example: linearization of a nonlinear model involving a nonlinear function of two variables

Example: Linearization of a nonlinear model involving a nonlinear function of two variables.

The dynamic behavior of the liquid level h in the conical storage tank system shown in the figure can be shown to be represented below, where now the cross section area of the tank is given by:

Fi

Hh

r

R

F

Ahc

AF

dtdh i

2/1

22

HhRrA

The tank model becomes

where α and β are parameters defined by:

cRH

,1 2

FFdtdhA i hcF

iFhcdtdhA

2/32

hhF

dtdh i

Dr. Kalyana Veluvolu

2/32

hhF

dtdh i

This process model has two types of nonlinear functions: Fih-2 , a product of two functions, and h-3/2. We shall have to linearize each of these functions seperately around the steady state (hs, Fis).

The linearization of f(h, Fi) = Fih-2 proceeds as follows:

termsorderhigherFFFfhh

hfFhfFhf isi

Fhis

Fhsisi

ississ

),(),(

,,

whereupon carrying out the indicated operations now gives:

isisssissisi FFhhhhFFhfFhf 232,,

If we ignore the higher order terms.

The steps involved in linearizing the second nonlinear term are no different from those illustrated in the previous example; the result is

sss hhhhh 2/52/32/3 )(23)(


We may now introduce these expressions in place of corresponding nonlinear terms in

2/32

hhF

dtdh i

Recalling that under steady-state conditions αFis=βhs1/2, and introducing the

deviation variables y=(h-hs) and u = (Fi-Fis), the approximate linear model is obtained upon further simplification as:

uKydtdy

where the steady-state gain, and time constant associated with this approximate linear model, are given by:

2/12/1 22ss h

chK

and

2/52sh


If we desire an approximate transform-domain transfer function model, Laplace transformation again gives

)(1

)( sUsKsY

Note that the approximate linear model for the conical tank has the same process gain, K, as for the cylindirical tank; but the time constant is a much stronger function of the liquid level hs.

Some important points to note about the results of these two illustrative examples are the following:

In each case, the approximate transfer function model is of the first-order process, with the time constants, and steady-state gain values dependent on the specific steady-state around which the system model was linearized.

Using such approximate transfer function models will give approximate results which are good in a small neighborhood around the initial steady-state; farther away from this steady-state the accuracy of the approximate results becomes poorer.

Using such approximate models, we are able to get a general (even if not 100% accurate) idea about the speed and magnitude of response to expect from any arbitrary nonlinear system. Numerical analysis can give more accurate information, but only about the specific input, starting from a specific operating condition for a specific set of parameters.


Let us conclude with a comparison of the responses of the nonlinear model and an approximate linear model for the conical tank of the Example. Suppose that α=2, β=1, c=0.5 so that the nonlinear model is:

2/32

2 hhF

dtdh i

while the approximate linear model has the transfer function

)(12

4)( 2/5

2/1

sUshhsYs

s

which in the time domain is

isisss

FFhthhhdt

dh 2/1

2/5 4)(2

1


clc;clear tss=4; [t,h]=ode45('tank1',0,tss,[0.5]); plot(t,h(:,1))

tank.m

function hdot=tank1(t,h) hs=0.5; dfi=0.15; hdot=[1/(2*hs^2.5)*(hs-h+4*hs^0.5*dfi)];

tank1.m

Nonlinear model

2/32

2 hhF

dtdh i

3536.02

5.02

20 2/32

sis

ss

is

hF

hhF

clc;clear tss=4; [t,h]=ode45('tanknl1',0,tss,[0.5]); plot(t,h(:,1))

function hdot=tanknl1(t,h) alfa=2;beta=1;Fs=0.3536;dfi=0.05;Fi=Fs+dfi; hdot=[alfa*Fi/h^2-beta*h^(-1.5)];

0 0.5 1 1.5 2 2.5 3 3.5 40.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Time (sec)

h (m

)

LinearNonlinear

dFi=0.15

dFi=0.05

The step response for these models is shown in the Figure for steps of two different magnitudes.

For small magnitude step inputs, the linear model is a good approximation of the nonlinear model.

For large amplitude inputs the linear model predictions deviate somewhat from the nonlinear model behavior.

However, in some situations this may be inadequate (e.g. for the control of highly nonlinear processes) so the development of nonlinear controllers has featured prominently in process control in the last decade.

Adaptive Control:A controller designed for the systems in the level control example in the previous example using the approximate transfer function model will function acceptably as long as the level, h, is maintaned at, or close to, the steady-state value hs. Under these conditions (h-hs) will be small enough to make the linear approximation adequate. However, if the level is required to change over a wide range, the farther away from hs the level gets, the poorer the linear approximation.

