# Internal Model Control

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Outline Part 1: Continuous SISO systems Introduction Control Objectives in the Presence of UncertaintyModeling Uncertainty Nominal Stability & Performance Robust Stability Robust Performance IMC Structure Stability and Performance Prefect Control IMC Design Procedure Stable systems Example1: System with time delay uncertainty 2

Outline IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises3

Part 1: Continuous SISO systems Introduction Control Objectives in the Presence of UncertaintyModeling Uncertainty Nominal Stability & Performance Robust Stability Robust Performance IMC Structure Stability and Performance Prefect Control IMC Design Procedure Stable systems Example1: System with time delay uncertainty 4

Introduction IMC is an effective method for designing and implementingrobust controllers. IMC structure is an alternative to the classic feedback structure. Its main advantages are: Simple and easy to understand designing procedure On-line tuning of IMC controllers are very convenient It can easily control plants in presence of actuatorconstraints. use uncertainty information in design procedure.5

Part 1: Continuous SISO systems Introduction Control Objectives in the Presence of UncertaintyModeling Uncertainty Nominal Stability & Performance Robust Stability Robust Performance IMC Structure Stability and Performance Prefect Control IMC Design Procedure Stable systems Example1: System with time delay uncertainty 6

lm (w)p (iw) la (iw) p(iw) la (iw) la ( w)p (iw)(1 lm (iw)) p(iw) lm (iw) lm ( w) p : p(iw) p (iw) la ( w) Uncertainty usually increases with frequency.NominalmodelActualPlant7

la (w)Control ObjectivesControllerModel Nominal Stability: The system is stabe with no modeluncertainty. Nominal Performance: The system satisfies the performancespecifications with no model uncertainty. Robust Stability: The system is stabe for all perturbed plants,here means for all family plants. Robust Performance: The system satisfies the performancespecifications for all family plants.8

Remember: Nominal Performancey ( s ) T ( s )r S ( s )d T ( s )ndrc pynthe sensitivity function S is a very good indicator of closed-loopperformance1 S ( j ) , wP ( j ) S ( j ) 1wP ( j )H optimal controller is obtained by soliving : min N ( K ) K9

la (w)ControllerRobust StabilityModelTheorem 1 : Assume that controller c stabilizes thenominal plant p . Then the system is stable for family p p (iw) p : lm ( w) p If and only ifT lm sup T lm 1 w Model uncertainty imposes a bound on T. Noise also tends to impose a bound on magnitude of Tbut usually the constraint imposed by model uncertaintytends to dominate.10

Robustness / Performance Tradeoff T lm sup T lm 1wy ( s ) T ( s )r S ( s )d T ( s )n we want to minimize S w p for nominal performance and T lm for robustness. This problem ic inherent in feedback control andcannot be removed by clever design.11

la (w)ROBUST PERFORMANCEControllerModelTheorem 2 : Assume that controller c stabilizes thenominal plant p, then the closed - loop system will meetthe performance specification wP ( j ) S ( j ) 1 if and only if T lm S w p 1 w p In practice our objective is to find a controller which solve min supT lm S w p c wIt is a difficult problem to solve. For general MIMO case noreliable solution are availableIMC easily provides a good approximation to the optimalsolution12

Part 1: Continuous SISO systems Introduction Control Objectives in the Presence of UncertaintyModeling Uncertainty Nominal Stability & Performance Robust Stability Robust Performance IMC Structure Stability and Performance Prefect Control IMC Design Procedure Stable systems Example1: System with time delay uncertainty 13

IMC StructuredryControllerPlantModelBecause in addition to the controller, It includes the plant modelexplicitly this feedback configuration is calledinternal model control (IMC) The feedback signal d expresses the uncertainty(model uncertainty and disturbance).14

IMC vs. Classic FeedbackCqc , 1 pqcq 1 pcSo Why IMC has so much advantages over classic feedback?!!15

pStabilizing Controllers c q1 pqAssume plant is stable, so the only requirement for nominal stability is stability of qpq1 pqy r d 1 q( p p )1 q( p p ) S 1 pq T pq T pc1 pc16

lm (w)Prefect Controlpq1 pqy r d 1 q( p p )1 q( p p )If we put q 1 then T 1 and S 0, so we can achievepprefect control.But there are three reasons that make the prefect control impossible:1) Nonminimum-phase(NMP) plants: q become noncasual orunstable, and nominal plant become unstable.2) Strictly proper plants: q become improper, but this can be solvedby adding some far poles.3) Model Uncertainty:S 0 lm 1 T lm 1 (1 S )lm 117

