Control System Design - McMaster University

1 ELEC ENG 4CL4: Control System DesignNotes for Lecture #2 Wednesday, January 7, 2004Dr. Ian C. BruceRoom: CRL-229 Phone ext.: 26984 Email: Graebe, Salgado , Prentice Hall 2000 Chapter 2 Chapter 2 Introduction to the Principlesof FeedbackTopics to be covered include: An industrial motivational example; A statement of the fundamental nature of the Control problem; The idea of inversion as the central ingredient in solvingcontrol problems; Evolution from open loop inversion to closed loop , Graebe, Salgado , Prentice Hall 2000 Chapter 2We will see that feedback is a key tool that can beused to modify the behaviour of a behaviour altering effect of feedback is a keymechanism that Control engineers exploitdeliberately to achieve the objective of acting on asystem to ensure that the desired performancespecifications are , Graebe, Salgado , Prentice Hall 2000 Chapter 2A motivating industrial exampleWe first present a simplified, yet essentiallyauthentic, example of an industrial Control example, taken from the steel industry, is of aparticular nature, however the principal elements ofspecifying a desired behaviour, modeling and thenecessity for trade-off decisions are , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Photograph of Bloom Caster Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Process schematic of an Industrial Bloom Caster Goodwin, Graebe, Salgado.

2 Prentice Hall 2000 Chapter 2 Continuous caster. Typical bloom (left) and simplifieddiagram (right)continuously withdrawn,semi-solid strandprimary coolingtundish withmolten steelcontrolvalvewmouldltGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Operators viewing the mould Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 The cast strip in the secondary cooling chamber Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Performance specificationsThe key performance goals for this problem are: Safety: Clearly, the mould level must never be in danger ofoverflowing or emptying as either case would result inmolten metal spilling with disastrous consequences. Profitability: Aspects which contribute to this requirementinclude: Product quality Maintenance ThroughputGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 ModelingTo make progress on the Control System Design problem,it is first necessary to gain an understanding of how theprocess operates. This understanding is typicallyexpressed in the form of a mathematical level of steel in mouldactual level of steel in mouldvalve positioncasting speedinflow of matter into the mouldoutflow of matter from the mould:)(:)(:)(:)(:)(:*tqtqttvthhoutin Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Model as simple tankTundishValveMolten SteelMould LevelCooling WaterGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Block diagram of the simplified mould level dynamics,sensors and actuatorsThese variables are related as shown below:+++measurementOutflow due tocasting speedMould levelMeas.

3 NoiseCasting speedcontrol valveInflow fromMeasured mould level Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Feedback and FeedforwardWe will find later that the core idea in Control is thatof inversion. Moreover, inversion can beconveniently achieved by the use of two keymechanisms (namely, feedback and feedforward).Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : Model of the simplified mould level Control with feedforward compensation for casting speed Note that this controller features joint feedback and a preemptive action (feedforward).Suggested Control Strategy:Mould levelMeasured mould levelcasting speedMeas. noiseOutflow due toCommandedmould level++ Casting speedmeasurementInflow fromcontrol valve K 1K ++++Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2A first indication of trade-offsOn simulating the performance of the above controlloop for K=1 and K=5, see Figure , we find thatthe smaller controller gain (K=1) results in a slowerresponse to a change in the mould level set-point.

4 Onthe other hand, the larger controller gain (K=5),results in a faster response but also increases theeffects of measurement noise as seen by the lesssteady level Control and by the significantly moreaggressive valve , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : A first indication of trade-offs: Increased responsiveness to set-point changes also increases sensitivity to measurement noise and actuator wear. level012345678910 1012345 Time [s]Valve commandK=1 K=5 K=1 K=5 Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 QuestionWe may ask if these trade-offs are unavoidable orwhether we could improve on the situation by suchmeasures as: better modelling more sophisticated Control System designThis will be the subject of the rest of our deliberations.(Aside: Actually the trade-off is fundamental as we shall see presently).Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Definition of the Control problemAbstracting from the above particular problem, we canintroduce:Definition :The central problem in Control is to find a technicallyfeasible way to act on a given process so that theprocess behaves, as closely as possible, to some desiredbehaviour.

5 Furthermore, this approximate behaviourshould be achieved in the face of uncertainty of theprocess and in the presence of uncontrollable externaldisturbances acting on the , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Prototype solution to the controlproblem via inversionOne particularly simple, yet insightful way ofthinking about Control problems is via inversion. Todescribe this idea we argue as follows: say that we know what effect an action at the input of a System produces at the output, and say that we have a desired behaviour for the System output, then one simply needs to invert the relationship between input and output to determine what input actionis necessary to achieve the desired output , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : Conceptual controller The above idea is captured in the following diagram:r+ f 1 uf ++ydConceptual controllerPlantGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2We will actually find that the inverse solution givenon the last slide holds very , all controllers implicitly generate an inverse ofthe process, in so far that this is feasible.

