Linear Time-invariant Multivariable Systems Research Articles

A PARAMETRIC SOLUTION TO THE MODAL CONTROLLERS DESIGN PROBLEM

The main purpose of this paper is to present a method for determining a parametric solution to the problem of modal-control of controllable and observable linear time-invariant multivariable systems via output-feedback. The approach followed is based, at first, on finding a general set of state-feedback modal controllers, then a subset of it is chosen such that it satisfies the necessary and sufficient conditions for the existance of an output-feedback modal-controller . This approach makes use of the controllable canonical form of the given system. A de$ign algorithm is established and it can be implemented with ease. Then, this algorithm is extended to the case of designing modal dynamic controllers with arbit-rarily fixed poles. Two numerical examples are provided to illustrate the theoritical aspects. The paper consists of two main parts: - the first part describes the design of static modal controllers; - the second part is related to the design of dynamic modal controllers.

Achieving robust stability by sensing additional system outputs

In this note we investigate the problem of robust stability in the context of linear time-invariant multivariable systems which are affected by structured parametric type uncertainty. We demonstrate that by generating additional outputs which are then made available for feedback, and thus using sensors for more measurements, robust closed-loop system stability can be achieved. We state and prove results that suggest when this is possible, and show how it can be done.

Control of a servomechanism problem for linear time-invariant multivariable systems with constraints on transition states

The control of a servomechanism problem, which is an extension of the results of Kaczorek (1976) and Porter and Bradshaw (1974) is considered. Also, the problem of tracking quality is presented.

A note on robust stabilization for systems with parameters

We consider the problem of stabilizing a linear time invariant multivariable system whose transfer function admits a factorization which contains parametric type uncertainties. The parameters are known to take values in some region Ω a . We first give a sufficient condition for robust stability when a proper feedback compensator is used. Here it is important to note that no assumption is made about the number of unstable plant poles. Following this we focus attention on how to compute an appropriate robustly stabilizing compensator. This is done in a two step procedure. Initially the solutions of a Diophantine equation generate a class of proper compensators, each of which stabilizes the ‘nominal’ closed loop system. This is followed by a Nyquist Theorem inspired minimization which yields the final choice. An illustrative physical example is given. The compensator thus constructed guarantees stability over Ω u . A discussion about how to compute the size of the attainable robust stability region Ω u is also included.

General solution of robust tracking problem in two-degree-of-freedom control systems

In this note, we study linear time-invariant multivariable systems with two-degree-of-freedom compensators. First, we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property. Second, utilizing this parametrization, we solve a general robust tracking problem including the case of nonsquare plants, and show that "tracking" and "robustness" can be independently specified in two-degree-of-freedom control systems.

Synthesis of robust tracking systems with specified transfer matrices

In this paper we study linear time-invariant multivariable systems in which the compensators consist of both feedforward and feedback parts. Firstly, a necessary and sufficient condition for the system to be an internally stable robust tracking system is obtained. Secondly, the class of attainable transfer matrices from reference signal to plant output of internally stable robust tracking systems is clarified. Thirdly, a simple method is proposed for specifying two feedback properties, i.e., sensitivity and stability margin. Finally, based on these results, we propose a synthesis procedure of robust tracking systems with desirable transient responses, sensitivity and stability margin

On the similarity transformation in the design of an adaptive observer for multivariable systems

The matrix inversion, that was required in the design of an adaptive observer pre sented by the authors in a previous paper (Hori et al. 1984 a) for a general class of linear time-invariant multivariable systems is eliminated by looking at the problem from a different point of view. In addition, a more simple proof of the transformation matrix than that mentioned in the previous paper is provided.

