Compensation of disturbance for state-delayed plants

Compensation of disturbance for state-delayed plants

Grigoriev A.A.

Astrakhan State Technical University

Chair of Technical Processes Automation

Tatisheva str.16, Astrakhan 414025, RUSSIA.

Abstract

In this paper I have extended the known results, represented in paper [1]. The target is to stabilize dynamic linear delayed plant. The system is undergoing the external non-measurable disturbance. Control signal is not delayed. State vector is fully measurable.  Observer of an external disturbance has been constructed, using the known results, obtained earlier.

1.  Introduction

It’s not accidental, that systems with time-delay are paid a great attention. Practically, many technological or other processes are described by differential equations with delay. The research of control systems for such plants represents a great practical interest.

This paper presents a extension of the results, obtained in the paper [1]. The method of observers for external determinant disturbances construction has been considered here. The method of inside model has been used - external disturbance was considering as output of some linear system (disturbance generator).

The problem of linear dynamic delayed plant stabilization through compensation of external non-measurable disturbance has been regarded in this paper.

2. Statement of the problem

We shall touch upon a class of stationary control plants with delay, which can be described by finite dimension differential equations. Those systems can be described by state-space equations:

,                                                                                      (1)

y=L^Tx,                                                                                                                         (2)

where vector  is fully accessible; output signal y, input control signal u  and external non-measurable disturbance are scalar values,  A, D are  n x n matrixes, b, d, L are vectors with proper dimensions.

Properties of plant and disturbance will be defined by following hypotheses:

1) Vector b is equal to vector d

2) Matrix D can be parameterized as , where сis some vector of constant coefficients.

3) Disturbance  can be represented as output of finite-dimension linear generator [1].

                                                                                                                        (3)

               ,                                                                                                                    (4)

where  is a state vector with initial state , Г is q х q matrix of constant coefficients, h is constant vector of proper dimension. Twain (h^T, Г) is fully observable.

In this context, disturbances representing a solution of linear differential equation are considered. The greatest practical interest is presented by limited periodical disturbances (roots characteristic equation is located on imaginary axis).

We shall set the problem of recollecting the external disturbance’s parameters (parameters of disturbance generator (3),(4)), based on measurements of state vector x and stabilization of a system via compensation of this disturbance. Target condition is following: .

3. Solution method

3.1 Canonical form of observer

|Canonical form of external disturbance observer has been defined in the paper [1]. This form plays an important role in forming of filter, which is necessary for defining disturbance generator parameters.

                                                                                                                (5)

                                                                                                                      (6)

where G is any given q x q stable matrix, and pair (G, l)  is fully controllable,  is constant vector, at the same time, state vector   is linked with vector of disturbance generator (3), (4)  by similarity . The existence such form is proved in the paper [1].  Thus, the indeterminateness of external disturbance consists in the indeterminateness of constant coefficients vector. It’s important to underline, that this observer can’t be physically realized, as it includes unknown input variable. So we have to solve problem of recollecting of vector  via values of vectors x or y.

An estimation of vector  can be defined as following form [1].

,                                                                                                                  (7)

where auxiliary vector  is generated by a dynamic filter:

,                                                                (8)

and matrix  N has to be chosen from matrix equation

               .                                                                                                                        (9)

Properties of observer (7,8) are defined by following theorem.

Theorem. (This theorem is an analog of theorem, proved in work [1])

Let hypothesis 3 is executed, then disturbance, affecting on the plant (1) can be considered as

               ,                                                                                                                             (10)