08 Explicit-Implicit STC - PolePlace

08 Explicit-Implicit STC - PolePlace - SYS635 Adaptive...

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SYS635 Adaptive Control Systems Ka C. Cheok 08 Explicit-Implicit STC - PolePlace 15 Oct ‘05 1 EXPLICIT AND IMPLICIT SELF-TUNING CONTROLLER (STC) for Closed-Loop Pole-Placement 1 Explicit Self-Tuning Pole Placement (STPP) Controller, a.k.a. Indirect STPP Controller An explicit self-tuning pole placement controller consists of a parameter estimator, controller designer and controller executor as illustrated in Figure 1. Steps for Explicit Self-Tuning Controller (you already know this) a. Specify desired closed-loop characteristic polynomial 32 12 3 () Cq q cq cq c = ++ + . b. Estimate parameters in 2 ˆ ˆˆ A qqa qa =+ + and 01 Bq bq b = + c. Design 1 Rq q r =+ and Sq sq s from ˆ ˆ ()() AqRq BqSq += . d. Implement 11 0 11 () () , .. , c kkk k Rquk Sq u k yk Sqek ie u ru se =− = = + + A general configuration of an explicit STPP control system is shown below. 2 + + + 1 s qs qr + + - 1 1 1 '' 1 1 1 [1 ] ˆ [] /( ) / kk k k k k k k k k k k k yy u u Ky KP P PI K P φ θθ θ φφφ λ φλ +− + + + + + = −− k y k u k e k r 1 1 1 2 1 2 2 13 22 ˆ 1 ˆ ˆ ˆ b rc a s abb ca sc ab    1 2 1 2 ˆ ˆ ˆ k a a b b c = 1 0 1 r s s k w k v 3 3 Specify desired closed-loop poles , and thus the desired closed-loop polynomial and zc λλ +
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SYS635 Adaptive Control Systems Ka C. Cheok 08 Explicit-Implicit STC - PolePlace 15 Oct ‘05 2 Note that the closed-loop system output responds to the input u c , disturbance w and noise v as follows () c A xB uw Ru Su Sy yxv =+ =− c c c c c SS Ax B u y w RR ARx BSu BS x v BRw AR BS x BSu BSv BRw BS BS BR BS BS xu vw AR BS AR BS BR yv u v w AR BS AR B AR B SB S S AR + ++ +  +   =−+ + +=− + −= + + cc BS AR BR BS AR BR yu u v w AR BS AR BS AR B CC S C += + + + where C A RB S . Only poles (hence, stability) of the closed loop systems are affected by the STPP design Note 1. The pair ( A, B ) must not have common polynomial factors so that C A S = + is a Diophantine equation, so that we can solve for R & S when C is given. Note 2: The effect of numerators BS, AR and BR should also be taken into account while designating the closed-loop polynomial C . Example, choose dominant poles, then play or adjust the remaining poles in simulation so that the response y is acceptable. This is usually validated using simulation. Potential problems that may be encountered in the explicit (indirect) STC: 1) The degrees of polynomials must be specified (known). 2) There must not be common polynomial factors in the estimated A and B. 3) Stability of closed-loop must be guaranteed (by bounding the parameters) 4) Persistent exciting signals are necessary to ensure parameter estimation convergence.
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08 Explicit-Implicit STC - PolePlace - SYS635 Adaptive...

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