Control System I

EE 411

State Space AnalysisState Space Analysis

Lecture 11

Dr. Mostafa Abdel-geliel

Course Contents

� State Space (SS) modeling of linear systems

� SS Representation from system Block Diagram

� SS from Differential equation� phase variables form

�Canonical form�Canonical form

�Parallel

�Cascade

� State transition matrix and its properties

� Eigen values and Eigen Vectors

� SS solution

State Space Definition

• Steps of control system design

– Modeling: Equation of motion of the system

– Analysis: test system behavior

– Design: design a controller to achieve the required – Design: design a controller to achieve the required specification

– Implementation: Build the designed controller

– Validation and tuning: test the overall sysetm

• In SS :Modeling, analysis and design in time domain

SS-Definition

• In the classical control theory, the system model is

represented by a transfer function

• The analysis and control tool is based on classical methods

such as root locus and Bode plot

• It is restricted to single input/single output system• It is restricted to single input/single output system

• It depends only the information of input and output and it

does not use any knowledge of the interior structure of the

plant,

• It allows only limited control of the closed-loop behavior using

feedback control is used

• Modern control theory solves many of the

limitations by using a much “richer” description

of the plant dynamics.

• The so-called state-space description provide the • The so-called state-space description provide the

dynamics as a set of coupled first-order

differential equations in a set of internal variables

known as state variables, together with a set of

algebraic equations that combine the state

variables into physical output variables.

SS-Definition

• The Philosophy of SS based on transforming the equation of motions of order n (highest derivative order) into an nequation of 1st order

• State variable represents storage element in the system which leads to derivative equation between its input and output; it could be a physical or mathematical variablescould be a physical or mathematical variables

• # of state=#of storage elements=order of the system

• For example if a system is represented by

• This system of order 3 then it has 3 state and 3 storage elements

SS-Definition

• The concept of the state of a dynamic system refers to a

minimum set of variables, known as state variables, that fully

describe the system and its response to any given set of

inputs

The state variables are an internal description of the system

which completely characterize the system state at any time t, and

from which any output variables yi(t) may be computed.

The State Equations

A standard form for the state equations is used throughout system dynamics. In the

standard form the mathematical description of the system is expressed as a set of n

coupled first-order ordinary differential equations, known as the state equations,

in which the time derivative of each state variable is expressed in terms of the state

variables x1(t), . . . , xn(t) and the system inputs u1(t), . . . , ur(t).

It is common to express the state equations in a vector form, in which the set of n

state variables is written as a state vector x(t) = [x1(t), x2(t), . . . , xn(t)]T, and the set of r

inputs is written as an input vector u(t) = [u1(t), u2(t), . . . , ur(t)]T . Each state variable is

a time varying component of the column vector x(t).

٩

In this note we restrict attention primarily to a description of systems that are linear and

time-invariant (LTI), that is systems described by linear differential equations with constant

coefficients.

where the coefficients aij and bij are constants that describe the system. This set of n

equations defines the derivatives of the state variables to be a weighted sum of the

state variables and the system inputs.

١٠

where the state vector x is a column vector of length n, the input vector u is a

column vector of length r, A is an n × n square matrix of the constant coefficients

aij, and B is an n × r matrix of the coefficients bij that weight the inputs.

A system output is defined to be any system variable of interest. A description of a

physical system in terms of a set of state variables does not necessarily include all of

١١

physical system in terms of a set of state variables does not necessarily include all of

the variables of direct engineering interest.

An important property of the linear state equation description is that all system

variables may be represented by a linear combination of the state variables

xi and the system inputs ui.

An arbitrary output variable in a system of order n with r inputs may be written:

١٢

١٣

where y is a column vector of the output variables yi(t), C is an m×n matrix of the

constant coefficients cij that weight the state variables, and D is an m × r matrix of the

constant coefficients dij that weight the system inputs. For many physical systems the

matrix D is the null matrix, and the output equation reduces to a simple weighted

combination of the state variables:

Example

Find the State equations for the series R-L-C electric circuit shown in

١٤

Solution:

capacitor voltage vC(t) and the inductor current iL(t) are state variables

)1(*)( Kdt

diLviRtv cs ++=

Prove

Appling KVL on the circuit

The relation of capacitor voltage and current

11dv

then dt

dvci c=

1

212

2

21

)](*[1

)(*

)1(

11

xvy

tuxRxL

x

tviRvdt

dix

equationfrom

xc

icdt

dvx

c

sc

c

==

+−−=

+−−==

===

&

&

&

Example

Draw a direct form realization of a block diagram, and write the

state equations in phase variable form, for a system with the

differential equation

Solution

we define state variables as,13,, 321 uyxandyxyx +=== &&&

١٦

we define state variables as

then the state space representation is

1

123

123

3

32

21

11713197

261319)13(7

261319713

13

xy

uxxx

uxxux

uyyyuyx

uxyx

xyx

=+−−−=

+−−−−=+−−−=−=

−====

&&&&&&&&

&&&

&&

[ ] )(]0[)(001)(

)(

117

13

0

)(

71913

100

010

)(

tutty

tutt

+=

−+

−−−=

x

xx&

Then the model will be

[ ] 0,001

117

13

0

,

71913

100

010

==

−=

−−−=

DC

BA

[ ] )(]0[)(001)(

where

Electro Mechanical System

١٨

State Space RepresentationThe complete system model for a linear time-invariant system consists of:

(i) a set of n state equations, defined in terms of the matrices A and B, and

(ii) a set of output equations that relate any output variables of interest to the state

variables and inputs, and expressed in terms of the C and D matrices.

The task of modeling the system is to derive the elements of the matrices, and to write The task of modeling the system is to derive the elements of the matrices, and to write

the system model in the form:

The matrices A and B are properties of the system and are determined by the

system structure and elements. The output equation matrices C and D are

determined by the particular choice of output variables.

Block Diagram Representation of Linear Systems

Described by State Equations

Step 1: Draw n integrator (S−1) blocks, and assign a state variable to the output of each

block.

Step 2: At the input to each block (which represents the derivative of its state variable)

draw a summing element.

Step 3: Use the state equations to connect the state variables and inputs to the

٢٢

Step 3: Use the state equations to connect the state variables and inputs to the

summing elements through scaling operator blocks.

Step 4: Expand the output equations and sum the state variables and inputs through a

set of scaling operators to form the components of the output.

Example 1

Draw a block diagram for the general second-order, single-input single-output

system:

The overall modeling procedure developed in this chapter is based on the following

steps:

1. Determination of the system order n and selection of a set of state variables

from the linear graph system representation.from the linear graph system representation.

2. Generation of a set of state equations and the system A and B matrices using a

well defined methodology. This step is also based on the linear graph system

description.

3. Determination of a suitable set of output equations and derivation of the

appropriate C and D matrices.

Consider the following RLC circuit

٢٦

We can choose state variables to be

Alternatively, we may choose

This will yield two different sets of state space equations, but both of them have the identical input-output relationship, expressed by

Can you derive this TF?

),(),( 21 tixtvx Lc ==).(ˆ),(ˆ 21 tvxtvx Lc ==

.1)(

)(2

0

++=

RCsLCs

R

sU

sV

Linking state space representation and

transfer function

� Given a transfer function, there exist infinitely many input-

output equivalent state space models.

� We are interested in special formats of state space

representation, known as canonical forms.

٢٧

We are interested in special formats of state space

representation, known as canonical forms.

� It is useful to develop a graphical model that relates the state

space representation to the corresponding transfer function.

The graphical model can be constructed in the form of signal-

flow graph or block diagram.

We recall Mason’s gain formula when all feedback loops are

touching and also touch all forward paths,

Consider a 4th-order TF

012

23

34

0

)()(

)(++++

==asasasas

b

sU

sYsG

gain loopfeedback of sum1gain path forward of Sum

11

−=

−=

∆

∆=

∑

∑∑

=

N

qq

kk

kkk

L

PPT

٢٨

We notice the similarity between this TF and Mason’s gain formula above. To represent the system, we use 4 state variables Why?

40

31

22

13

40

0123

1

)(

−−−−

−

++++=

++++

sasasasa

sb

asasasassU

Signal-flow graph model

This 4th-order system

can be represented by

40

31

22

13

40

1)()(

)( −−−−

−

++++==

sasasasa

sb

sU

sYsG

٢٩

How do you verify this signal-flow graph by Mason’s

gain formula?

Block diagram model

Again, this 4th-order TF

can be represented by the block diagram as shown

40

31

22

13

40

012

23

34

0

1

)()(

)(

−−−−

−

++++=

++++==

sasasasa

sb

asasasas

b

sU

sYsG

٣٠

can be represented by the block diagram as shown

With either the signal-flow graph or block diagram of

the previous 4th-order system,

we define state variables as ,,,, xxxxxxbyx &&& ====

٣١

we define state variables as

then the state space representation is

,,,, 3423120

1 xxxxxxbyx &&& ====

10

433221104

43

32

21

xby

uxaxaxaxax

xx

xx

xx

=+−−−−=

===

&

&

&

&

Writing in matrix form

we have

)()()(

)()()(

ttt

ttt

DuCxy

BuAxx

+=+=&

٣٢

[ ] 0,000

1

0

0

0

,1000

0100

0010

0

3210

==

=

−−−−

=

Db

aaaa

C

BA

When studying an actual control system block diagram, we wish to

select the physical variables as state variables. For example, the block

diagram of an open loop DC motor is

155 −+ s 1−s 16 −s

٣٣

We draw the signal-flow diagraph of each block separately and then connect them. We selectx1=y(t), x2=i(t) and

x3=(1/4)r(t)-(1/20)u(t) to form the state space representation.

