Contents lecture 4. Automatic Control III. Summary of lecture 3 (II/II) Summary of lecture 3 (I/II) Lecture 4 Controller structures and control design

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1 Content lecture 4 Automatic Control III Lecture 4 Controller tructure and control deign Thoma Schön Diviion of Sytem and Control Department of Information Technology Uppala Univerity. thoma.chon@it.uu.e, www: uer.it.uu.e/~thoc Summary of lecture 3 2. Which control deign method do we have? 3. Who hould control what? Relative Gain Array (RGA) a) The pairing problem b) Decentralized control c) Decoupled control 4. Internal Model Control (IMC) 1 / 23 T. Schön, / 23 T. Schön, 215 Summary of lecture 3 (I/II) Bode relationhip provide an upper bound on the phae, which depend on the derivative of the amplitude curve. Hence, Bode relationhip provide a fundamental limit by revealing a certain coupling between the amplitude and the phae. 3 / 23 T. Schön, 215 Summary of lecture 3 (II/II) S once, and only in an optional reading of an unaigned chapter in one of the claical textbook. Thi integral urfaced for me for the econd time in the mid 197, referenced in a paper The econd integral did not urface for me until 1983, in a talk by Jim Freudenberg at an IEEE Conference on Deciion and Control in San Antonio [3]. If memory erve, omeone pointed out at the time that thi reult wa jut a verion of Jenen theorem, well known in mathematic for a long time. Perhap thi hitorical reference reduced the value of the reult in the mind of ome litener, but it hould not have, becaue the integral explain o much about the difficultie of controlling untable Bode integralat ytem. A Bode Integral Interpretation I like to think of Bode integral a conervation law. They tate preciely that a certain quantity the integrated value of the log of the magnitude of the enitivity function i conerved under the action of feedback. The total amount of thi quantity i alway the ame. It i equal to zero for table plant/compenator pair, and it i equal to ome fixed poitive amount for untable one. Since we are talking about the log of enitivity magnitude, it follow that negative value are good (i.e., enitivitie le than unity, better than open loop) and poitive value are bad (i.e., enitivitie greater than unity, wore than open loop). log S(iω) dω =. 1 So for open-loop table ytem, the average enitivity improvement a feedback w loop achieve over frequency i ex actly offet by it average enitivity deterioration. For open-loop untable ytem, thing are wore becaue the average deterioration i alway larger than the improvement. Thi applie to every controller, a mound i depoited omewhere ele. Thi fact i mot evident to the ditch digger, becaue he i right there to ee it happen. by Iaac Horowitz titled On the Superiority of Tranfer Functionytem over State-Variablewe Method. derived... Itappeared Bode a a per- integral theorem tating In the ame pirit, I can alo illutrate job of a more ac- For table pective paper in IEEE Tranaction on Automatic Control that amid a certain amount of controvery [2]. ademic control deigner with more abtract tool uch a linear quadratic Gauian (LQG), H, convex optimization, and the like, at hi dipoal. Thi deigner guide a powerful ditch-digging machine by remote control from the afety of hi worktation (Figure 4). He et parameter (weight) at hi tation to adjut the contour of the machine digging blade to get jut the right hape for the enitivity function. He then let the machine dig down a far a it can, and he ave the reulting compenator. Next, he fire up hi automatic code generator to write the implementation code for the compenator, ready to run on hi target microproceor. no matter how it wa deigned. Senitivity improvement in one frequency range mut be paid Formal Deign for with enitivity deterioration in another i=1 Re(p i). frequency range, and the price i higher Automatic if the plantcontrol III, Lecture 4 Controller tructure and control deign.g 4 / 23 T. Schön, 215 i open-loop untable. It i curiou, omehow, that our field ha not Formal Synthei adopted a name for thi quantity being conerved (i.e., the integrated log of Machine enitivity Log Magnitude 1 1. Seriou Deign Frequency Figure 3. Senitivity reduction at low frequency unavoidably lead to enitivity increae at higher frequencie. For untable ytem (with M pole {p i } M i=1 in the RHP): log S(iω) dω = π M.g 2.

