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Chapter 8 Polynomial Approach

Chapter 8 Polynomial Approach. I/O Model. where and are polynomials in forward-shift operator q. Basic assumptions i) deg B ( q ) < deg A ( q ) ii) A ( q ) and B ( q ) do not have any common factors. (coprime)

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Chapter 8 Polynomial Approach

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  1. Chapter 8Polynomial Approach

  2. I/O Model where and are polynomials in forward-shift operator q. Basic assumptions i) deg B(q) < deg A(q) ii) A(q) and B(q) do not have any common factors. (coprime) iii) The polynomial of A(q) is monic. (normalized for uniqueness) Note: Pulse transfer function B(z)/A(z)

  3. Controller where R(q), T(q)and S(q) are polynomials in forward-shift operator. R(q) can be chosen so that the coefficient of the term of the highest power in q is unity. Notes: deg R(z) deg T(z) deg R(z) deg S(z) causal controller

  4. The characteristic polynomial of the closed-loop system if there is a time delay in the control law of one sampling period

  5. Pole Placement Design Algebraic problem of finding polynomials R(z) and S(z) that satisfy (4) for given A(z), B(z) and Acl(z)

  6. It is natural to choose the polynomial T(z) so that it cancels the observer polynomial Ao(z). where to is the desired static gain of the system.

  7. ex) Control of a double integrator

  8. Diophantine Equation Let A, B, and C be polynomials with real coefficients and X and Y unknown polynomials. Then the above equation has a solution iff the greatest common factor of A and B divides C. Notes: i) The Diophantine equation has many other names in literature, the Bezout identity or the Aryabhatta’s identity. ii) iii) The extended Euclidean algorithm is a straightforward method to solve the Diophantine equation.

  9. Regulator Design by Pole Placement – state space approach

  10. Regulator Design by Pole Placement - polynomial equation approach

  11. Pole Placement Design- More Realistic Assumptions .

  12. Pulse transfer function(uc to y) Remarks: • Causality • deg R  deg T • deg R  deg S • deg A  deg B • Uniqueness • deg A > deg S, deg B > deg R • iii) The cancelled factors must correspond to stable modes.

  13. Causality Solution

  14. ex) DC motor with cancellation of process zero

  15. ex) DC motor with no cancellation of process zero

  16. Optimal Design

  17. Minimum Variance Control - system with stable inverse

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