Lcs2

1

INTRODUCTION

2

LINEAR CONTROL SYSTEMS

3

4

ECI 660 LINEAR CONTROL SYSTEMS 3(3, 0)Course Contents

System Definition, Control System Archetechture, Input/Output System Models, Basic System Properties, Continuous and Discrete-Time Systems, System Modeling, State Space Representation, State Equations, System Analysis, Equivalence, Canonical Forms, Realizations, Stability, Sensitivity, Disturbance Rejection, Linearization, Controllability and Observability, Rank Tests, Linear Feedback Control, Fixed-Order Compensators (System Augmentation), Controller / Compensator Applications, Design Techniques: Root Locus, Frequency Response and Pole Placement, LQ Control, Luenberger Observers, Separation Principle (Estimated State Feedback).

5

TEXT AND REFERENCE BOOKS

Text Books Chi-Tsong Chen : Linear System Theory and Design, 3rd edition, Oxford

University Press

Charlles L. Phillips, Royce D. Harbor : Feedback Control Systems, 4th edition, New Jersey: Prentice Hall

Gene F. Franklin, J. David Powell: Feedback Control of Dynamic Systems, 5th edition, New Jersey: Prentice Hall

Reference Books Katsuhiko Ogata : Modern Control Engineering, 5th edition, PHI Learning

Callier, F. M., Desoer, C. A: Linear System Theory, New York, Spronger-Verlag

Rugh, W : Linear System Theory, 2nd edition, New Jersey: Prentice Hall

6

CONTROL SYSTEM

Definition

A control System is an interconnection of components to

provide a desired function. The portion of the system to be

controlled is called the plant (process, system), and the part

doing the controlling is called the controller (compensator,

filter). A control system designer has little or no design

freedom with the plant; as it is fixed. The designer’s task is to

develop a controller that will control the given plant

acceptably.

7

ARCHITECTURE OF CONTROL SYSTEMS

Feed-forward or open-looped control system

Input Output

Feedback or closed-looped control system

Input Output

(unity-gain feedback)

Controller

Plant

Controller

Plant

8



Input Output

(non-unity-gain feedback)

Controller

Plant

9

DIGITAL CONTROL SYSTEMS

Feedback control system using Digital Controller

Disturbances

(Desired response) (error Signal) (Response)

Input r(t) e(t) c(t) Output

(Sensor output)

Disturbances


DigitalControll

erPlant

Sensor

A/D D/A

10

CONTROL SYSTEM DEFINITIONS

SISO System A system with only one input and one output is called single-input single-

output system.

MIMO System A system with two or more input terminals and two or more output

terminal is called multivariable system or multi-input multi-output system.

Continuous-time System A system is called continuous-time system if it accepts continuous-time

signals as input and generates continuous-time signals as its output.

Discrete-time System A system is called discrete-time system if it accepts discrete-time signals

as input and generates discrete-time signals as its output.

11

CONTROL SYSTEM DEFINITIONSMemory less System A system is called memory less system if its output y(t0) depends only on

the input applied at t0; it is independent of the past and future input (input applied before or after t0).

Casual System A system is called casual or non-anticipatory if its current output depends

on past and current inputs and not on future inputs. Every physical system is casual.

State of a System The state x(t0) of a system at time t0 is the information at t0 that, together

with input u(t), for t ≥ t0, determines uniquely the output y(t0) for all t ≥ t0.

Lumped System A system is called lumped if its number of state variables is finite.

Disturbed System A system is called disturbed if its number of state variables are infinite.

12

LINEAR SYSTEM

Linear System

A system is called a linear system if for every t0 any state-input-output pairs

xi (t0)

ui (t)

for i = 1, 2, we have

x1(t0) + x2(t0)

u1(t) + u2(t), t ≥ t0

and

αxi (t0)

αui (t), t ≥ t0

yi (t), t ≥ t0

y1(t) + y2(t), t ≥ t0 (additivity)

αyi (t), t ≥ t0 (homogeneity)

13

LINEAR SYSTEM

α1x1(t0) + α2x2(t0)

α1u1(t) + α2u2(t), t ≥ t0

for any real constants α1 and α1, and is called the superposition

property. A system is called a nonlinear system if the superposition property does not hold.

