A Review on Controllability of Dynamical Systems

Presented to : Academic Weekly Sessions of Bagher Menhaj Students

2010, Nov 22

Presented by:S. M. Dibaji, IEEE MemberB. Sc. Student of of Applied MathB. Sc. in Electrical Engineering(Control)M.Sc. Student of Electrical Engineering(Control)Supervisors : Prof. Amir Abolfazl SuratgarProf. Mohammad Bagher Menhaj

What is Controllability?

• Controllability is one of the system’s characteristics .

• It expresses whether the system dynamic has the ability to steer from one state to another state by settling the admissible controls!?

• In other words, Is it possible to do a transfer the system states from assumed initial state?

The Most Philosophical Question From the System

Is it possible?Maybe you can’t believe but controllability analysis asks this essential question from the system. If the system is controllable it is possible to expect it to do our desired behaviour. When we say it is locally controllable, in fact, it can do special jobs by available oppurtunites.

Controllability problems has inherent complexity

• Since the controllability question is about the existence of admissible control(input) to be able to steer the state of the systems to desired region, It can be the most difficult problem in the world of mathematics rather than world of control. ( There are some overlaps between these two wide worlds.)

• The question of controllability like stability refers to mathematical nature of system dynamic

دانشکده مهندسی برق دانشگاه صنعتی امیرکبیر

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Controllability Concept

Add uAdd u

Add u

Add u

System

x2

x3

x1

Start

Goal

x0

xf

کنترل پذیری سیستم های تصادفی

A More Realistic ExampleConsider a box with several balls in it. A baby

roles in a box as a control (input), and balls role as states. Everey new situation of balls positions is affected only baby is obviously is hands and legs( controls).

We say the box and balls System is controllable if we Can always fine an admissibleSequence of baby movements to new situation

of balls(states) in the box.

Rudolf Emil KalmanControllability Idea:Rudolf Emil Kalman

”On the general theory of control systems”

First International Congress on Automatic Control, Moscow 1960

“This paper initiates study of pure theory of control “

Theoratical Framework

Linear Systems

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Linear Time Invariant Systemsx Ax Buy Cx Du

A n nB n pc r nD r p

1( ) [ ; ;...; ]( ) ( 1)

kP k B AB A BP k P k

( )P n n

C.T.Chen and Niedereslki 1966

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Rosenbrock Matrix

• Rosenbrock Matrix:

(A,B) is controllable iff R has full row rank

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PBH Criterion

• PBH( Popov-Belevitch-Hautus) Citerion:

– (A,B) is controllable iff P has full row rank for all s.

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Controllability of LTI Properties

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• (A,B) is controllable iff (A+BK,B) is controllable• (A,B) is controllable iff the equivalent system

is controllable :• Duality with Observability• Check with Matlab:

Ctrb(A,B) is the controllability matrix of A, B

Problem with Rank

• All the criterion needs to check the rank of matrix,

• R.D. Lemma:• The matrix M has full row rank iff MM’ is

nonsingular.

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Linear Time Varying

• The Controllability depends on the nonsingularity of the below matrix:

If D=0, P is nonsingular iff (A,B) is controllable in (t0,tf)If D is not zero, the nonsingularity of below matrix is necessary and sufficient:

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CSLHS

• What is CSLHS?• It stands for Controlled Switching Linear

Hybrid System

Yang Controllability Conditions for CSHLS (2002)

If the above controllability matrix has full row rank, the CSLHS is controllable.

Theorem 1: Sufficiency

References[1]Zhenyu Yang “An Algebraic Approach Towards the Controllability of Controlled

Switching Linear Hybrid Systems” Automatica 38 (7), pp 1221-1228,2002 [2] Kazuo Tanaka, Hua O. Wang, Fuzzy Control Systems Design and Analysis: A Linear

Matrix Inequality Approach, John Wiley & Sons, Inc. 2001.

Your Ideas, Questions , and Guidance

Thanks for your warmly attention