A universal tool for solving different control problems Ulle - - PowerPoint PPT Presentation

A universal tool for solving different control problems

¨ Ulle Kotta

kotta@cc.ioc.ee

Institute of Cybernetics at TUT

A universal tool for solving different control problems – p. 1/28

Contents

• Three control problems
• Algebraic formalism
• Solutions and examples

A universal tool for solving different control problems – p. 2/28

Three control problems

• Analysis

Controllability (Accessibility)

• Synthesis

Linearization by static state feedback

• Modelling

Realization in classical state-space form

A universal tool for solving different control problems – p. 3/28

Accessibility

• System
• ✁

is said to be accessible if it does not have any nonzero autonomous element

• Function

is said to be an autonomous element if there exists an integer

and a nonzero meromorphic function

such that

A universal tool for solving different control problems – p. 4/28

Linearization

• ✁

A universal tool for solving different control problems – p. 5/28

State-space realization

• ✝✟✞

• ✝

• ✝

• ✝

NB!!!

A universal tool for solving different control problems – p. 6/28

Extended system

Associate to an extended state-space system

• ✝

• ✝

• ✁

• ✠

• ✓✕✔

A universal tool for solving different control problems – p. 7/28

The forward-shift operator

• Let

denote the field of meromorphic functions in a finite number of variables

• ✆

• The forward-shift operator

is defined by

• ✒

is one-to-one, if the system is submersive.

A universal tool for solving different control problems – p. 8/28

Submersivity

The system is said to be submersive, if

• ✁

• ✁

A universal tool for solving different control problems – p. 9/28

Submersivity - example

• ✁

• ✂

• ☎

• ✁

• ✁

The system is not submersive

A universal tool for solving different control problems – p. 10/28

Submersivity - example

• 1. Consider the following function
• ✝

• ✝

• ✝

2.

• ✆

is not defined

A universal tool for solving different control problems – p. 11/28

Difference field

If

is one-to-one then pair

is a difference field

A universal tool for solving different control problems – p. 12/28

Inversive closure

• It is not guaranteed that each element in

has a pre-image. Example

• ✁

• ✂

• ☎

• ✂

• ✁

But for

and

there are no pre-images in .

A universal tool for solving different control problems – p. 13/28

Construction of the inversive closure

• The idea is to construct a backward shift operator

in

• rder to enlarge domain of state-equation to the negative

time instances

.

• ✁

• ☛

• ☛

• ✌

• ☛

• ☛

A universal tool for solving different control problems – p. 14/28

Construction of the inversive closure

• Define

• ✂

such that

Example - continuation

A universal tool for solving different control problems – p. 15/28

Vector space

• ver
• ☎

• ☛

Forward shift operator

induces in

• a

forward shift operator

• ✄

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Relative degree

The relative degree

• f a one form

• ☛

• ✄

• ☛

If such an integer does not exist set

• ☎

A universal tool for solving different control problems – p. 17/28

• ✁

• is said to be exact if

• ✁

• is said to be closed if

• Exact one-form is always closed.

Closed form is not always exact globally but always exact locally.

• The subspace of differential one-forms is

said to be completely integrable if it has a basis which consists of closed differential

• ne forms only.

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Frobenius theorem

• Let
• ✝

is the subspace of

.

• is

completely integrable iff

• If Frobenius theorem is satisfied then

• ☎

• ✝

such that

• ✁

• ✝

• ☎

• ✝

A universal tool for solving different control problems – p. 19/28

The sequence of subspaces

• ✁

• ☛

• ✁

• ✁

• ✠

contains only one-forms of relative degree equal or greater than

• .

• ✁

• ✂

A universal tool for solving different control problems – p. 20/28

Solutions

• The system

is accessible iff

• ✕

.

• The system

is linearizable by static state feedback iff

• ☛

are completely integrable.

• The system

admits classical state-space realization iff

• ✁

are completely integrable.

A universal tool for solving different control problems – p. 21/28

Accessibility - example

• ✁

• ✂

• ☎

• ✆

• ☛

The system is not accessible.

A universal tool for solving different control problems – p. 22/28

Linearization - example

• ✝

• ✝

• ✝

A universal tool for solving different control problems – p. 23/28

Linearization - example

does not depend on

• ✁

• ✁

• ☎

A universal tool for solving different control problems – p. 24/28

Linearization - example

Define

• ✁

• ✂

• ☎

• ☛

• ✂

• ✄

A universal tool for solving different control problems – p. 25/28

Realization - example

• ✝✟✞

• ✝✟✞

• ✝

• ☛
• ✝✟✞

• ✝

• ✝✟✞

is not integrable the system is not realizable.

A universal tool for solving different control problems – p. 26/28

Realization - example

• ✝✟✞

• ✝✟✞

• ✝

• ☛
• ✝

• ✝

• ✝

is integrable The system is realizable.

• ✁

• ✝✟✞

A universal tool for solving different control problems – p. 27/28

Thank you!!!

A universal tool for solving different control problems – p. 28/28