Solution of Linear State- Space Equations
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Solution of Linear State- Space Equations
description
Solution of Linear State- Space Equations. Outline. • Laplace solution of linear state-space equations . • Leverrier algorithm. • Systematic manipulation of matrices to obtain the solution. Linear State-Space Equations. 1. Laplace transform to obtain their solution x ( t ) . - PowerPoint PPT Presentation
Transcript of Solution of Linear State- Space Equations
Solution of Linear State-Space Equations
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Outline
• Laplace solution of linear state-space equations.• Leverrier algorithm.• Systematic manipulation of matrices to obtain the solution.
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Linear State-Space Equations
1. Laplace transform to obtain their solution x(t).2. Substitute in the output equation to obtain
the output y(t).
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Laplace Transformation• Multiplication by a scalar (each matrix entry).• Integration (each matrix entry).
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State Equation
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Matrix Exponential
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Zero-input Response
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Zero-state Response
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Solution of State Equation
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State-transition Matrix
• LTI case φ (t − t0) = matrix exponential• Zero-input response: multiply by statetransition matrix to change the systemstate from x(0) to x(t).
• State-transition matrix for time-varyingsystems φ (t, t0)– Not a matrix exponential (in general).– Depends on initial & final time (notdifference between them).
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Output
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Example 7.7
x1 = angular position, x2 = angular velocityx3 = armature current. Find:a)The state transition matrix.b)The response due to an initial current of 10 mA.c)The response due to a unit step input.d)The response due to the initial condition of (b) together with the input of (c)
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a) The State-transition Matrix
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State-transition Matrix
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Matrix Exponential
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b) Response: initial current =10 mA.
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c) Response due to unit step input.
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Zero-state Response
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d) Complete Solution
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The Leverrier Algorithm
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Algorithm
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Remarks
• Operations available in hand-held calculators(matrix addition & multiplication, matrix scalarmultiplication).• Trace operation ( not available) can be easily22
p ) yprogrammed using a single repetition loop.• Initialization and backward iteration starts with:Pn-2 = A + an-1 In an-2 = − ½ tr{Pn-2 A}
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Partial Fraction Expansion
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Resolvent Matrix
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Example 7.8
Calculate the matrix exponential for thestate matrix of Example 7.7 using theLeverrier algorithm.
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Solution
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(ii) k = 0
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Check and Results
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Partial Fraction Expansion
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Constituent Matrices
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Matrix Exponential
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Properties of Constituent Matrices
