NUU meiling CHENModern control systems1 Lecture 01 --Introduction 1.1 Brief History 1.2 Steps to study a control system 1.3 System classification 1.4 System.

NUU meiling CHEN Modern control systems 1

Lecture 01 --Introduction

1.1 Brief History

1.2 Steps to study a control system

1.3 System classification

1.4 System response

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Brief history of automatic control (I)• 1868 First article of control ‘on governor’s’ –by Maxwell

• 1877 Routh stability criterion

• 1892 Liapunov stability condition

• 1895 Hurwitz stability condition

• 1932 Nyquist

• 1945 Bode

• 1947 Nichols

• 1948 Root locus

• 1949 Wiener optimal control research

• 1955 Kalman filter and controlbility observability analysis

• 1956 Artificial Intelligence

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Brief history of automatic control (II)

• 1957 Bellman optimal and adaptive control

• 1962 Pontryagin optimal control

• 1965 Fuzzy set

• 1972 Vidyasagar multi-variable optimal control and Robust control

• 1981 Doyle Robust control theory

• 1990 Neuro-Fuzzy

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Three eras of control • Classical control : 1950 before

– Transfer function based methods • Time-domain design & analysis

• Frequency-domain design & analysis

• Modern control : 1950~1960 – State-space-based methods

• Optimal control

• Adaptive control

• Post modern control : 1980 after – H∞ control

– Robust control (uncertain system)

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Control system analysis and design • Step1: Modeling

– By physical laws– By identification methods

• Step2: Analysis – Stability, controllability and observability

• Step3: Control law design – Classical, modern and post-modern control

• Step4: Analysis • Step5: Simulation

– Matlab, Fortran, simulink etc….

• Step6: Implement

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Time systemInput signals Output signals

Signals & systems

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Signal Classification

• Continuous signal

• Discrete signal

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System classification• Finite-dimensional system (lumped-parameters system

described by differential equations)

– Linear systems and nonlinear systems– Continuous time and discrete time systems– Time-invariant and time varying systems

• Infinite-dimensional system (distributed parameters system described by partial differential equations)

– Power transmission line– Antennas– Heat conduction– Optical fiber etc….

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Some examples of linear system• Electrical circuits with constant values of circuit

passive elements• Linear OPA circuits• Mechanical system with constant values of k,m,b

etc• Heartbeat dynamic• Eye movement• Commercial aircraft

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Linear system

)(1 tx

A system is said to be linear in terms of the system input x(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity.

Superposition ：

Homogeneity ：

)(1 ty )(2 tx )(2 ty

)()( 21 txtx )()( 21 tyty

)(1 tx )(1 ty )(1 tax )(1 tay

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Example

)1()()( txtxty

)()1()()1()()(

)()(

)1()()(

)()(

12

112

11

1

111

1

tyatxtxataxtaxty

taxtxlet

txtxty

txtxlet

)()( 1 tayty Non linear system

)(tx )(ty

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example

)(3)()()(2)( txtxtytyty

The system is governed by a linear ordinary differential equation (ODE)

Linear time invariant system

)(tx )(ty

)]()([])()([2])()([

)]()(2)([)]()(2)([

)](3)([)](3)([

)(3)(3)()()]()([3])()([

)(3)()()(2)(

)(3)()()(2)(

212121

222111

2211

21212121

22222

11111

tbytaytbytaytbytay

tytytybtytytya

txtxbtxtxa

txbtxatxbtxatbxtaxtbxtax

txtxtytyty

txtxtytyty

linearity

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Examples : Circuit system

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Examples Discrete system

Time delay

][ku ][ky]1[ ku

A

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Properties of linear system :

(1)

(2)

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Time invarianceA system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal.

