Formula Sheets: State Space Analysis | Control Systems - Electrical Engineering (EE) PDF Download

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Control S ystems F ormula Sheet: State
Space Analysis
1. State Space Representation
• State Space Model (Linear Time-Invariant S ystem):
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
– x(t) : State vector (n×1 ).
– u(t) : Input vector (m×1 ).
– y(t) : Output vector (p×1 ).
– A : State matrix (n×n ).
– B : Input matrix (n×m ).
– C : Output matrix (p×n ).
– D : F eedthrough matrix (p×m ).
• Discrete-Time State Space Model :
x(k +1) =A
d
x(k)+B
d
u(k)
y(k) =C
d
x(k)+D
d
u(k)
2. Solution of State Equations
• State Tr ansition Matrix :
F(t) =e
At
=L
-1
[
(sI-A)
-1
]
• Time-Domain Solution (Continuous):
x(t) = F(t)x(0)+
?
t
0
F(t-t)Bu(t)dt
y(t) =CF(t)x(0)+C
?
t
0
F(t-t)Bu(t)dt +Du(t)
1
Page 2


Control S ystems F ormula Sheet: State
Space Analysis
1. State Space Representation
• State Space Model (Linear Time-Invariant S ystem):
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
– x(t) : State vector (n×1 ).
– u(t) : Input vector (m×1 ).
– y(t) : Output vector (p×1 ).
– A : State matrix (n×n ).
– B : Input matrix (n×m ).
– C : Output matrix (p×n ).
– D : F eedthrough matrix (p×m ).
• Discrete-Time State Space Model :
x(k +1) =A
d
x(k)+B
d
u(k)
y(k) =C
d
x(k)+D
d
u(k)
2. Solution of State Equations
• State Tr ansition Matrix :
F(t) =e
At
=L
-1
[
(sI-A)
-1
]
• Time-Domain Solution (Continuous):
x(t) = F(t)x(0)+
?
t
0
F(t-t)Bu(t)dt
y(t) =CF(t)x(0)+C
?
t
0
F(t-t)Bu(t)dt +Du(t)
1
• Discrete-Time Solution :
x(k) =A
k
d
x(0)+
k-1
?
i=0
A
k-1-i
d
B
d
u(i)
y(k) =C
d
A
k
d
x(0)+C
d
k-1
?
i=0
A
k-1-i
d
B
d
u(i)+D
d
u(k)
3. Tr ansfer Function from State Space
• Tr ansfer Function :
G(s) =C(sI-A)
-1
B +D
• Char acteristic Equation : Obtained from the determinant of (sI-A) :
det(sI-A) = 0
• Eigenvalues : Roots of the char acteristic equation, determining system stabil-
ity .
4. Controllability and Observability
• Controllability : S ystem is controllable if the controllability matrix has full
r ank (n ):
C = [B AB A
2
B ··· A
n-1
B]
Rank(C ) =n .
• Observability : S ystem is observable if the observability matrix has full r ank
(n ):
O =
?
?
?
?
?
?
?
C
CA
CA
2
.
.
.
CA
n-1
?
?
?
?
?
?
?
Rank(O ) =n .
5. Stability Analysis
• Continuous-Time Stability : S ystem is stable if all eigenvalues ofA have neg-
ative real parts (lie in the left half-plane).
• Discrete-Time Stability : S ystem is stable if all eigenvalues ofA
d
have magni-
tude less than 1 (lie inside the unit circle).
2
Page 3


Control S ystems F ormula Sheet: State
Space Analysis
1. State Space Representation
• State Space Model (Linear Time-Invariant S ystem):
? x(t) =Ax(t)+Bu(t)
y(t) =Cx(t)+Du(t)
where:
– x(t) : State vector (n×1 ).
– u(t) : Input vector (m×1 ).
– y(t) : Output vector (p×1 ).
– A : State matrix (n×n ).
– B : Input matrix (n×m ).
– C : Output matrix (p×n ).
– D : F eedthrough matrix (p×m ).
• Discrete-Time State Space Model :
x(k +1) =A
d
x(k)+B
d
u(k)
y(k) =C
d
x(k)+D
d
u(k)
2. Solution of State Equations
• State Tr ansition Matrix :
F(t) =e
At
=L
-1
[
(sI-A)
-1
]
• Time-Domain Solution (Continuous):
x(t) = F(t)x(0)+
?
t
0
F(t-t)Bu(t)dt
y(t) =CF(t)x(0)+C
?
t
0
F(t-t)Bu(t)dt +Du(t)
1
• Discrete-Time Solution :
x(k) =A
k
d
x(0)+
k-1
?
i=0
A
k-1-i
d
B
d
u(i)
y(k) =C
d
A
k
d
x(0)+C
d
k-1
?
i=0
A
k-1-i
d
B
d
u(i)+D
d
u(k)
3. Tr ansfer Function from State Space
• Tr ansfer Function :
G(s) =C(sI-A)
-1
B +D
• Char acteristic Equation : Obtained from the determinant of (sI-A) :
det(sI-A) = 0
• Eigenvalues : Roots of the char acteristic equation, determining system stabil-
ity .
4. Controllability and Observability
• Controllability : S ystem is controllable if the controllability matrix has full
r ank (n ):
C = [B AB A
2
B ··· A
n-1
B]
Rank(C ) =n .
• Observability : S ystem is observable if the observability matrix has full r ank
(n ):
O =
?
?
?
?
?
?
?
C
CA
CA
2
.
.
.
CA
n-1
?
?
?
?
?
?
?
Rank(O ) =n .
5. Stability Analysis
• Continuous-Time Stability : S ystem is stable if all eigenvalues ofA have neg-
ative real parts (lie in the left half-plane).
• Discrete-Time Stability : S ystem is stable if all eigenvalues ofA
d
have magni-
tude less than 1 (lie inside the unit circle).
2
6. State F eedback Control
• Control Law : u(t) =-Kx(t) , whereK is the feedback gain matrix.
• Closed-Loop S ystem :
? x(t) = (A-BK)x(t)
• Char acteristic Equation with F eedback :
det(sI-(A-BK)) = 0
• Pole Placement : Choose K to place closed-loop poles at desired locations b y
solving for desired char acteristic polynomial.
7. State Observer
– Observer Equation :
?
ˆ x(t) =Aˆ x(t)+Bu(t)+L(y(t)- ˆ y(t))
ˆ y(t) =Cˆ x(t)+Du(t)
where ˆ x(t) is estimated state,L is observer gain matrix.
– Error Dynamics : e(t) =x(t)- ˆ x(t) :
? e(t) = (A-LC)e(t)
– Observer Stability : ChooseL such that eigenvalues of (A-LC) have nega-
tive real parts.
3