3. State Vaiable Analysis of Discrete-data Systems

8/2/2019 3. State Vaiable Analysis of Discrete-data Systems

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Note: the matrices G and H in the discrete version can be

computed from its contimuous-time analog A and B. For that we

have to compute the matix exponentialAt

e This can be computed easily using Caylay Hamilton theorem.

_____________________________________________Caylay Hamilton theorem

Theorem states that every square matrix satisfies its owncharacteristic equation.

Steps to compute f(A)

1. Evaluate the eigen values 1,2,...n of A from

2. Obtain the corresponding scalar function f(i)

Comments:

If the eigen values are getting repeated to get the additionalequation in step4,the equation should be differentiated bothsides.

Iff(A)is a function of tthen the coefficients in step4 will befunctions of t, otherwise constants.

_________________________________________________

0ASI

g(A)f(A)asf(A)cmputeThen5.

tscoefficientheforsolveand)f()g(Equate4.

)g(Let3.

iii

1n

i1n

3

i3

2

i2i10i


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Example

tttt

tttt

tttt

tt

tt

tt

At

tt

tt

t

t

At

eeee

eeee

eeee

ee

ee

ee

eAIAf

ee

ee

eg

e)g(

g

givesAI

AifeCompute

22

22

22

2

2

2

10

20

21

210

10

10

21

222

2

3322

0

20

02

)(

2

2)2(

1

)(

210

32

10

Example

Given the continuous-time system

x(t)ty;tux(t)tx 01)()(1

0

50

10)(

Eigen values are 50, 21

Obtain the discrete-time model.


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Ans.

ComputeAt

e using Cayley-Hamilton theorem

t

tAt

eee5

5

51

0)1(1

T

TAT

e

eeG

5

5

51

0

)1(1

)1(

)2

1(

5

51

5

51

0 T

T

T A

e

eTBdeH

For T=0.1 sec.

0.60650

0.07871G ;

0.0787

0.0043H

and the discrete data system is

x(t)y(t)u(t) ;.

.x(t)

.

.)x(k 01

07870

00430

606500

0787011

_________________________________Matlab code

A=[0 1;0 -2]

B=[0;1]

T=1

[G,H]=c2d(A,B,T)


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__________________________________

Example

Consider the system

54

1)(

2

35.0

ss

sesT s

with input delay 0.35 second. To discretize this system using the triangle

approximation with sample time 0.1 second, type

_________________________________Matlab code

H = tf([1 -1],[1 4 5],'inputdelay',0.35)

Hd = c2d(H,0.1,'foh')

step(H,'-',Hd,'--')

_________________________________ Specifying Discrete-Time Models

Recognizing Discrete-Time Systems

sys = ss(.5,1,.2,0,0.1)

step(sys)
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2. State variable analysis of digital control systems

The dynamics of the linear time-invariant system is described bythe equations

)(

)(

)(

)(

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

2

1

0

kx

kx

kx

kx

kDukCxkyxx;kHukGxkx

n

3. Solution of the state difference equation

Method 1:Directly by recursion

The solution can be obtained directly by recursion as follows:

1

0

1

021

23

2

)()0(

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

)2()1()0()0()3()1()0()0(G

Hu(1)Hu(0)]G[Gx(0)

)1()1()2(

)0()0()1(

k

i

ikk

kkk

iHuGxG

kHuGHuGHuGxGkx

HuGHuHuGxGxHuGHux

HuGxx

HuGxx

0x(0)x);()()1( kHukGxkx


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Clearly the solution consists of two parts. One due to initialconditionx(0) and the other due to the input u(i); i=0,1,2, (k-1)

So the state transition equation will become

1

0

10k

i

i)Hu(i)(k)(k)x(x(k)

For LITV systems KGK )( iK

GiK 1)1(

Method 2: By using z-transforms

Taking the z-transforms of the discrete state equation

DHGzI

zI-GC

DHGzIC

zUDHGzICxzGzICzY

zHUGzIxzGzIzX

zHUzxzXGzI

zHUzGXzxzzX

)(

)(U(z)

Y(z)

havewe,conditionsinitialtheNeglecting

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

)()()0()()(

)()0()()(

)()()0()(

1

11

11

Now the solution can be obtained by taking the inverse z-transforms.

)()()0()()()()()0()()(

1111

11

zHUGzIZxzGzIZkx

zHUGzIxzGzIzX


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An example

Consider the matrix

116.0

10G

kkkk

kkkk

k

z

z

z

z

z

z

z

zz

z

z

z

z

z

z

z

L

zGzIZGk

)8.0(3

4)2.0(

3

1)8.0(

3

8.0)2.0(

3

8.0)8.0(3

5

)2.0(3

5

)8.0(3

1

)2.0(3

4

8.03

4

2.03

1

8.03

8.0

2.03

8.08.03

5

2.03

5

8.03

1

2.03

4

])[()(

1

11

ExerciseObtain the above result using Cayley Hamilton method.

ExampleObtain the discrete-time state and output equations and pulsetransfer function of the following continuous-time system given by

)2(

1

)(

)()(

sssU

sYsG assume the time period T=1sec.