Part 1: Continuous SISO systems Introduction Control Objectives in the Presence of UncertaintyModeling Uncertainty Nominal Stability & Performance Robust Stability Robust Performance IMC Structure Stability and Performance Prefect Control IMC Design Procedure Stable systems Example1: System with time delay uncertainty 18

IMC Design For Stable Systems Final Objective : min supc T lm S w p wIMC design procedure consists of two steps:Step 1: Nominal Performanceq is selected to yield a good performance without regardfor constraints and model uncertainty min e min S w min (1 pq ) wq 2q 2q 2Step 2: Robust Stability and Performance19

Step 1:Nominal PerformanceTheorem3 (H 2 optimal control) : assume p is stable,factor p and input w into an allpass portion p and a MPAportion pM , p pA pM .so that pA includes all the RHP zeros and delays of p and p (iw) 1 w , The controller q which solves above equation is :A q ( pM wm ) 1 p A 1wm *Where the operator . denotes that after a partial fraction expansion ofthe operand all terms involving the poles of p A 1 are omitted.20

s 2s 1 Example : p p A , vm s 2ss s 2 s 1 s 16 s 1 q . 1 s 1 s 2 s s 1 s s 2 s 1 In general the optimal controller q in not proper. min S wq T lm 2 sup T lm 1w21

Step2: Robust Stabilityand Performance S 1 pq T pq T q q f pc1 pcFor robust stability q is augmented by a low - pass filter fto simplify t he design tas k.Low-pass filter can improve robustness.What about performance?22

we fix the filter structure and search over a small number1m 1of filter parameters, f ( m 1s . 1s 1).n( s 1)Here n is selected large enough for q to be proper.To havezero steady state error, the condition on f to be satisfied are:1Type1 : f (0) 1 f ( s ) ( s 1) ndfn s 1Type 2 : f (0) 1 and lim 0 f (s) s 0 ds( s 1) n is the adjustable filter parameters, Increasing willincrease robustness but it decrease the performace.IMC makes the on - line tuning so easy and intuitive.23

Example 1: System with time delay uncertainty- s2ePlant : p ,0.5s 1-2s2eModel : p 0.5s 1Input: StepUncertainty: Time delay uncertainty lm (iw) e iw 1 2 , 0.5 so : lm ( w) e iw0.5 1 0.5Step1 : Nominal Performance :lm ( w) 2 p e 2 s, q p 1 0.5(0.5s 1)Aw 2 w 2 mStep2 : Robust stability and Performance :1f s 124

Example 1 T lm sup T lm 1w 0.21.421.21.50.8lmTlm110.60.50.40.20-1100101 5

2e-2.45sassume real plant is : p 0.5s 1Example 1 0.2 0.6Step Responselambda 0.230Step Responce, Lambda 0.5Step Response1.4Step 0.2-3001020Time (sec)304050q0.5(0.5s 1)c 1 pq 0.6s 1 e 2 s600051015Time (sec)202530How to implement?!!!!26

IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises27

Introduction By far the most widely used controllers in the industryprocesses are the two-term PI and the three-term PIDcontroller. Because IMC is clearly more general and therefore morepowerful it is worthwhile to explore the relationships betweenIMC, PI and PID in order to gain insight into the tuning ofthese simple controllers, their performance and robustness.28

General relationships Remember: General relationship between the classic feedbackcontroller c and the IMC controller q are:qc 1 pq,cq 1 pcf the plant m odel . So The com plexity of the equivalent classic controller c is The order of q obtained in this way is generally higher than theorder of the plant model. So The complexity of the equivalentclassic controller c is determined by the complexity of themodel.Simple modelssimple controllers29

Example 2: PID Design In the previous example , our model was as fellows:2e 2 s p 0.5s 1 By the use of “Pade” approximation our model is obtained asfollows:1 s2 p .1 s (.05s 1) pA pM Step 1: Nominal Performance q ( pM wm ) 1 p A 1 wm *30

Example 2 11 1 s 1 0.5s 1 2q . . 2 .05s 1 s 1 s s *Step 2: Robust stability and performance:q qf0.5s 1 1q .2 s 1So:0.25s 0.5q(s 1)(0.25s 0.5) s 1c 1 s0.25s 0.51 pq 1 s ( s 2 ).(.05s 1)(s 1) s 131