If the input u(t) is identically zero for all t ≥ t0 , the output will be exited exclusively by the initial state x(t0). This output is called the zero-input response and is denoted by yzi

or

x(t0)

u(t)= 0, t ≥ t0 yzi (t), t ≥ t0

α1y1(t) + α2y2(t), t ≥ t0

14

LINEAR SYSTEM

If the initial state x(t0) is zero, the output will be exited exclusively by the input u(t). This output is called the zero-state response and is denoted by yzs

or x(t0) = 0

u(t) , t ≥ t0

The additive property implies

x(t0) x(t0)

u(t) , t ≥ t0 u(t) = 0 , t ≥ t0

x(t0) = 0 u(t) , t ≥ t0

Response = zero-input response + zero-state response

The two responses can be studied separately and their sum yield the complete response. This is not true for nonlinear system, where the complete response can be different.

yzs (t), t ≥ t0

Output due to = output due to

+ output due to

15

LINEAR SYSTEM

Input-output Description

Zero-state response of a linear system: Consider a SISO linear system. Let δΔ(t-t1) be the pulse as shown:

It has a width Δ and height 1/ Δ and is located

at time t1.

Then every input u(t) can be approximated by a sequences of pulses as shown:

t t1 t1+ Δ

Δ

1/ Δ

u(ti) δΔ(t-t1) Δ

u(ti)

ti

t

16

LINEAR SYSTEM

If the pulse has a height of 1/ Δ then δΔ(t-t1) Δ has a height 1 and the left-most pulse with height u(ti) can be expressed as u(ti) δΔ(t-t1) Δ. Consequently, the input can be expressed as:

Let gΔ(t, ti) be the output at time t excited by the pulse

u(t) = δΔ(t-t1) applied at time ti. Then we have

Thus the output y(t) excited by the input u(t) can be approximated by

U(t) ≈ ∑ u(ti) δΔ(t-ti) Δ

δΔ(t-ti) gΔ(t, ti)

δΔ(t-ti) u(ti) Δ gΔ(t, ti) u(ti) Δ (homogeneity)

∑ δΔ(t-ti) u(ti) Δ ∑ gΔ(t, ti) u(ti) Δ (additivity)

y(t) ≈ ∑ gΔ(t,ti) u(ti) Δ

17

LINEAR SYSTEM

now if Δ approaches zero, then the pulse δΔ(t-ti) becomes an impulse at ti, denoted by δ(t-ti), and the corresponding output will be denoted by g(t, ti). As Δ approaches zero, the approximation in

becomes an equality, the summation becomes an integration, the discrete ti becomes a continuum and can be replaced by ּז , and Δ can be written as d ּז. Thus the output can be expressed as:

Note that the impulse response g(t, ּז) is a function of two variables. The second variable denotes the time at which the impulse input is applied; the first variable denotes the time at which the output is observed.

y(t) = ʃ g(t, ּז) u(ּז) d ּז-∞

∞


18

LINEAR SYSTEM

For a casual system the output will not appear before an input is applied.

Thus g(t, ּז) = 0 for t < ּז

If the system is relaxed, its initial state at t0 is 0. Therefore, for a linear system that is casual and relaxed at t0, the upper limit can be replaced by t and the lower limit by t0. The output can be expressed as:

If a linear system has r input and p output terminals then out is expressed as:

y(t) = ʃ g(t, ּז) u(ּז) d ּז for t < ּז t0

t

y(t) = ʃ G(t, ּז) u(ּז) d ּז for t < ּז t0

t

19

LINEAR SYSTEM

where G(t, ּז) is called the impulse response matrix of the system.

g11(t, ּז) g12(t, ּז) …. g1r(t, ּז)

g21(t, ּז) g22(t, ּז) …. g2r(t, ּז)

G(t, ּז) = . . .

. . .

gp1(t, ּז) gp2(t, ּז) …. gpr(t, ּז)

gij(t, ּז) is the response at time t at the ith output terminal due to an impulse applied at time ּז at the jth input terminal, the input at other terminals being identically zero.

20

LINEAR SYSTEM

State-space description: Every linear lumped system can be defined by a set of first order coupled equations of the form:

(set of n differential equations)

(set of p algebraic equations)

For r input and q output system, u is a r x 1 vector and y is a q x 1

vector. In the system has n state variables then x is an n x 1 vector. A, B, C and D must be n x n, n x r, p x n and p x r matrices.

x(t) = A(t) u(t) + B(t) u(t).

y(t) = C(t) u(t) + D(t) u(t)

21

LINEAR SYSTEM

Linear Time-Invariant (LTI) Systems: A system is said to be time invariant if for every state-input-output pair

and for any T, we have

It means that if the initial state is shifted to time t0 + T and the same waveform is applied from t0 + T instead of t0 , then the output waveform will be the same except that it starts to appear from time t0 + T. In other words, if the initial state and the input are same, no matter at what time they are applied, the output waveform will always be the same.

x(t0) u(t), t ≥ t0

y(t), t ≥ t0

x(t0+T) u(t -T), t ≥ t0 + T

y(t - T), t ≥ t0 + T (time shifting)

22

LINEAR SYSTEMInput-output description for LTI Systems: For time invariant system

g(t, ּז) = g(t + T, ּז + T) = g(t - 0, ּז ) = g(t - ּז)

for any T

where we have replaced t0 by 0 and the above integration is called convolution integral. Unlike the time varying case, where g is a function of two variables, g is a function of single variable in time invariant case. g(t) = g(t - 0) is the output at time t due to impulse in put at time 0.