15155

−++

s

s121 −+ s

s131

6−+ s

s

Physical state variable model

٣٤

The corresponding state space equation is

x

xx

]001[

)(

1

5

0

500

2020

063

=

+

−−−

−=

y

tr&

Electro Mechanical System

٣٥

Control Flow

٣٨

State-Space Representations in Canonical Forms.

1- Controllable Canonical Form

٤٢

Special Case

Assume

then

٤٤

General Case

٤٥

٤٦

Controllable Canonical Form General case

٤٧

2- Observable Canonical Form

ubudt

dbu

dt

dbu

dt

dbyay

dt

day

dt

day

dt

dnn

n

n

n

n

n

nn

n

n

n

n

n

++++=++++−

−

−

−

−

−

−

−

..........2

2

21

1

102

2

21

1

1

)(.....)()( 222

2

111

1

0 yaubyaubdt

dyaub

dt

du

dt

dby

dt

dnnn

n

n

n

n

n

n

n

−++−+−+=−

−

−

−rearrange

Integrate both side n times

Prove

٤٨

dtyaubdtyaubdtyaububyn

nn∫∫∫ −++∫∫ −+∫ −+= )(.....)()( 22110

Integrate both side n times

ʃ-

+

+

+

y

xn

bo

ʃ-

+

+Xn-1

b1

+

b2

ʃ

an

-

+

bn

x1

u

a2

-

a1

General Form

3- Diagonal Canonical Form

٥٠

General Form

٥١

4- cascade Form

155 −+ s 1−s 16 −s

٥٢

15155

−++

s

s121 −+ s

s131

6−+ s

s

٥٣

The corresponding state space equation is

x

xx

]001[

)(

1

5

0

500

2020

063

=

+

−−−

−=

y

tr&

1- Consider the system given by

Obtain state-space representations in the controllable canonical form,

observable canonical form, and diagonal canonical form.

Controllable Canonical Form:

Examples

٥٤

Controllable Canonical Form:

Observable Canonical Form:

Examples

٥٥

Diagonal Canonical Form:

0

Examples

٥٦

0

1

10 10

Examples

٥٧

0

1

5 . 24. 0

Eigenvalues of an n X n Matrix A.

The eigenvalues are also called the characteristic roots. Consider, for example,

the following matrix A:

٥٨

The eigenvalues of A are

the roots of the

characteristic equation,

or –1, –2, and –3.

Jordan canonical form

If a system has a multiple poles, the state space

representation can be written in a block diagonal form,

known as Jordan canonical form. For example,

Three poles are equal

٥٩

are equal

State-Space and Transfer Function

The SS form

Can be transformed into transfer function

٦٠

Tanking the Laplace transform and neglect initial condition then

)()0()()(

then

(2)

(1) and )()()0()(

sBsAss

(s)D(s)C (s)

sBsAss

UxXX

UXY

UXxX

+=−

+=+=−

( ))()()(

thencondition intial neglectingby

)()0()()(

1

sBsAsI

sBsAss

UX

UxXX

−−=

=−

+=−

( ) )()( 1 sBAsIs UX −−=

( )( ) DBAsIsGss

sDsBAsI

+−==

+−=−

−

1

1

C)()(/)(

)()(CY(s)

2in sub

UY

UU

State-Transition Matrix

We can write the solution of the homogeneous state equation

Laplace transform

٦٢

The inverse Laplace transform

Note that

Hence, the inverse Laplace transform of

٦٣

State-Transition Matrix

Note thatWhere

If the eigenvalues of the matrix A are distinct, than

will contain the n exponentials

٦٤

Properties of State-Transition Matrices.