Observe also from the approximate transfer function model that the apparent steady state-state gain and time constant are dependent on the steady-state around which the linearization was carried out. This implies that in going from one steady state to the other, the approximate process parameters will change.


The main problem with applying the classical approach under these circumstances is that it ignores the fact that the characteristics of the approximate model must change as the process moves away from hs if the approximate model is to remain reasonably accurate. The immediate implication is that any controller designed on the basis of this changing approximate model must also have its parameters adjusted if it is to remain effective.

In the adaptive control scheme, the controller parameters are adjusted (in an automatic fashion) to keep up with the changes in the process characteristics. We know intuitively that, if properly designed, this procedure will be a significant improvement over the classical scheme.

There are various types of adaptive control methods, differing mainly in the way the controller parameters are adjusted. The three most popular methods are: scheduled adaptive control, model reference adaptive control, and self-tuning controllers.


Scheduled Adaptive Control

A scheduled adaptive control method is one in which, as a result of a priori knowledge and easy quantification of what is responsible for the changes in the process charactersitics, the commensurate changes required in the controller parameters are programmed (or scheduled) ahead of time. This type of adaptive controller, sometimes referred to as gain scheduling, is illustrated by the block diagram in the following figure.

Parameter Adjustment

Figure. Scheduled adaptive controller.


Continuing with the example of liquid level system, the figure represents control of linearised approximation of the original nonlinear system. The approximation is in the sense that the process transfer function G(s) only holds good for a particular level of the liquid in the tank. If the level h changes drastically, the transfer function G(s) may fail to represent the original system even approximately. Then there is a need to change the transfer function G(s) to account for the change in level h. One way of doing this is through Scheduled adaptive controller shown in previous page. From the system transfer function

)(1

)( sUsKsY

it is seen that both K and τ are affected by the liquid level h. The effect on K is more prominent since it affects the system gain. The effect on the time constant is less apparent and not so prominent. Thus we only compansate for the effect of liquid level h on K. This is done in a way that the overall system gain KKc is constant. Notice that

ch

A s2

ch

K s2

ch

K s2


So, to keep KKc = K0 (a constant)

sc h

cKKKK

200

This means that the controller gain Kc has to be adjusted inversely as the square root of the liquid level h. This is what will be done by the block represented as “Parameter adjustment” in the figure.

More generally, if K = K(t), that is, the process gain changes with time, then to keep the product KKc constant it is necessary to adjust Kc inversely with respect to K(t), i.e.

)(/1 tKKc

K(t) itself can be determined periodically by inserting a signal u into the process and monitoring the process output y.


In practice, it is often possible to find measured variables that correlate well with changes in process dynamics. These variables can be used to change the controller parameters, using a precalculated schedule. The controller parameters are computed off-line for several operating conditions and stored in memory.

It is difficult to give general rules. Each case must be treated individually. The key question is to determine the auxiliary variables to be used as scheduling variables. It is necessary to have good insight into the dynamics of the process if gain scheduling is going to be used. The controller can be automatically tuned at a different operating points and the resulting tuning parameters can be saved and a schedule created.

Example: The figure given below shows different valve characteristics, if the scheduling variable in a level control system is the opening of a control valve, then a schedule may look as in the Table:

Position

FlowQuick opening

Linear

Equal percentage

Table: Scheduled control of Valve

Valve opening Kp Ti Td

0.0-0.15 1.7 95 23

0.15-0.22 2.0 89 22

0.22-0.35 2.9 82 21

0.35-1.0 4.4 68 17


Example: The nonlinearity of the valve is assumed to be:

4)( uufv

PI f-1(c) f(u) G(s)Command signal uc

+

-Nonlinear Valve Process

Output y

Contol signal u

Let f-1 be an approximation of the inverse of the valve characteristics. To compensate for the nonlinearity, the output of the controller fed through this function before it is applied to the valve. This gives the relation:

cffufv 1)(

where c is the output of the PI controller


Assume that f(u)=u4 is approximated by two lines as shown in the figure. One from (0,0) to (1.3,3) and the other from (1.3,3) to (2,16).

