MODERN CONTROL SYS-LECTURE VII.pdf

MODERN CONTROL SYSTEMS ENGINEERING

COURSE #: CS421

INSTRUCTOR: DR. RICHARD H. MGAYA

Date: October 25th, 2013

Optimal Control System Design

Consider a state differential equation:

The system can be represented in a general form:

(i)

where

x(t) states of the system

u(t) control effort

Optimal control:

Seeks to maximize the return from a system at minimum cost

Finding the control u which causes the system (i) to follow an optimal trajectory x(t) that minimizes the performance criterion or cost function

(ii)

Dr. Richard H. Mgaya

)()( tButAxx

ttutxgx ),(),(

1

0

),(),(

t

t

ttutxhJ


Types of Optimal Control Problems

Terminal control problems: Deals with bringing the system as close as possible to a given terminal state within a given period of time

Example an automatic aircraft landing system

Optimal policy-Minimizing errors in state vector at the point of landing

Minimum-time control problem: Reaching the terminal state in shortest time possible

Example Bang-bang control policy

Control is set to umax initially, switching to umin at some specified time

Minimum energy control problem: Transferring of the system from initial state to final state with minimum expenditure of control effort

Example Satellite control



Types of Optimal Control Problems

Regulator control problems: System initially displaced from the equilibrium, will return to equilibrium state in such a manner so as

to minimize a given performance index

Example an autopilot for aircraft, shipping vessel or yacht

Optimal policy-Minimizing errors between actual course and desired course

Tracking control problem: Cause the state of the system to track as close as possible some desired state time history in such a manner so

as to minimize a given performance index

Example missile tracking a target



Selection of Performance Index

Decision on the type of performance index depends on the nature of the control problem

Example: Autopilot system - yacht

Goal Maintain course

Minimize the error e , between desired course d , and the actual course a , with minimum transient period

Source of disturbance wind, waves and currents

Requirements

Minimize distance off-track, ye(t) wandering off track increases the distance

Minimize course or heading error e

Minimize radar activity a minimize control energy Minimize forward speed loss ue(t) yaw movement increases angle of

attack resulting to increased drag and forward speed loss



Selection of Performance Index

General performance index:

Control variable:

x1 = ye(t) , x2 = e , x3 = ue(t) , and u = a

Performance index equation

If the state and control variables are square then the performance index becomes quadratic performance index


1

0

)(),(),(),(

t

t

aeee dtttuttyhJ

1

0

1333222111

t

t

dturxqxqxqJ

1

0

t

t

dtRuQxJ


Quadratic Performance Index

A linear system with quadratic performance index has a solution that yield a linear control law

Therefore,

General form

Q and R are control and weight matrix respectively, J is a scalar quantity


1

0

321

2

1

2

33

2

22

2

11

t

t

dturxqxqxqJ

)()( tKxtu

1

0

1

33

2

1

33

22

11

321

00

00

00t

t

dturu

x

x

x

q

q

q

xxxJ

1

0

t

t

TT dtRuuQxxJ


Quadratic Performance Index

When is J finite?

Can converge to a limit or diverge to infinity

Note:

If (A, B) is stabilizable, then the uncontrollable eigen values are of A have negative real part

If (A, C) is detectable, then the unobservable eigen values are of A have negative real part

Suppose (A, B) is stabilizable and (A, M) is detectable where Q = MTM and u(t) = -Kx(t), then the cost function J is finite for every x(0) Rn if and only

if Re[(A + BK)] < 0

Note:


0)]([ ARe


Linear Quadratic Regulator, LQR

Provides optimal control law for a linear system with quadratic performance index

Continuous Form

Functional equation:

from eqn. i and ii a Hamilton-Jacobi can be expressed as follows


1

0

),(min),(

t

t

u dtuxhtxf

0),(

))0((),(

1

0

txf

xftxf

),(),(min uxg

x

fuxh

t

fT

u



Continuous Form

Substitution

Introduce the relationship of the form:

where P is square and symmetric, Riccati matrix


1

0

)(:),(

:),(

t

t

TT dtRuuQxxJuxh

BuAxxuxg

Pxxtxf T),(

)(min BuAx

x

fRuuQxx

t

fT

TT

u



Continuous Form

Substitution:

