be obtained for the distortions, or significant simplifications can be made to the equations, thus removing or reducing the considerable computational overhead inherent in the method. The results of the paper could find application not only in the area of linear modelling (for example in compartmental modelling) but also in model reduction and linearisation, where the method offers a means of testing the validity of the linear or reduced-order model.

Control Engineering Data Structures for Concurrent Engineering

Abstract Concurrent Control Engineering aims at multidisciplinary integration of control engineering methods with the data and procedures of complementary engineering disciplines in view of holistic system optimization. This implies an intradisciplinary abstraction of relevant control engineering data structures as well as an interdisciplinary export of control engineering data objects. We show how control engineering datatypes can be specified and realized in this view using the class-concept and especially the inheritance mechanism of object-oriented data models. The control engi-neering datatypes to be considered are: general nonlinear dynamic systems, time-invariant state space linear systems, transfer matrices, real functions such as time functions, and complex functions such as frequency response. Feasible operations on these datatypes are: listing or graphical visualization, connection of objects within a class, transformation within a class, transformation into another class, and generation of partial objects.

Computing transfer function zeros of a state space system

Abstract A new algorithm is developed for computation of the transfer function zeros of an SISO linear state space system, based on an improved Sandberg and So algorithm and Householder transformations which reduce the problem to that of finding the standard eigenvalues of a matrix. Techniques are introduced which take full advantage of sparsity to improve accuracy. Where possible a reducing transformation is performed which does not cause any round-off error. Methods for improving the accuracy of the zeros are included.