2 PID The mot ucceful controller ever: PID Boulton och Watt 1788: peed control of team engine, mechanical implementation. Aume one input ignal and one output ignal If you have everal input and output ignal you mut pair them in two Interpretation in Bode plot: lead and lag Hydraulic and pneumatic implementation: late 18 Tuning uing intuition and experience reult in not more than mediocre performance (except in very imple cae). Electronic: 193. Tuning uing ytematic analyi (pole, zero, S, T,...) can give controller with very good performance. Computer: 195. PID-on-a-chip : 199. The firt ytematic approach (uing pole): Maxwell Application: All. Maxwell, J.C. On Governor, Proceeding of the Royal Society, no. 1 (1868). Seam engine: PID-on-a-chip: 5 / 23 T. Schön, / 23 Where PID i not enough 1. Tall ytem (more output than input ignal): G= Internal model control (IMC) Minimization of quadratic criteria: LQ, LQG. Model predictive control (MPC) Sytematic haping of tranfer function: H2, H. All output ignal cannot be controlled perfectly prioritize. Nonlinear method 2. Fat ytem (more input ignal than output ignal) G= The foundation of all modern control method i to make ue of model. T. Schön, 215 MIMO who hould control what? (Typically mean that the ytem i multivariable and/or nonlinear) 7 / 23 T. Schön, 215 How hould the actuation be ditributed among the control ignal? 8 / 23 T. Schön, 215

3 Several input and output ignal interaction Interaction/cro coupling (I/II) If there are many input and output ignal we can make the control deign much eaier by breaking down the ytem in ub-ytem with little interaction between each other. Two-handle mixer, a ytem with a tough cro coupling Several input ignal (ignificantly) affect an output ignal. Several output ignal are (ignificantly) affected by an input ignal. The relative gain array (RGA) i a way of meauring the level of cro coupling or interaction in a ytem. Vinkel kallvattenvred Vinkel varmvattenvred Temperatur Vinkel kallva VattenflödeVinkel varmva 9 / 23 T. Schön, / 23 T. Schön, 215 Interaction/cro coupling (II/II) Decentralized control Mixer with one handle, a ytem with a nice cro coupling. Each input ignal affect (almot) only one output ignal. Each output i affected (almot) only by one input ignal. Vinkel temperatur Vinkel flöde TemperaturVinkel temperatur Vattenflöde Idea (decentralized control): Build a controller for a MIMO ytem where one output ignal control one input ignal. The reult i a et of ingle variable loop Vinkel flöde u i = F i rr j F i yy j, where the individual controller are all independent ( they do not know of each other ). F y i a quadratic tranfer matrix. If the number of input and G11 output ignal i different ome of them Temperatur are imply dicarded. The le cro coupling there are, G22 Vattenflöde the better thi trategy work We want to pair the input and output ignal that have the tronget connection, the pairing problem. How do we determine the coupling between the variou input and output ignal? 11 / 23 T. Schön, / 23 T. Schön, 215