Example 2Structure of the PID controller is defined as follows:kI1K (s ) (k P k D s ).s F s 1The controller is so simple and come to the form of PID controller:c 0.25s 2 0.75s 0.5(2 ) s( 2 s 1)By equating K c , PID parameters is obtained from the aboveIMC controller. As we see, all of the parameters are just dependon .kP0.750.50.25 , kI , kD , F 2 2 2 2 32

Example 2Step response for different values of is plotted.Both robustness and performance is achieved. 10 0.33 1 330.2

IMC based PID for different plants The IMC controllers shown in next slide Table, is designed viathe standard procedure developed in the previous example.IMC leads to PID controllers for virtually all models commonin industrial practice. Note that the table includes systems withpure integrators and RHP zeros. Occasionally, the PIDcontrollers are augmented by a first-order lag.c k c (1 D s 2 211), I s F s 1 234

IMC based PID for different plants35

IMC based PID for different plants36

IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises37

Example 3: Model with high uncertaintyAcutal plant :Plant model :2e-2.45sp 0.5s 11p s (s 5) Checking the uncertaintyNyquist Diagram for pNyquist Diagram for p38

Example3Step 1 (Nominal Performance): q (pMw m ) 1 p A 1w m * s (s 5)Step 2 (Robust Stability & Performance):f 1( s 1)2,q qf s (s 5)( s 1)2qs (s 5)c 1 pq ( s 1)2 139

Example 3As you see in these figures, step response for different values of wereplotted. Because of high uncertainty, good performance did not achieve. 2.5 3.3 5 1040

IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises41

IMC Structure- MIMO Case In MIMO case, we deals with transfer function matrices.drControllerdyPlantModelQC Q (I PQ ) 1Q C (I PC ) 142

IMC Structure- MIMO Case Sensitivity function and Complementary sensitivity functiony Tr SdQS (I PQ )(I (P P )Q ) 1T PQ (I (P P )Q ) 1 If the model is perfect ( P P ) S 1 PQ, T PQ43

Internal Stability for stable plants Theorem: Assume that the model is perfect ( P P ),Then the IMC system is Internally stable IFF both the plant Pand controller Q are stable.Q44

General Internal Stability Remember: All elements of the below matrix have to be stable PC (I PC ) 1 C (I PC ) 1 (I PC ) 1 P C (I PC ) 1 P In IMC Structure for P PBy substituting C Q (I PQ ) 1QWe have PQ Q(I PQ )P QP 45

General Internal Stability PQ Q(I PQ )P QP This implies that Q has to be stable and that in the elements ofabove matrix the factor Q and I-PQ have to cancel any unstablepoles of P. Thus both Q and I-PQsmust have RHP zeros at theplant RHP poles. Special care has to be taken to cancel these 1common RHP zeros when the controller C Q (I PQ )isconstructed. Minimal or balanced realization software can beused to accomplish that.46

IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises47

Inner-Outer factorization Theorem: let G (s ) C (sI A ) 1 B D be a minimal realization ofthe square transfer matrix G(s) , and let G(s) have no zeros on the 1G(s) N(s)M(s)iw-axis including infinity. Then we havewhere N,M are stable and N (i )H N (i ) I , andN (s ) (C QF )(sI (A BR 1F )) 1 BR 1 QM (s ) 1 F (sI A ) 1 B RwhereF Q T C (BR 1 )T Xwith X the stabilizing [i.e., it makes (A BR 1F ) stable] real symmetric solutionof the following algebraic Riccati equation (ARE):(A BR 1Q T C )T X X (A BR 1Q T C ) X (BR 1 )(BR 1 )T X 0So that :PA NandPM M 148

IMC Design – MIMO case IMC design procedure consists of two steps Step1: Nominal PerformanceQ is selected to yield a good performance for inputs, without regard forconstraint and model uncertainty. Step2: Robust Stability & PerformanceThe Q obtained in step 1 is detuned to satisfy the robustness requirements.For that purpose, Q is augmented by a filter F of fixed structure.Q Q (Q , F )i .e : Q QF49

Step1: Nominal Performanceo The plant P can be factored as PA is stable allpass portionPM is MP portionP PA PMThe procedure for carrying out this factorization is:“Inner – Outer Factorization”Define the square matrix V(s){set of n inputs vi }V ( s ) (v1 ( s ), v 2 ( s ), , v n ( s ))v i ( s ) is a vector that describes expected direction andfrequency content of i th input.Similarly : V VM V A50

