Introduction to Robust Control [PDF]

Introduction to Robust Control

Dr Abraham T Mathew

What is a Control System? Is it the physical system?

Is it the mathematical system?

ENVIRONMENT

SYSTEM BOUNDARY

INPUT

INPUT

CONTROL SYSTEM

OUTPUT

INPUT SUCCESS IN CONTROL DESIGN IS SAID TO BE BASED ON THE SUCCESS IN IDENTIFYING THE SYSTEM BOUNDARY, INPUTS,OUTPUTS & THE ENVIRONMENT

Model Based Control Design- Issues  Analytical or computational models cannot truly characterize and

emulate the phenomenon.  A model, no matter how detailed, is never a completely accurate representation of a real physical system

Control Design-classical way  Normally, in the conventional control design for SISO

system, the stability margin is specified to ensure stability in the presence of model uncertainties  But, the uncertainties or perturbations are not quantified, nor performance was not taken into account in terms of disturbance, noise etc.  For MIMO systems, many of the SISO methods cannot be scaled up

Robust Control Design a controller such that some level of performance of the controlled system is guaranteed irrespective of the changes in the plant dynamics/process dynamics within a predefined class and  the stability is guaranteed

Control design targets  Stability

 Disturbance rejection  Sensor(measurement) noise rejection  Avoidance of actuator saturation

 Robustness- the process/plant performance should not

deteriorate to unacceptable level if there occurs the changes due to the uncertainties All these targets cannot be achieved simultaneously and perfectly. So there has to be some compromise or tradeoffs, because of various reasons

Modeling in the context of robust control

We consider a simple example !!

Modeling a DC Servo  We consider a DC servo mechanism consisting of a DC

motor, gear train, and the load shaft  It is required to control the angular displacement and speed using a voltage signal applied across the armature

Motor Load

Linear Model of the DC Servo-Physical Equations of dynamics      0   0  L       i     0 

1 0  NK m La

    0 0     0    1 NK m         0 v(t )   TL    1 Je Je      Ra   i   L   0   a La 

    1 0 0      0 1 0       i   

TF FORM  (s)

NK m J e La

a0   2 2 2 v( s )  2 Ra N K m  s s  b1s  b2  s s  s La J e La  





Nominal Model  Km=0.05 Nm/A, Ra=1.2 ohms, La=0.05H

 Jm=8x10-4 kgm2 , J=0.020 kgm2  N=12  Je=J+N2Jm=0.1352 kgm2

Uncertainty  Let the parameters are subject to changes as follows

0.04≤Km ≤0.06 6x10-4 ≤Jm ≤ 10-3 0.01≤J ≤0.05

Model with Uncertainty (as an Interval System)

[74.22, 99.58] G( s)  2 s s  12s  [47.8, 53.4]





Abstracting a

Control System Structure

Control System Structure Disturbance w(t) wm

Noise v(t)

Sensor

S1 Input yd

Controller

C

u

Plant/Process

P

y

Sensor

S

Output ym

System Equations  If the Plant is LTI the zero state linearity dictates that y is a linear

combination of effects of the two plant inputs u and w  That is

y(s)  P(s)u(s)  P (s)w(s) (1)  Quite often it is convenient to work with the disturbance d(s) at the plant output given as w

d (s)  P (s)w(s)

 Then, we have

w

y(s)  P(s)u(s)  d (s)

(2)

(3)

System Equations  Sensor is assumed to have two inputs, plant output y and the

measurement noise v. So, we have

y ( s )  P ( s ) y ( s )  v( s )  Ideally Ps(s)=1 and v(t)=0 so that ym=y m

s

(4)

(this is achieved if sensor bandwidth is larger than system bandwidth or we say the sensor is fast and accurate)

Now, look at the Controller

Disturbance w(t) wm

Noise v(t)

Sensor

S1 yd

Controller

C

u

Plant/Process

P

y

Sensor

S

ym

Contd…  Controller gets three inputs ym, yd and wm

 Here wm is the disturbance measured using suitable sensor  Let the controller be LTI. Then

s) all Fdthe (s) ythree ( s)  Fm (s)need ym (sto ) be Fwused (s)wmalways (s) here. Several (5) d u(Not inputs

control structures are defined according to whether ym, yd or wm is used to produce u or not . Accordingly we will have different schemes of control

