V. Practical Optimization

Transcription

1 V. Practical Otimization Scaling Practical Otimality Parameterization Otimization Objectives Means Solutions ω c -otimization Observer based design V-

2 Definition of Gain and Frequency Scales G ( s) kg () s G ( s) G ( s/ ω ) k = = ss ( +.4) s s s s ( + ) ( + ).4.4 ω ω k =.67 ω =.4 V-

3 Unit-Gain Unit-Bandwidth Plants,,,,,,... s+ s s + ξs+ s(s+) s s + ξ s + ξ s+ 3 s+ s + ξ zs+ 3 ξ ξ ξ,,... s + s+ s + s + s+ Controllers for UGUB lants are redetermined V-3

4 Scaling of a controller for a new lant Sto reinventing the wheel A class of rocesses/lants, differ only in dc-gain and bandwidth Normalization of lants: unit-gain unit-bandwidth Controller design for UGUB lants Controller is obtained by scaling UGUB controllers {G (s), G c (s) } {kg (s/ω ), G c (s/ω )/k} V-4

5 Scaling of PID ki { G(), s Gc() s = k+ + ks d } s ω s { kg( s/ ω ), Gc() s = ( k + ki + kd )/ k} s ω k kiω kd k =, ki =, kd = k k kω V-5

6 Scaling Examle G () s = s + s+.5 ste resonse original PID scaled PID k =3, k i =, and k d =. G () s = s s ( ) + + k = 3, k =, k =. i d zoomed in control signal time second V-6

7 Practical Otimality Otimization of Control Law: To maximizing a function of given erformance measures subject to the hysical limitations of the design Quality of Control: Seed, Accuracy, Disturbance Rejection Constraints: Samling Rate, Sensor Noise, Uncertainty, Actuation Smoothness, Saturations V-7

8 w c -Otimization maximizing the loo gain bandwidth, ω c, subject to the hysical limitations of the design. V-8

9 w c - Parameterization arameterization of the controllers, G c (s,ω c ), for the unit gain and unit bandwidth that result in the loo gain bandwidth to be ω c, or closed-loo transfer function to be ω ω ω,,,... s+ s+ s+ 3 c c c 3 ωc ( ωc) ( ωc) V-9

10 w c - Parameterization Examles G (s) s + s s + ξ s+ s ss+ ( ) G c (s,ω c ) ω c ( s + ) s ω c ω s + ξ s+ c ss ( + ωc) ω c ( s + ) s + ω c ωc s s + ω c V-

11 Parameterization of Loo Shaing Controller db G ( jω) G ( jω) c Command Following Disturbance Rejection - db/dec ω ω c ω Sensor Noise Unknown Dynamics ω s + ω Ls () = G () sg () s = s s + s ω + c ω c n m m s + ω c() () n G s = G s s s + s ω + c ω V-

12 Assumtions The lant is minimum hase; G (s) is given; G c (s) is ω c -arameterized A transient rofile is defined; A simulator is available V-

13 Otimization Procedure. Find lant frequency and gain scales, ω and k;. Select controller G c (s, ω c ) 3. Scale G c (s, ω c ) to obtain G c (s/ω, ω c )/k 5. Digitization and imlementation in simulator; 6. Set an initial value of ω c 7. Gradually increase ω c until a. Control signal becomes too noisy and/or to uneven b. Indication of instability (oscillatory behavior) V-3

14 Examle y = (.4y + 3. T ) + 3.u d G k () s =, k=.67, ω =.4 s s ( + ) ω ω G () s = ss ( + ) G cl ( s) = ωc ( s + ω ) c V-4

15 Control Law u = k ( r y) + k ( y) k = ω and k = ω c d c d k ω ω = =.86 and = =.6( ) c c ωc kd ωc k kω aroximate differentiator: s s ( ) ω + c V-5

16 Secs Settling time: sec Noise is control signal (u) not exceeding mv Torque disturbance: 3%, ste Sensor noise:.% white noise V-6

17 Simulation.5.5 Ste Resonse ste resonse bandwidth: 4 rad/sec bandwidth: rad/sec error With Transient Profile transient rofile and outut bandwidth: 4 rad/sec bandwidth: rad/sec transient rofile error control 3signal time second control 3signal time second V-7

18 Otimization of Observer Based Design Otimization of the observer Otimization of the control law Combined otimization V-8

19 A Generalized Disturbance Observer ( n) ( n ) ( n ) y f(, t y, y,, y =, uu,,... u, w) + bu x = x x = x 3... x n = xn+ + bu x n+ = h y = x z = z β( z yt ()) z = z3 β( z yt ())... z n zn+ βn( z yt ()) bu = + z n+ = βn+ ( z yt ()) z t yt z t yt z t y t ( n ) () (), () (),, n() () zn + t f t y y y uu u w ( n ) ( n ) () (,,,,,,,..., ) V-9

20 w o -otimization Observer bandwidth: ω o s + β s β s+ β = ( s+ ω ) n n n n n o Otimization: Maximize ω o until noise reaches threshold V-

21 Control Law u = z + n+ u b ( n) y = ( f zn+ ) + u u u = k ( r z ) k z... k z n d d n s + k s k s+ k = ( s+ ω ) n n n d d c n V-

22 Design Procedure Ste : Design a arameterized observer and feedback controller; Ste : Design a transient rofile with the equivalent bandwidth of ω ct ; Ste 3: Select an ω o to 5- times larger than ω ct ; Ste 4: Set ω c = ω o and increase both by the same amount until the noises levels and/or oscillations in the control signal and outut exceed the tolerance; Ste 5: Incrementally increase or decrease ω c and ω o individually, but kee the sum the same. V-

23 Examle y = (.4y + 3. T ) + (3. 4) u+ 4u = f + 4u d z 3ωo 3ωo u = 3ωo z+ 4 3ω o y 3 3 ωo ω o z yz, yand, z f =.4y + 3. T +(3.-4)u, as t 3 d u u 3 = d 4 z u = k ( r z ) kz k = ξω, ξ =, and k = ω d c c V-3

24 Simulation outut PD Controller Disturbance Observer Based Controller osition y z error control 3signals time second velocity dy/dt z - disturbance and unknown 3 dyanmics f z time second V-4

25 Self-Tuning ω o Automatically resonse to the noise level ω c Bandwidth scheduling Scaling Factors (k,ω ) Adatation to lant dynamics variations V-5

26 CAD Package Bluerint User Enters Plant Info and Design Sec lant model or resonse hysical limitations design secifications No Feasible? No Yes find a arameterized solution otimize the solution via simulation V-6

27 Summary Three new concets Scaling Parameterization Practical Otimization Alications From art to science Stos reinventing the wheels Otimization in real world V-7