Adaptive Control Chapter 11: Direct Adaptive Control 1 Adaptive - PowerPoint PPT Presentation

Adaptive Control Chapter 11: Direct Adaptive Control 1 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Chapter 11: Direct Adaptive Control 2 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Adaptive Control – A Basic Scheme SUPERVISION Adaptation loop Plant Performance Controller Model specifications Design Estimation Adjustable Plant + Controller - - Indirect adaptive control - Direct adaptive control (the controller is directly estimated) 3 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Outline • Digital control systems •Tracking and regulation with independent objectives (known parameters) • Adaptive tracking and regulation with independent objectives (direct adaptive control) • Pole placement (known parameters) • Adaptive pole placement (indirect adaptive control) 4 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Digital Control System The control law is implemented on a digital computer e(k) u(k) u(t) y(t) y(k) PLANT r(k) DAC DIGITAL + COMPUTER + Actuator Process Sensor ADC ZOH - CLOCK ADC: analog to digital converter DAC: digital to analog converter ZOH: zero order hold 5 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Digital Control System CLOCK e(k) u(k) u(t) y(t) y(k) r(k) DAC + + PLANT COMPUTER ADC ZOH - DISCRETIZED PLANT - Sampling time depends on the system bandwidth - Efficient use of computer resources 6 Adaptive Control – Landau, Lozano, M’Saad, Karimi

The R-S-T Digital Controller CLOCK r(t) u(t) D/A y(t) Computer PLANT + A/D (controller) ZOH Discretized Plant r(t) y(t) u(t) B − + 1 q d B T m A S A m - Plant Model R Controller = − − q 1 y ( t ) y ( t 1 ) 7 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Discrete time model – General form n n ∑ A ∑ B = − − + − − y ( t ) y ( t i ) u ( t d i ) (*) a b i i = = i 1 i 1 d –delay (integer multiple of the sampling period) n ∑ A − − − − − − − + n 1 i 1 1 * 1 * 1 1 + = = + = + + + 1 a q A ( q ) 1 q A ( q ) A ( q ) a a q ... a n q ; A i 1 2 A = i 1 n ∑ B − − − − − − − + i 1 1 * 1 * 1 1 n 1 = = = + + + b q B ( q ) q B ( q ) B ( q ) b b q ... b n q ; B i 1 2 B = i 1 − − − 1 d 1 = (*) A ( q ) y ( t ) q B ( q ) u ( t ) − − 1 1 + = (*) A ( q ) y ( t d ) B ( q ) u ( t ) (Predictive form) − − d 1 q B ( q ) − 1 − = 1 (*) y ( t ) H ( q ) u ( t ) = H ( q ) ; - pulse transfer operator − 1 A ( q ) − − z 1 − → z q B ( z ) − 1 1 − 1 q = H ( z ) - transfer function − 1 A ( z ) 8 Adaptive Control – Landau, Lozano, M’Saad, Karimi

First order systems with delay Fractional delay − τ s G e = H ( s ) Continuous time model τ = + < < d . T L ; 0 L T + s s 1 T s − − − − − − d 1 2 d 1 1 + + z ( b z b z ) z ( b b z ) Discrete time − 1 1 2 1 2 = = H ( z ) − − 1 1 model + + 1 a z 1 a z 1 1 − L T T L T s s − s − = − = − a e T T T b G ( 1 e ) b Ge ( e 1 ) = − T 1 2 1 b > ⇒ > ⇒ − 2 > Remark: For L 0 . 5 T b b unstable zero ( 1 ) s 2 1 b 1 +j b 2 <1 z b 1 o o x x o - zero 1 x - pole -a 1 b 2 >1 -j b 1 9 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Tracking and regulation with independent objectives It is a particular case of pole placement (the closed loop poles contain the plant zeros)) It is a method which simplifies the plant zeros Allows exact achievement of imposed performances Allows to design a RST controller for: • stable or unstable systems • without restrictions upon the degrees of the polynomials A et B • without restriction upon the integer delay d of the plant model • discrete-time plant models with stable zeros!!! Remarks: • Does not tolerate fractional delay > 0.5 T S (unstable zero) •High sampling frequency generates unstable discrete time zeros ! 10 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Tracking and regulation with independent objectives The model zeros should be stable and enough damped Zero Admissible Zone 1 0.3 0.2 0.8 f 0 /f s = 0.4 0.6 0.1 0.4 0.2 ζ = 0.1 ζ = 0.2 Imag Axis 0 -0.2 -0.4 0.4 -0.6 0.1 -0.8 0.3 0.2 -1 -1 -0.5 0 0.5 1 Real Axis Admissibility domain for the zeros of the discrete time model 11 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Tracking and regulation with independent objectives + d + y * ( t 1 ) y ( t ) u ( t ) − r ( t ) B q d 1 B + m T A S A m - R − ( + d 1 ) q P − d ( + 1 ) q − ( + d 1 ) q B m A m − − − 1 1 1 = P ( q ) P ( q ) P ( q ) D F − 1 B ( q ) * m Reference signal (tracking): + + = y ( t d 1 ) r ( t ) − 1 A ( q ) m − − − = + + − Controller: 1 1 * 1 S ( q ) u ( t ) T ( q ) y ( t d 1 ) R ( q ) y ( t ) 12 Adaptive Control – Landau, Lozano, M’Saad, Karimi -1 (q )