(iii)

In order to minimize u then


Pxx

f

Pxx

f

Pxt

xt

f

T

T

T

2

2

)(2min BuAxPxRuuQxxxt

Px TTTu

T

022

PBxRuu

tf

TT



Optimal control law:

(iv)

where K = -R-1BTP

Substitution: eqn. (iv) to eqn. (iii)

But

then

(v)

Equation (v) belong to a class known as matrix Riccati equations


PxBRu Topt1

xPBPBRPAQxxPx TTT )2( 1

xPAPAxPAxx TTT )(2

Kxuopt

PBPBRQPAPAP TT 1



Discrete Form

Discrete solution to state equation:

or

Discrete quadratic performance index:

Solution of matrix Riccati is solved recursively for P and K


)()()()()1( kTuTBkTxTATkx

1

0

))()()()((N

k

TT TkRukukQxkxJ

)()()()()1( kuTBkxTAkx

)()()()()()()1( 1 TAkNPTBTBkNPTBTRkNK TT



Recursive equation for P:

Boundary condition:

Optimal regulator:


BuAxx x

r

-

+ uC y

K

)1(()()()()1(()()(

))1(())1(()1(

kNKTBTAkNPkNKTBTA

kNTRKkNKTQkNP

T

T

0)( NP



Example: The plant and performance index are given as follows:

Determine Riccati matrix P, state feedback matrix K and the

closed-loop eigenvalues

Soln:


xy

ux

x

x

x

01

1

0

21

10

2

1

2

1

dtuxxJ T

0

2

21

10

PBPBRQPAPAP TT 1




222122

121112

2221

1211

2

2

21

10

ppp

ppp

pp

ppPA

2

222122

22122112

2221

22

12

2221

1211

2221

12111

1011

0

ppp

pppp

ppp

p

pp

pp

pp

ppPBPBR T

22122111

2221

2221

1211

2221

10

pppp

pp

pp

ppPAT



Four Simultaneous Equations:

P is symmetric, i.e., p21 = p12


010

02

222

2

2

222122

22122112

22122111

2221

222122

121112

ppp

pppp

pppp

pp

ppp

ppp

0122

02

02

02

2

2222122212

2212121122

2212221211

2

121212

ppppp

ppppp

ppppp

ppp



Solving simultaneous equations:

P is symmetric, i.e., p21 = p12

p12 = p21 = 0.732

From the last simultaneous eqn.:

p22 = 0.542

From second simultaneous eqn:


022 122

12 pp

732.2 732.0 2112 andpp

0464.24 222

22 pp

0 142 2222212 ppp

542.4 542.022 andp

403.2

0)542.0732.0(542.0)732.02(

11

11

p

p



Ricatti matrix P:

Feedback Matrix K

Closed-loop eigen values:


542.0732.0

732.0403.2

2221

1211

pp

ppPA

542.0732.0

732.0403.21011 PBRK T

542.0732.0K

0542.0732.01

0

21

10

0

0

s

s

341.0271.1,0732.1542.2,0542.2732.1

1

0542.0732.0

00

21

1

2,1

2 jssss

s

s

s

0 BKAsI



Generally, for 2nd order systems if Q is diagonal and R is scalar then the elements for Riccati matrix are given as:


r

qpaa

b

rp

r

bqaa

b

rpp

apapppr

bp

22122

22222

2

22

2

2112

21212

2

2112

221221222212

2

211

2



Tracking Problem:

Control effort to drive the plant state vector x(t) to follow a desired trajectory r(t) in optimal manner

Continuous Form

Performance index to be minimized:

State tracking equation:

(vi)

Boundary conditions for tracking vector s:


1

0

)()(

t

t

TT dtRuuxrQxrJ

QrsPBBRAs TT )( 1

0)( 1 ts



Tracking Problem:

Optimal control law:

