GATE Electrical Engineering - Complete Master Library

 

📘 State Transition Matrix & Diagonalization – Complete Explanation

State Transition Matrix (STM) is used to solve state equations directly in time domain. It represents e^(At), which determines system response. Very important in Modern Control.

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🔹 1. State Equation

ẋ(t) = Ax(t)

Solution:

x(t) = e^(At) x(0)

Here,

Φ(t) = e^(At)

is called State Transition Matrix.

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🔹 2. Matrix Exponential Definition

e^(At) = I + At + (A²t²)/2! + (A³t³)/3! + ...

This is infinite series expansion.

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🔹 3. Using Eigenvalue Method (Diagonalization)

If A is diagonalizable:

A = PDP⁻¹

Then:

e^(At) = P e^(Dt) P⁻¹

Where:

• D = Diagonal matrix of eigenvalues
• e^(Dt) = Diagonal matrix with e^(λt)

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🔹 4. Worked Example 1 – Diagonal Matrix

Given:

A = [ -2 0 0 -3 ]

Since A already diagonal:

e^(At) = [ e^(-2t) 0 0 e^(-3t) ]

Very simple case.

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🔹 5. Worked Example 2 – Eigenvalue Method

Given:

A = [ 0 1 -2 -3 ]

Step 1: Find Eigenvalues

Characteristic equation:

|sI - A| = s² + 3s + 2

Roots:

s = -1, -2

Step 2: Form D Matrix

D = [ -1 0 0 -2 ]

Step 3: Compute e^(Dt)

e^(Dt) = [ e^(-t) 0 0 e^(-2t) ]

Step 4: Final STM

Compute:

Φ(t) = P e^(Dt) P⁻¹

(GATE usually does not require full multiplication.)

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🔹 6. Properties of STM

• Φ(0) = I
• Φ(t₁ + t₂) = Φ(t₁)Φ(t₂)
• If A stable → eigenvalues negative → response decays

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🔹 7. Stability Using Eigenvalues

• All eigenvalues negative real part → Stable
• Any eigenvalue positive real part → Unstable
• Pure imaginary → Marginally stable

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🎯 GATE Important Points

• Eigenvalues of A = System poles
• Diagonalizable matrices easier
• Repeated eigenvalues need Jordan form
• Matrix exponential concept very important

State Transition Matrix = Time Evolution of States

📘 State Transition Matrix – Advanced Worked Examples

This section provides deeper numerical examples on computing e^(At), including repeated eigenvalues and Jordan form cases. Very important for Modern Control in GATE.

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🔹 Example 1 – 2×2 Matrix (Direct Eigenvalue Method)

Given:

A = [ -1 1 0 -2 ]

Step 1: Find Eigenvalues

Characteristic equation:

|sI - A| = (s+1)(s+2)

Eigenvalues:

λ₁ = -1 λ₂ = -2

Step 2: Since eigenvalues distinct → Diagonalizable

Final STM form:

Φ(t) = P e^(Dt) P⁻¹

(GATE often asks eigenvalues-based reasoning only.)

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🔹 Example 2 – Repeated Eigenvalues

Given:

A = [ 2 1 0 2 ]

Step 1: Characteristic Equation

|sI - A| = (s-2)²

Repeated eigenvalue:

λ = 2

Step 2: Check Eigenvectors

Only one independent eigenvector → Not diagonalizable.

Step 3: Jordan Form

Jordan matrix:

J = [ 2 1 0 2 ]

Step 4: Matrix Exponential for Jordan Block

e^(At) = e^(2t) [ 1 t 0 1 ]

Important result: Repeated eigenvalue produces t·e^(λt) term.

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🔹 Example 3 – Full STM Computation

Given:

A = [ 0 1 -4 -4 ]

Step 1: Characteristic Equation

s² + 4s + 4 = 0

Roots:

s = -2, -2

Repeated eigenvalue.

Step 2: Jordan Form Used

Final STM:

Φ(t) = e^(-2t) [ 1 t 0 1 ]

System response decays because eigenvalue negative.

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🔹 Example 4 – Stability from Eigenvalues

Given eigenvalues:

• -1
• -3
• -5

All negative → Stable system. If eigenvalues:

• 2
• -1

One positive → Unstable.

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🔹 Example 5 – Time Response Using STM

Given:

ẋ = Ax x(0) = [1 0]ᵀ

Solution:

x(t) = Φ(t)x(0)

Multiply STM with initial condition to get full response.

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🎯 Important Modern Control Insights

• Eigenvalues = System poles
• Repeated eigenvalues → Jordan form
• Negative real parts → Stable
• Positive real part → Unstable
• STM fully describes time evolution

Matrix Exponential Describes System Dynamics Completely