State-space description for LTI Systems:

y(t) = ʃ g(t - ּז) u(ּז) d ּז = ʃ g(ּז) u(t - ּז) d ּז 0

t

0

t

x(t) = A u(t) + B u(t).

y(t) = C u(t) + D u(t)

23

CONTROL PROBLEMA physical system or process is to be controlled, so that the output (response) is adjusted as required by the error signal. The error signal is a measure of the difference between the system response as determined by the sensor and the desired response (input).

The controller is required to process the error signal such that: To track (or follow) the reference input r(t) To reject (or not respond to) the disturbances To reduce steady state errors To obtain desired transient response. We hope that the sensor has a transfer function close to 1, so often we ignore sensor

dynamics If the sensor dynamics are significant, they should be included in the analysis If r(t) is always 0, the special case is called a regulator . The block labeled digital controller can be implemented with any sort of hardware:

Computer running real-time operating system Digital signal processor (DSP) or other special-purpose digital hardware

24

ANALYSIS OF PHYSICAL SYSTEMS

Empirical Methods Applying various inputs to a physical system and measuring

its response. If performance is not satisfactory, some of its parameters are adjusted or connected to a compensator to improve its performance.

Analytical Methods The analytical study of a physical system consists of four

parts: Modeling Development of mathematical description Analysis (quantitative & qualitative) Design (controller or compensator)

25

EFFECT OF GAIN ON SYSTEM PERFORMANCE

26


27

SYSTEM MODELING

Rf

Ri

Cf

Ri

ViVf

Rf sCf

Ri

VfVi

1 0 orVf

Vi

Rf

Ri sRiCf

1

Vf

Vi

Kps

Ki

Electrical CircuitPI Controller

28

SYSTEM MODELING

i(t) = C dvc

dt

v(t) = R i(t) + L + vc(t)dt

di

Electrical CircuitPID Controller

v vc

LR C

i

V(s) = R I(s) + sL I(s) - LI(0) + Vc(s)I(t) = sCVc(s) - CVc(0)

sC

1= R + sL +

I(s)

V(s)V(s)

I(s)H(s) = =

sC

1 R + sL +

1

29

STATE SPACE MODEL

i(t) = C dvc

dt


di


= - - vc(t) - v(t)

di dt

d dt

R L

1 L

1 L

= i(t) dvc

dt 1 C

i(t)

1 L dvc

dt

1 C

R L

di dt

1 L

0

i(t)

vc(t)

+= v(t)

0

- -

-

Set of first order differential equations

30

ECI 660 LINEAR CONTROL SYSTEMS 3(3, 0)Course Contents

Input/Output System Models, Basic System Properties, Continuous and Discrete-Time Systems, System Modeling, State Space Representation, State Equations, System Analysis, Equivalence, Canonical Forms, Realizations, Stability, Sensitivity, Disturbance Rejection, Linearization, Controllability and Observability, Rank Tests, Linear Feedback Control, Fixed-Order Compensators (System Augmentation), Controller / Compensator Applications, Design Techniques: Root Locus, Frequency Response and Pole Placement, LQ Control, Luenberger Observers, Separation Principle (Estimated State Feedback).

31

TEXT AND REFERENCE BOOKS

Text Books Chi-Tsong Chen : Linear System Theory and Design, 3rd edition, Oxford

University Press

Charlles L. Phillips, Royce D. Harbor : Feedback Control Systems, 4th edition, New Jersey: Prentice Hall

Gene F. Franklin, J. David Powell: Feedback Control of Dynamic Systems, 5th edition, New Jersey: Prentice Hall

Reference Books Katsuhiko Ogata : Modern Control Engineering, 5th edition, PHI Learning

Callier, F. M., Desoer, C. A: Linear System Theory, New York, Spronger-Verlag

Rugh, W : Linear System Theory, 2nd edition, New Jersey: Prentice Hall

32

CONTROL SYSTEM

Definition

A control System is an interconnection of components to

provide a desired function. The portion of the system to be

controlled is called the plant (process, system), and the part

doing the controlling is called the controller (compensator,

filter). A control system designer has little or no design

freedom with the plant; as it is fixed. The designer’s task is to

develop a controller that will control the given plant

acceptably.