Let

(vii)

and

Then

If r(t) is known then the system can be designed to follow the command v(t) from eqn. (vi) and (vii)


sBRv T1

sBRPxBRu TTopt11

PBRK T1

Kxvuopt



Optimal Controller for Tracking System

Discrete Form

Discrete quadratic performance index


BuAxx xr

-

+ optuC

y

K

QrsPBBRAs TT )( 1TBR 1

s v

Tracking vector Command vector

1

0

)()()()()()(N

k

TT TkRukukxkrQkxkrJ



Discrete Form

Discrete state tracking equation:

where F(T) and G(T) are control and transition matrices

Boundary condition:

Command vector v:

Optimal control at time (kT):

Thus,


)()()()()1( kNrTGkNsTFkNs

0)( Ns

)(1 kNsBRkNv T

)()()()( kTxkTKkTvkTuopt

)()()()()1( kTuTBkTxTATkx opt

Robust Control System Design

Deals with finding the control law which maintains system response and error signals within prescribed tolerances despite

the effect of uncertainties on the system

Form of uncertainties:

Disturbance effect on the plant

Measurement noise

Modeling error due to nonlinearity

Model uncertainties due to time varying parameter

Generally, physical system are nonlinear

Linearized for small perturbation about an operating point

Controller design for that operating point



If we consider a family of operating points the uncertainty in the nominal model is taken into account by considering all

possible variations in the family of models

Controller is said to have robust stability if it can stabilize all plants within the family

Controller is said to have robust performance if all the plants within the family meet a given performance specification

Trade-off between conflicting requirements is also taken into account



Consider a feedback control system bellow

N(s) and D(s) are disturbance and measurement noise

(viii)

Substituting U(s) and B(s) in Y(s)

(ix)


)]()()[()(

)()()(

)()()()(

sBsRsCsU

sNsYsB

sDsUsGsY

)()(1

)()()(

)()(1

)(

)()(1

)()()()(

sCsG

sNsCsG

sCsG

sD

sCsG

sRsCsGsY


Introduce sensitivity function S(s) relates Y(s) and D(s) when R(s) = N(s) = 0

Complementary sensitivity function:

If N = 0 eqn. (viii) becomes:

If T(s) = 1 and S(s) = 0, then perfect set point for tracking and disturbance rejection

If N(s) 0 then

Therefore,


)()(1

1)(

)(

)(

sCsGsS

sD

sY

)()()()()( sDsSsRsTsY

)()(1

)()()(1)(

sCsG

sCsGsSsT

)()()()()()()( sNsTsDsSsRsTsY

rejectionNoisejTTrackingjT ,0)( , ,1)(


Model Uncertainties

Unstructured model uncertainties relates to the unmodelled effects such as plant disturbance and are related to the nominal

plant as either additive or multiplicative

Additive uncertainties, (a):

(x)

Multiplicative uncertainties, (b):

(xi)


)()()( slsGsG an

)( )(1)( sGslsG nm

)(sla

)(sGn+

+

)(slm

)(sGn+

+

)(b)(a


Model Uncertainties

Structured model uncertainties relates to parametric variation in the plant dynamic

Uncertain variation in coefficients of the differential equation

Normalized System Input:

All inputs to the control loop are normalized

Inputs , i.e., changes in set-point or disturbance, are represented by V(s)

where V1(s) and W(s) are bounded input and input transfer function, .i.e.,

the input weight, respectively


)()()()( 1 sWsWsVsV

ssWsV

sWsV

1)(1)( :Step

1)(1)(:Impulse

1

1


Normalized System Input:

Set of bounded input : 2-norm of the input signals

Transformation to frequency domain Parsevals Theorem


1)(

)(

2

1)(

22

2

1

d

jW

jVtv

0

212

2

1 1)()( dttvtv


Linear Quadratic H2 Optimal Control

Consider the performance index of the linear quadratic tracking problem

In scalar for with u not constrained = Integral Square Error

H2 - Optimal control problem:

Find a controller c(t) such that the 2-norm of ISE is minimized for a specific input

Transformation using Parsevals Theorem:


1

0

)(2t

t

dtteISE

1

0

)()(

t

t

TT dtRuuxrQxrJ

djEte cc22

2)(

2

1min)(min


Linear Quadratic H2 Optimal Control

H2 - Optimal control problem:

From set of eqn. (viii)

