Algorithm of adaptive control, based on the modified parametrization of the plant’s equation

ALGORITHM OF ADAPTIVE CONTROL, BASED ON THE MODIFIED PARAMETRIZATION OF THE PLANT’S EQUATION

Igor B. Furtat

Astrakhan State Technical University

 Tatischeva av. 16, Astrakhan, 414025, RUSSIA

Tel: +7(512) 559248, E-mail: cainenash@mail.ru

Abstract - In the paper was solved the problem of following of an output of control plant for the reference signal in the conditions of priori uncertainty. Only scalar input and scalar output of the plant are accessible to measuring. Solution of problem is founded on the new modified parameterization, which get the possibility to lead the plant to the SPR-function at once. The further  syntheses in founded on the use of the modified algorithm of adaptation of the high order. These parameters in the sum get the possibility to lower the order of the closed-loop system.

1. Introduction

At present moment in a class of the tasks of adaptive systems were received many approaches and methods [1-10] for control of linear plant on an output in the conditions of priory uncertainty. These methods conditionally can be divided into two class: adaptive control with the extended error [1,4] and adaptive control with algorithm adaptation of the high order [2-5, 9]. The concept of the extended error consist of reception of simple scheme an extending signal, what the sum of an tracking error and the generator of extension.

Another approach is the method of algorithm adaptation of the high order. These algorithms also can be divided into two class by the principle of realization: algorithms with an estimation of derivative from an tracking error, for what different observers are used. The order of the closed-loop system in the second case is less than in the first case.

In this paper the algorithm of adaptation of the high order, in which the derivative of an tracking error are estimated, is offered. Thus the algorithm was received on the base of new parameterization, which get the possibility to present parametrical uncertainties of model as additive indignations, linear on unknown parameter and the model without additional transformation, has SPR-function (strict positive realness). It gets the possibility to lower the order of the closed-loop system, because the auxiliary filters are expelled and to simplify the technical realization of adaptive system.

2. Statement of the problem

Let’s consider the plant of control, in which the dynamic processes are described by the equation

.                                              (1)

Here , - differential operators with the constant unknown coefficients, where  and - are monic polynomials;  ; - scalar plant output; - scalar control signal; - disturbance influence; - operator of differential.

The reference model is described by the equation

,                                                 (2)

where ; ; - bounded reference signal.

Thus the standard problem control with the reference model on an output signal with the demand to construct the control system, providing for any entry conditions the boundedness of all signals of the system and also it provides the fulfillment of the target condition is formed

,                                            (3)

where - some, small enough number, which can be diminished.

Assumption. А1. Orders of the polynomials , , , , , where  a complex variable in Laplas transformation; - relative degree of plant. А2. Polynomial  is Hurwitz. А3. ; ; . А4. Reference  input  and its  derivatives, and also disturbance influence is bounded function of time. А5. Functions: , , ,  are accessible to measuring. Using of derivative of these quantities in a control system is not admitted.

3. Synthesis of algorithm of adaptive control

Let’s decompose the operators ,  on  addendums ,  , where ,  are operators with well-known coefficients, so that polynomials  and - Hurwitz and it have orders  and  accordingly, then ,  are operators with unknown coefficients on orders  and  accordingly. Using such parameterization will receive SPR-function  at once without additional transformations, as for example in [1-10], and additionally choose  and  so that . Then, using procedure of  “operator’s division” the equation (1) can be written down as

,                                  (4)

where - bounded functions in the force of Hurwitz of polynomials  and assumption A4. Let’s realize the decay of the operator  on the following components