1Discrete-Time State-Space EquationsM. Sami Fadali

Professor of Electrical EngineeringUNR

2

Outline Discrete-time (DT) state equation from

solution of continuous-time state equation. Expressions in terms of constituent

matrices. Solution of DT state equation. Example.

3

Solution of State Equation Analog systems with piecewise constant inputs over

a sampling period: relate state variables at the end of each period by a difference equation.

Obtain difference equation from the solution of the analog state, over a sampling period .

Solution of state equation for initial time , and final time

= state vector at time

Piece-wise Constant Input1. Move input outside the integral.2. Change the variable of integration

Discrete-time state equation

4

State & Input Matrices discrete state matrix discrete input matrix

(same orders as their continuous counterparts). Discrete state matrix = state transition matrix of

the analog system evaluated at the sampling period .

Properties of the matrix exponential: integral of the matrix exponential for invertible matrix

5 6

Constituent Matrices Use expansion of the matrix exponential in

terms of the constituent matrices. Eigenvalues of discrete state matrix related

to those of the analog system.

7

Input Matrix

Scalar integrands: easily evaluate integral.

1

, 0

1

, 0

Assume distinct eigenvalues (only one zero eigenvalue)8

Discrete-time State-space Representation

Discrete state & output equation. Discrete-time state equation:

approximately valid for a general input vector provided that the sampling period is sufficiently short.

Output equation evaluated at time

Example 7.15 Obtain the DT state equations for the system of

Example 7.7

for a sampling period T=0.01 s. Solution: From Example 7.7, the state-transition

matrix is

10 11 10 0 00 0 0

10

0 10 10 10 10 10 1

9

0 1 10 10 100 100 100

90

9

Discrete state matrix

. .

.

Simplifies to

10

11

Discrete-time Input matrix

Simplifies to

01.01001.0

01.0103

01.021

1901100

101

19101

11

0001.0

101101.0

ee

eBZeBZBZBd

12

MATLABMATLAB command to obtain (Ad , Bd , C, D)

form (A, B, C, D) pd = c2d(p,0.01)Alternatively the matrices are obtained using

the MATLAB commands ad = expm(a * 0.05) bd = a\ (ad-eye(3) )* b

13

Solution of DT State-Space Equation

DT State Equation: state at time k in terms of the initial condition vector x(0) and the input sequence u(k), k = 0, 1, 2,..., k1.

At k = 0, 1, we have

)1()0()0()1()1()2()0()0()1()()()1(

2 uuxuxxuxxuxx

dddd

dd

dd

dd

BBAA

BABA

kBkAk

14

Solution by Induction

State-transition matrix for the DT system. State-transition matrix for time-varying DT system:

not a matrix power dependent on both time k and initial time k0 .

Solution=zero-input response+ zero-state response

1

0

1

12

0

122

)()0()(

)()0()2(k

id

ikd

kd

id

idd

iBAAk

iBAA

uxx

uxx

x x u( ) ( ) ( )k A k A B idk k dk i di k

k

00

01

1

15

Output Solution Substitute in output equation

)()()0(

)()()(1

0

1 kDiBAAC

kDkCkk

id

ikd

kd uux

uxy

16

Z-Transform Solution of DT State Equation

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

1 zBzAIzz

zBzAzzz

ddn

dd

UxXUXxX

......

1

221

11

iidddn

dndn

zAzAzAI

Az

IzAIz

z-transforming the discrete-time state equation

17

Inverse z-transformInverse z-transform [z In Ad ]1z

0

211

2211

,...,...,,,......

kkd

idddndn

iidddndn

A

AAAIzAIz

zAzAzAIzAIz

Z

Analogous to the scalar transform pair

0kkdd aazz Z

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Matrix Inversion Evaluate using the Leverrier algorithm. Partial fraction expansion then multiply by z.

n

ii

i

nnn

nn

dn

Zz

z

zzazaazPzPzPzAIz

1

1110

12

101...

...

ki

n

ii

kd ZA

1

19

DT State Matrix

Parentheses: pertaining to the CT state matrix A.Equality for any sampling period T and any matrix A

Same constituent matrices for DT state matrix & CT state matrix A DT eigenvalues are exponential functions of the CT eigenvalues times the sampling period.

A Z Z A ed ii

n

i ii

nA Ti

1 1

Z Z A

e

i i

iA Ti

20

Zero-state Response

Known inverse transform for {.} term. Multiplication by z 1: delay by T.Convolution theorem: inverse of product is the

convolution summation

1

0

1

1

)()(k

id

ikd

kTAn

ii

kd

iBAt

eZA

ZS

i

ux

X UZS z z I A z z B zn d d( ) ( ) 1 1

21

Alternative Expression

n

j

k

i

iTATkAdj

k

id

TikAn

jj

ieeBZt

iBeZt

jjZS

jZS

1

1

0

1

1

0

1

1

)()(

)()(

ux

ux

Useful when the summation over i can be obtained in closed form.

22

Example 7.16(a) Solve the state equation for a unit step input

and the initial condition vector x(0) = [1 0]T(b) Use the solution to obtain the discrete-time state

equations for a sampling period of 0.1s. (c) Solve the discrete-time state equations with the

same initial conditions and input as in (a) and verify that the solution is the same as that of (a) evaluated at multiples of the sampling period T.

uxx

xx

10

3210

2

1

2

1

23

Example Solution

( )ss

s

ss

s s

s

s s

1

2 3

3 123 2

1 00 1

3 12 0

1 21

2

2

21

121

21

11

211

sssss

ssss

(a) The resolvent matrix

Partial fraction expansions

24

State-transition Matrix

22211

112

122

0213

10012

10213

1001

)(

ss

sss

( )t e et t

2 12 1

1 12 2

2

25

Zero-input Response

tt

tt

ee

eetZI

2

2

21

22

01

2211

1212)(

x

26

Zero-state Response

tete

teet

tt

ttZS

12111

1

110

2211

1212)(

2

2

x

221

11

021

21

2111

1)(2

2

tt

tt

ee

eetZS

x

27

Total Response

221

11

021

221

11

021

21

22)(

2

22

tt

tttt

ee

eeeet

x

)()()( ttt ZSZI xxx

At the sampling points: t = multiples of 0.1s28

(b) Discrete-time state equations A e ed

( . )

. .. .