33


Feed-forward or open-looped control system

Input Output


Input Output

(unity-gain feedback)

Controller

Plant

Controller

Plant

34



Input Output


Controller

Plant

35

DIGITAL CONTROL SYSTEMS

Feedback control system using Digital Controller

Disturbances

(Desired response) (error Signal) (Response)

Input r(t) e(t) c(t) Output

(Sensor output)

Disturbances


DigitalControll

erPlant

Sensor

A/D D/A

36

CONTROL SYSTEM DEFINITIONS

SISO System A system with only one input and one output is called single-input single-

output system.

MIMO System A system with two or more input terminals and two or more output

terminal is called multivariable system or multi-input multi-output system.

Continuous-time System A system is called continuous-time system if it accepts continuous-time

signals as input and generates continuous-time signals as its output.

Discrete-time System A system is called discrete-time system if it accepts discrete-time signals

as input and generates discrete-time signals as its output.

37

CONTROL SYSTEM DEFINITIONSMemory less System A system is called memory less system if its output y(t0) depends only on

the input applied at t0; it is independent of the past and future input (input applied before or after t0).

Casual System A system is called casual or non-anticipatory if its current output depends

on past and current inputs and not on future inputs. Every physical system is casual.

State of a System The state x(t0) of a system at time t0 is the information at t0 that, together

with input u(t), for t ≥ t0, determines uniquely the output y(t0) for all t ≥ t0.

Lumped System A system is called lumped if its number of state variables is finite.

Disturbed System A system is called disturbed if its number of state variables are infinite.

38

LINEAR SYSTEM

Linear System

A system is called a linear system if for every t0 any state-input-output pairs

xi (t0)

ui (t)

for i = 1, 2, we have

x1(t0) + x2(t0)

u1(t) + u2(t), t ≥ t0

and

αxi (t0)

αui (t), t ≥ t0

yi (t), t ≥ t0

y1(t) + y2(t), t ≥ t0 (additivity)

αyi (t), t ≥ t0 (homogeneity)

39

LINEAR SYSTEM

α1x1(t0) + α2x2(t0)

α1u1(t) + α2u2(t), t ≥ t0

for any real constants α1 and α1, and is called the superposition

property. A system is called a nonlinear system if the superposition property does not hold.

If the input u(t) is identically zero for all t ≥ t0 , the output will be exited exclusively by the initial state x(t0). This output is called the zero-input response and is denoted by yzi

or

x(t0)

u(t)= 0, t ≥ t0 yzi (t), t ≥ t0

α1y1(t) + α2y2(t), t ≥ t0

40

LINEAR SYSTEM

If the initial state x(t0) is zero, the output will be exited exclusively by the input u(t). This output is called the zero-state response and is denoted by yzs

or x(t0) = 0

u(t) , t ≥ t0

The additive property implies

x(t0) x(t0)

u(t) , t ≥ t0 u(t) = 0 , t ≥ t0

x(t0) = 0 u(t) , t ≥ t0

Response = zero-input response + zero-state response

The two responses can be studied separately and their sum yield the complete response. This is not true for nonlinear system, where the complete response can be different.

yzs (t), t ≥ t0

Output due to = output due to

+ output due to

41

LINEAR SYSTEM

Input-output Description

Zero-state response of a linear system: Consider a SISO linear system. Let δΔ(t-t1) be the pulse as shown:

It has a width Δ and height 1/ Δ and is located

at time t1.

Then every input u(t) can be approximated by a sequences of pulses as shown:

t t1 t1+ Δ

Δ

1/ Δ

u(ti) δΔ(t-t1) Δ

u(ti)

ti

t

42

LINEAR SYSTEM

If the pulse has a height of 1/ Δ then δΔ(t-t1) Δ has a height 1 and the left-most pulse with height u(ti) can be expressed as u(ti) δΔ(t-t1) Δ. Consequently, the input can be expressed as:

Let gΔ(t, ti) be the output at time t excited by the pulse

u(t) = δΔ(t-t1) applied at time ti. Then we have

Thus the output y(t) excited by the input u(t) can be approximated by

U(t) ≈ ∑ u(ti) δΔ(t-ti) Δ

δΔ(t-ti) gΔ(t, ti)

δΔ(t-ti) u(ti) Δ gΔ(t, ti) u(ti) Δ (homogeneity)

