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Control Engineering

Lecture# 629thMarch,2008

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State Variable Equations Computer-aided analysis and design of state

variable models are done more easily on computers

for high-order systems (differential equations of high

orders.)

More information (internal variables) are involved

hence better control when needed.

Design procedures that give the best control

system are often based on state variable models.

State variable models are required for digital

simulation.

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State Variable Equations

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High order D.Eq. => T.F. => Signal Flow Graph

(Simulation diagram) => State Variable Eqs.

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Review:

State Variable Models

This equation gives the position of y(t)as a function of force f(t).Suppose we

also want information about the velocity.

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Review:

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Review:

Where X(t) = state vector = (n by 1) vector of the

states of an n-th order system.

X(t) is the time derivative of the vector X(t)

A = (n by n) system matrix

B = (n by r) input matrix

u(t) = input vector = (r by 1) vector of the systeminput functions

Y(t) = output vector = (p by 1) vector of the

defined outputs C = (p by n) output matrix

D = (p by r) matrix representing direct coupling

between input and output.

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State Diagram

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Review:

Simulation diagram:

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Review:

Signal flow graph from the above graph:

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More Example

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State variable equations:

CXty

BuAXX

xbty

uxaxaxaxax

xx

xxxx

)(

'

)(

'

'

''

10

433221104

43

32

21

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Review:

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Solutions of state equations:

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23

1

)(

)()(

2

sssU

sYsG

An example:

Note 1: The same transfer function can give different signal flowgraphs (see the signal flow graph here and the one 3 slides back.)

Note 2: Different signal flow graphs may give different differential

equations (compare the d.eqs. below with the d.eqs. 3 slides back.)

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Note 3: Although with different signal flow graphs anddifferent d.eqs. the final outcome (y(t)) is the same(compare the above solution for y(t)= here with the

solution for y(t) given before, assuming all initial conditionsare zero.)

)(1 tx

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More Examples:

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4321

1

43211

34321143232

343213

1432

2321

43211

)(

1

1

GGGGsG

GGGGM

HGGGGHGGHGG

HGGGGL

HGGL

HGGL

GGGGP

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Another Example:

N i l S l ti f th St t

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Numerical Solutions of the State

Equations

State equation: x(t) = Ax(t) + Bu(t)

Let h= integration increment (or time-step)

x(kh) = x[(k-1)h] + hx[(k-1)h] (1)

x[(k-1)h] = Ax[(k-1)h] + Bu [(k-1)h] (2)

Algorithm:

1. k= 1

2. Evaluate x[(k-1)h] as in (2)

3. Evaluate x(kh) as in (1)

4. k = k+1

5. Go to step 2.

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