chapter 2_ state space representation

8/2/2019 Chapter 2_ State Space Representation

1/32

Copyright 2008 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Modern Control Systems, Eleventh EditionRichard C. Dorf and Robert H. Bishop

EEE 5223

Control System Design

Chapter 2:

State Variable Models

Prepared by Chew S.P

01/03/2012


2/32




System block diagram.


3/32




Dynamic system.


4/32




An RLCcircuit.


5/32




RLCnetwork. (a) Signal-flow graph. (b)Block diagram.


6/32




The Isim function for calculating the output andstate response.


7/32




Computing the time response for nonzero initial conditions and zero input usingIsim.


8/32




The Design of State Variable FeedbackSystems

Controllability and Observability Full- State Feedback Control Design

Observer Design


9/32




State Variable Design Controlling system with a control signal u(t),

function of several measurable statevariables.

State variable design: Assume all state variables are measurable

and utilize in full state feedback control law Construct an Observer to estimate states Connect observer to full state feedback

control law ( full state law + observer=compensator)


10/32




Controllability and Observability

State variable compensator employing full-state feedback in

series with a full-state observer.


11/32




Controllability and Observability

In order to achieve a system which is controllable andobservable, placing poles at desired locations to meetperformance specifications

Full-state feedback design relies on pole placement

techniques

observer.


12/32




A system is completely controllable ifthere exists an unconstrained control u(t)

that can transfer any initial state x(to) toany desired location x(t) in a finite time

Matrix A (nxn) & Matrix B (nx1)

If the determinant of Controllability matrixPc is non zero, the system is controllable.


13/32

Copyright 2008 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.


E.g. 11.1 (Page 859)


14/32



Observability

All poles of the closed-loop system can be placedarbitrarily in the complex plane if the system is observableand controllable.

Observability refers to the ability to estimate a state

variable. A system is completely observable if and only if there

exists a finite time T such that the initial state x(0) can bedetermined from the observation history y(t) given thecontrol u(t)

observer.


15/32



Cxy

BuAxx

Observability

If the determinant of observability matrixPo is non zero, the system is observable.

1

:

n

o

CA

CA

C

P


16/32



E.g. 11.4 (Page 862)


17/32



Full-State Feedback Control Design

Full-state variable feedback to achievedesired pole locations of the closed looppoles

Assume all states are available forfeedback. The system input u(t) is givenby

Kxu


18/32



Full-state feedback block diagram

f


19/32



Plant without/ with state variable feedback

ll f db k C l


20/32



Full-state feedback Control Design

Kxu

BuAxx

Control feedback is given by

State variables model is given by

Thus, closed loop system:

xBKABKxAxx )(


21/32



0))(det( BKAI

The closed loop system is stable if all the

roots of the C.E lie in the left half plane The addition of a reference input can be

written as

Characteristics equation is given by

)()()( tNrtKxtu

P l Pl t F Pl t i Ph V i bl


22/32



Pole Placement For Plants in Phase VariableForm

Steps for pole placement methodology: Represent the plant in phase variable form Feed back each phase variable to the input

of the plant through gain ki Find the CE for the closed loop sys.

Represented in Step 2. Decide upon all closed loop pole locations

and determine an equivalent CE Equate like coefficients of the CE from Step

3 & 4 and solve for ki

P l Pl t F Pl t i Ph V i bl F


23/32



Pole Placement For Plants in Phase Variable Form

Wh t i h i bl i l f ?


24/32



What is phase variable canonical form?

Recall Masons rule:

factorsloopfeedbacktheofsum1

factorspath-forwardtheofsum1

)(1

q

N

q

kk

L

PsG

A k F l


25/32



Ackermanns Formula

Another method of determining the state

variable feedback matrix nkkkK ...21

o

n

n

nq

...)( 11

)(10...00 1 AqPK c

Desired characteristic equation

State feedback gain equation


26/32



Refer to E.g. 11.7 (Page 868)

R f t E 12 1 (N P 640)


27/32



Refer to E.g. 12.1 (Norman Page 640)

Given the plant as below, design the phase

variable feedback gains to yield 9.5%overshoot and settling time of 0.74

)4)(1(

)5(20)(

sss

ssG


28/32



Step 1: Phase variable representation


29/32




Step 2: Find desired CE Step 3: Find the compensated system CE

Step 4: Compare to obtain feedback gainmatrix

Observer Design


30/32




Observer Design

Needs of sensors to measure directly all

states- not practical and costy If the system is completely observable with

a given sets of outputs, then it is possible

to determine/ estimate the states that arenot directly measures.

The Full State Observer


31/32




The Full-State Observer

)(

xCyLBuxAx

Cxy

BuAxx


32/32

Copyright 2008 by Pearson Education, Inc.Upper Saddle River New Jersey 07458Modern Control Systems, Eleventh Edition

Refer to E.g. 11.8 (Page 870)

chapter 2_ state space representation

Documents