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7/29/2019 Modern Control Lec2
1/19
7/29/2019 Modern Control Lec2
2/19
Introduction to State space
Definition of Space
The state of the system at time t0
is the minimum information neededto uniquely specify the system response given the input variable over thetime interval [t0,]
State space representation
The state space equations of thesystem is
x = Ax+ Bu
y = Cx+ Du
wherex IRn ; u IRm ; y IRp
A IRnn B IRnm
C IRpn D IRpm
State space analysis
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Example
State space model of a circuit
Applying KVL to the circuit,
Vin = Ri+ Ldi
dt+
1
C
idt
Vout =1
C
idt
Let x1 =idt
x2 =dx1dt
= x1 = iThen,
x1 = x2x2 =
VinL
Rx2L
x1LC
State space analysis
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Example
Continuation of example
In the standard form,
X =
x1x2
X = 0 1 1LC RL
x1x2 + 0
1LVin
y =
1C
0 x1
x2
+ [0]Vin
x = Ax+ Buy = Cx+ Du
where,
A=
0 1
1LC
RL
B=
01L
C=
1C
0
D=[0]
State space analysis
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Time Domain analysis
Response of a system
Total response = Zero input response + Zero state response
Zero-input state response
u(t) 0x(t0) x(t) =?
Zero-input system response
u(t) 0x(t0) y(t) =?
State space analysis
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Time Domain analysis
Response of a system
Total response = Zero input response + Zero state response
Zero-input state response
u(t) 0x(t0) x(t) =?
Zero-input system response
u(t) 0x(t0) y(t) =?
State space analysis
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Time Domain analysis
Zero-state state response
x(t0) 0
u(t) x(t) =?
Zero-state system response
x(t0) 0
u(t) y(t) =?
State space analysis
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Time Domain analysis
Zero-state state response
x(t0) 0
u(t) x(t) =?
Zero-state system response
x(t0) 0
u(t) y(t) =?
State space analysis
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Time Domain analysis
System response
System response = Zero input system response + Zero state systemresponse
where,Zero input response= eAtX(0)
Zero state response=t
0 eA(t)Bu()d
Impulse response,
h(t)= CeAtBu(t)= k1 (1 Amplitude)
The total response isy(t) = eAtX(0) +
t0eA(t)Bu()d
State space analysis
E l 1
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Example-1
A mechanical system
Md2x
dt2 + B
dx
dt
= f(t)
Let x1 = x ; x2 =dxdt
x1 = x2x2 =
f(t)M
BMx2
x1x
2 =
0 1
0 BM +
01
M u(t)
y =
1 0x1
x2
+ (0)u(t)
Let M=0.5 kg ; B=3 Ns/m
A =
0 10 6
; B =
02
; C =
1 0
; D=(0)
State space analysis
E l 1
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Example-1
Continuation of example..
The solution is
x(t) = eAtx(0) +t
0eA(t)Bu()d
Analysis of system responses
Zero-input state response
x(t) =
1 16
16e
6t
0 e6t
x1(0)x2(0)
x1(t) = x1(0) +x2(0)
6 x2(0)
6e6t
x2(t) = x2(0)e6t
Zero-input system response
y(t) = x1(t) = x1(0) +x2(0)
6 x2(0)
6 e6t
State space analysis
E l 1
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Example-1
Analysis of system responses
Zero-state state responsex(t) =
t0eA(t)Bu()d
Let us consider the impulse input,t
0u()d = 1
x(t) =t
0eA(t)Bd
=t
0
1
1
6
1
6e6(t)
0 e6(t)
02d
=
t0
13 t
13e
6(t)dt
02e6(t)d
x(t) = 1
3
t 1
18
(1 e6t)13 (1 e
6t)
The total output response of the system is
y(t) = x1(0) +x2(0)
6 (1 e6t) + 13 t
118 (1 e
6t)
State space analysis
T f f ti t St t S
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Transfer function to State Space
Example-1
Consider a transfer function, G(s) =5s2+7s+9
s3+8s2+6s+2
Let y(s)u(s) =
y(s)x(s)
x(s)u(s) =
5s2+7s+9s3+8s2+6s+2
Thus,
y(s)
x(s)
= 5s2 + 7s+ 9 (1)
x(s)
u(s)=
1
s3 + 8s2 + 6s+ 2(2)
From eq(2), u(s) = x(s)[s3 + 8s2 + 6s+ 2]
u(t) =d3x
dt3+ 8
d2x
dt2+ 6
dx
dt+ 2x
d3x
dt3= u(t) 8
d2x
dt2 6
dx
dt 2x
State space analysis
T sf f ti t St t S
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Transfer function to State Space
Example-1
Let x1 = x ; x2 = dxdt ; x3 = d2xdt2
x1x2x3
=
0 1 00 0 12 6 8
x1x2x3
+
00
1
u(t)
From eq(1), y(s) = x(s)[5s2 + 7s+ 9]
y(t) = 5d2x
dt2+ 7
dx
dt+ 9x
y(t) = 5x3 + 7x2 + 9x1
y =
9 7 5 x1x2
x3
+ [0]u
State space analysis
Transfer function to State Space
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Transfer function to State Space
Example-2Consider a transfer function, G(s) = 5s
2+7s+9s2+2s+15
Let
x(s)
u(s)
=1
s2
+ 2s+ 15d2x
dt2= 15x 2
dx
dt+ u(t)
Let x1 = x ; x2 =dxdt
x1x2
=
0 115 2
x1x2
+
01
State space analysis
Transfer function to State Space
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Transfer function to State Space
Example-2
y(s)
x(s)= 5s2 + 7s+ 9
y(t) = 5d2x
dt2 + 7dx
dt + 9x
= 5(15x 2dx
dt+ u(t)) + 7
dx
dt+ 9x
= 66x 3dx
dt+ 5u(t)
y(t) =66 3
x1x2
+ 5u(t)
State space analysis
Similarity Transformation
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Similarity Transformation
The state space equations of a system is given as
x = Ax+ Buy = Cx+ Du
Choose any non-singular matrix T such thatz = Tx x = T1z and x = T1z
Substituting in the system equations
z = TAT1z+ TBu
y = CT1z+ Du
It can be written as
z = Azz+ Bzu
y = Czz+ Dzu
where,
Az = TAT1
; Bz = TB ; Cz = CT1
; Dz = DState space analysis
State Space to Transfer function
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State Space to Transfer function
The state space equations of a system is given as
x = Ax+ Bu
y = Cx+ Du
It can be written as
sx(s) = Ax(s) + Bu(s)
y(s) = Cx(s) + Du(s)
Substituting for x(s) in y(s), we get
y(s) = C[sI A]1Bu(s) + Du(s)y(s)
u(s)= C[sI A]1B+ D
State space analysis
State Space to Transfer function
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State Space to Transfer function
State Space to Transfer function is unique
y(s)
u(s)= Cz(sI Az)
1Bz + Dz
= CT1(sI TAT1)1TB+ D
= CT1[TsIT1 TAT1]1TB+ D
= CT1T[sI A]1T1TB+ D
y(s)
u(s)
= C[sI A]1B+ D
State space analysis
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