Contents lecture 4. Automatic Control III. Lecture 4 Controller structures and control design

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1 Contents lecture 4 Automatic Control III Lecture 4 Controller structures and control design Thomas Schön Division of Systems and Control Department of Information Technology Uppsala University. thomas.schon@it.uu.se, www: user.it.uu.se/~thosc Summary of lecture 3 2. Which control design methods do we have? 3. Who should 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, 215 Automatic Control III, Lecture 4 Controller structures and control design 2 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

2 Summary of lecture 3 (I/II) Bode s relationship provides an upper bound on the phase, which depends on the derivative of the amplitude curve. Hence, Bode s relationship provides a fundamental limit by revealing a certain coupling between the amplitude and the phase. 3 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design Summary of lecture 3 (II/II) S once, and only in an optional reading of an unassigned chapter in one of the classical textbooks. This integral surfaced for me for the second time in the mid 197s, referenced in a paper The second integral did not surface for me until 1983, in a talk by Jim Freudenberg at an IEEE Conference on Decision and Control in San Antonio [3]. If memory serves, someone pointed out at the time that this result was just a version of Jensen s theorem, well known in mathematics for a long time. Perhaps this historical reference reduced the value of the result in the minds of some listeners, but it should not have, because the integral explains so much about the difficulties of controlling unstable Bodes integralsats systems. A Bode Integral Interpretation I like to think of Bode s integrals as conservation laws. They state precisely that a certain quantity the integrated value of the log of the magnitude of the sensitivity function is conserved under the action of feedback. The total amount of this quantity is always the same. It is equal to zero for stable plant/compensator pairs, and it is equal to some fixed positive amount for unstable ones. Since we are talking about the log of sensitivity magnitude, it follows that negative values are good (i.e., sensitivities less than unity, better than open loop) and positive values are bad (i.e., sensitivities greater than unity, worse than open loop). log S(iω) dω =. 1 So for open-loop stable systems, the average sensitivity improvement a feedback w loop achieves over frequency is ex actly offset by its average sensitivity deterioration. For open-loop unstable systems, things are worse because the average deterioration is always larger than the improvement. This applies to every controller, a mound is deposited somewhere else. This fact is most evident to the ditch digger, because he is right there to see it happen. by Isaac Horowitz titled On the Superiority of Transfer Functionssystems over State-Variablewe Methods. derived... Itappeared Bode s as a per- integral theorem stating In the same spirit, I can also illustrate job of a more ac- For stable spectives paper in IEEE Transactions on Automatic Control that amid a certain amount of controversy [2]. ademic control designer with more abstract tools such as linear quadratic Gaussian (LQG), H, convex optimization, and the like, at his disposal. This designer guides a powerful ditch-digging machine by remote control from the safety of his workstation (Figure 4). He sets parameters (weights) at his station to adjust the contours of the machine s digging blades to get just the right shape for the sensitivity function. He then lets the machine dig down as far as it can, and he saves the resulting compensator. Next, he fires up his automatic code generator to write the implementation code for the compensator, ready to run on his target microprocessor. no matter how it was designed. Sensitivity improvements in one frequency range must be paid for with sensitivity deteriorations in another i=1 Re(p i). frequency Formal Design 4 / 23 range, and the price is higher Automatic if the plantcontrol III, Lecture 4 Controller structures and control design s.g T. Schön, 215 is open-loop unstable. It is curious, somehow, that our field has not adopted a name for this quantity being conserved Formal Synthesis Machine (i.e., the integrated log of sensitivity magnitude), to put it on a par with some of the great quantities of physics such as mass, momentum, or energy. But since it has not, we are 1. free to choose a name right now. Let me propose that we simply call it dirt. It is stuff we would rather not have around; the less we have, the better. I want to choose this name because it lets me liken the job of a serious control designer to that of a ditch digger, as illustrated in.1 Figure 3. He moves dirt from one place to another, using appropriate tools, but he never gets Frequency rid of any of it. For every ditch dug somewhere, Figure 4. Sensitivity shaping automated by modern control tools. Log Magnitude Log Magnitude 1 1. Serious Design Frequency Figure 3. Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies. For unstable systems (with M poles {p i } M i=1 in the RHP): log S(iω) dω = π M s.g 2. August 23 IEEE Control Systems Magazine 15