)(ˆ uf)(uf

1 2

1510

0

Figure. A crude approximation of valve characteristics.Time

Then we have:

163139.10538.030433.0

)(ˆ 1

cccc

cf


Model Reference Adaptive Controller (MRAC)

The key component of the MRAC scheme is the reference model that consists of a reasonable closed-loop model of how the process should respond to a set-point change. This could be as simple as a reference trajectory, or it could be more detailed closed-loop model. The reference model output is compared with the actual process output and the observed error ᵋm is used to drive some adoptation scheme to cause the controller parameters to be adjusted so as to reduce ᵋm to zero. The adoptaion scheme could be some control parameter optimization algorithm that reduces the integral squared value of ᵋm or some other procedure. This is an adaptive control technique where the performance specifications are given in terms of a model. The model represents the ideal response of the process to a command signal. The controller has two loops:

The inner loop, which is an ordinary feedback loop consisting of the process and the controller.

The outer loop, which adjust the controller parameters in such a way that the error e = y- ym is small (not trivial)


myye Tracking error:

Introduce the cost function J:

2

21 eJ

Where θ is a vector of controller parameters. Change the parameters in the direction of the negative gradient of e2.

eeJ

dtd

e is called the sensitivity derivative. It indicates how the error is

influenced by the adjustable parameters θ.

Example:

Process: buyadtdy

Model:cmmm

m ubyadtdy

Controller: yuu c 21 Closed loop system:

cc ubybayubaybuyadtdy

1221

MIT Rule

Ideal controller parameters for perfect model-following

baa

bb mm

02

01 ;

Derivation of adaptive law

Error: myye

where

cubasby

2

1

c

c

ubybasubybays

12

12

Sensitivity derivatives

cubasbe

21

y

basbu

basbe

c2

22

12

2

cmmcmm ubyaubbby

baaba

dtdy

ysdtdy

Approximate

masbas 2

where

euasa

seu

asb

t cm

mc

m

1

1

eyasa

sey

asb

t m

m

m

2

2

mab

• System

MRAC of Pendulum

TdmgdcJ c 1sin

cmgdcsJsd

sTs

2

1

)()(d2

d1dc

T

77.100389.089.1

)()(

2

sssTs

Keith SevcikKeith Sevcik

• Controller will take form:

Controller

Model

AdjustmentMechanism

Controller Parameters

ymodel

u yplant

uc

77.100389.089.1

2 ss


MRAC of Pendulum

• Following process as before, write equation for error, cost function, and update rule:

modelplant yye

)(21)( 2 eJ

eeJ

dtd

sensitivity

derivative


MRAC of Pendulum

• Assuming controller takes the form:

cplant

plantcpplant

cmpmodelplant

plantc

uss

y

yuss

uGy

uGuGyye

yuu

22

1

212

21

89.177.100389.089.1

77.100389.089.1

MRAC of Pendulum

plant

c

c

cmc

yss

uss

e

uss

e

uGuss

e

22

22

21

2

2

22

1

22

1

89.177.100389.089.1

89.177.100389.089.1

89.177.100389.089.1

89.177.100389.089.1

MRAC of Pendulum

• If reference model is close to plant, can approximate:

plantmm

mm

cmm

mm

mm

yasas

asae

uasas

asaeasasss

012

01

2

012

01

1

012

22 89.177.100389.0

MRAC of Pendulum

• From MIT rule, update rules are then:

eyasas

asaeedtd

euasas

asaeedtd

plantmm

mm

cmm

mm

012

01

2

2

012

01

1

1

MRAC of Pendulum

• Block Diagram

ymodel

e

yplantuc

Π

Π

θ1

Reference Model

Plant

s

77.100389.089.1

2 ss

Π

+

-

mm

mm

asasasa

012

01

mm

mm

asasasa

012

01

mm

m

asasb

012

s

Π

-+

θ2


MRAC of Pendulum

• Simulation block diagram (NOTE: Modeled to reflect control of DC motor)

am

s+amam

s+am

-gamma

sgamma

s

Step

Saturation

omega^2

s+amReference Model

180/pi

Radiansto Degrees

4.41

s +.039s+10.772

Plant

2/26

Degreesto Vol ts

35

Degrees

y m

Error

Theta2

Theta1

y


MRAC of Pendulum

0 200 400 600 800 1000 1200-100

-50

0

50

100

150

ymg=.0001

Simulation with small gamma = UNSTABLE!Simulation with small gamma = UNSTABLE!


• Solution: Add PD feedback


MRAC of Pendulum

• Simulation results with varying gammas

0 500 1000 1500 2000 25000

5

10

15

20

25

30

35

40

45

ymg=.01g=.001g=.0001

707.sec3

:such that Designed

56.367.256.3

2

s

m

T

ssy


MRAC of Pendulum

example: linearization of a nonlinear model involving a nonlinear function of two variables

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