1. 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
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The behavior of x(t) et y(t) :
Homogeneous solution of x(t)
Non-homogeneous solution of x(t)
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Homogeneous solution
State transition matrix
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Properties
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5. Non-homogeneous solution
Convolution
Homogeneous
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Zero-input response
Zero-state response
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Example 1
Ans:
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Using Maison’s gain formula
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How to find
State transition matrix
Methode 1:
Methode 2:
Methode 3: Cayley-Hamilton Theorem
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Methode 1:
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Method 2: Diagonalization
Example 4.5
diagonal matrix
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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
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where
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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
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Example
Find eigenvector
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(4.2)
Solution of (4.1):
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In the case of A matrix is phase-variable form and
Vandermonde matrix
for phase-variable form
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Example:
depend
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Case 3:
Jordan form
Generalized eigenvectors
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Example:
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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
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any
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Example:
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Example:
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Example:
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Example:
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