3 PID The most successful controller ever: PID Boulton och Watt 1788: speed control of steam engines, mechanical implementation. Assumes one input signal and one output signal If you have several input and output signals you must pair them in twos Interpretation in Bode plots: lead and lag Hydraulic and pneumatic implementation: late 18s Tuning using intuition and experience results in not more than mediocre performance (except in very simple cases). Electronic: 193s. Tuning using systematic analysis (poles, zeros, S, T,...) can give controllers with very good performance. Computer: 195s. PID-on-a-chip : 199s. The first systematic approach (using poles): Maxwell Applications: All. Maxwell, J.C. On Governors, Proceedings of the Royal Society, no. 1 (1868). Seam engine: PID-on-a-chip: 5 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 6 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

4 Where PID is not enough MIMO who should control what? (Typically means that the system is multivariable and/or nonlinear) Internal model control (IMC) Minimization of quadratic criteria: LQ, LQG. Model predictive control (MPC) Systematic shaping of transfer functions: H 2, H. Nonlinear methods The foundation of all modern control methods is to make use of models. 1. Tall system (more output than input signals): G = All output signals cannot be controlled perfectly prioritize. 2. Fat system (more input signals than output signals) ] [ G = How should the actuation be distributed among the control signals? 7 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 8 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

5 Several input and output signals interaction Interaction/cross coupling (I/II) If there are many input and output signals we can make the control design much easier by breaking down the system in sub-systems with little interaction between each other. Two-handle mixer, a system with a tough cross coupling Several input signals (significantly) affects an output signal. Several output signals are (significantly) affected by an input signal. The relative gain array (RGA) is a way of measuring the level of cross coupling or interaction in a system. Vinkel kallvattenvred Vinkel varmvattenvred Temperatur Vinkel kallva VattenflödeVinkel varmva 9 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 1 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

6 Interaction/cross coupling (II/II) Decentralized control Mixer with one handle, a system with a nice cross coupling. Each input signal affects (almost) only one output signal. Each output is affected (almost) only by one input signal. Vinkel temperatur Vinkel flöde TemperaturVinkel temperatur Vattenflöde Idea (decentralized control): Build a controller for a MIMO system where one output signal controls one input signal. The result is a set of single variable loops Vinkel flöde u i = F i rr j F i yy j, where the individual controllers are all independent ( they do not know of each other ). F y is a quadratic transfer matrix. If the number of input and G11 output signals is different some of them Temperatur are simply discarded. The less cross couplings there are, G22 Vattenflöde the better this strategy works We want to pair the input and output signals that have the strongest connections, the pairing problem. How do we determine the couplings between the various input and output signals? 11 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 12 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

7 Temperature control T 1 T 2 PID U 1 U 2 Two rooms with a separating inner wall. The temperatures T 1, T 2 are states, which are both measured. Both rooms can be both heated and cooled by U 1 and U 2. [ ] [.5s e 5 ] [ ] T1 = s s+.225 s s+.225 U e 5.5s T 2 U s s+.225 s s / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design Which sensor should be used in controlling which heating/cooling source (i.e. the pairing problem)? Temperature control Decentralized PI control T 1 is used for U 1. T 2 is used for U 2. [ ] F (s) = s s Decentralized PI control T 2 is used for U 1. T 1 is used for U 2. [ ] F (s) = s s After 1 hours 1 people enters room 1. Whooops... Is there any way in which we can predict this problem analytically? Effekt störning [W] Temperatur [grad. C] Effekt [W] T1 T Tid [h] x Tid [h] Tid [h] 14 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design Temperatur [grad. C] Effekt [W] Effekt störning [W] x x

