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**Time Response and State Transition Matrix**

Lecture 05 Analysis (I) Time Response and State Transition Matrix 5.1 State Transition Matrix 5.2 Modal decomposition --Diagonalization 5.3 Cayley-Hamilton Theorem Modern Control Systems

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**Modern Control Systems**

The behavior of x(t) et y(t) : Homogeneous solution of x(t) Non-homogeneous solution of x(t) Modern Control Systems

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Homogeneous solution State transition matrix Modern Control Systems

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Properties Modern Control Systems

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**Non-homogeneous solution**

Convolution Homogeneous Modern Control Systems

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Zero-input response Zero-state response Modern Control Systems

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Example 1 Ans: Modern Control Systems

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Using Maison’s gain formula Modern Control Systems

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How to find State transition matrix Methode 1: Methode 2: Methode 3: Cayley-Hamilton Theorem Modern Control Systems

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Methode 1: Modern Control Systems

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Method 2: Diagonalization Example 4.5 diagonal matrix Modern Control Systems

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Diagonalization via Coordinate Transformation Plant: Eigenvalue of A: Assume that all the eigenvalues of A are distinct, i.e. Then eigenvectors, are independent. Coordinate transformation matrix Modern Control Systems

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where Modern Control Systems

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New coordinate: (4.1) Solution of (4.1): The above expansion of x(t) is called modal decomposition. Hence, system asy. stable ⇔ all the eigenvales of A lie in LHP Modern Control Systems

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Example Find eigenvector Modern Control Systems

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(4.2) Solution of (4.1): Modern Control Systems

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In the case of A matrix is phase-variable form and Vandermonde matrix for phase-variable form Modern Control Systems

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Example: depend Modern Control Systems

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Case 3: Jordan form Generalized eigenvectors Modern Control Systems

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Example: Modern Control Systems

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Cayley-Hamilton Theorem Method 3: Theorem: Every square matrix satisfies its char. equation. Given a square matrix A, Let f(λ) be the char. polynomial of A. Char. Equation: By Caley-Hamilton Theorem Modern Control Systems

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any Modern Control Systems

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Example: Modern Control Systems

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**Modern Control Systems**

Example: Modern Control Systems

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**Modern Control Systems**

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Example: Modern Control Systems

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**Modern Control Systems**

Example: Modern Control Systems

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