Regulation (computation of R(q -1 ) and S(q -1 ) ) T.F. of the closed loop without T: − + − − + − + − d 1 * 1 d 1 d 1 * 1 q B ( q ) q q B ( q ) − 1 = = = H ( q ) CL − − − + − − − − − 1 1 d 1 * 1 1 1 * 1 1 + A ( q ) S ( q ) q B ( q ) R ( q ) P ( q ) B ( q ) P ( q ) The following equation has to be solved : − − − + − − − − (*) 1 1 d 1 * 1 1 * 1 1 + = A ( q ) S ( q ) q B ( q ) R ( q ) B ( q ) P ( q ) − − − − − 1 1 n * 1 ′ 1 S should be in the form: = + + + = S ( q ) s s q ... s q B ( q ) S ( q ) S 0 1 n S After simplification by B*, (*) becomes: − − − + − − 1 1 d 1 1 1 + = (**) A ( q ) S ' ( q ) q R ( q ) P ( q ) Unique solution if: n P = deg P ( q -1 ) = n A +d ; deg S' ( q -1 ) = d ; deg R ( q -1 ) = n A -1 − − − − − − − 1 1 n 1 1 1 d = + + = + + R ( q ) r r q ... r q S ' ( q ) 1 s ' q ... s ' d q A − 0 1 n 1 1 A 13 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Regulation ( computation of R(q -1 ) and S(q -1 ) ) x = M -1 p (**) is written as: Mx = p n A + d + 1 1 0 0 a 1 1 . a 2 a 1 0 . : : 1 . a d a d-1 ... a 1 1 . n A + d + 1 a d+1 a d a 1 1 . a d+2 a d+1 a 2 0 . . . . 0 0 0 ... 0 a n A 0 0 1 d + 1 n A T T ′ ′ = = p [ 1 , p , p ,..., p , p ,..., p ] x [ 1 , s ,..., s , r , r ,..., r ] + + − 1 2 n n 1 n d 1 d 0 1 n 1 A A A Use of WinReg or predisol.sci(.m) for solving (**) Insertion of pre specified parts in R and S is possible 14 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Tracking (computation of T(q -1 ) ) Closed loop T.F.: r y − + − − − − + ( d 1 ) 1 1 1 ( d 1 ) q B ( q ) B ( q ) T ( q ) q − 1 m m = = H ( q ) BF − − − 1 1 1 A ( q ) A ( q ) P ( q ) m m Desired T.F. It results : T(q -1 ) = P(q -1 ) Controller equation: − − − 1 = 1 * + + − 1 S ( q ) u ( t ) P ( q ) y ( t d 1 ) R ( q ) y ( t ) − − 1 * 1 + + − P ( q ) y ( t d 1 ) R ( q ) y ( t ) = u ( t ) − 1 S ( q ) [ ] 1 − − − 1 * * 1 1 = + + − − − u ( t ) P ( q ) y ( t d 1 ) S ( q ) u ( t 1 ) R ( q ) y ( t ) (s 0 = b 1 ) b 1 − 1 B ( q ) Reference signal (tracking) : * + + = m y ( t d 1 ) r ( t ) − 1 A ( q ) m 15 Adaptive Control – Landau, Lozano, M’Saad, Karimi

Tracking and regulation with independent objectives A time domain interpretation y* (t+d+1) u(t) y(t) 1 + − ( + r(t) B d 1 ) q B * P m S A A m - − + ( d 1 ) q R = + + y ( t ) y * ( t d 1 ) − 1 P ( q ) − ( + d 1 ) q P − − + = + + 1 ( d 1 ) P ( q ) y ( t ) q y * ( t d 1 ) − d ( + + P 1 ) q - − − + − + + = 1 ( d 1 ) P ( q ) y ( t ) q y * ( t d 1 ) 0 ε ≡ 0 ( t ) 0 (in case of correct tuning) Reformulation of the “design problem”: Find a controller which generate u(t) such that: [ ] ε + + = + + − + + = 0 ( t d 1 ) P y ( t d 1 ) y * ( t d 1 ) 0 16 Adaptive Control – Landau, Lozano, M’Saad, Karimi