Lecture – Basic Control Systems 1.1 Brief History 1.2 Steps to study a control system 1.2 Steps to study a control system 1.3 Classification of Systems 1 4 System classification 1.4 System classification 1.5 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 criterien  1892  Liapunov stability condition  1895  Hurwitz stability condition  1932  Nyquist  1945  Bode   1947 Nichols 1947  Nichols  1948  Root locus  1949 Wiener optimal control research 1949  Wiener optimal control research  1955  Kalman filter and controlbility observability analysis 1956  Artificial Intelligence     Modern control systems

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Brief history of automatic control (II) 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  9 2 id li i bl i l l d Robust control 1981 Doyle Robust control theory • 1981  Doyle Robust control theory  • 1990  Neuro‐Fuzzy   • • • •

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Three eras of control Three eras of control   • Classical control : 1950 before – Transfer function based methods  • Time‐domain design & analysis  • Frequency Frequency‐domain domain design & analysis   design & analysis

• Modern control : 1950~1960  – State‐space‐based methods  p • Optimal control  • Adaptive control 

• Post modern control : 1980 after   P d l 1980 f – H∞ control  – Robust control (uncertain system) Robust control (uncertain system) Modern control systems

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Control system analysis and design  y y g • Step1: Modeling  – By physical laws – By identification methods

• Step2: Analysis  – Stability, controllability and observability 

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

• Step4: Analysis  Step5: Simulation • Step5: Simulation  – Matlab, Fortran, simulink etc…. 

• Step6: Implement  p p Modern control systems

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Si l & t Signals & systems Output signals 

Input signals Time system

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Signal Classification Signal Classification  •• Continuous signal  Continuous signal

• Discrete signal 

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System classification y • 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) t d ib d b ti l diff ti l ti )

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

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Some examples of linear system p y • Electrical circuits with constant values of circuit  passive elements passive elements • Linear OPA circuits • Mechanical system with constant values of k,m,b  M h i l t ith t t l fk b etc • Heartbeat dynamic Heartbeat dynamic • Eye movement • Commercial aircraft C i l i ft

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Linear system y 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. homogeneity

Superposition：

y1 (t )

x1 (t )

x1 (t ) + x2 (t )

y2 (t )

x2 (t ) y1 (t ) + y2 (t )

Homogeneity：

x1 (t )

y1 (t ) Modern control systems

ax1 (t )

ay1 ((tt ) 10

Example

x((t )

let

y (t ) = x(t ) x(t − 1)

y (t )

x(t ) = x1 (t )

y1 (t ) = x1 (t ) x1 (t − 1) let

x(t ) = ax1 (t )

y (t ) = ax1 (t )ax1 (t − 1) = a 2 x1 (t ) x1 (t − 1) = a 2 y1 (t )

y (t ) ≠ ay1 (t )

Non linear system

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example The system y is g governed by y a linear ordinary y differential equation q ((ODE))

y ′′(t ) + 2 y ′(t ) + y (t ) = x′(t ) + 3 x (t ) x(t )

Linear time invariant system

y (t )

y1′′(t ) + 2 y1′ (t ) + y1 (t ) = x1′ (t ) + 3 x1 (t ) y2′′ (t ) + 2 y 2′ (t ) + y 2 (t ) = x2′ (t ) + 3 x2 (t ) [ ax1 (t ) + bx2 (t )]′ + 3[ ax1 (t ) + bx2 (t )] = ax1′ (t ) + bx2′ (t ) + a3 x1 (t ) + b3 x2 (t ) = a[ x1′ (t ) + 3 x1 (t )] + b[ x2′ (t ) + 3 x2 (t )] = a[ y1′′(t ) + 2 y1′ (t ) + y1 (t )] + b[ y2′′ (t ) + 2 y 2′ (t ) + y2 (t )] = [ ay1 (t ) + by 2 (t )]′′ + 2[ ay1 (t ) + by 2 (t )]′ + [ ay1 (t ) + by 2 (t )]

Modern control systems

linearity

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Properties of linear system :

(1)

(2)

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Time invariance A 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.

x(t )

y ((t ) Time invariant system

y (t − t0 )

x(t − t0 )

t0

t0 Modern control systems

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Example 1.18

x(t ) y (t ) = R(t )

x(t )

y (t )

x1 (t ) y1 (t ) = R(t ) x2 (t ) = x1 (t − t0 ) x2 (t ) x1 (t − t0 ) ⇒ y2 (t ) = = R(t ) R(t ) x1 (t − t0 ) but y1 (t ) = R(t − t0 ) y1 (t − t0 ) ≠ y2 (t ),

ffor t0 ≠ 0 Time varying system

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LTI System y representations p Continuous-time LTI system 1. O 1 Order-N d NO Ordinary di Diff Differential ti l equation ti 2. Transfer function (Laplace transform) 3. State equation (Finite order-1 differential equations) )

Discrete-time LTI system 1. Ordinary Difference equation 2. Transfer function (Z transform) 3. State equation (Finite order-1 difference equations)

