📘 State Space Analysis – Complete Theory & Worked Examples
State Space Analysis is a modern method of representing dynamic systems. Unlike classical control, it works directly in time domain. Very important for GATE and competitive exams.
🔹 1. What is State?
State variables are minimum number of variables required to describe system completely.
General form:
ẋ = Ax + Bu y = Cx + Du
Where:- x = State vector
- A = System matrix
- B = Input matrix
- C = Output matrix
- D = Feedthrough matrix
🔹 2. Converting Differential Equation to State Form
Given:
d²y/dt² + 3dy/dt + 2y = u
Step 1: Define State Variables
x₁ = y x₂ = dy/dt
Step 2: Write State Equations
ẋ₁ = x₂ ẋ₂ = -2x₁ - 3x₂ + u
Step 3: Matrix Form
[ ẋ₁ ] = [ 0 1 ] [ x₁ ] + [ 0 ]u [ ẋ₂ ] [ -2 -3 ] [ x₂ ] [ 1 ]
Output:y = [1 0]x
🔹 3. Transfer Function from State Space
Formula:
G(s) = C(sI - A)⁻¹B + D
🔹 4. Controllability
System is controllable if controllability matrix rank = n.
Q = [ B AB A²B ... Aⁿ⁻¹B ]
If rank(Q) = n → Controllable.🔹 5. Observability
System observable if observability matrix rank = n.
O = [ C CA CA² ... ]
🔹 6. Worked Example – Controllability
Given:
A = [ 0 1 ; -2 -3 ] B = [ 0 ; 1 ]
Compute:AB = [ 1 ; -3 ]
Controllability matrix:Q = [ 0 1 1 -3 ]
Determinant ≠ 0 → Rank = 2 → Controllable.🔹 7. Worked Example – Observability
Given:
C = [1 0]
Compute:CA = [0 1]
Observability matrix:O = [ 1 0 0 1 ]
Rank = 2 → Observable.🎯 GATE Important Points
- Rank test frequently asked
- Conversion from differential equation important
- Transfer function relation very important
- Eigenvalues of A = System poles
State Space = Modern Foundation of Control Systems
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