7/29/2019 Homework 2012

1/12

Uppsala UniversityDepartment of Information TechnologySystems and ControlOctober 26, 2012

Automatic control III

Homework assignments, 2011

Deadlines

Assignment 1.1 Friday November 16, 24.00

Assignment 1 Monday December 10, 24.00

Assignment 2 Tuesday January 8 (2013), 24.00

Both assignments are compulsary and form an important part of the ex-amination.

Assignment 1 is to be solved in groups, with up to 4 students. The solutionis to be given in written form, as a pdf file. All group members are responsiblefor the full report. There will also be an oral examination, in December 79,where each group is allocated a period of time. Each group member will begiven a few individual questions related to the assignment and the report.

Part 1.1 of Assignment 1 is to be delivered on November 16, and somefeedback on that part of the report will be given.

Instructions for how to prepare the report for Assignment 1 are given inSection 3 in the end of this document.

Assignment 2 is to be solved individually, and in written form.

7/29/2019 Homework 2012

2/12

1 Homework assignment 1 Linear controldesign

Consider a simplified model of the inverted pendulum, as depicted in thefigure below.

(

)

( )

-

-

(

)

(

)

(

)

(

)

Here,

denotes the control input,

is the true output,

is measurementnoise,

is the measured (noisy) output, and

is the feedback controller.

In this homework assignment, we assume that the system to be controlledhas the transfer function (from

to

)

(

) =

(

+ 1)

2 1

(1)

which captures the fact that there is one stable and one unstable pole. Set

= 0

1.

1.1 Problem 1

(a) Write the system (1) in state space form. Be sure to use controllercanonical form. Then sample it (use zero-order-hold), using the sam-pling interval

= 0

1. Represent the sampled model in state spacefrom as

( + ) = ( ) + ( )

(

) =

(

)

(2)

Give the matrices

,

and

. Determine the poles and zeros of thesampled-data system (2).

(b) Consider the effect of the measurement noise

. Assume that discrete-time measurements take place, and that the noise has spectrum

(

) =20

8

2

8 cos(

)

(3)

Determine the variance of the measurement noise

(

).

2

7/29/2019 Homework 2012

3/12

(c) Determine, by using discrete-time LQG design, a feedback controller,

that makes the output variance E 2( ) as small as possible.

Treat two different cases:

Let there be a delay in the controller, so that

(

) depends onprevious outputs, that is ( ) ( 2 ) .

Let there be no delay in the regulator, so that a direct term canbe included. Thus, in this case

(

) depends on

(

) as well as

(

)

(

2

)

.

For both these cases you should answer the following questions:

(i) What is the (discrete-time) transfer operator

(

) of the con-troller?

(ii) What is the minimum variance of the true output

(

)?

(iii) What is the variance of the input signal ( ) when this controlleris used?

(d) Simulate the system, using an appropriate model for the noise ( ).Use the controllers designed in part (c). What values are obtained forthe variances of the noise

, the output

and the input

from the

simulated data?

1.2 Problem 2

Consider continuous-time 2 control of the system with transfer function

(

), given in (1).

(a) Set the weighting of the transfer functions as

(

) =1

(

) = 1

(

) = 1

(4)

Describe why one can expect this choice of weighting to lead to a con-troller with integral action.

(b) Determine the extended model of the system. Check if all conditionsrelated to the matrices

,

and innovation form are fulfilled. Ifthese conditions are not fulfilled, how should the standard recipe forthe determining the optimal

2 controller be modified?

(c) Determine the optimal 2 controller. What is the feedback transfer

function

(

) in this case? Does it have integral action?

3

7/29/2019 Homework 2012

4/12

(d) Plot the Bode diagrams of the weighted transfer functions

(

)

(

),

( ) ( ) and ( ) ( ).

1.3 Problem 3

Consider the control of the a system with the transfer function

(

) =1

(

+ 1)(5)

and assume that a feedback controller

(

) =

+ 1

+ 1 (6)

has been designed.