4 Temperature control T 1 T 2 PID U 1 U 2 Two room with a eparating inner wall. The temperature T 1, T 2 are tate, which are both meaured. Both room can be both heated and cooled by U 1 and U 2. [ ] [ e 5 ] [ ] T1 = U e T 2 U / 23 T. Schön, 215 Which enor hould be ued in controlling which heating/cooling ource (i.e. the pairing problem)? RGA again Temperature control Decentralized PI control T 1 i ued for U 1. T 2 i ued for U 2. [ ] F () = Decentralized PI control T 2 i ued for U 1. T 1 i ued for U 2. [ ] F () = After 1 hour 1 people enter room 1. Whooop... I there any way in which we can predict thi problem analytically? Effekt törning [W] Temperatur [grad. C] Effekt [W] T1 T x / 23 T. Schön, 215 Temperature control RGA Temperatur [grad. C] Effekt [W] Effekt törning [W] x x Ue RGA to decide which input ignal to pair with which output ignal. The two main rule in deigning a decentralized controller uing RGA. Pair meaurement ignal and control ignal uch that the diagonal element in 1. RGA(G(iω c )) are cloe to 1 in the complex plane. 2. RGA(G()) are poitive (if they are negative thi can lead to intability) Pairing implie a change of the poition of row and column in the RGA-matrix. RGA in Matlab: RGA(A) = A.*pinv(A. ) (pinv = peudoinvere, handle non-quare matrice). Deign rule from previou lide i different word: 1. Select input-output pair o that the diagonal element of RGA(iω c ) are cole to 1 2. Avoid pairing that give negative diagonal element of RGA(G()). In the temperature control example: ( ) 1 RGA(G(i8)) RGA(G()) = 1 The pairing where T 2 i ued for U 1 and T 1 i ued for U 2, break both of thee rule (the row change place)! ( 1.17 ) / 23 T. Schön, / 23 T. Schön, 215

5 Decoupled control Temperature control decoupled Sound good, but how do we chooe W 1 and W 2? In order to obtain a completely decoupled virtual ytem we would need -dependent wright matrice W 1 and W 2. Thi i in general not poible, ince it would lead to a complicated and/or non-proper controller. Intead we chooe one frequency where the ytem become decoupled: ω = ω = ω c (G (ω c ) i often approximated to get rid of complex valued element). The choice W 1 = G () and W 2 = I reult in decoupling in tationarity. With the right choice of W 1 and W 2 we can make the two-handle mixer behave a a one-handle mixer. Eaier to control!! After 1 hour 1 people enter room 1. Decoupled control uing W 1 = G (), and W 2 = I. Effekt törning [W] Temperatur [grad. C] Effekt [W] x T1 T / 23 T. Schön, / 23 T. Schön, 215 Content lecture 4 Internal Model Control (IMC) Feedback only uing the new information y Gu. 1. Summary of lecture 3 2. Which control deign method do we have? 3. Who hould control what? Relative Gain Array (RGA) a) The pairing problem b) Decentralized control c) Decoupled control 4. Internal Model Control (IMC) r + u y F r Q G + G Reult in (if G = G) G c = GQ F r, S = I GQ, T = GQ 19 / 23 T. Schön, / 23 T. Schön, 215

6 How do we chooe Q? (I/II) How do we chooe Q? (II/II) The ideal cae Q = G would reult in S =, G c = I, but it i infeaible ince F y =. Hence, we have to approximate! Some rule of thumb: 1. If G ha more pole than zero: The invere of G cannot be realized. Ue Q() = 1 (λ + 1) n G (). Chooe n uch that Q() i proper (# pole = # zero). Chooe λ to adjut the bandwidth of the cloed loop ytem. 2. If G() non-minimum phae G ha an untable zero and contain a factor ( β + 1), β >. Two alternative a) Omit the factor when Q = G i formed. b) Replace the factor ( β + 1), β > with (β + 1), β > when Q = G i formed. 3. If G contain a time delay, i.e. a factor e τ : a) Omit the factor when Q = G i formed. b) Approximate the factor uing e τ 1 τ/1 1 + τ/2 21 / 23 T. Schön, / 23 T. Schön, 215 A few concept to ummarize lecture 4 Cro coupling: The key difficulty in controlling MIMO ytem i that there are cro coupling between input and output ignal. If we change one input ignal thi affect everal output ignal. Relative gain array (RGA): The relative gain array i a meaure of the amount of cro coupling in a matrix (RGA(A) = A. (A ) T ). Decentralized control: Let every input be determined by feedback from one ingle output. The pairing problem: The pairing problem i to elect which input-output pair that hould be ued for the feedback. Decoupled control: Decoupled control make ue of a change of variable uch that uitable pairing of meaurement and control ignal become eaier to ee. Internal model control (IMC): Chooe Q G (y = GQr) and let the new information in the form of y Gu be fed back to affect u. 23 / 23 T. Schön, 215