1. Single Degree of Freedom controller  When Fm=-Fd and Fw=0, we have

u(s)  F ( s)[ y ( s)  y (s)] d

d

m

 The figure shows 1DoF Control Structure realizing this

equation yd

+ ym

Fd

u

Two Degree of Freedom Controller  If we have a structure of the form given below, designer will

have freedom to independently select Fm and Fd we will have the TDoF Feedback Controller structure

yd

Fd

+ +

u

Fm

ym

Feedback Control Scheme

v

w Fw Pw yd

Fd

+

+

d u

P

+

+ Fm

+ y

Ps

+

+

ym

Problem formulation  System enclosed in the dotted box is seen to have three

inputs and one output  By assuming linearity, we can say that plant output y(t) is produced as a superposition of the effects of these three signals coming to the output port through three transfer channels  That is

y(s)  H (s) y (s)  H (s)w(s)  H (s)v(s) d

d

w

v

Tracking problem  Let the error e(t) be defined as e(t)=yd-y

 That is,

e( s)  y ( s )  H ( s) y ( s )  H ( s) w( s)  H ( s )v( s) d

d

d

w

v

Or

e( s)  1  H ( s)y ( s)  H ( s) w( s )  H ( s )v( s) d

d

w

v

 The central design problem is to obtain Hd, Hw, and Hv with

desirable properties using appropriate methods or criteria

Look at it again

v

w Fw Pw yd

Fd

+

+

d u

P

+

+ Fm

+ y

Ps

+

+

ym

Emphasis for output disturbance  In cases where it is desirable or convenient to work with the

output disturbance d rather than w, we have

y(s)  H (s) y (s)  H (s)d (s)  H (s)v(s) d

d

wd

v

e( s )  y ( s )  H ( s ) y ( s )  H ( s ) d ( s )  H ( s ) v ( s ) d

d

d

wd

v

Or

e( s)  1  H ( s)y ( s )  H ( s)d ( s )  H ( s)v( s) d

d

wd

v

Tracking performance  For the system to ideally track the reference, the error must

be zero  To achieve this for all possible yd,v,w and d, we would require Hd(s)=1 and Hw(s)= Hwd(s)=Hv(s)=0  In the practical setting, as we see more in detail, we can see that this condition cannot be satisfied perfectly for the entire bandwidth or entire region of system perturbations  Some design tradeoffs, optimality conditions and so on would have to be called for as we have already noted.

Admissible/acceptable designs  In order to do the adjustment/tradeoff for obtaining an

admissible or acceptable design and discriminate between acceptable and unacceptable departures from the ideal performance, we need to have the specifications  These specifications give rise to different control structures like open loop, feedforward, feedback, etc.  We may differentiate between SISO and MIMO and start with SISO and generalize the notations for MIMO, subsequently

Control System Performance  From a system’s perspective, the performance specification

for control system starts with “Stability”  Followed by Sensitivity, Disturbance Rejection, Noise Rejection etc. where needed.

Stability  When it comes to stability, in the modern settings of design,

we consider two classes of stability, namely  Input-output stability  Internal stability

 Internal stability is of paramount importance in the MIMO

system framework, both in Matrix Transfer function form and State variable/transfer function forms

Internal Stability  A system to be internally stable means all the transfer functions

associated with all the transfer channels connecting exogenous input to the output(including set point, disturbance & noise) shall be stable  In reality, it is possible for a system to be internally unstable and yet to have a stable “set point to output” channel transfer functions  Under this circumstance, we say that system has unstable hidden modes  Therefore, internal stability must be ensured before the transfer function that define the response to the system inputs are considered

Design Model be a set of all plants that each member of set P is an admissible model, given the uncertainty region (interval)

 Let P

 P0 in

P is one model with the nominal value of the

parameters  If P0 is used for the robust designs, then let us call P0 as Design

model (for the sake of convenience!!)

Model Uncertainty & Internal stability  If the plant is expected to deviate from the design

model(nominal model), it is better represented by a set of models centered on the design model(nominal model)  For a control system to be acceptable, the design must be internally stable for every model in the set  This property is known as robust stability  Once stability & robustness are assured, we can shift the attention to “response”

Summary  A model of the physical system is only an approximation of

the real phenomenon/process  Control system output is the measurement showing the status or effectiveness of control  Inputs, in a general framework will include set point, disturbance and measurement noise

Summary contd…  Models are subjected to various uncertainties

 Nominal model in the set of uncertain models can be used as

Design model  Internal Stability and robust stability are starting points for good control system design  Once stability is assured, other performance measures can be specified

Design Dilemma  It will not usually be possible(which we will see in detail) to

have good set point tracking, and disturbance rejection and noise rejection uniformly effectively for all functions of yd, v, w and d  Also, emphasis on sensitivity on one may negatively affect the other