∑ δΔ(t-ti) u(ti) Δ ∑ gΔ(t, ti) u(ti) Δ (additivity)


43

LINEAR SYSTEM

now if Δ approaches zero, then the pulse δΔ(t-ti) becomes an impulse at ti, denoted by δ(t-ti), and the corresponding output will be denoted by g(t, ti). As Δ approaches zero, the approximation in

becomes an equality, the summation becomes an integration, the discrete ti becomes a continuum and can be replaced by ּז , and Δ can be written as d ּז. Thus the output can be expressed as:

Note that the impulse response g(t, ּז) is a function of two variables. The second variable denotes the time at which the impulse input is applied; the first variable denotes the time at which the output is observed.

y(t) = ʃ g(t, ּז) u(ּז) d ּז-∞

∞


44

LINEAR SYSTEM

For a casual system the output will not appear before an input is applied.

Thus g(t, ּז) = 0 for t < ּז

If the system is relaxed, its initial state at t0 is 0. Therefore, for a linear system that is casual and relaxed at t0, the upper limit can be replaced by t and the lower limit by t0. The output can be expressed as:

If a linear system has r input and p output terminals then out is expressed as:

y(t) = ʃ g(t, ּז) u(ּז) d ּז for t < ּז t0

t

y(t) = ʃ G(t, ּז) u(ּז) d ּז for t < ּז t0

t

45

LINEAR SYSTEM

where G(t, ּז) is called the impulse response matrix of the system.

g11(t, ּז) g12(t, ּז) …. g1r(t, ּז)

g21(t, ּז) g22(t, ּז) …. g2r(t, ּז)

G(t, ּז) = . . .

. . .

gp1(t, ּז) gp2(t, ּז) …. gpr(t, ּז)

gij(t, ּז) is the response at time t at the ith output terminal due to an impulse applied at time ּז at the jth input terminal, the input at other terminals being identically zero.

46

LINEAR SYSTEM

State-space description: Every linear lumped system can be defined by a set of first order coupled equations of the form:

(set of n differential equations)

(set of p algebraic equations)

For r input and q output system, u is a r x 1 vector and y is a q x 1

vector. In the system has n state variables then x is an n x 1 vector. A, B, C and D must be n x n, n x r, p x n and p x r matrices.

x(t) = A(t) u(t) + B(t) u(t).

y(t) = C(t) u(t) + D(t) u(t)

47

LINEAR SYSTEM

Linear Time-Invariant (LTI) Systems: A system is said to be time invariant if for every state-input-output pair

and for any T, we have

It means that if the initial state is shifted to time t0 + T and the same waveform is applied from t0 + T instead of t0 , then the output waveform will be the same except that it starts to appear from time t0 + T. In other words, if the initial state and the input are same, no matter at what time they are applied, the output waveform will always be the same.

x(t0) u(t), t ≥ t0

y(t), t ≥ t0

x(t0+T) u(t -T), t ≥ t0 + T

y(t - T), t ≥ t0 + T (time shifting)

48

LINEAR SYSTEMInput-output description for LTI Systems: For time invariant system

g(t, ּז) = g(t + T, ּז + T) = g(t - 0, ּז ) = g(t - ּז)

for any T

where we have replaced t0 by 0 and the above integration is called convolution integral. Unlike the time varying case, where g is a function of two variables, g is a function of single variable in time invariant case. g(t) = g(t - 0) is the output at time t due to impulse in put at time 0.

State-space description for LTI Systems:

y(t) = ʃ g(t - ּז) u(ּז) d ּז = ʃ g(ּז) u(t - ּז) d ּז 0

t

0

t

x(t) = A u(t) + B u(t).

y(t) = C u(t) + D u(t)

49

CONTROL PROBLEMA physical system or process is to be controlled, so that the output (response) is adjusted as required by the error signal. The error signal is a measure of the difference between the system response as determined by the sensor and the desired response (input).

The controller is required to process the error signal such that: To track (or follow) the reference input r(t) To reject (or not respond to) the disturbances To reduce steady state errors To obtain desired transient response. We hope that the sensor has a transfer function close to 1, so often we ignore sensor

dynamics If the sensor dynamics are significant, they should be included in the analysis If r(t) is always 0, the special case is called a regulator . The block labeled digital controller can be implemented with any sort of hardware:

Computer running real-time operating system Digital signal processor (DSP) or other special-purpose digital hardware

50

ANALYSIS OF PHYSICAL SYSTEMS

Empirical Methods Applying various inputs to a physical system and measuring

its response. If performance is not satisfactory, some of its parameters are adjusted or connected to a compensator to improve its performance.