Ans.The continuous-time representation of the above equation is

2

1

2

1

2

1

01

1

0

20

10

x

xy

ux

x

x

x


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T

TAT

e

eeTG

2

2

21

0

)1(1)(

)1(2

1

)2

1

(

1

0)1()(2

2

21

02

221

0 T

T

T

T

TT A

e

e

Td

e

edBeTH

With T=1sec.

kx

kxy(t)

;ku.

.

kx

kx

.

.

kx

kx

)(

)(01

)(43230

28380

)(

)(

135300

432301

)1(

)1(

2

1

2

1

2

1

The pulse transfer function can be obtained as

DHGzI

zI-GC

DHGzICzf

)(

)(U(z)

Y(z)

)(1


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Liapunov stability analysis

In this section, we extend the Liapunov stability analysis to the

autonomous discrete time system given by

001 )f(f(x(k));)x(k

If there exist a scalar function a scalar function V(x(k)) which, for

some real number 0 satisfies the following properties for all x

in the region x :

(1) V(x(k)) is positive definite

(2) V(x(k)) has continues partial derivatives w.r.t all its

components

Then the equilibrium state at the origin is

(a) stable if the difference V(x(k))]-1))V(x(k[V(x(k)) is a

negative semi-definite scalar function;(b) Assypmtotically stable, if V(x(k)) is a negative definite scalar

function;

OR

If V(x(k)) is a negative semi-definite scalar function;

and V(x(k)) does not vanish along on the trajectories of

the system other than at 0x(kT)

(c) Globally asymptotically stable, if the system is asymptotically

stable, in addition,

xas)x(V ; i.e., V(x) radially bounded.


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Stability of LTIV systems

Consider LTIV autonomous system

P)GP(GQWhere

x(k)Q(k)-x

P)x(k)PG(G(k)x

x(k)P(k)xGx(k)PX(k))(G

x(k)Px(k)1)x(kP1)(kx

V(x(k))-1))V(x(kV(x)

defferenceThe

PxxV(x)

asfunctionLiapunovpossibleachooseWeGx(t)1)x(k

T

T

TT

TT

TT

T

Hence for asymptotic stability of the origin it is sufficient that Q

be positive definite. So we can choose any positive definite matrix

P and see whether or not Q is positive definite.

Instead of choosing P first and checking for the positive

definiteness of Q, It is possible to select any +ve definite matrix Q

and solve the equation for P. This will make the condition

necessary and sufficient.

P)PG(GQ T


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Theorem

Consider the system described by

Gx(k)1))x(K

Where x(k) is the n-dimensional state vector and G is the nxn non-

singular system matrix. A necessary and sufficient condition so

that the equilibrium state at the origin x=0 is ASIL is that given any

+ve definite real symmetric matrix Q, there exists a positive

definite real symmetric matrix P such that

The scalar function V(x)= PxxT is a Liapunov function for this

system.

The following remarks are in order

1.If the origin of the linear system is stable it is automatically ASIL.

2.If x(k)Q(k)xV(x)T

does not vanish identically along any

trajectory, Q chosen may positive semi definite. It can be shown

that if Q does not vanish along any trajectory if

nAAArank

T

n

121

221

21

21

QQQQ

3.Since the end result does not depend on choice of Q we need

select Q=I, the identity matrix.

4.Only n(n+1)/2 equation must be solved to solve the equation

P)PG(GQ T

P)GP(GI T


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5.In all above, T(.) should be replaced by (.) if the matrices

involved are complex ones.

ExampleConsider the system

)(

)(

15.0

10)1(

2

1

kx

kxkx

Determine the stability of the origin.Ans.

10

01

15.0

10

11

5.00

2212

1211

2212

1211

pp

pp

pp

pp

Solving forp,

0;0524

58

58511

511

524

58

58

511

2212

1211

pp

pp

Now, applying Sylvesters criterion the matrixp is positive definite

and the system is stable.

Exercise

In the above example assume

00

01q ,

a positive semi-definite matrix and obtain the Liapunov function and

show that the trajectories does not vanish other than at the origin.


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ProblemProve that if all solutions of

Gx(t)1)x(k tend to zero as k , then all the solutions of

Hu(k)Gx(t)1)x(k are bounded provided that the input vector u(k) is bounded.

Ans.Since u(k) is bounded, there exists a positive constant c such that

cku )( for all k

The solution of the state equation is given by

1

0

1)()0()(

k

i

ikkiHuGxGkx

Hence,

1

0

1

1

0

1

lim)0(

)0()(

k

i

ik

k

k

k

i

ikk

HGcxG

HGcxGkx

Since the origin of the homogeneous system is asymptoticallystable, there exists positive constants a and b (0 < b < 1) such that

kabkG )(0

Then

1

0

1

0

11

1

1

limlim

k

i

k

i

ik

k

ik

k babacGc

Therefore

baHcxakx

1

1)0()(

Thus )(kx is bounded.

3. State Vaiable Analysis of Discrete-data Systems

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