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Solution of Linear State- Space Equations

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 ) .

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Solution of Linear State- Space Equations

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  1. Solution of Linear State-Space Equations

  2. Outline • Laplace solution of linear state-space equations. • Leverrieralgorithm. • Systematic manipulation of matrices to obtain the solution.

  3. 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).

  4. Laplace Transformation • Multiplication by a scalar (each matrix entry). • Integration (each matrix entry).

  5. State Equation

  6. Matrix Exponential

  7. Zero-input Response

  8. Zero-state Response

  9. Solution of State Equation

  10. State-transition Matrix • LTI case φ (t − t0) = matrix exponential • Zero-input response: multiply by state transition matrix to change the system state from x(0) to x(t). • State-transition matrix for time-varying systems φ (t, t0) – Not a matrix exponential (in general). – Depends on initial & final time (not difference between them).

  11. Output

  12. Example 7.7 x1 = angular position, x2 = angular velocity x3 = 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)

  13. a) The State-transition Matrix

  14. State-transition Matrix

  15. Matrix Exponential

  16. b) Response: initial current=10 mA.

  17. c) Response due to unit step input.

  18. Zero-state Response

  19. d) Complete Solution

  20. The Leverrier Algorithm

  21. Algorithm

  22. Remarks • Operations available in hand-held calculators (matrix addition & multiplication, matrix scalar multiplication). • Trace operation ( not available) can be easily 22 p ) y programmed using a single repetition loop. • Initialization and backward iteration starts with: Pn-2 = A + an-1 In an-2 = − ½ tr{Pn-2 A}

  23. Partial Fraction Expansion

  24. Resolvent Matrix

  25. Example 7.8 Calculate the matrix exponential for the state matrix of Example 7.7 using the Leverrier algorithm.

  26. Solution

  27. (ii) k = 0

  28. Check and Results

  29. Partial Fraction Expansion

  30. Constituent Matrices

  31. Matrix Exponential

  32. Properties of Constituent Matrices

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