Robust Control System  A system is said to be robust when  It is durable, hardy and resilient  It has low sensitivities in the system passband  It is stable over the range of parameter variations

 The performance continues to meet the specifications in the

presence of a set of changes in the system parameters

Robustness is the sensitivity to the effects that are not considered in the analysis and designfor example, the disturbances, measurement noise, and unmodeled dynamics

Sensitivity & Sensitivity Analysis

Sensitivity  It is the percentage change in system transmission or

response or some quantity of interest with respect to the percentage change in another quantity  In control theory we use  Parameter Sensitivity  System Sensitivity  Root Sensitivity  Eigenvalue Sensitivity

Parameter Sensitivity  Let T be the system function which depends on a parameter

  Then, the parameter sensitivity ST of T with respect to  s defined as T

T

 ln T T  T  S    ln   T





System Sensitivity  Let T be the system closed loop transfer function which

depends on the open loop transfer function G  Then sensitivity of T w.r.t G is given as T

T

 ln T T  T  S  G  ln G G G G T

G

Root Sensitivity  Let T be the system closed loop transfer function with the ith

root given as i and the parameter of interest is say K  Root sensitivity is the sensitivity in terms of the position of the roots of the characteristic equation on the (, j) plane(root locus plane)

Significance of Root Sensitivity  Roots of the characteristic equation represents the

dominant(visible) modes of the transient response  The effect of parameter variation on the position of the root and the direction of shift of the root are important and useful measures to say about the sensitivity  Can be combined with Root Locus Method for Control Designs

Definition of Root Sensitivity  The root sensitivity of the system T(s) is defined as

  S    ln K K i

i

i

K

 Let

K

m

T ( s) 

K ( s  z ) 1

j

j 1

n

( s   ) i 1

i

Contd…  Let K be a parameter that influences the location of the roots i

and the gain K1  Then the root sensitivity is related to the system sensitivity to K and is given as(if zeros of T(s) are not dependent)

 ln K  1  S   ln K  ln K ( s   ) n

T

1

K

i

i 1

i

 In the event of gain K1 independent of K, we have

 1 1  S     S  ln K ( s   ) (s   ) T

K

n

n

i

i

i 1

i 1

i

K

i

Eigenvalue Sensitivity  Let us assume that we have the relation(A is from the state

space equation)

A    i

i

i

 Differentiating with respect to the element akj of A we will

have

A     A    a a a a i

i

i

kj

i

i

kj

kj

i

kj

Contd…  Premultiplying with i , the left eigenvector we have ii=1

and i (A-i I)=0  Then, we get

 A    a a

i

i

i

kj

kj

Contd…  All elements in

which will be 1  Therefore we get

A will a be zero except the (k,j)th element, kj

   a i

ik

kj

 This is the eigenvalue sensitivity

ji

Sensitivity Analysis of transfer functions  Consider a closed loop system as shown in Figure

yd

+

u -

T

G 1 G

T  ln T T  G T  1 S    ln G G T G 1  G G T

G

G

y

Waterbed effect  Now, add T and S

We get T+S =1

System with cascade compensator  We consider the following system yd

+

K

u

G

y

-

T

GK 1  GK

T  ln T T  G T  1 S    ln G G T G 1  GK G T

G

Check T+S

System with feedback compensator  Consider the following system

yd

+

u

G

y

-

H T

G 1  GH

T  ln T T  G T  1 S    ln G G T G 1  GH G T

G

Check T+S

Sensitivity & Complimentary Sensitivity Functions  In the Robust Control Literature, Sensitivity Function plays a

crucial role  Let S(s) be the Sensitivity Function  Then T(s) is the Complimentary Sensitivity Function such that S+T=1 for SISO and S+T=I for MIMO

Open Loop Control

Open Loop Control  It is the simplest control structure

 Limited in performance  Usually reserved for special applications where feedback

control is either impossible or unnecessary  It is a good starting point for control design  It helps to appreciate the advantages of feedback control  Stability, performance etc are relatively in simpler forms to understand

Open Loop Structure d

yd

F

u

+ P

+ y

-

e

+

Input-Output Relations  In open loop control input yd is usually a synthesized signal

for the given application and u is derived from that as shown  Open loop control requires no measurements.  Now, from Figure above, we write as

y  FPy  d d

and e  (1  FP) y  d d

H ( s)  F ( s) P( s) d

and H ( s)  1 wd

Tracking Performance  Perfect tracking of yd occurs if

H ( s )  F ( s ) P( s )  1 d

 That is, if

F ( s )  P( s )