. .01 2 12 11 1

2 20 9909 0 086101722 0 7326

0 1 2 0 1

B e e

e e

d

2 12 1 1

1 12 2

12

01

1 20

11

12 2

0 00450 0861

0 12 0 1

0 10 2

..

.. .

.

CT system response to a step input of duration one sampling period, is the same as the response of a system due to a piecewise constant input

)0()0()1( uxx dd BA

29

Zero-input Response A k e ed

k k k

( . ) . .01 2 12 11 1

2 20 1 0 2

x xZI k k

e e

e e

k k

k k

( ) ( . ) ( )

. .

. .

01 02 12 1

1 12 2

10

22

12

0 1 0 2

0 1 0 2

Same as the zero-input response of the continuous-time system at all sampling points k = 0, 1, 2, ...

tt eetZI2

21

22)(

x 30

z-transform of the zero-state response

( ) . .zz

z ez

z e

2 12 1

1 12 20 1 0 2

121100635.9

111105163.9

10861.00045.0

2211

1212

)()()(

2.02

1.02

12.01.0

1

zezz

zezz

zzz

ezz

ezz

zUBzzz dZSX

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Partial Fractions

zz e z e

zz

zz e

zz

zz

0 1 0 1 0 111

1 11

105083 11

0 9048

. . .

. .

8187.01

15167.5

111

11 2.02.02.0

zz

zz

ezz

zz

ezezz

32

Expand zero-state response 2.01.0 2

15.011

105.0)(

ezz

ezz

zzzzsX

kk eekzs2.01.0

215.01

105.0)(

x

Identical to the zero-state response for the continuous system at time t = 0.1 k, k = 0,1,2,...

221

11

021)(

2tt eetZS

x

33

Zero-state response

a aa

ik

i

k

110

1

x ZS jj

j

j

j

k Z B eee

Z Bee

j dA k T

A kT

A Tj

n

j d

A kT

A Tj

n

( )

1

1

1

11

11

n

j

k

i

iTATkAdj ieeBZt jjZS

1

1

0

1 )()( ux

z-Transfer Function

For zero initial conditions

Substitute

34

Impulse Response & Modes

Inverse transform of the transfer function

Z

, 1, 0

Substitute in terms of constituent matrices

35

0,1,)(1)(

1

11 kD

kBZCkGDz

BZCzGkid

n

ii

id

n

ii

Z

36

Poles and Stability poles= eigenvalues of discrete-time state

matrix = exponential functions of i(A) (continuous-time state matrix A).

For stable A, i(A) have negative real parts and i have magnitude less than unity.

Discretization yields a stable DT system for a stable CT system.

37

Minimal Realizations Product C Zj B can vanish & eliminate eigenvalues from the transfer function: if C Zj =0, Zj B =0, or both. If cancellation occurs, the system is said to have an

C Zj =0: output-decoupling zero at j Zj B =0: an input-decoupling zero at j C Zj =0 Zj B =0: an input-output-decoupling zero at j

Poles of the reduced transfer function are a subset of the eigenvalues of the state matrix Ad .

A state-space realization that leads to pole-zero cancellation is said to be reducible or nonminimal.

If no cancellation occurs, the realization is said to be irreducible or minimal.

38

Decoupling Modes Output-decoupling zero at j: the forced system response does not include the mode jk. Input-decoupling zero: the mode is decoupled

from or unaffected by the input. Input-output-decoupling zero: the mode is

decoupled both from the input and the output. These properties are related to the concepts of

controllability and observability discussed later in this chapter.

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Example 7.17 Obtain the z-transfer function for the position

control system of Example 7.16

(a) With x1 as output. (b) With x1 + x2 as output.

The resolvent matrix and input matrix were obtained in Example 7.16.

40

Solution( ) . .z

zz e

zz e

2 12 1

1 12 20 1 0 2

B ee

d

1 2

011

12 2

0 00450 0861

0 10 2

.. .

.

C D 1 0 0

2.0

2

1.0

2

2.01.0

100635.9105163.90861.00045.01

22111

121201)(

ezez

ezezzG

(a) Output x1

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(c) Output x1+x2

2.0

2

1.0

2.01.0

100635.900861.00045.01

22111

121211)(

ezez

ezezzG

1. Output-decoupling zero at . since 2. System response to any input does not include

the decoupling term.

42

Step Response

8187.01

15.01

11100635.9

1100635.9)(

2.02.0

2

2.0

2

zz

zz

ezz

zz

e

zezzzY

Z-transform inverse.

43

z-Transfer Function: MATLAB Let T =0.05s P = ss(Ad, Bd, C, D, 0.05) g = tf(P) % Obtain z-domain transfer function zpk(g) % Obtain transfer function poles and zeros The command reveals that the system has a zero

at 0.9048 and poles at (0.9048, 0.8187) with a gain of 0.09035.

With pole-zero cancellation, the transfer function is the same as that of Example 7.17(b).

minreal(g) % Cancel poles and zeros