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CHE 391

CHE 391. T. F. Edgar Spring 2012. Response of Linear system. Solution : = ( Transition Matrix ) Usually Computationally not feasible for general use  need analytical solution. Take Laplace Transform: Take inverse Laplace Transform As (note scalar analog: ). Example. =

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CHE 391

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  1. CHE 391 T. F. Edgar Spring 2012

  2. Response of Linear system • Solution: = ( Transition Matrix) Usually Computationally not feasible for general use need analytical solution

  3. Take Laplace Transform: • Take inverse Laplace Transform As (note scalar analog: )

  4. Example = (Here, ) Invert, partial fraction decomposition

  5. Note that Also, denominator of is Roots of are the same as the factors of characteristic equationeigenvalues() of Eigenvalue Equation:

  6. Eigenvector/eigenvalue solution • Example: 2nd order Equating terms: If , satisfy eigenvalue, eigenvector equations For nth order model,

  7. Define= matrix of eigenvectors (also called similarity transform, modal matrix, eigenvector assembly matrix)  Relates , , and

  8. Example • , , , • Note that , must be linearly independent (at , solve for , ) For , we have , Then, Note that when is very large, neglect the 2nd term.

  9. If is in canonical form, • (Vandermonde Matrix) • Note, if is complex, is complex.

  10. Another way to calculate using , , let (transformation), then we have  The solution for the ODE is Where (a diagonal matrix) hence (“modes” of the system) The analytical solution to is therefore and If are complex or repeated, modify the above procedure. (see Ogata)

  11. Add controller Substitute , we have Let (Modal feedback. ) Multivariable root locus (some limitations in MIMO as in SISO – closed-loop response depends on eigenvectors as well as eigenvalues.) Shifts eigenvalues

  12. “Autonomous” model , not functions of ; , deviation variables. Take L.T., If () (output states or controlled variables) Then the output is Transfer function matrix

  13. Example: , Shift pole to higher value • However, if is such that , shifting has no effect on response

  14. Stability Requirements • Continuous: all • Discrete: eigenvalues of , (for complex , all must lie within unit circle)

  15. Analytical Solution: Pre-multiply by (integrating factor)  Integrating between and : Homogenous Solution, Particular Solution

  16. If  convolution integral Note that depends only on . Let Sampled data system: () Define (, are functions of only ) Generalize to any time step  • Calculation of , : use infinite series expression

  17. Where • Use finite number of terms No integration error for sampled data control (zero-order hold)

  18. Discrete-time State Controllability is the order of the system. We need to be able to generate any from using , , etc. Example: is a scalar, , , is • equations, and unknowns (rank must be n, determinant ≠ 0. No rows, columns are linearly dependent.) Use Gaussian elimination to check controllability

  19. Example: , (not linear independent) • Controllability Matrix: Discrete-time: Continuous-time:

  20. Observability , , Example: Suppose is (unknown vector), is (scalar) After time steps, can we reconstruct using values of (, , …, )? In this case is equations, unknowns Must have rank

  21. Observability Matrix • Discrete: must have rank • Continuous: , must have rank

  22. Example Linearize Equations -> deviation variables

  23. Case 1: Measure A, C, D () 4 linear independent columns  Observable

  24. Case 2, 3: Measure (A,B,D) , (B,C,D) Observable • Case 4: Measure A, B, C  Not observable Physical interpretation: since D does not affect any other outputs (does not appear in any rate equations), we don’t know D(0) unless it is measured directly • Note: if adding doesn’t increase the rank of the sequence , you don’t need to check any further. (Same for controllability).

  25. Other Controllability Considerations • Modal equation: must not have a null row (must be able to control all modes) (2) must have no common factor with (3) For non-distinct eigenvalues, rules must be modified; (4) Output controllability of system , (usually , , )

  26. MIMO Model Linearization • Nonlinear model: : state variable () : decision variable () • Linearization define deviation variables: (Requires iterative solution of )

  27. Use 1st order Taylor Series (invalid for large , ) • Scalar case: • Vector case: Analytical vs. numerical

  28. Minimal State Vector Representation(MIMO Systems)  , Make , , then the transfer function is Question: Given , how do we find which gives a model of minimal order? (or given step response for all inputs/outputs)

  29. Example: (Second order) (2 poles => 2nd order, )

  30. Expand by partial fraction expansion Matching coefficients, Let , then , , Let , then , ,

  31. Therefore (1) (2) • Assume , calculate in Eq. (2); • Using , find in Eq.(1); • . Note that there is an infinite number of realizations to yield

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