8 RGA again Temperature control RGA Use RGA to decide which input signals to pair with which output signals. The two main rules in designing a decentralized controller using RGA. Pair measurement signals and control signals such that the diagonal elements in 1. RGA(G(iω c )) are close to 1 in the complex plane. 2. RGA(G()) are positive (if they are negative this can lead to instability) Pairing implies a change of the position of rows and columns in the RGA-matrix. RGA in Matlab: RGA(A) = A.*pinv(A. ) (pinv = pseudoinverse, handles non-square matrices). Design rules from previous slide is different words: 1. Select input-output pairs so that the diagonal elements of RGA(iω c ) are colse to 1 2. Avoid pairing that gives negative diagonal elements of RGA(G()). In the temperature control example: ( ) 1 RGA(G(i8)) RGA(G()) = 1 The pairing where T 2 is used for U 1 and T 1 is used for U 2, breaks both of these rules (the rows change places)! ( 1.17 ) / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 16 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

9 Decoupled control Temperature control decoupled Sounds good, but how do we choose W 1 and W 2? In order to obtain a completely decoupled virtual system we would need s-dependent wright matrices W 1 and W 2. This is in general not possible, since it would lead to a complicated and/or non-proper controller. Instead we choose one frequency where the system becomes decoupled: ω = ω = ω c (G 1 (ω c ) is often approximated to get rid of complex valued elements). The choice W 1 = G 1 () and W 2 = I results in decoupling in stationarity. With the right choice of W 1 and W 2 we can make the two-handle mixer behave as a one-handle mixer. Easier to control!! After 1 hours 1 people enters room 1. Decoupled control using W 1 = G 1 (), and W 2 = I. Effekt störning [W] Temperatur [grad. C] Effekt [W] Tid [h] x Tid [h] 1 5 T1 T Tid [h] 17 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 18 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

10 Contents lecture 4 Internal Model Control (IMC) Feedback only using the new information y Gu. 1. Summary of lecture 3 2. Which control design methods do we have? 3. Who should 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 Results in (if G = G) G c = GQ F r, S = I GQ, T = GQ 19 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 2 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

11 How do we choose Q? (I/II) How do we choose Q? (II/II) The ideal case Q = G 1 would result in S =, G c = I, but it is infeasible since F y =. Hence, we have to approximate! Some rules of thumb: 1. If G has more poles than zeros: The inverse of G cannot be realized. Use Q(s) = 1 (λs + 1) n G 1 (s). Choose n such that Q(s) is proper (# poles = # zeros). Choose λ to adjust the bandwidth of the closed loop system. 2. If G(s) non-minimum phase G has an unstable zero and contains a factor ( βs + 1), β >. Two alternatives a) Omit the factor when Q = G 1 is formed. b) Replace the factor ( βs + 1), β > with (βs + 1), β > when Q = G 1 is formed. 3. If G contains a time delay, i.e. a factor e sτ : a) Omit the factor when Q = G 1 is formed. b) Approximate the factor using e sτ 1 sτ/1 1 + sτ/2 21 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design 22 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design

12 A few concepts to summarize lecture 4 Cross couplings: The key difficulty in controlling MIMO systems is that there are cross couplings between input and output signals. If we change one input signal this affects several output signals. Relative gain array (RGA): The relative gain array is a measure of the amount of cross couplings in a matrix (RGA(A) = A. (A ) T ). Decentralized control: Let every input be determined by feedback from one single output. The pairing problem: The pairing problem is to select which input-output pairs that should be used for the feedback. Decoupled control: Decoupled control makes use of a change of variables such that suitable pairings of measurements and control signals becomes easier to see. Internal model control (IMC): Choose Q G 1 (y = GQr) and let the new information in the form of y Gu be fed back to affect u. 23 / 23 T. Schön, 215 Automatic Control III, Lecture 4 Controller structures and control design