Non-proper Systems Research Articles

General computational approach for optimal fault detection

We propose a new computational approach to solve the optimal fault detection problem in the most general setting. The proposed procedure is free of any technical assumptions and is applicable to both proper and non-proper systems. This procedure forms the basis of an integrated numerically reliable state-space algorithm, which relies on powerful descriptor systems techniques to solve the underlying computational subproblems. The new algorithm has been implemented into a Fault Detection Toolbox for Matlab.

On computing achievable fault signatures

The knowledge of achievable fault signatures is a valuable information in designing residual generators providing structured residual sets for fault detection and isolation. We propose an efficient computational approach to determine the achievable fault signatures for a given additive fault model. The proposed procedure relies on recently developed numerically reliable nullspace updating techniques involving orthogonal reductions to Kronecker-like forms. The new procedure is general, being applicable to both proper as well as non-proper systems, and is significantly more efficient than an exhaustive search based approach.

Pole assignment in a disk for linear systems by static output feedback

A systematic approach for pole assignment in a disk for continuous or discrete-time linear systems using a static output control law is proposed. It is shown that if the open-loop system satisfies certain structural conditions, a static output feedback gain can be easily computed, using a formula that simply involves the original system matrices and the parameters that define the disk. Among the conditions the system has to satisfy, the key one relies on a minimum phase argument. Square and non-square, proper and non-proper systems are considered. The main results are derived invoking the limiting behaviour of the optimal linear quadratic regulator for discrete-time systems. Some other implications are also presented.

Robust Fault-Tolerant Control for Descriptor Systems

A new architecture for fault tolerant controllers is proposed for the generic class of descriptor systems. It is based on coprime factorization of nonproper systems and on the Youla parameterization of stabilizing controllers. Noticing that the Youla controllers include a so called residual signal, fault tolerant control is achieved. Nominal control and robust fault tolerance are addressed separately. Moreover, fault tolerant control can be improved with a scheme integrating fault diagnosis. The design of the diagnosis and fault tolerant control filters reduce to a standard H/sub /spl infin//-control problem of usual state-space system.

Behavioral Approach Over A Lipschitz Bounded Set

In this article we discuss on a behavioral approach restricted over a set of functions whose second derivative are bounded to constant C. The condition is equavalent to the Lipschitz condition to the first derivative of the functions. At first characterization of the bounded set restricted by Lipschitz consatant is discussed. Several kinds of examples are introduced and the usefulness of the set is indicated. Secondly Levant VS differentiator is introduced. The principle of the differentiator is a high gain feedback with integrator in the feedback path. By choosing an appropriate parameters and switching speed, variable structure system can realize a high gain feedback system and practical differentiator can be composed. The choice of the parameters are closely related to the Lipschitz constant of the functions. Finally the invariant property of the function is discussed. The Levant differentiator can release the restriction of the properness of the system. Here we examine the conditions which ensure the application of non-proper systems. Discussion in the article is a challenge of the application of the behavioral approach to practical engineering problems. Appropriate restriction of the set of function might lead the behavioral approach from theoretical cite to practical engineering.

Optimal control of proper and nonproper descriptor systems

In recent years, the analysis and synthesis of control systems in the descriptor form have been established. This general description of dynamical systems by differential/algebraic equations is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. The present contribution deals with the design of optimal control for such dynamic systems. This means that the calculus of variations and Pontryagin's maximum principle have to be generalized for descriptor systems. An important role in the applicability of usual design algorithms plays the properness of the systems. The distinction between proper and nonproper systems is necessary in determining descriptor systems which are exclusively governed by the control inputs or additionally by their higher-order time-derivatives. In case of nonproper systems, quite different optimal control problems appear. For both cases, necessary conditions for optimal control are presented.

Some aspects of the optimal control of non-linear descriptor systems

The modification of the algorithms of the calculus of variations and Pontryagin's maximum principle required for them to be applicable to non-linear descriptor control systems is demonstrated. The classical calculus of variations is still applicable in optimization without constraints on the control, but when such constraints are imposed, the application of Pontryagin's maximum principle in its standard or extended form requires a distinction to be made between proper and non-proper descriptor systems. In the non-proper case, the solution depends on higher-order time derivatives of the control inputs. For the correct description of the problem and for Pontryagin's maximum principle to be applicable additional phase variables and corresponding integrator chains have to be introduced. The optimal control thus obtained becomes dynamic. To simplify the notation, the index of the algebraic equations of the constraints is assumed to be uniform. In principle, the results remain valid for a non-uniform index also, i.e. when different constraint equations have different indices and different components of the control occur with different maximum orders of the time derivatives. The results are somewhat complicated, particularly in the case of constrained optimization of non-proper systems.

Linear-quadratic optimal control of descriptor systems

In recent years the analysis and synthesis of (mechanical) control systems in descriptor form has been established. This general description of dynamical systems is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear mechanical descriptor systems and its control design with respect to a quadratic performance criterion. Here, the notion of properness plays an important role whether the standard Riccati approach can be applied as usual or not. Properness and non-properness distinguish between the cases if the descriptor system is exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design has to be considered. Both cases will be solved completely.

Linear control design of linear descriptor systems

In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential algebraic equations is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear descriptor systems and its control design with respect to a quadratic performance criterion and based on a generalized Lu-enberger observer. Here, properness plays an important role whether usual design algorithms can be applied or not. The distinction of proper and non-proper systems is necessary to distinguish descriptor systems which are exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design and observer realization has to be considered. Both cases will be solved completely.

The determination of structural properties of a linear multivariable system by operations of system similarity 2. Non-proper systems in generalized state-space form†

Abstract This paper considers structural properties of non-proper linear multivariable systems in Rosenbrock's generalized state-space form. A particular form of the system matrix is shown to yield various relevant structural properties. The relationship with the theory of singular pencils of matrices and with geometric properties is investigated and provides a full explanation of Rosenbrock's definitions of the zeros of a system. New definitions are proposed for the poles, zeros and decoupling zeros at infinity of a non-proper system. The relevance and the properties of the concepts proposed here are investigated in detail and lead to generalizations of the work of Rosenbrock and Vergliese. The redundancy represented in the polynomial part of a system is established and this yields in two different ways a generalization of Kalman's canonical decomposition of the state-space. Finally, the relationship with invertibility properties is investigated.

Asymptotic multivariable root-locus behaviour for non-proper systems

The study of some of the fundamental geometric properties of the Markov parameters of linear, time-invariant and continuous systems has recently led to the extension of the classical root locus technique to the multivariable case. Most of the results derived however are restricted to the case of strictly proper systems. In the present paper an extension of the results concerning the asymptotic behaviour of the root loci to systems with a direct pass from inputs to outputs is given ; the results are derived as a natural generalization of already existing results when the Markov parameter chain is augmented in order to include the feed-forward operator of non-proper systems. A design procedure for the adjustment of the asymptotic behaviour is suggested and a simple numerical example is included in order to illustrate the effectiveness of the presented technique.