1

State-Space:Controllably & Observability

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds)

Lecture 23

elec3004@itee.uq.edu.auhttp://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

May 23, 2013

Today in Linear Systems…

ELEC 3004: Systems 24 May 2013 - 2

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

1222-MayState-Space

24-MayControllability & Observability13

29-MayInformation Theory/Communications & Review31-MaySummary and Course Review

2

Today:• State-Space

• Compensator Design

Friday:• Controllability

• Observability

ELEC 3004: Systems 24 May 2013 - 3

Goals for the Week

Affairs of state• Introductory brain-teaser:

– If you have a dynamic system model with history (ie. integration) how do you represent the instantaneous state of the plant?

Eg. how would you setup a simulation of a step response, mid-step?

t = 0t

start

24 May 2013 -ELEC 3004: Systems 4

3

Introduction to state-space• Linear systems can be written as networks of simple dynamic

elements:

2

7 1224

13

7

1

12

2

u y

24 May 2013 -ELEC 3004: Systems 5

Introduction to state-space• We can identify the nodes in the system

– These nodes contain the integrated time-history values of the system response

– We call them “states”

7

1

12

2

u yx1 x2

24 May 2013 -ELEC 3004: Systems 6

4

• u: Input:

• x: State:

• y: Output

ELEC 3004: Systems 24 May 2013 - 7

State-Space Terminology

ELEC 3004: Systems 24 May 2013 - 8

State-Space Terminology

5

• If the system is linear and time invariant, then A,B,C,D are constant coefficient

ELEC 3004: Systems 24 May 2013 - 9

LTI State-Space

• If the system is discrete, then x and u are given by difference equations

ELEC 3004: Systems 24 May 2013 - 10

Discrete Time State-Space

6

• Series:

ELEC 3004: Systems 24 May 2013 - 11

Block Diagram Algebra in State Space

• Parallel:

ELEC 3004: Systems 24 May 2013 - 12

Block Diagram Algebra in State Space

7

State-space representation• State-space matrices are not necessarily a unique

representation of a system– There are two common forms

• Control canonical form– Each node – each entry in x – represents a state of the system

(each order of s maps to a state)

• Modal form– Diagonals of the state matrix A are the poles (“modes”) of the

transfer function

24 May 2013 -ELEC 3004: Systems 13

Control canonical form

• CCF matrix representations have the following structure:

⋯1 0 0 0 00 1⋮ ⋱ ⋮

1 0 00 0 ⋯ 0 1 0

Pretty diagonal!

24 May 2013 -ELEC 3004: Systems 14

8

State variable transformation• Important note!

– The states of a control canonical form system are not the same as the modal states

– They represent the same dynamics, and give the same output, but the vector values are different!

• However we can convert between them:– Consider state representations, x and q where

x = Tq

T is a “transformation matrix”

24 May 2013 -ELEC 3004: Systems 15

State variable transformation• Two homologous representations:

and

We can write:

Therefore, and

Similarly, and

24 May 2013 -ELEC 3004: Systems 16

9

Controllability matrix

• To convert an arbitrary state representation in F, G, H and J to control canonical form A, B, C and D, the “controllability matrix”

⋯must be nonsingular.

Why is it called the “controllability” matrix?

24 May 2013 -ELEC 3004: Systems 17

Controllability matrix

• If you can write it in CCF, then the system equations must be linearly independent.

• Transformation by any nonsingular matrix preserves the controllability of the system.

• Thus, a nonsingular controllability matrix means x can be driven to any value.

24 May 2013 -ELEC 3004: Systems 18

10

Why is this “Kind of awesome”?• The controllability of a system depends on the particular set of

states you chose

• You can’t tell just from a transfer function whether all the states of x are controllable

• The poles of the system are the Eigenvalues of F, ( ).

24 May 2013 -ELEC 3004: Systems 19

State evolution• Consider the system matrix relation:

The time solution of this system is:

If you didn’t know, the matrix exponential is:12!

13!

⋯

24 May 2013 -ELEC 3004: Systems 20

11

Stability• We can solve for the natural response to initial conditions :

∴

Clearly, a system will be stable provided eig 0

24 May 2013 -ELEC 3004: Systems 21

Characteristic polynomial

• From this, we can see or, I 0

which is true only when det I 0Aka. the characteristic equation!

• We can reconstruct the CP in s by writing:det I 0

24 May 2013 -ELEC 3004: Systems 22

12

Great, so how about control?• Given , if we know and , we can design a

controller such thateig 0

• In fact, if we have full measurement and control of the states of , we can position the poles of the system in arbitrary locations!

(Of course, that never happens in reality.)

24 May 2013 -ELEC 3004: Systems 23

Example: PID control• Consider a system parameterised by three states:

– , ,– where and

=1

12

0 1 0 0

is the output state of the system; is the value of the integral;

is the velocity.

24 May 2013 -ELEC 3004: Systems 24

13

• We can choose to move the eigenvalues of the system as desired:

det1

12

All of these eigenvalues must be positive.

It’s straightforward to see how adding derivative gain can stabilise the system.

24 May 2013 -ELEC 3004: Systems 25

Just scratching the surface

• There is a lot of stuff to state-space control

• One lecture (or even two) can’t possibly cover it all in depth

Go play with Matlab and check it out!

24 May 2013 -ELEC 3004: Systems 26

14

Discretisation FTW!• We can use the time-domain representation to produce

difference equations!

Notice is not based on a discrete ZOH input, but rather an integrated time-series.We can structure this by using the form:

,

24 May 2013 -ELEC 3004: Systems 27

Discretisation FTW!• Put this in the form of a new variable:

Then:

Let’s rename and

24 May 2013 -ELEC 3004: Systems 28

15

Discrete state matricesSo,

1

Again, 1 is shorthand for

Note that we can also write as:

where

2! 3!⋯

24 May 2013 -ELEC 3004: Systems 29

Simplifying calculation• We can also use to calculate

– Note that:

Γ1 !

itself can be evaluated with the series:

≅2 3

⋯1

24 May 2013 -ELEC 3004: Systems 30

16

State-space z-transformWe can apply the z-transform to our system:

which yields the transfer function:

24 May 2013 -ELEC 3004: Systems 31

State-space control design

• Design for discrete state-space systems is just like the continuous case.– Apply linear state-variable feedback:

such that detwhere is the desired control characteristic equation

Predictably, this requires the system controllability matrix ⋯ to be full-rank.

24 May 2013 -ELEC 3004: Systems 32

17

• Final Exam:Saturday, June 15 at 9:30 AM (sorry!)

• Problem Set 2 is due this Friday!

ELEC 3004: Systems 24 May 2013 - 33

Announcements:

!

• AKA: I can see that. Yes, I can control that!

ELEC 3004: Systems 24 May 2013 - 34

Next Time in Linear Systems ….1

27-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

1222-MayState-Space

24-MayControllability & Observability13

29-MayInformation Theory/Communications & Review31-MaySummary and Course Review