On the destabilizing effect of cross-coupling in decentralized adaptive control

The stability of square multivariable linear time-invariant (LTI) systems coupled with an adaptive controller for which the process cross-coupling terms have been neglected is studied in this paper. Our aim is to find conditions over the reference signals richness and the process transfer matrix frequency response such that, for sufficiently small adaptation rates, the stability of the adaptive system is insured. It is shown that if the adaptive system without interconnections is stable then stability of the overall system is preserved if either: (1) The frequency bands of the reference signals are non-overlapping. (2) When there is a spectrum overlap each output of the LTI system to input signals in this spectrum has an average energy where the contribution from its corresponding input is larger than the one due to the other inputs. Global results are obtained for a simple single-parameter adaptation scheme. The extension to general adaptive controllers is straightforward in a local context [5].

A DESIGN PROCEDURE FOR PI OUTPUT-FEEDBACK MODAL CONTROLLERS

In this paper the problem of pole-assignment in linear time invariant multi-variable systems is generalized to include the case of systems incorporating integral control action. In some control system design poroblems, integral feedback is introduced in conjunction with proportional one in order to ele-minate steady-state errors arising from sustained disturbances. Most published work deals with the introduction of integral as well as pro-portional feedback of some or all of the state variables of the system under control. This paper establishes a new design procedure of proportional plus-integral controllers (PI-controllers) based on the feedback of system output variables. The controllability of the compensated system is investigated and a necessary and sufficient condition for the augmented system to be controllable is deri-ved. The existance problem of such type of modal-controllers is discussed and a necessary condition for complete pole-assignment of the closed-loop system is derived. A pratical design algroithm of this type of controllers is developed, and it is used to design a PI-controller for a VSTOL aircraft.

Gain margin and time lag tolerance constraints applied to the stabilization problem and robust servomechanism problem

Definitions of gain margin tolerance and time lag tolerance for a multivariable linear time-invariant system are given which correspond to a generalization of the notions of gain and phase margin of classical control system theory. Existence conditions are then given to solve the stabilization problem and the robust servomechanism problem with gain margin and time lag tolerance constraints. In particular, the following is shown for a system described in terms of a state-state model: 1) Generically, there exists a controller to stabilize an unstable m -input, r -output system so that the resultant closed-loop system possesses a nonzero gain margin, if and only if min (r, m) = 1 . 2) For stable plants, there always exists a controller to solve the robust servomechanism problem so that it has an arbitrary specified gain margin/time lag tolerance constraint, if and only if there exists a solution to the robust servomechanism problem for the plant. A controller synthesis method is then given which solves the robust servomechanism problem so that it has a specified gain margin/time lag tolerance. Various examples of the proposed algorithm are given to illustrate the type of results that may be obtained.

Multivariable state-feedback self-tuning controllers∗

This paper describes a state-space approach for self-tuning control of a class of multivariable stochastic systems having the same number of inputs as outputs. A multivariable state-feedback self-tuning controller, based on pole-assignment concepts, is derived. The developed multivariable self-tuner can be applied to stable/unstable and minimum/non-minimum phase linear time-invariant multivariable systems. A multivariable reduced-order self-tuner and a state-feedback minimum-variance self-tuner are also derived. The simplicity and flexibility of the proposed state-space approach facilitate the practical applications of self-tuning control concepts to real systems

On the development of equiobservable forms for linear multivariable systems

Three types of equiobservable forms are developed here for linear time-invariant multivariable systems. These forms are developed using a nonminimal realization of a system and have properties that are useful in the analysis and design of state observation and parameter identification problems. By employing the third form, the state equation can be expressed by using filtered values generated from inputs and outputs in an algebraic form.

On Synthesis of Robust Servo Systems

In this paper we study linear time-invariant multi variable systems including both feedforward and feedback compensators. Firstly, a condition for the system to be an internally stable robust tracking system is obtained. Secondly, the class of attainable transfer matrices from reference signal to plant output of internally stable robust tracking systems is clarified. Thirdly, measures of the sensitvity and the stability margin are given, and a simple method is proposed for setting these quantities to desirable ones Finally, based on these results, we propose a synthesis procedure of robust tracking systems with desirable transient responses, sensitivity and stability margin.

