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Chapter 7 Multivariable and Optimal Control

Chapter 7 Multivariable and Optimal Control. Time-Varying Optimal Control - deterministic systems. LQ problem (Linear Quadratic)–Finite time problem. Using Lagrange multipliers. Control gains vs. time. 10. Q2 = 0.01. 8. 6. K theta. 4. Q2 = 0.1. 2. Q2 = 1.0. 0. 0. 5. 10. 15. 20.

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Chapter 7 Multivariable and Optimal Control

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  1. Chapter 7Multivariable and Optimal Control

  2. Time-Varying Optimal Control - deterministic systems

  3. LQ problem (Linear Quadratic)–Finite time problem Using Lagrange multipliers

  4. Control gains vs. time 10 Q2 = 0.01 8 6 K theta 4 Q2 = 0.1 2 Q2 = 1.0 0 0 5 10 15 20 25 30 35 40 45 50 k 5 Q2 = 0.01 4 3 Q2 = 0.1 K theta dot 2 Q2 = 1.0 1 0 0 5 10 15 20 25 30 35 40 45 50 k

  5. LQR (Linear Quadratic Regulator) -Infinite time problem • ARE(Algebraic Riccati Equation) • analytic solution is impossible in most cases. • numerical solution is required.

  6. Remark : Using reciprocal root propertiesin p372

  7. Symmetric root locus 1.5 1 0.5 Imag Axis 0 -0.5 -1 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Real Axis

  8. Eigenvector Decomposition

  9. outside the unit circle inside the unit circle

  10. Cost Equivalents

  11. Least Squares Estimation n1 unknown vector pn matrix p1 measurement error vector p1 measurement vector

  12. Sales fit and prediction 14 12 10 8 Sales ($1000) 6 4 2 0 0 5 10 15 20 25 Months

  13. Weighted Least Squares

  14. Recursive Least Square

  15. old estimate covariance of old estimate new estimate old estimate

  16. Sometime, it is a scalar. That is ifwe use just one new information.

  17. Stochastic Models of Disturbance • We have dealt with well-known well-defined, ideal systems. • disturbance (process, load variation) • measurement noise sample point in sample space(event) s 0 real line new (range) sample space

  18. X F(X, t1) X F(X, t3) F(X, t2) X(•, 3) X(•, 2) X(•, 1) t 1 0 0 t2 t3 t1

  19. Remark:

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