1

 The practical objective is to make

in the system passband

F ( j ) P( j )  1

Disturbance rejection  Since

open H ( s)  1 loop control does nothing to attenuate wd

the effects of disturbance inputs nor does it amplify them either

Sensitivity  The sensitivity of

H with (s) respect to P(s) is calculated as d

follows

H  F ( P  P)  FP  FP 0

d

FP S  H

P

P

0

FP

0

1

P

0

 A sensitivity 1 implies that a given percent change in P

translates into the equal percent change in the transmission function H d ( j )  Open loop control does not affect sensitivity

Stability Conditions  We modify the block diagram of the Open loop control

system as shown here v

yd

+ F

+

z

u

P

y

Analysis  In any system, any addition or deletion of some of the input lines 

  

or some output lines won’t alter the internal stability We shall add inputs and outputs and view this as injecting test inputs into the system and taking extra measurements, neither of which is expected to change the stability properties of the system The test inputs and and outputs are chosen so that the resulting system is controllable and observable For such a fully controllable and observable system there shall not be any hidden modes So, internal stability is then guaranteed by input-output stability

Fig.1

yd

F

u

y

P v

Fig.2

yd

+ F

+

u

P

z

The system, in Fig 1 and Fig 2 are same but with additional input v and one additional output z in Fig 2

y

Controllability/Observability/Stability  System in Fig.2 is controllable and observable if both F(s) and P(s)

are controllable and observable  System in Fig 2 is internally stable if and only if the both F(s) and P(s) are stable. See below

Y ( s )  FPy ( s )  Pv( s ) d

z ( s )  Fy ( s ) d

Or  y( s )  FP P   y ( s )  z ( s )    F 0   v( s )       d

Analysis contd…  Because the realization is controllable and observable, it is

internally stable if, and only if, it is input-output stable.  That is, if all elements of the matrix transfer function above are stable  Thus F(s), P(s) and F(s)P(s) must have only LHP poles  If P is of non-minimum phase type, then F cannot be used to cancel the RHP zeros of P, because then F will become unstable.

Feedforward Control  Feedforward control is a variation of open loop control.

 It is applicable when the disturbance input is measured  The open lop controller F is chosen, to make the output to follow the

reference, in spite of the disturbance w Pw

u

+ P

y’ +

z

d y

w

Pw Pw

d

F

u

Here, to realize Feedforward control:

+ P

y’ +

z

1. d has to be obtained by proper measurements 2. F is chosen such that y’ is close to –d 3. Or FP is almost unity

d y

Closed loop control-1 DoF

Closed loop control-1 DoF  Consider the following system

d

e

yd +

F

u

+ P

+

+

-

y

e

ym

Ps + +

v

Analysis  We have

FP 1 y( s )  y (s)  d (s) 1  FP 1  FP d

1 1 e( s )  y ( s )  y( s )  y (s)  d (s) 1  FP 1  FP d

d

With Sensor noise/Measurement Noise  If yd =d=0 and v0, then

y( s )   FP( y  v )  y( s )  

FP v( s )  T ( s )v( s ) 1  FP

and e( s )  y  y( s )  T ( s )v( s ) d

Norms are Performance Measures

Signal forms and Signal Norms  Norm based approach for control design gives a sound

platform for robust control designs  Different types of norms are used in control systems  Use would be depending on the mathematical approaches used to define the norm

Norms of signals and systems  Euclidean Norm or l2 norm for vector x is given as

  n

x  x 2

1

2

i

i 1

2

 ( x x) T

1

2

 For a vector signal x(t), l2 norm is

x   x (t ) x(t )dt  

2

T

1

2



 This norm is the square root of the energy in each

component of the vector  If norm exists x(t)  l2

Norms of signals and systems  For power signals, we may use the root mean square

value(rms) norm

1   rms( x )   lim  x (t ) x(t )dt  2T   T

T 

T

T

1

2

Frobenius Norm  For an mxr matrix A, the Frobenius norm is defines as



m

r

A  a 2

 It can be shown that

2

i 1 j 1



1

2

i, j

2

A 2  tr ( A A)  tr ( AA ) T

T

System Norm  LTI systems are generalization of matrices-

 A matrix operates on a vector to produce another vector  An LTI system operates on a signal to produce another signal  So, analogous to Frobenius norm, we can define the system

norm

L2 Norm for LTI systems  Let G(s) be an mxr matrix transfer function

 Then the L2 norm for G(s) is defined as

 1  G    tr G (  j)G( j)d   2  

2

1

2

T



 ||G||2 exists if an only if each element of G(s) is strictly

proper. For SISO we have a scalar TF which need to be strictly proper. There should not any poles on the imaginary axis for either case.  Then we say G L2