Analytical Methods The analytical study of a physical system consists of four

parts: Modeling Development of mathematical description Analysis (quantitative & qualitative) Design (controller or compensator)

51


52


53

SYSTEM MODELING

Rf

Ri

Cf

Ri

ViVf

Rf sCf

Ri

VfVi

1 0 orVf

Vi

Rf

Ri sRiCf

1

Vf

Vi

Kps

Ki

Electrical CircuitPI Controller

54

SYSTEM MODELING

i(t) = C dvc

dt


di


v vc

LR C

i

V(s) = R I(s) + sL I(s) - LI(0) + Vc(s)I(t) = sCVc(s) - CVc(0)

sC

1= R + sL +

I(s)

V(s)V(s)

I(s)H(s) = =

sC

1 R + sL +

1

55

STATE SPACE MODEL

i(t) = C dvc

dt


di


= - - vc(t) - v(t)

di dt

d dt

R L

1 L

1 L

= i(t) dvc

dt 1 C

i(t)

1 L dvc

dt

1 C

R L

di dt

1 L

0

i(t)

vc(t)

+= v(t)

0

- -

-

Set of first order differential equations

56

MODELING MECHANICAL SYSTEMS

Mechanical elements (linear)

Mass:

Spring:

Friction:

f = K x

dxdtf = B v = B

f = M a = M2

dx

dt2 M

f

xK

f

x

B x

f

57

LINEAR MECHANICAL SYSTEM

Bx

f

K

MM + B + K x = f

2dx

dt2

dxdt

= - v(t) - x(t) + f(t)dvdt

KM

1M

BM

dxdt

= v(t)

v(t)BM

KM

1M

dxdt

=dvdt

-+

0 1

-

x(t) 0f(t)

output equation y(t) = x(t)

y(t) = 01 x(t)

State equations

State Space Model

58

TRANSFER FUNCTION

M + B + K x = f

2dx

dt2

dxdt

Taking Laplace Transform

s M X(s) - s M x(0) – M x(0) + s B X(s)– B x(0) + K X(s) = F (s)

2

With zero initial conditions

s2 M X(s) + s B X(s) + K X(s) = F (s)

The system transfer function:

X(s)F(s)

=H(s) = 1

M s + B s + K

2

59

ROTARY MECHANICAL SYSTEM

K

Ƭ , ɵ

J

J + B + K ɵ = Ƭ

2dɵ

dt2

dɵdt

θ(s)Ƭ(s)

=H(s) = 1

J s + B s + K 2

System Dynamics:


60

MODELING ELECTROMECHANICAL SYSTEMS

ef

iL

eg ea

LaRa

ZL

ifRf

Lf

Field Circuit LoadArmature Circuit

DC Generator is driven mechanically by a prime mover. The shaft excite the field winding

The equation for the field circuit is:

ef = Rf if + Lf dt

dif Ef(s) = (s Lf + Rf) If(s)or A

61


DC Generator

The equation for the armature circuit is:

eg = Ra ia + La + eadt

dia

The armature voltage vg is generated through field flux as shown by the equation:

eg = K ɸdt

dɵ

The flux ɸ is directly proportional to the field current, as shown by the equation:

eg = Kg if

Eg = [s La + Ra + ZL(s)] Ia(s)or

ea = ia ZL

C

D

B

Ea = Ia(s) ZL(s)

Eg(s) = Kg if(s)

K is a parameter determined by physical structure of the generator & angular velocity of the armature is assumed to be constant

62


DC Generator

From equations A, B, C, and D


G(s) = Ef(s)

Ea(s)

(sLf + Rf) [s La + Ra + ZL(s)]

Kg ZL(s)=

Ea(s)Ia(s)Ef(s) 1[s La + Ra + ZL(s)]

1(s Lf + Rf)

Kg ZL(s) If(s) Eg(s)

The system block diagram is:

63


Servomotor (DC Motor)• Apply a dc source to the armature• Excite the field (sets up air-gap flux)

Stationary field winding, orPermanent magnets

•Commutator works as an inverter, converts dc terminal voltage to ac voltage on rotating armature winding

The voltage generated in the armature coil because of the motion of the coil in the motor’s magnetic field is called the back emf

em(t) = KΦdθdt

, θ

es

BLaRa

Jea em

Rs

64


Servomotor

The equation for the armature circuit is:

es(t) = (Rs + Ra) ia(t) + La + em(t)dt

dia

Where K is a motor parameter, Φ is filed flux and θ is the angle of motor shaft. If we assume that the flux Φ is constant , then

Es (s) = [s La + Rs + Ra] Ia (s) + Em (s)

2

1Em(s) = Km s Θ(s)

em(t) = Km

dθdt

Ia(s) =Es (s) - Em (s) s La + Rs + Ra

65


3

= Ki Φ ia(t)(t) or = K ia(t)(t)

The torque is proportional to the flux and the armature current.