If B(s) and Y(s) are substituted and U(s) be written as C(s)E(s), then

Since, V(s) = W(s) for specific input

Thus,

Therefore, H2 optimal controller minimizes the average magnitude of sensitivity function S(j) weighted by W(j)


)()()()(

)()()()()(1

1)(

sNsDsRsS

sNsDsRsCsG

sE

djWjSte cc22

2)()(

2

1min)(min


Linear Quadratic H Optimal Control

Assumption: The input V(j) belong to a set of bounded function with weight W(j)

Each input V(j) in a set will result in a corresponding error, E(j)

H - Optimal control problem:

Minimize the worst error that can arise from any input in the set, i.e., the final result will have the least upper bound

Therefore, H optimal controller minimizes the maximum magnitude of weighted sensitivity function S(j) over a

frequency range . Dr. Richard H. Mgaya

)()(supmin)(min jWjSte cc

1)(

)(

2

1)(

22

2

1

d

jW

jVtv


Robust Stability

Let Gm(j) be a nominal model belong to a family of plants

If G(j) is modelled exactly, i.e., disturbance D(s) and noise N(s) are zero, G(j) = Gm(j)

If la() is the bound of additive uncertainties, then G(j) belonging in the

family is given as

Multiplicative uncertainties bound: from eqn. (x) and (xi)

Note: G(j)C(j) is the open-loop forward transfer function


)()(

)()(

jCjG

ll

m

am

)()()(: am ljGjGG

)()()()( mma ljCjGl


Robust Stability

Closed-loop uncertainty bound:

or

Robust stability:

If all plants G(s) in the family have same number of RHP poles then

controller C(s) stabilizes the nominal plant iff the complementary

sensitivity function satisfy the following condition

lm() is referred as the weighted function for T(j)

Note: Robust stability provides minimum requirements, .i.e., stability


1)()()(1

)()(

m

m

ml

jCjG

jCjG

1)()( mljT

1)()(sup)()(

mm ljTljT


Robust Performance

Control system is said to have robust performance if the controller can minimize the worst plant error in the family

Recall:

If a bound is placed on the sensitivity function S(j) such that:

Then,


)()(sup)()( jWjSjWjS

1)()(

jWjS

)( allfor 1)()(sup jGjWjS


Example: Given the following control system

(a) Plot the bode magnitude for sensitivity function S(j) and

complementary sensitivity function T(j), for k = 10 and

comment on their values

(b) For step input, let W(s) = 1/s and produce a bode magnitude

plot for for k = 10, 50, and 100 and identify the

optimal value for using both H2 and H criteria


)()( jWjS


Soln:

(a) Sensitivity function:

Complementary sensitivity function:


)1(32

132

)21)(1(1

1

)()(1

1)(

2

2

Kss

ss

ss

KsCsGsS

)1(32

)1(32

1321)(1)(

2

2

2

Kss

K

Kss

sssSsT


Soln:

(a) Bode plot:

The system has approximately set-point tracking error of -0.8dB and disturbance rejection of -20dB for up to 1 rad/s



Soln:

(b) Weighted sensitivity function for step input:

The H2-norm reduces as k increases, thus k = 100 is the be value

H-norm: Maximum magnitude of the weighted sensitivity function occurs at the lowest frequency. Thus, the least upper bound is is 0dB at 0.01rad/s

where k = 100


)}1(32{

132)()(

2

2

Ksss

sssWsS


Example: The nominal forward path transfer function for a

closed-loop system is given as follows

Let the bound for multiplicative model uncertainty be

What is the maximum value of K for a robust stability?

Soln:


)42()()(

2

sss

KsCsGm

)25.01(

)1(5.0)(

s

sslm

1)()(sup)()(

mm ljTljT

)()(1

)()()(

sCsG

sCsGsT

m

m

Ksss

KsT

42)(

23


Soln:

Marginal stability occurs at k = 3.5, thus maximum value for robust stability


)42)(25.01(

)1(5.0)(

23 Kssss

sKlsT m


If N(s) 0 then

Therefore,


)()()()()( sDsSsRsTsY

)()()()()()()( sNsTsDsSsRsTsY

rejectionNoisejTTrackingjT ,0)( , ,1)(

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MODERN CONTROL SYS-LECTURE VII.pdf