6 However,the details of controllers will differ with respect tothe mechanism used to generate the requiredapproximate , Graebe, Salgado , Prentice Hall 2000 Chapter 2 High gain feedback and inversionWe next observe that there is a rather intriguing propertyof feedback, namely that it implicitly generates anapproximate inverse of dynamic transformations, withoutthe inversion having to be carried out loop implements an approximate inverse of f , u = f r , ifr - h-1 u rFigure : Realisation of conceptual controllerz+ uyPlantrf h Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Specifically,ufrhzrhu = =orufruh = 1 Hencerfuhrfu111 =Provided is small, is high gain. uh1 hGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 The above equation is satisfied if h-1 u is large. Weconclude that an approximate inverse is generatedprovided we place the model of the System in a highgain feedback , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Assume that a plant can be described by the modeland that a Control law is required to ensure that y(t)follows a slowly varying way to solve this problem is to construct aninverse for the model which is valid in the lowfrequency region.

7 Using the architecture in , we obtain an approximate inverse, provided thath has large gain in the low frequency )()(2)(tutydttdy=+Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : Tank level Control using approximate inversion Simulating the resultant controller gives the results [s]Ref. and plant outputr(t) y(t) Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 From open to closed looparchitecturesUnfortunately, the above methodology will not leadto a satisfactory solution to the Control problemunless: the model on which the Design of the controller has beenbased is a very good representation of the plant, the model and its inverse are stable, and disturbances and initial conditions are are thus motivated to find an alternative solutionto the problem which retains the key features butwhich does not suffer from the above , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : Open loop Control with built-in inverse Figure.

8 Closed loop Control Ay(t)Open loop controllerModelu(t)Feedback gainPlant +r(t)A +y(t)r(t)Plantu(t)e(t)Feedback gainGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 The first thing to note is that, provided the modelrepresents the plant exactly, and that all signals arebounded ( the loop is stable), then both schemes areequivalent, regarding the relation between r(t) and y(t).The key differences are due to disturbances and differentinitial conditions. In the open loop Control scheme the controller incorporatesfeedback internally, a signal at point A is fed , Graebe, Salgado , Prentice Hall 2000 Chapter 2 In the closed loop scheme, the feedback signal depends onwhat is actually happening in the plant since the true plantoutput is will see later that this modified architecture hasmany advantages including: insensitivity to modelling errors; insensitivity to disturbances in the plant (that are notreflected in the model).Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Trade-offs involved in choosingthe feedback gainThe preliminary insights of the previous two sectionswould seem to imply that all that is needed to generatea controller is to put high gain feedback around theplant.

9 This is true in so far that it goes. However,nothing in life is cost free and this also applies to theuse of high gain example, if a plant disturbance leads to a non-zeroerror e(t), in Figure , then high gain feedback willresult in a very large Control action u(t). This may lieoutside the available input range and thus invalidate , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Another potential problem with high gain feedback isthat it is often accompanied by the very substantial riskof instability. Instability is characterised by selfsustaining (or growing) oscillations. As an illustration,the reader will probably have witnessed the high pitchwhistling sound that is heard when a loudspeaker isplaced too close to a microphone. This is a manifestationof instability resulting from excessive feedback manifestations of instability include aircraftcrashes and the Chernobyl disaster in which a runawaycondition , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Yet another potential disadvantage of high loop gainwas hinted at in the mould level example.

10 There wesaw that increasing the controller gain lead toincreased sensitivity to measurement noise.(Actually, this turns out to be generically true).Goodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2In summary, high loop gain is desirable from manyperspectives but it is also undesirable when viewedfrom other perspectives. Thus, when choosing thefeedback gain one needs to make a conscious trade-off between competing , Graebe, Salgado , Prentice Hall 2000 Chapter 2 The previous discussion can be summarised in thefollowing statement:High loop gain gives approximate inversion which isthe essence of Control . However, in practice, thechoice of feedback gain is part of a complex web ofdesign trade-offs. Understanding and balancingthese trade-offs is the essence of Control , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Measurements Finally, we discuss the issue of measurements ( what itis we use to generate the feedback signal). A more accurate description of the feedback Control loopincluding sensors is shown in Figure , Graebe, Salgado , Prentice Hall 2000 Chapter 2 Figure : Closed loop Control with sensors A ym(t)+ y(t)u(t)r(t)Measurement and signaltransmission systemPlantControllerGoodwin, Graebe, Salgado , Prentice Hall 2000 Chapter 2 Desirable attributes of sensors Reliability.