)(txTime invariant

system

)(ty

)( 0ttx

0t

)( 0tty

0t

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Example 1.18

)(

)()(

tR

txty

0),()(

)(

)()(

)(

)(

)(

)()(

)()(

)(

)()(

0201

0

011

0122

012

11

tfortytty

ttR

ttxtybut

tR

ttx

tR

txty

ttxtx

tR

txty

Time varying system

)(tx )(ty

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)()sin()(

)cos1()(

)()()sin()(

)cos1()(

)()(

)(

00

00

titLRdt

tditLtv

tRititLdt

tditLtv

tRidt

dLi

dt

tdiLtv

i

i

i

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LTI System representations

1. Order-N Ordinary Differential equation2. Transfer function (Laplace transform)3. State equation (Finite order-1 differential equations) )

1. Ordinary Difference equation2. Transfer function (Z transform)3. State equation (Finite order-1 difference equations)

Continuous-time LTI system

Discrete-time LTI system

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)()()()(

2

2

tutydt

tdyRC

dt

tydLC

constantsOrder-2 ordinary differential equation

Continuous-time LTI system

1

1

)(

)(

)()()()(

2

2

RCsLCssU

sY

sUsYsRCsYsYLCs

Transfer function

Linear system initial rest

1

12 RCsLCs

)(sU )(sY

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)(1

0

)(

)(10

)(

)(

2

1

12

1 tutx

tx

tx

tx

LR

LC

dt

tdytx

tytxlet

)()(

)()(

2

1

)()(1

)()(

)()(

122

21

tutxLC

txL

Rtx

txtx

)(tx )(tx)(tu

A

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System response: Output signals due to inputs and ICs.

1. The point of view of Mathematic:

2. The point of view of Engineer:

3. The point of view of control engineer:

Homogenous solution )(tyh Particular solution )(ty p＋

＋

＋ Zero-state response )(ty zsZero-input response )(ty zi

Natural response )(tyn Forced response )(ty f

Transient response Steady state response

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1)0(

,1)0(,0,)(3)(

4)( 2

2

2

dt

dyytety

dt

tdy

dt

tyd t

Example: solve the following O.D.E

(1) Particular solution: )()]([ tutyp

tp

pp etydt

tdy

dt

tyd 22

2

)(3)(

4)(

tp etylet 2)(

tp

tp etyetythen 22' 4)(2)(

13)2(44 2222 tttt eeee

tp etyhavewe 2)(

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(2) Homogenous solution: 0)]([ tyh0)(3)(4)( tytyty hhh

tth BeAety 3)(

)()()( tytyty hp have to satisfy I.C. 1)0(

,1)0( dt

dyy

1)0()0(1)0(

1)0()0(1)0(

ph

ph

yydt

dy

yyy

tth eety 3

2

1

2

5)(

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(3) zero-input response: consider the original differential equation with no input.

1)0(,1)0(0,0)(3)(4)( zizizizizi yyttytyty

0,)( 321 teKeKty tt

zi

21

21

3)0(

)0(

KKy

KKy

zi

zi

1

2

2

1

K

K

0,2)( 3 teety ttzi

zero-input response

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(4) zero-state response: consider the original differential equation but set all I.C.=0.

0)0(,0)0(0,)(3)(4)( 2 zizi

tzszszs yytetytyty

tttzs eeCeCty 23

21)(

023)0(

01)0(

21

21

CCy

CCy

zs

zs

2

12

1

2

1

C

C

tttzs eeety 23

2

1

2

1)(

zero-state response

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(5) Laplace Method:

1)0(

,1)0(,0,)(3)(

4)( 2

2

2

dt

dyytety

dt

tdy

dt

tyd t

2

1)(3)0(4)(4)0()0()(2

ssYyssYysysYs

12

5

2

1

32

1

342

15

)(2

ssssss

ssY

ttt eeesYty

2

5

2

1)]([)( 231

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Complex response

Zero state response Zero input response

Forced response(Particular solution)

Natural response(Homogeneous solution)

Steady state response Transient response

ttt eeety

2

5

2

1)( 23

tttzs eeety 23

2

1

2

1)( 0,2)( 3 teety tt

zi

ttt eeety

2

5

2

1)( 23

tp ety 2)( tt

h eety 3

2

1

2

5)(