Adaptive Control Chapter 7: Digital Control Strategies 1 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Chapter 7: Digital Control Strategies 2 Adaptive Control – Landau,Lozano, M’Saad, Karimi
The R-S-T Digital Controller r(t) y(t) B u(t) − 1 d + q B m T A S A m - Plant Model R Controller q B ( q ) q B * ( q ) − − − − − d 1 d 1 1 = = = G ( q − ) H ( q − ) Plant Model: 1 1 A ( q ) A ( q ) − − 1 1 = + + + = + + = − − − − − − − − A ( q 1 ) 1 a q 1 ... a n q n B ( q 1 ) b q 1 ... b q n q 1 B * ( q 1 ) A B 1 1 n A B = + + − − − − R-S-T Controller: 1 1 1 S ( q ) u ( t ) T ( q ) y * ( t d 1 ) R ( q ) y ( t ) Characteristic polynomial (closed loop poles): = + − − − − − − P ( q 1 ) A ( q 1 ) S ( q 1 ) q d B ( q 1 ) R ( q 1 ) 3 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Pole placement The pole placement allows to design a R-S-T controller for • stable or unstable systems • without restriction upon the degrees of A and B polynomials • without restrictions upon the plant model zeros (stable or unstable) It is a method that does not simplify the plant model zeros The digital PID can be designed using pole placement 4 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Structure p(t) + y(t) r(t) - d + 1 + - 1 q B ----------- - q T( ) ) -------- - - 1 S (q ) A - PLANT - 1 R(q ) - d - 1 q B(q ) ------------------- - - 1 P (q ) − − d 1 q B ( q ) − 1 = Plant: H ( q ) − 1 A ( q ) − − − − − − − − − 1 1 n 1 1 2 n 1 * 1 = + + + = = + + + A ( q ) 1 a q ... a n q B ( q ) b q b q ... b q q B ( q ) A B 1 1 2 n A B 5 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Pole placement Closed loop T.F. (r y) (reference tracking) − − − − − − d 1 1 d 1 1 q T ( q ) B ( q ) q T ( q ) B ( q ) − 1 = = H ( q ) BF − − − − − − 1 1 d 1 1 1 + A ( q ) S ( q ) q B ( q ) R ( q ) P ( q ) − − − − − − − − 1 1 1 d 1 1 1 2 = + = + + + P ( q ) A ( q ) S ( q ) q B ( q ) R ( q ) 1 p q p q .... 1 2 Defines the (desired )closed loop poles Closed loop T.F. (p y) (disturbance rejection) A ( q − ) S ( q − ) A ( q − ) S ( q − ) 1 1 1 1 = = S ( q − ) 1 + yp A ( q ) S ( q ) q B ( q ) R ( q ) P ( q ) − − − − − − 1 1 d 1 1 1 Output sensitivity function 6 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Choice of desired closed loop poles (polynomial P ) − − − 1 1 1 = P ( q ) P ( q ) P ( q ) D F Dominant poles Auxiliary poles Choice of P D (q -1 )(dominant poles) Specification discretization − 2 nd order ( ω 0 , ζ ) 1 in continuous time P ( q ) D T (t M , M) e ≤ ω ≤ 0 . 25 T 1 . 5 0 e ≤ ζ ≤ 0 . 7 1 Auxiliary poles • Auxiliary poles are introduced for robustness purposes • They usually are selected to be faster than the dominant poles 7 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Regulation( computation of R(q -1 ) and S(q -1 ) ) − − − − − − 1 1 d 1 1 1 + = (Bezout) A ( q ) S ( q ) q B ( q ) R ( q ) P ( q ) (*) ? ? A and B do not have − − 1 1 = = n B deg B ( q ) n A deg A ( q ) common factors unique minimal solution for : − 1 = ≤ + + − n deg P ( q ) n n d 1 P A B − − 1 1 = = + − = = − n deg S ( q ) n d 1 n deg R ( q ) n 1 S B R A − − − − − n 1 1 1 1 = + + = + S ( q ) 1 s q ... s q 1 q S * ( q ) S 1 n S − − − 1 1 n = + + R ( q ) r r q ... r n q R 0 1 R 8 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Computation of R(q-1) and S(q-1) Equation (*) is written as: Mx = p x = M -1 p T T = = x [ 1 , s ,..., s , r ,..., r ] p [ 1 , p ,..., p ,..., p , 0 ,..., 0 ] 1 n 0 n 1 i n S R P n B + d n A 1 0 ... 0 0 ... ... 0 a 1 1 . b' 1 a 2 0 b' 2 b' 1 1 . b' 2 n A + n B + d a 1 . . a n A a 2 b' n B . 0 0 . . . . 