1Solution of State Equations

M. S. FadaliProfessor EE

University of Nevada, Reno

2

Outline Time domain solution for zero-input

response. Properties of matrix exponential. Complete solution using integrating factor. Laplace transform solution of state

equations. Transfer function.

LTI SystemsState-space model

Solve differential equation. Homogeneous equation: time domain series

solution + integrating factor for the complete solution.

Laplace transform solution.

3 4

Time Domain Approach Zero-input (homogeneous) state equation

Differentiate and substitute

Taylor Series (t = 0)

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Example: Diagonal State Matrix

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Numerical Example

Determine the matrix exponential for the matrix

Solution

7 8

Properties of e At Identity at

Proof !

Product Proof

0

2

9Properties of (Cont.) Inverse for any is

Proof:

Transpose

Proof:

Derivative of

Proof

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Commutative Multiplication

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Proof: Assume then

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Proof:

Subtract from both sides

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Complete Solution Multiply by an integrating factor

Integrate both sides from to then

Output Response

Zero-state (output) response (zero ICs)

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Impulse Response

Response due to

Impulse response matrix cols of (resp.)

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Laplace Transform State equation

Use Laplace transform of derivative_

_

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Solve for X(s)

Resolvent Matrix:

We need to inverse Laplace transform the resolvent matrix to obtain the solution.

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Matrix Exponential(state-transition matrix)

1

1

1 1

1

1

_

By analogy with scalar exponential

_

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Complete Solution Inverse Laplace transform (use convolution)

Solution for non-zero initial time

Substitute in the output equation

Example: Matrix Exponential

is in companion form

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Partial Fractions

Partial Fractions

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Partial Fractions: Matrix Coeffts.

Inverse Laplace transform

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Example: Zero-input Response

Example: Zero-state Response

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Exercise

Find the matrix exponential and the response to

You can show that: 1, 4 2 18 4 24 2 1

,

3, 4 3 212 9 68 6 4

, 5, 1 1 14 4 44 4 4

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Linearity Zero-input linearity

Zero-state linearity

Additivity 28

Transfer Function Use formula if given only.

Transform of solution with zero initial conditions

Example: Transfer Function

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0 18 6 01

1 1 Can obtain transfer function by inspection (special case)

1 12 0.54 1 2

1 0.54 2 4

01

0.5 2 1.5 4

1 6 8

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Example: Zero-state Response Obtain output from transfer function

MATLAB>> A=[0,1;-8,-6];B=[0;1];C=[1,1];>> poly(A) % characteristic polynomialans =

1 6 8>> p = ss(A, B, C, 0); >> g= tf(p) % can also use g=zpk(p)s + 1

-------------s^2 + 6 s + 8

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Resolvent Matrix>> n=size(A);

>> resolvent=tf(ss(A, eye(n), eye(n), 0))

Justification:

For

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MATLAB Simulation

>> p = ss(A, B, C, 0); step(p) % (output) step response>> step(ss(A,B,eye(n),0)) %Make C=eye(n) to plot x

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0

0.05

0.1

T

o

:

O

u

t

(

1

)

0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

T

o

:

O

u

t

(

2

)

Step Response

Time (seconds)

A

m

p

l

i

t

u

d

e

Impulse Response Matrix_ _

Example:

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Modes

d ii

i

d i

n

iid

n

j

tj

ji

i

n

i

ti

At

n

iid

n

jijj

i

i

n

ii

in

nnej

tZ

eZe

nnZs

Zs

AIs

1

1

01

1

1 1

11

tymultiplicidistinct,!

distinct

tymultiplicidistinct,1

distinct1

System Modes: or (some books )

Example Repeated eigenvalue

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Modal Vectors Eigenvectors of :

is an eigenvector of with eigenvalues

State stays on eigenvector is it is the initial state.38

Example: Modal Vectors

Similarly, for the second modal vector as the response involves the second mode only.

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Complex Conjugate EigenpairsFor a real matrix , if is an eigenpair, then is an eigenpair.

denotes the complex conjugate

eigenpair:

Complex conjugate:

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MATLAB Example>> a=[0 1;-10,-2]a =

0 1-10 -2

>> [v,d]=eig(a)v =

-0.0953 - 0.2860i -0.0953 + 0.2860i0.9535 0.9535

d =

-1.0000 + 3.0000i 0 0 -1.0000 - 3.0000i

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Complex Modal Vectors

If is in plane then so is

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References1. P. Belanger, Control Engineering, Saunders, Fort

Worth, 1995.2. N. S. Nise, Control Systems Engineering, Wiley,

Hoboken, NJ, 2011.3. R. L. Williams & D. A. Lawrence, Linear State-

Space Control Systems, J. Wiley, Hoboken, NJ, 2011.