nuu meiling chenmodern control systems1 lecture 01 --introduction 1.1 brief history 1.2 steps to...
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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)(