(

)

(

)

1

(

)

(

)

(

)

(

)

The parameter values are

= 4

74

= 0

25

= 0

667

(a) Determine the cross-over frequency

and the phase margin

for theloop gain. What is the stationary value of the control error

(

) whenthe reference signal

(

) is a step? What is the value if

(

) is a ramp?

(b) Assume that the system is controlled digitally. Sample the system

(

), (assuming the input

(

) is constant during the sampling inter-

vals).Approximate

the designed controller

(

) with a discrete-timecontroller, testing three principles:

zero-order-hold, i.e. assume that

(

) (input to

(

)) is constantbetween the sampling instants

backward difference, i.e.

1

Tustins approximation, i.e.

2( 1) ( +1)

4

7/29/2019 Homework 2012

5/12

Set the sampling interval to

= 0

1. Determine the cross-over fre-

quency and the phase margin. What is the stationary value of thecontrol error

(

) when the reference signal

(

) is a step? What is thevalue if

(

) is a ramp?

(c) Repeat part (b) for the case when the sampling interval is = 0 3.

5

7/29/2019 Homework 2012

6/12

1.4 Hints and comments

1.4.1 Problem 1c The optimal discrete-time LQG controller

System model

(

+

) =

(

) +

(

) +

(

)

(7)

(

) =

(

) +

(

)

(8)

where

(

),

(

) are white noise sequences with zero mean and covariances

E

(

)

(

)

(

)

(

)

=

1 12

12

2

(9)

Note for Problem 1 that the state space model (8) has to describe boththe input-output effect in (2) and the noise effect in (3).

Criterion, loss function, performance index

E

(

) 1 ( ) +

(

) 2 ( )

(10)

For the optimal controller, it is necessary to specify the structure, howthe information is used. Distinguish between the two cases

A delay in the controller, so

(

) is a function of

(

)

1 = . The

optimal LQG controller is

(

) =

(

)

(11)

(

+

) =

(

) +

(

) +

[

(

)

(

)]

(12)

where

=

+ 2

1

(13)

= + 1

+ 2

1

(14)

=

+ 12

+ 2

1

(15)

=

+ 1

+ 12

+ 2

1

+

12

(16)

The solutions to (14), (16) must be nonnegative definite, and are inmost cases positive definite.

6

7/29/2019 Homework 2012

7/12

No delay in the controller, so

(

) is a function of

(

)

= . As-

sume in this case that 12 = 0 holds. The optimal LQG controlleris

(

) =

(

)

(17)

where (12) and (13) (16) still apply, and

(

) =

(

) +

[

(

)

(

)]

(18)

=

+ 2

1

(19)

In both cases, it is possible to write the optimal controller in state space

form as

(

+

) = reg ( ) + reg ( ) (20)

(

) = reg ( ) reg ( ) (21)

(The negative signs in (21) are meant to emphasize that negative feedbackis used, i.e. that the control law is

(

) =

(

)

(

). Thus, whencomputing

(

) the negative signs should be omitted.)The closed loop system can be written in the form

(

+

) ( + )

=

tot

(

) ( )

+

tot

(

) ( )

(22)

(

)

(

)

= tot

(

)

(

)

+ tot

(

)

(

)

(23)

The matrices reg, tot, etc depend on the open loop system and the con-

troller. They are also influenced if there is a delay in the controller or not.

The matrices tot, tot, are introduced just to allow an expression for how

the signals

(

) and

(

) depend on the state variables in (23).

The following command in Matlab and the Control Systems Toolbox areuseful when solving the problem:

dare, tf, ssdata, zpkdata, dlyap, dlsim, ss

Note that the command lqr cannot be used directly for the present case, asit requires a strictly positive input penalty

2.

7

7/29/2019 Homework 2012

8/12

1.4.2 Problem 2 2 control

As mentioned in the textbook, the extended model is not fully stabilizable,and therefore standard routines for computing the corresponding LQG reg-ulator may fail. A possible remedy is to change the weighting

(

) to

(

) = 1

(

+

), where

is a small, positive number. Also be careful whenexamining the innovation form.