٦٥

Obtain the state-transition matrix of the following system:

٦٦

٦٧

٦٨

and premultiplying both sides of this equation by

the NON- homogeneous state equation

٦٩

Integrating the preceding equation between 0 and t gives

or

unit-step function

٧٠

٧١

10

٧٢

Prove Transfer function of the given ss

Solution

DBAsICsG +−= −1)()(

Relation of Different SS Representations of the

Same System

For a given system G(s) has two different ss representations

)()()(

)()()(:M:Rep.1

11

111

tDtCt

tBtAt

uxy

uxx

+=+=

)()()(

)()()(:M:Rep.2

22

222

tDtCt

tBtAt

uzy

uzx

+=+=

Let Z=T x

Where T is the transformation matrix between x and z

For example

yyyx

yyyx

yyx

take

&&&&&

&&

+==+==

==

13

22

11

z ,

z ,

z ,

yyyx

yyyx

yyx

take

&&&&&

&&

+==+==

==

33

22

11

z ,

z ,

z ,

122

11

z

z

xx

x

then

+==

233

122

z

z

xx

xx

+=+=

xz

z

T

x

x

x

z

z

z

=

=

=

3

2

1

3

2

1

110

011

001

=110

011

001

T

Sub. By z=Tx in rep. 2

)()()(

)()()()(

Tby mutiply )()()()(

22

2

1

2

111

-1

22

tDtTCty

tBTtTATtTTtzT

tBtTAtTtz

ux

uxx

uxx

+=+==

+==−−−−

&&

&

)()()( 22 tDtTCty ux +=

Compare with M1;rep.1

)()()(

)()()(:M:Rep.1

11

111

tDtCt

tBtAt

uxy

uxx

+=+=

then

21

21

2

1

1

2

1

1

DD

TCC

BTB

TATA

====

−

−

21

1

12

12

1

22

DD

TCC

TBB

TTAA

====

−

−

State-Space Diagonalization Function

Eign values and eign vectorsDefinition: for a given matrix A, if ther exist a real (complex)

λ and a corresponding vector v≠0, such that

vv λ=A

٧٦

Then λ is called eign value and v is the eign vector

i.e.

And since v≠0

Then

i.e

vv λ=A

0)( =− vIA λ

0)( =− IA λ0)det( =− IA λ

Eigenvalues of an n X n Matrix A.

The eigenvalues are also called the characteristic roots. Consider, for example,

the following matrix A:

٧٧

The eigenvalues of A are

the roots of the

characteristic equation,

or –1, –2, and –3.

Example

−=

28

10A

0)2(8

1

0A-I ofsolution theis eign value then the

=+−

−=−

=

λλ

λ

λ

AI)2(8 +− λ

24

)2)(4(082

21

2

=−=

−+==−+

λλ

λλλλ

and

then

−=∴

=

−−−−

==

028

14

..

0A)-I(

-4

12

11

11

v

v

ei

v

at

λλ

=

−−

==

048

12

..

0A)-I(

2

22

21

22

2

v

v

ei

v

at

λλ

Eign vectors are obtained as

−=

−=∴

4

1

4

1

1112

vlet

vv

=

=∴

2

1

2

2

222

22

vlet

vv

][ 21 nV vvv L=Eign vector matrix

For all eign values and vectors

niA iii ,,1,0; K== vv λThese equations can be written in matrix form

Λ= VVAwhere

n ][ 21 L= vvvV

{ }nidiag i

n

n

,,2,1,

00

00

00

][

2

1

21

L

L

MOMM

L

L

L

==

=Λ

=

λ

λ

λλ

vvvV

thus

1−Λ= VVA

VV A1−=Λ

thus

1

22 ...

!2)(

−Λ=

+++==

VVee

tAAtIte

tAt

At φ

),...,2,1,(

...!2

)(

2

1

22

niediage

e

e

ttIte

t

t

t

t

t

i ==

=

+Λ+Λ+==

−Λ

Λ

λλ

λ

φ

O

e tn

λ

O

Then for a given system has a system matrix A and a state vector X

The diagonal system matrix Ad and state Xd

21

1

1

1

1

1

1

;

DD

TCC

BVB

AVVA

matrixvectoreignVT

TT

ATTA

d

d

d

dd

d

====

====

=Λ=

−

−

−

−

xxxx

Example 2: find the transformation into diagonal form and the state transition matrix of example1

1

2

4

0

0

20

04

−Λ

−Λ

=

=

−=Λ

VVee

e

ee

tAt

t

t

t

= VVee

−

−=

−

−=

−

−−

14

12

0

0

24

11

6

1

24

11

0

0

24

11

2

4

1

2

4

t

t

At

t

t

At

e

ee

e

ee

Discus how to obtain the transformation matrix between two representation

Diagonal Canonical Form

٨٣

Alternative Form of the Condition for Complete State Controllability.

If the eigenvectors of A are distinct, then it is possible to find a transformation matrix

٨٤

If the eigenvectors of A are distinct, then it is possible to find a transformation matrix

P such that