1. Robotics Research Laboratory 1 Chapter 8 Polynomial Approach

2. Robotics Research Laboratory 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. Robotics Research Laboratory 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 3 Controller

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

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

6. Robotics Research Laboratory 6 It is natural to choose the polynomial T(z) so that it cancels the observer polynomial A o (z). where t o is the desired static gain of the system.

7. Robotics Research Laboratory 7 ex) Control of a double integrator

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9. 9 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. Diophantine Equation

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14. Robotics Research Laboratory 14 Regulator Design by Pole Placement – state space approach

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16. Robotics Research Laboratory 16 Regulator Design by Pole Placement - polynomial equation approach

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19. Robotics Research Laboratory Pole Placement Design - More Realistic Assumptions. 19

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21. Robotics Research Laboratory 21 Pulse transfer function (u c to y) Remarks: i)Causality deg R  deg T deg R  deg S deg A  deg B ii)Uniqueness deg A > deg S, deg B > deg R iii)The cancelled factors must correspond to stable modes.

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23. Robotics Research Laboratory 23 Causality Solution

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25. Robotics Research Laboratory 25 ex) DC motor with cancellation of process zero

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28. Robotics Research Laboratory 28 ex) DC motor with no cancellation of process zero

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31. Robotics Research Laboratory 31 Optimal Design

32. Robotics Research Laboratory 32 Minimum Variance Control - system with stable inverse

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