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Continuous-time LTI system

d 2 y (t ) dy (t ) LC + RC + y (t ) = u (t ) 2 dt dt Order-2 ordinary differential equation constants

LCs 2Y ( s ) + RCsY ( s ) + Y ( s ) = U ( s ) Y (s) 1 = U ( s ) LCs 2 + RCs + 1

U (s )

Linear system y Æ initial rest

Transfer function

1 LCs 2 + RCs + 1

Modern control systems

Y (s )

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let

x1 (t ) = y (t ) dy (t ) x2 (t ) = dt

x1 (t ) = x2 (t ) R 1 x2 (t ) = − x2 (t ) − x1 (t ) + u (t ) L LC

⎡ x1 (t ) ⎤ ⎡ 0 ⎢ x (t )⎥ = ⎢− 1 ⎣ 2 ⎦ ⎣ LC

u (t )

x (t )

1 ⎤ ⎡ x1 (t ) ⎤ ⎡0⎤ + ⎢ ⎥u (t ) ⎥ R ⎥⎢ − L ⎦ ⎣ x2 (t )⎦ ⎣1⎦

x(t )

∫ A

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System response: Output signals due to inputs and ICs. 1. The point of view of Mathematic:  Homogenous solution y h (t ) ＋ Homogenous solution       

Particular solution      Particular solution y p ((t )

2. The point of view of Engineer:  Natural response        y n (t )

＋

Forced response       y f (t )

3. The point of view of control engineer: 

y zi (t ) Zero‐input Zero input response          response Transient response 

＋

Zero‐state Zero state response        response y zs ((t ) Steady state response

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Example: solve the following O.D.E

d 2 y (t ) dy (t ) − 2t + 4 + 3 y ( t ) = e , t ≥ 0, 2 dt dt (1) Particular solution: (1) Particular solution: 

d 2 y p (t ) dt 2

y (0) = 1,

dy (0) =1 dt

A[ y p (t )] = u (t ) +4

dy p (t ) dt

+ 3 y p (t ) = e − 2t

y p (t ) = αe −2t

let then

y ' p (t ) = −2αe −2t

y′p′ (t ) = 4αe −2t

4αe −2t + 4(−2)αe −2t + 3αe −2t = e −2t ⇒ α = −1

we have

y p (t ) = −e −2t

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A[ yh (t )] = 0

(2) Homogenous solution: 

yh′′ (t ) + 4 yh′ (t ) + 3 yh (t ) = 0

yh (t ) = Ae − t + Be −3t y (t ) = y p (t ) + yh (t )

have to satisfy   I.C. 

y (0) = 1⇒

y (0) = 1 ,

dy (0) =1 dt

y h ( 0) + y p ( 0) = 1

dy (0) d = 1⇒ dt

yh′ (0) + y′p (0) = 1

5 − t 1 − 3t yh (t ) = e − e 2 2 Modern control systems

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

y ′zi′ (t ) + 4 y ′zi (t ) + 3 y zi (t ) = 0,

t≥0

y zi (0) = 1,

y ′zi (0) = 1

y zi (t ) = K 1e − t + K 2 e −3t , t ≥ 0 y zi (0) = K 1 + K 2 y ′zi (0) = − K 1 − 3K 2

K1 = 2 K 2 = −1

y zi (t ) = 2e − t − e −3t , t ≥ 0 zero‐input response

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

y ′zs′ (t ) + 4 y ′zs (t ) + 3 y zs (t ) = e −2t ,

t≥0

y zi (0) = 0 ,

y ′zi (0) = 0

y zs (t ) = C1e − t + C 2 e −3t − e −2t

y zs (0) = C1 + C 2 − 1 = 0 y ′zs (0) = −C1 − 3C 2 + 2 = 0

1 2 1 C2 = 2 C1 =

1 − t 1 − 3t y zs (t ) = e + e − e − 2t 2 2 zero‐state response Modern control systems

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(5) Laplace Method:

d 2 y (t ) d (t ) dy − 2t + 4 + 3 ( ) = , t ≥ 0, y t e 2 dt dt

y (0) = 1,

dy (0) d =1 dt

1 s Y ( s ) − sy (0) − y ′(0) + 4 sY ( s ) − 4 y (0) + 3Y ( s ) = s+2 2

1 1 5 s+5+ − −1 s + 2 2 + = + 2 Y (s) = 2 s + 3 s + 2 s +1 s + 4s + 3 − 1 − 3t 5 −t − 2t y (t ) = A [Y ( s )] = e −e + e 2 2 −1

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Complex response

Zero state response

y zs (t ) =

1 − t 1 − 3t e + e − e −2t 2 2

Forced response (Particular solution) 

− 1 − 3t 5 e − e − 2 t + e −t 2 2

Zero input response

y zi (t ) = 2e − t − e −3t , t ≥ 0 Natural response (Homogeneous solution) 

y p (t ) = −e −2t Steady state response Steady state response

y (t ) =

yh (t ) =

5 − t 1 − 3t e − e 2 2

Transient response Transient response

− 1 − 3t 5 −t − 2t y (t ) = e −e + e 2 2 Modern control systems

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