G(s) plane in H2  When G L2 we can write the norm with respect to

complex s plane as 1 G   tr (G (  s )G( s ))ds 2j 2



2



T

1   tr (G (  s )G( s ))ds 2j T

 Contour of integration for the last integral is along the

entire imaginary axis and the infinite semicircle in the LHP or RHP  Since G(s) is strictly proper, it is easily shown that the integral vanishes over the semicircle  If G L2 and in addition, G is stable, then we say that G H2  H2 is the Hardy Space defined with the 2-norm

Exercise  Calculate the L2 norm of G(s) given as:

( s  3 )  ( s  2 )  1 G( s )    s  3s  2   2 ( s  2)  2

Answer  3s  21 tr G (  s )G( s )  ( s  1)( s  2)(  s  1)(  s  2) 2

T

Every term in G(s) is strictly proper Contour is Imaginary axis + LHP semicircle with radius 

L2 norm of G(s) =(3/2)

Induced norm  Induced norm is a different type of norm which applies to

operators and is essentially a type of “maximum gain”  For a matrix, the induced Euclidean norm is

A  max Ad 2i

 =sqrt(eigen(ATA))

d

2

1

  ( A)

 is the max( ) and is min(  )

2

Induced norm for LTI system  To obtain induced norm for an LTI system, consider first a

stable, strictly proper SISO system  Then, if the input u(.)  l2 , then the output y(.)  l2  By Parseval’s theorem

1 y   G( j) u( j) d 2 2



2



2

2

(A)

 Clearly

1 y  sup G( j)  u( j) d 2 2



2

2

2





 Or

y  sup G( j) u 2 2

2



2 2

(B)

 We argue that the RHS of the inequality in (B) can be

reached arbitrarily closely for a fixed value of ||u||2 that is chosen to be 1 with no loss of generality

 Suppose |u(j)|2 approach an impulse of weight 2 in the

frequency domain at = 0  Then the integral of Eq(A)

1 y   G( j) u( j) d 2 will approach G( j ) 2



2



2

2

2

0

(A)

 If

G( j)has a maximum at some finite value of , we may

choose 0 to be that frequency  If not, then G( j must ) approach a supremum as .  We can make 0 as large as we like and will G( jbe 0as) close to the supremum as we wish  The RHS of inequality in (B) can be reached arbitrarily closely and we get

sup y  sup G( j) u 2 1

2



Hinfinity Norm  The norm calculated last is also the infinity norm given by 

G  lim(  G( j) ) 

p

p

1

p



 The infinity norm of G(s) exists if and only if G is proper

with no poles on the j axis  In that case we write G L  If in addition, G is stable, then we say G H H is the Hardy Space defined with the -norm

Norms for Multivariable systems

H norm for Multivariable systems  For multivariable systems, we have

1 y   G( j)u( j) d 2 

2 2

2



 This can be written as

1 y   [ (G( j))] u( j) d 2 2 2



2

2



 Further, we may write as

1 y  sup (G( j)  u( j) d 2 2 2

 Or



2









y  sup [G( j)] u 2 2

2



2 2

2

2

Contd…  The factor ||u(j)||2 in the integrand refers to the 2-norm

of the vector u(j)  In SISO, the equivalent term refers to the 2-norm of a signal  We argue that the RHS of the last inequality





y 2  sup j)] by u propoer [G( closely, can be approached arbitrarily choice of 2  u(j) 2

2

2

 Essentially we pick u(j) to be the eigenvector of

G*(j)G(j) corresponding to the largest eigenvalue, and we concentrate the spectrum of u(j) at the frequency where is the largest (or for some frequencythat is arbitrarily large, if has no maximum, but a supremum. Therefore 

sup y  sup [G( j)] u 2 1

2



MIMO H norm As a continuation of the development, we define

G  sup [G( j)] 



Disturbance Rejection

Disturbance Rejection  Disturbance rejection is a performance measure

 Effect of disturbance is studied in two ways  Input disturbance  Output disturbance

Rejection of Input disturbance d

yd

+

+

u

G

-

H

y

Analysis T (s )  yd

G 1  GH

and T (s )  d

G 1  GH

To suppress disturbance, we want |Td|