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State Variable Approach
State Model
X = AX + BU State equations
Y = CX + DU Output equations
And both equations combined together is called state model.
X  State vector
U  Input vector
Y  Output Vector
A  System matrix
B  Input matrix
C  Output matrix
D  Transmission matrix
Advantages of State Variable Method
REPRESENTATION OF STATE MODEL
Note: State model of a system is not unique property but transfer function of the system is unique.
Physical Variable Representation
Example: Find out state model for the network shown below:
Solution: Applying KVL,
........(i)
= f(x_{1}, x_{2}, u)
From n/w,
Also, V_{output} = V_{o}
Y = x_{2}
= (0)x_{1} + (1)x_{2} + (0) u ...(ii)
writing equation (i), (ii), (iii) in matrix form,
Phase Variable Representation
Put y = x_{1}
From equation we have,
Canonical Representation
X = A'X + B'u
Y = C'X + D'u
A' → Diagonal matrix
Note:
Transfer function from state model
X = AX + BU
Y = CX + DU
Y → single output
U → single input
Initial conditions X(0) = 0 (for transfer function)
Taking Laplace of equation
sX(s) = AX)s) + BU(s)
sX(s) – AX(s) = BU(s)
[sI – A] = BU(s)
Taking Laplace of equation
Y(s) = CX(s) + DU(s)
Putting expression of X(s) from we have,
Y(s) = {C[sI – A]–1 B + D} U (s)
SOLUTION OF STATE EQUATIONS
(i) Homogeneous System (U = 0) :
(ii) Nonhomogeneous System (U ≠ 0):
X = AX + BU
Taking Laplace transform we have
Solution of State Equation in Time Domain
Integrating both both sides we get,
CONTROLLABILITY & OBSERVABILITY
Definition of Controllability
Kalman’s Test for Controllability
Consider,
According to Kalman’s Test
System is completely controllable if rank of matrix Q_{c} is equal to order of the system i.e. (r = n)
System will not be completely controllable (uncontrollable) if rank of matrix Q_{c} is less than order of the system i.e. (r < n)
then number of uncontrollable states = (n – r)
number of controllable states = r
Rank of matrix Qc = n i.e. r = n
if det.Q_{c} ≠ 0
Observability
Kalman's Test for Observability
Q_{0} =[C^{T} A^{T} C^{T} (A^{T})^{2}C^{T}....(A)^{n1} C^{T}]
n → order of system
Q_{0} → Observability testing matrix
X = AX + BU
Y = CX + DU
Number of unobservable states = n – r
Number of observable states = r
Note: r = n if Q_{0} ≠ 0
Duality
JORDAN MATRIX
Matrix =
2 videos75 docs40 tests

2 videos75 docs40 tests