Servomotor

For the mechanical load the torque equation isƬ(s) = K Ia(s)

J + B = (t)

d2θ

dt2

dθdt

Ƭ(s) = [s2 J(s) +s B] Θ(s) 4

Equations 1,2,3 and 4 will give us the system block diagram

66


Block Diagram of Servomotor

Ia(s)Es(s)

H(s)=s Km

G1(s)= 1s La + Rs + Ra

G2 (s)= S2 J + s B

1K

Em(s)

Θ(s)Ƭ(s)Es(s) - Em(s)

Ia(s)Es(s)

H(s) = Km

1s La + Rs + Ra S J + B

1K

Em(s)

Θ(s)Ƭ(s)Es(s) - Em(s)

1s

Θ(s).

67

MODELING ELECTROMECHANICAL SYSTEMS Transfer function of Servomotor

G(s) = s3J La + s2 (J Rs +J Ra + B La) + s ( B Rs + B Ra + Km K )

K

Approximation can be made by ignoring the armature inductance

G(s) = s3 K1+ s2 K2+ s K3

KG(s) = s(s2 K1 + s K2 + K3)

K

G(s) = s2(J Rs + J Ra ) + s ( B Rs + B Ra + Km K )

K

G(s) = s2J R + s ( B R+ Km K )

K

G(s) =Es(s)

Θ(s) G1(s) K G2(s) 1 + K G1(s) G2(s) H(S)

=

68

STATE SPACE MODELDefinition:

State Variable or State Space Model is a set of first order coupled differential equations, usually written in vector matrix form. It is a mathematical model of a physical system as a set of input, output and state variables related by first-order coupled differential equations, which preserve the input- output relationship.

Advantages• In addition to the input- output characteristics, the internal characteristics of the system are represented.

• Provides a convenient and compact way to model and analyze systems

with multiple inputs and outputs (MIMO).

• Computer aided analysis and design of state models are performed

more easily on digital computer for higher order systems.

• We can feedback more information(internal variables) about the plant

to perform more or complete control of the system.

• This model is required for simulation of complex systems.

69

EXAMPLE OF STATE SPACE MODELLinear Mechanical translational system:

f(t)

M

K B

y(t)

The differential equation model is

M + B + K y = f

2dy

dt2

dydt

= - - y + f(t)

2dy

dt2

dydt

BM

KM

1M

The transfer function model is

Y(s)F(s)

=G(s) = 1

M s + B s + K

2

This model gives a description of position y(t) as a function of force f(t). If we also want information of velocity, the state variable model give the solution by defining two state variables as

X1(t) = y(t)

dydt

X2(t) =2

dy

dt2X2(t) =

.

70

EXAMPLE OF STATE SPACE MODELLinear Mechanical translational system:

is positionx1(t) = y(t)

dydt

x2(t) =is velocity

x1(t) = x2(t) .

2dy

dt2X2(t) =

.= - - + f(t)

BM

KM

1Mx2(t) x1(t)

y(t) = x1(t)

1

2

3

1 and 2 are first order state equations and 3 is the output equation, represent the second order system. These equations are usually written in vector matrix form (standard form), are called state equations of the system, which can be manipulated easily.

71

EXAMPLE OF STATE SPACE MODELState Space Model

--BM

KM

1M

= +0 1 x1(t) 0

f(t)

x2(t)

x1(t).

x2(t).

y(t) = 01x1(t)

x2(t)

The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, n, is usually equal to the order of the system's defining differential equation, or is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction.

72

STANDARD FORM OF STATE SPACE MODEL

x(t) = A x(t) + B u(t) .

y(t) = C x(t) + D u(t)

Where

x(t) = state vector (n × 1) vector of the states of an nth-order system

u(t) = input vector (r × 1) vector composed of the system input functions

y(t) = output vector (p × 1) vector composed of the defined outputs of the system

A = (n × n) system matrix

B = (n × r) input matrix

C = (p × n) output matrix

D = (p × r) feed-forward matrix (usually it is zero)

http://en.wikipedia.org/wiki/File:Typical_State_Space_model.png

73

SOLUTION OF STATE EQUATIONS

x(t) = A x (t) + B u(t) .

