Forming special observer structures

forming special observer structures

Mikhail А. Gorkavy

Department of Control and Informatics in technical systems

Komsomolsk-on-Amur State Technical University,

Lenina av. 27, Komsomolsk-on-Amur, 681013, RUSSIA

Tel: +8(4217)531327. E-mail: mix_komsa@mail.ru

Abstract

This paper discusses extra possibilities of observing devices, operating in structure of closed-loop determined systems with a modal regulator. Usually observers are considered to be devices for estimating immeasurable state variables. In certain situations they can give invariant properties  to system with modal regulator when external uncontrolled influence is applied. The equations given below explains the original properties

An example of control system synthesis and research for the 4-order state object gives practical explanation about system invariant properties. These invariant properties appear due to observer structure.

This work doesn’t give common approach for forming special observer structures. It only shows the possibility of determining such structures for concrete projecting systems.

1. introduction

Control systems with modal regulators [1, 2, 3] possess necessary properties for input influence. But the reaction of such systems on external uncontrolled influences almost isn’t satisfactory. These reactions are usually characterized by considerable dynamic and static errors. Using observers for estimating immeasurable state variables results in system reaction more unpredictable. It is caused by existence estimate errors of state variables.

In one case an additional integrator to remove unsatisfied static errors is put into a system. I.e. system with modal PI-regulator [2] is created. In another case an invariant canal for estimated external uncontrolled influence is formed. Estimation is carried out by expended observer [1]. Above mentioned actions have some advantages and disadvantages in certain situations. But in any case additional integrators appear in the system, it increases the closed-loop system order.

2. the main results

Let’s consider a control system model with modal regulator. Designations are used below accepted in [3]. Equation of object may be written as:

                                    (1)

                                                                              (2)

Where  - state vector;  - control signal vector;  - vector of external unknown influences,  - output vector.  - matrixes of object, input, output, external influence.

The equation of regulator may be written as:

                                                                (3)

                                                                          (4)

Where  - input vector;  - output regulator vector,  - coefficient regulator matrix. Without input influence  there is  - modal control law.

Equation of closed-loop system may be written as:

                          (5)

                                                                                   (6)

(5) shows, that the influence isn’t controlled.

Let as assume in object (1)  measurable state-variables, that make vector , and () immeasurable state-variables, that form vector . Then state vector is .

According to Luenberger’s method of projecting reduce-order observers [3], the object can be divided into two sub objects , :

                              (7)

.                               (8)

Then the observer equation may be written as:

             (9)

where  - an  observer’s matrix, that is chosen according to equality characteristic  observer equation with standard characteristic form [3]. If , in matrix  there will be some coefficient superfluity for setting observer at necessary quickness. Due to arbitrary choice of matrix  coefficients it is possible to form different observer structures to achieve determine aims. For example for forming estimate errors to compensate external uncontrolled influence on the system.

If we subtract equation (9) from equation (8), we’ll receive equation for immeasurable state-variable estimate errors from external uncontrolled influence:

                                (10)

where                                                                (11)

or in operator for:

         (12)