The following command in Matlab and the Control Systems Toolbox areuseful when solving the problem:

lqr, lqe, ss, tfdata, series, feedback, bode

1.4.3 Problem 3 Discretization and sampling effects

In continuous-time, the final value theorem reads (assuming that the limitsexist)

lim

(

) = lim

0

(

)

(24)

In discrete-time, the corresponding result, using

-transform, reads

lim

(

) = lim

1(

1)

(

)

(25)

The following command in Matlab and the Control Systems Toolbox areuseful when solving the problem:

c2d, margin

8

7/29/2019 Homework 2012

9/12

2 Homework assignment 2 Nonlinear feed-back systems

2.1 Problem 1

A well-known example of a second order system with a limit cycle is the Vander Pol oscillator, governed by the differential equation

2

2

(1

2)

+

= 0

(26)

where

0. Introduce the state variables 1 = and 2 = . The system

(26) is then equivalent with the state space representation

1 = 2

2 = (1

21) 2 1

(a) Perform a phase plane analysis of the Van der Pol oscillator, i.e. deter-mine and characterize all stationary points.

(b) Use Matlab to plot the phase portrait for

= 0

1,

= 1

0 and

= 4

0respectively.

(c) Let = 3. Show that the system

2

2

+

=

3

(27)

is equivalent with (26) for this particular choice of

.

(d) The system (27), with = 3, can be represented as the feedbacksystem in the blockdiagram below.

(

)

(

)

1

0

(

)

(

)

(

)

What is

(

) and

(

) in this particular case?

9

7/29/2019 Homework 2012

10/12

(e) Determine the sector and the circle in the circle criterion corresponding

to this particular ( ). Also show that the circle criterion is not fulfilledin this case.

(f) Show that the describing function method indicates a (stable) limitcycle for the Van der Pol oscillator. Determine the amplitude

and thefrequency

indicated by the describing function for the cases

= 0

1,

= 1

0 and

= 4

0 respectively. Compare with the real values fromsimulations.

2.2 Problem 2

A DC motor

(

) =1

(

+ 1)

(

)

is used as an actuator in a position servo. The input

is the voltage overthe motor, and

is the angle of the motor axis. A gear box is used totransform the rotational motion to linear motion. Due to an imperfection inthe manufacture there is a backlash in the gear box.

Thus the linear position is

=

(

), where

represents a backlash with

=

= 0

02, and its asociated describing function (for

0

02) is

Re

( ) =1

2+ arcsin

1 0

04

+ 2

1 0

04

0

02

1 0

02

Im

(

) =

0

08

1

0

02

(a) Assume that proportional control is used, i.e. ( ) = ( ( ) ( )).How large values of the gain

can be used if a limit cycle is to beavoided according to the describing function method? Compare withsimulations. If the results do not agree, try to explain why.

(b) Assume that the following requirements should be fulfilled:

In the step response (from

to

) the rise time should be

0

1seconds, and the overshoot should be

20%,

the controller

(

) must have integral action,

any latent oscillation in stationarity should have a frequency

5rad/s and an amplitude

0

025 at

.

10

7/29/2019 Homework 2012

11/12

Your task is to do one of the following two alternatives:

(i) Give specifications for the loop gain ( ) ( ) (and thereby im-plicitely for the controller

(

)), based on your knowledge in con-trol theory in general and in the describing function method inparticular, in order to meet these requirements.

(ii) Design a controller that meets the requirements (show this in simu-lations). Also analyse your obtained loop gain using the describingfunction method and compare these results with your simulations.

11

7/29/2019 Homework 2012

12/12

3 Some instructions for how to prepare thereport

The project report should be written carefully, in order to be understood bya person without prior knowledge of the project. The theory that you useshould also be clearly referenced. Summarize important findings in tablesand illustrative plots. Make sure to describe what variables are plotted andin what units. Also try to make the figures readable (eg making curves withdifferent type of lines) on non colored printouts.

Relevant Matlab code should also be provided in electronic form prefer-

ably in an Appendix to the report. Avoid sending Matlab code in a separateemail.

Some general guidelines on how to write reports can be found in thedocument Att skriva en teknisk rapport en kort instruktion (in Swedish)which is available as a pdf-file at the Student Portal.

12