The standard form of state equation is given by

The Laplace transform in matrix form can be written as:

Where

x(0) = [ x1(0) x2(0) . . . xn(0) ] T

---------------- 1

The inverse Laplace transform will give the solution of state equation, the state

vector x(t).

sX(s) - x(0)= A X(s) + B U(s)

sX(s) - A X(s) = x(0) + B U(s)

cx

a 1

(sI – A) X(s) = x(0) + B U(s)

X(s) = (sI – A)-1 [ x(0) + B U(s) ]

74

SOLUTION OF STATE EQUATIONS

The matrix (sI – A)-1 is called the resolvant of A and is written as:

Φ(s) = (sI – A)-1

The inverse Laplace transform of this term is defined as the state transition matrix:

φ(t) = £-1 [(sI – A)-1]

This matrix is also called the fundamental matrix and is (n×n) for nth order system.

the state matrix can be written as:

X(s) = Φ(s) x(0) + Φ(s) B U(s) ]

The inverse Laplace transform of the 2nd term in this equation can be expressed as a convolution integral.

x(t) = φ(t) x(0) + φ(t) B u(t - ) d ---------- 2

Both equations 1& 2 can be used for the solution of state equations.

0

t

# xx

75

SOLUTION OF STATE EQUATIONS Properties of state transition matrix φ(t) :

φ(0) = I (identity matrix)

φ(t) is nonsingular for finite elements in A

φ-1(t) = φ(-t)

φ(t1 – t2 ) φ(t2 – t3) = φ (t1 – t3)

φ(T) φ(T) = φ (2T)

The state transition matrix φ(t) satisfies the homogenous state equation,

Thus

Let eAt is the solution then

Therefore, the state transition matrix φ(t) is also defined as:

dx(t) dt = Ax(t)

dφ(t) dt = Aφ(t)

deAt

dt = AeAt

φ(t) = eAt = I + At + A2t2 + A3t3 + . . . 1 3!

1 2!

A simulation diagram is a type of either block diagram or signal flow diagram that is

constructed to have a specified transfer function or to model specified set of

differential equations. It is useful for construction computer simulation of a system.

It is very easy to get a state model from the simulation diagram.

The basic element of the simulation diagram is the integrator.

If

y(t) = x(t) dt

The Laplace Transform of this equation is

Y(s) = X(s)

76

SIMULATION DIAGRAMS

8

8y(t)x(t) Y(s)X(s) 1

sx(t)x(t)

.1s

1s

From system differential equationsThe transfer function of the device that integrate is , if output of the integrator is

y(t) then the input is . Similarly, if input is then out put of the integrator will

be .

Lets take the differential equation of mechanical translational system.

The simulation diagram can be constructed from the differential equation by

combination of integrators, gain and summing junction as:

77

SIMULATION DIAGRAMS

y(t).

y(t)..

.2dy

dt2 = - - + f(t)

BM

KM

1My(t) y(t)

y(t).

1s

y(t).

y(t)f(t)

BM

KM

1M

y(t)..

1s

1s

If simulation diagram is constructed from the differential equations then it will be

unique, but if it is constructed from system transfer function then it not unique. The

general form of system transfer function is:

Two different type of simulation diagrams can be constructed from the general form

of transfer function, for example if n = 3

(a) Control canonical form

(b) Observer canonical form

78

SIMULATION DIAGRAMS

From system transfer functions

bn-1 sn-1 +bn-2 sn-2 + ……. b0

sn + an-1 sn-1 +an-2 sn-2 + ……. a0

G(s) =

b2 s2 + b1 s + ……. b0

s3 + a2 s2 +a1 s + ……. a0

G(s) =

79

SIMULATION DIAGRAMS

Control Canonical Form

x2

.

a0

y(t)f(t) 1s

1s

1s

a1

a2

b1

b0

b2

x1

x1

.

x3 x2

x3

.

80

SIMULATION DIAGRAMS

Observer Canonical Form

x2

.x1

.x3

.y(t)

a0

u(t)

1s

1s

1s

a1a2

b1b0 b2

x1x3 x2

Once simulation diagram is constructed, the state model of the system can easily be obtained by assigning a state variable to the out put of each integrator and write equation for each state and system output.

81

STATE MODEL FROM SIMULATION DIAGRAMS

State model of the control canonical form

x =

.

-a0

u1

-a1 -a2

x +

0

0

0

00

01

1

y = xb1b0 b2

.x = -a0

1-a1

-a2

0 0

0

01

ux + b1

b0

b2

0 01y = x

State model of the observer canonical form

82

ANY QUESTION

83

THANK YOU

Lcs2

Education

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