Equivalent deterministic inputs for random processes

Given a linear time-invariant multivariable system with a prescribed set of stationary random inputs, we wish to determine the correlations among the outputs. A theorem is presented that permits this determination to be made in the time domain by integrating the response to certain deterministic transient inputs derived from the spectrum matrix of the prescribed random inputs. The method is especially useful when a mathematical model of the system is not available. It is tested for a two-input system and is shown to give the same numerical results as integrating the response spectra.

On the Development of Equiobservable forms with Application to Adaptive Control

Three types of equiobservable forms are presented for linear time-invariant multivariable systems. These forms are developed using a nonminimal realization of a system and have properties that are useful in the analyis and design of state observation and parameter identific ation problems. By employing the third form, the state equation can be expressed by using filtered values generated from inputs and outputs in an algebraic form, and a new state observer can be obtained. To illustrate tbe application of tbe third form, a multivariable adaptive observer is also discussed. This observer is used to design an two-level adaptive Hight controller for the pitch attitude control system of an aircraft. This approach appears to overcome some of the limitations that are ioberent in the design of tbe Iinear optimal control and conventional adaptive control methods. In particular, the feedback gains of the optimal controller are updated using the estimation provided by the adaptive observer. Tbe firstrlevel of control consists of updated feedforward and optimal controllers. The second-level control is an error servo which provides fine compensation. Extensive simulation studies were carried out and some results are provided for the pitch attitude control of a STOL aircraft under two different flight conditions.

A discrete adaptive observer A general linear multivariable case

A method of designing a discrete adaptive observer is proposed for a general class of linear time-invariant multivariable systems. This method is an extension of the procedure described by Kanai and Degawa (1979) for single-input single-output systems. A parameter updating algorithm is derived based upon a recursive least squares method and the number of iterations required for parameter convergence is examined.

Pole assignment of two-dimensional systems for separable characteristic equations

Abstract This paper considers the problem of pole assignment of linear time-invariant multivariable two-dimensional systems. The desirable closed-loop characteristic function is considered to be separable. An algorithm is presented for the solution of the problem using state feedback. The necessary condition for the problem to have a solution is established and an illustrative example is given.

Eigenvalue-eigenvector sensitivity analysis of linear time-invariant singular systems

This note refers to the eigenvalue-eigenvector sensitivity analysis of linear time-invariant multivariable singular systems. Procedures that can be easily programmed on a digital computer are presented for the determination of the deviation of the eigenvalues and the corresponding right and left eigenvectors which are caused by the system's parameter changes about their nominal values. The method is illustrated by two examples.

Organization of the non-uniqueness of a canonical structure of linear multivariable systems

This paper is concerned with organizing the non-uniqueness of the canonical structure of a class of linear time-invariant multivariable systems in such a way as to provide an efficient tool in control system design. A phase-variable block form of state equations is used to describe the system class in the transformed coordinates. The transformation procedure is characterized by an arbitrary parameter (in the form of a submatrix) which adds an attractive degree of freedom in the solution of practical problems. As a demonstration, the procedure is applied to synthesize a minimum-time deadbeat controller for linear discrete-time systems. A family of controllers, not identifiable by earlier approaches, is developed and that controller which additionally minimizes a control energy criterion is determined. A numerical example is provided to illustrate the different aspects of this work.

The problem of guaranteeing robust disturbance rejection in linear multivariable feedback systems

Abstract This paper considers a linear time invariant lumped multivariable feedback system consisting of a plant, which is subjected to perturbations, a precompensator and a feedback compensator. Given that, the nominal feedback system is exponentially stable and achieves considerable disturbance rejection, this paper establishes necessary and sufficient conditions on the nominal feedback system so that, over the prescribed class of additive/multiplicative perturbations, all perturbed feedback systems preserve stability and considerable disturbance rejection. The generalization to distributed systems and the extension to discrete systems are indicated.