0 0 a n A 0 0 0 b' n B ... n A + n B + d b' i = 0 pour i = 0, 1 ...d ; b' i = b i -d pour i > d Use of WinReg or bezoutd.sci(.m) for solving (*) 9 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Structure of R(q -1 ) and S(q -1 ) R and S include pre-specified fixed parts (ex: integrator) − − − − − − 1 1 1 1 1 1 = = R ( q ) R ' ( q ) H ( q ) S ( q ) S ' ( q ) H ( q ) S R H R , H S , - pre-specified polynomials − − − 1 1 n − − − = + + 1 1 n = + + S ' ( q ) 1 s ' q ... s ' n q R ' ( q ) r ' r ' q ... r ' n q S ' R ' 1 0 1 S ' R ' •The pre specified filters H R and H S will allow to impose certain properties of the closed loop. •They can influence performance and/or robustness 10 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Fixed parts ( H R , H S ). Examples Zero steady state error ( S yp should be null at certain frequencies) − − − 1 1 ′ 1 A ( q ) H ( q ) S ( q ) − 1 = S S ( q ) yp − 1 P ( q ) − − 1 1 = − H S ( q ) 1 q Step disturbance : − − 1 2 = + α + α = − ω Sinusoidal disturbance : H 1 q q ; 2 cos T S s Signal blocking ( S up should be null at certain frequencies ) − − − 1 1 ′ 1 A ( q ) H ( q ) R ( q ) − 1 = − R S ( q ) up − 1 P ( q ) − − 1 2 = + β + β = − ω Sinusoidal signal: H 1 q q ; 2 cos T R s − 1 n Blocking at 0.5f S: = + = H ( 1 q ) ; n 1 , 2 R 11 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Tracking (computation of T(q -1 ) ) Ideal case - 1 − − 1 1 q B m r (t) y* (t) y * q B ( q ) − 1 m = H ( q ) ----------- - m − 1 A m A ( q ) m r Tracking reference − − 1 1 = + + desired t B ( q ) b b q ... m m 0 m 1 model (H m ) trajectory for y (t) − − − 1 1 2 = + + + A ( q ) 1 a q a q ... m m 1 m 2 Specification discretization − 2 nd order ( ω 0 , ζ ) 1 in continuous time H m ( q ) T (t M , M) s ≤ ω ≤ 0 . 25 T 1 . 5 0 s ≤ ζ ≤ 0 . 7 1 The ideal case can not be obtained (delay, plant zeros) Objective : to approach y*(t) − + − ( d 1 ) 1 q B ( q ) * = m y ( t ) r ( t ) − 1 A ( q ) m 12 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Tracking (computation of T(q -1 ) ) − 1 B ( q ) Build: * m + + = y ( t d 1 ) r ( t ) − 1 A ( q ) m Choice of T(q -1 ) : • Imposing unit static gain between y* and y • Compensation of regulation dynamics P(q -1 ) ≠ ⎧ 1 / B ( 1 ) si B ( 1 ) 0 = G T(q -1 ) = GP(q -1 ) ⎨ = 1 si B ( 1 ) 0 ⎩ − * 1 − + − ( d 1 ) 1 B ( q ) q B ( q ) − 1 F.T. r y : m = ⋅ H ( q ) BF − 1 B ( 1 ) A ( q ) m ⎧ P ( 1 ) ⎪ ≠ si B ( 1 ) 0 Particular case : P = A m − 1 = = T ( q ) G ⎨ B ( 1 ) ⎪ = 1 si B ( 1 ) 0 ⎩ 13 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Pole placement. Tracking and regulation * y (t+d+1) u(t) y(t) -d B B m + 1 r(t) q T A S A m - R -1 -(d+1) B*(q ) q P(q -1 ) -(d+1) -1 q B*(q ) B(1) -1 -1 -(d+1) B m (q ) B*(q ) q -1 A m (q ) B(1) − − − 1 1 1 + = + + S ( q ) u ( t ) R ( q ) y ( t ) T ( q ) y * ( t d 1 ) 14 Adaptive Control – Landau,Lozano, M’Saad, Karimi
Pole placement. Control law − − 1 * 1 + + − T ( q ) y ( t d 1 ) R ( q ) y ( t ) = u ( t ) − 1 S ( q ) − − − − 1 1 1 * 1 * + = + + = + + S ( q ) u ( t ) R ( q ) y ( t ) GP ( q ) y ( t d 1 ) T ( q ) y ( t d 1 ) − − − 1 1 * 1 = + S ( q ) 1 q S ( q ) − − − 1 * * 1 1 = + + − − − u ( t ) P ( q ) Gy ( t d 1 ) S ( q ) u ( t 1 ) R ( q ) y ( t ) − 1 B ( q ) * m + + = y ( t d 1 ) r ( t ) − 1 A ( q ) m − − − 1 1 * 1 = + A ( q ) 1 q A ( q ) m m − − * * 1 1 + + = − + + y ( t d 1 ) A ( q ) y ( t d ) B ( q ) r ( t ) m m − − − − − 1 1 1 1 2 = + + = + + + B ( q ) b b q ... A ( q ) 1 a q a q ... m m 0 m 1 m m 1 m 2 15 Adaptive Control – Landau,Lozano, M’Saad, Karimi
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