Step1: Nominal Performance To achieve nominal performance, the controller is: Q PM 1W 1 W PA 1 V Mof the operand, all term s involving the poles of *V M 1SSS are om itted.Where the operator . * denotes that after a partial fractionexpansion of the operand, all terms involving the poles ofPA 1 are omitted.51

Step2: Robust Stability & Performance The controller Q is to be detuned through a lowpass filterF , to satisfy robust stability and performance. So the tunedcontroller is:Q QF The filter F(s) is chosen to be a diagonal rational function thatsatisfy:F (s ) diag f 1 (s ) , , f n (s ) o The controllerQ QF must be propero Internal stabilityo Asymptotic setpoint tracking/disturbance rejection.52

Step2: Robust Stability & PerformanceF (s ) diag f 1 (s ) ,, f n (s ) Experience has shown that the following structure isreasonable:f l (s ) av 1 1,l s v 1 1 Where a1,l s a0,l( s 1)v v l 1 v is pole-zero excess. K is the number of open RHP poles of Pm 0l is the largest multiplicity of such pole in any element of the lth rowof V. v l m 0l k53

Step2: Robust Stability & Performancef l (s ) av 1 1,l s v 1 1 a1,l s a0,l( s 1)v v l 1 The numerator coefficients can be computed from solving asystem of v l linear equations with v l unknowns. f l ( i ) 1, j d f (s ) 0, ds j ls 0 where :i 0,1,j 1,,k, m 0l 1 i (i 1,., k ) are open RHP poles of P 0 054

IMC based PIDIntroduction General Relationships Example 2: PID Design Example 3: Model with high uncertainty Part 2: Continuous MIMO systems IMC Structure - MIMO CaseInternal stability for stable plants General Internal stability IMC Design Procedure - MIMO Case Inner-Outer factorization Nominal Performance Robust Stability & Performance Exercises55

Exercise 1: Robust controller design 4 s2e s 3 model : p s 1 s 2 Input : StepDesign an IMC controller based on plant model.Assume that the real plant is :2e 4.6 s s 3 Plant :p s 1 s 2 1.find filter parameter ( ) range that stabilize the system?2. find in order to achieve 10% over shoot. plot step response and find rise time3. what is the relationship between filter pole locatoin,performance and robustness? why?56

Exercise 2: Poor modeling impact2e 4.6 s s 3 Assume that the real plant is :p s 1 s 2 Input : step- 1.5(0.2s 1)(0.3s 1)1. find filter parameter range that stabilize the system.2. find in order to achieve 10% over shoot.plot step response and find rise time.3. compare results with Exercise 1, propose better models andcompare the results, what are the advantages of better system identification?Assume that we have modeled this plant with : p 57

Exercise 3: PID designDesign a PID controller through IMC :2Plant Model : p (0.4 s 1)(0.2 s 1)Input : Step1 - first find q , q , c then find PID and lag compensatorparameters in terms of .2 - Install the PID controller on nominal plant,plot step response for 10does any makes the system unstable? why?58

References[1] C. E. Garcia and M. Morari, "Internal model control. A unifying review and some new results,"Industrial & Engineering Chemistry Process Design and Development, vol. 21, pp. 308-323,1982.[2] C. E. Garcia and M. Morari, "Internal model control. 2. Design procedure for multivariablesystems," Industrial & Engineering Chemistry Process Design and Development, vol. 24, pp.472-484, 1985.[3] C. E. Garcia and M. Morari, "Internal model control. 3. Multivariable control law computationand tuning guidelines," Industrial & Engineering Chemistry Process Design and Development,vol. 24, pp. 484-494, 1985.[4] M. Morari and E. Zafiriou, Robust Process Control. New Jersey: Prentice-Hall, Inc., 1989.[5] A. Porwal and V. Vyas, "Internal model control (IMC) and IMC based PID controller,"Bachelor of Technology, Department of Electronics & Communication Engineering, NationalInstitute of Technology, Rourkela, 2010.[6] D. E. Rivera, Internal Model Control: A Comprehensive View. Tempe, Arizona: Arizona StateUniversity, 1999.[7] D. E. Rivera, et al., "Internal model control: PID controller design," Industrial & EngineeringChemistry Process Design and Development, vol. 25, pp. 252-265, 1986.59

IMC is an effective method for designing and implementing robust controllers. IMC structure is an alternative to the classic feedback structure. Its main advantages are: Simple and easy to understand designing procedure On-line tuning of IMC controllers are very convenient

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