Model Predictive Control Lecture VIa: Impulse Response Models

Transcription

1 Moel Preictive Control Lectre VIa: Implse Response Moels Niet S. Kaisare Department of Chemical Engineering Inian Institte of Technolog Maras

2 Ingreients of Moel Preictive Control Dnamic Moel Ftre preictions Preicte ftre otpts Effect of crrent state memor Ftre inpt ajstments Feeforwar measrement Feebac correction Objectives an Constraints Optimization State Upate Receing Horizon Implementation

3 General Setp Previos State in Memor Dnamic Moel State: Compact representation Of the past inpt recor State Upate New Inpt Move Jst Implemente Preiction Crrent State? Measrement Correction Ftre Inpt Moves To Be Determine Preiction Moel for Ftre Otpts Feebac / Feeforwar Measrements To Optimization

4 Moel Tpes Options Finite Implse Response Moel or Step Response Moel State-Space Moel Linear or Nonlinear Measrement Correction To the preiction base on open-loop state calclation To the state throgh state estimation Objective Fnction Linear or Qaratic Constraine or Unconstraine

5 Options for Preiction Moels 1. Finite Implse Response Moel 2. Step Response Moel 3. State-Space Moel

6 Sample-Data Compter Control Moel relates: Inpts Otpts { v, v, v, } 0 {,,, } v MV or measre DV

7 Finite Implse Response Moel Assmptions: No immeiate effect: H 0 0 Stable process settles after time n: H 0 for j n 1 j

8 Sperposition Principle

9 v Finite Implse Response Moel: Sperposition Principle { v0, v1, v2, v3, } { 0, 1, 2, 3, n, n 1, {0, Hv0, Hv0, 1 2 Hv 0, 3 Hv0, 0, n 0, {0, Hv1, Hv1, 1 2 H 1, n 1 v Hv n 1, 0, {0, Hvn 1, Hvn 2, 1 2 n Hv0 H v1 Hvn 1 n n 1 1

10 Finite Implse Response Moel: Sperposition Principle H1v 1 H nv n

11 Step Response Moel Assmptions: No immeiate effect: S 0 0 Stable process settles after time n: S S for j n 1 j n

12 Step Response: Relationship with Implse Responses Implse: v { 1, 0, 0, 0, } { } 0, H, H, H,, H n, 0, Step: v { 1, 1, 1, 1, } { H1 H2 H3 H n } { 0, H, H,, H, H, 0, } 0,,,,,, 0, 0, 0, 1 2 n 1 { 0, H,, H, H, H, } 1 n 2 n 1 n n { 0, H, H, } 1 2

13 Step Response Moel: Sperposition Principle v vi i 0

14 Step Response Moel: Sperposition Principle n 1 v i S v S i 1 i n v 1 S1 v 2 S2 v n Sn v n 1 Sn v0 Sn v ns n

15 Implse vs. Step Response vs. State Space Implse response i 1 Inpts v-1, v-2,, v-n appear in the moel Step response vi appear in the moel instea of vi State space n H v i n i 1 i S v i S v n i ˆ ˆ x 1 Ax ˆ Bv Cx ˆ Dv n State-space is more parsimonios representation since the other two moels reqire n states to be store in the memor

16 Extension to Mltivariate Sstems Extension to MIMO sstems is straightforwar n i 1 H v i i Lets consier n inpts an n otpts is a n *1 vector is a n *1 vector H i is a n *n matrix H i p,q is the implse response coefficient for the effect of q th inpt on p th otpt.

17 Extension to Mltivariate Sstems

18 Mlti-Step Preiction Main interest for control: preicting ftre otpt behavior. Ftre otpt fnction of past inpt, hpothesize ftre inpt Dnamic state: memor abot the effect of past inpt Ftre otpt fnction of namic state, ftre inpt

19 General Setp Revisite Previos State in Memor Dnamic Moel State: Compact representation Of the past inpt recor State Upate New Inpt Move Jst Implemente Preiction Crrent State? Measrement Correction Ftre Inpt Moves To Be Determine Preiction Moel for Ftre Otpts Feebac / Feeforwar Measrements To Optimization

20 State Definition Effect of past inpts is ept in memor as the sstem state Possible choices of state: Past n inpt moves, or Ftre n otpt response assming v is hel constant at v

21 1. Finite Implse Response Moel State Definition For FIR moel, it is convenient to se past n inpts as the state v 1 x v n Compact notation in vector form: T T x v 1,, v n T Sstem otpt: v 1 [ H1 H n ] Cx v n Note that v incles MVs an measre DVs

22 1. Finite Implse Response Moel State Upate State Upate Replace previos state in the memor with the crrent state x 1 M 0 x Τ v shift insertion Qestion: Write M 0 in matrix form v v 1 v n 2 v n 1 insert rop v 1 v 2 v n 1 v n

23 1. Finite Implse Response Moel Moel Preictions Moel preiction of Moel preiction error H1v 1 H n v n Cx e m Moel preiction error: effect of nmeasre istrbances an moel ncertainties v Maniplate Variables Measre Distrbances i H1v i 1 Hiv Hi 1v 1 Hnv i n Effect of crrent/ftre inpts e i 1 e Bias correction of moel error Effect of past inpts

24 Dnamic Matrices mae of implse response coefficients Preicte ftre otpt samples 1 p Ψ 1 1. Finite Implse Response: Mlti-Step Preiction 1 Ψ 1 Ψ2 n p 1 1 e Ψ 2 n p 1 e Feeforwar term: Past istrbance samples store in memor Past inpt samples store in memor Ftre inpt samples to be ecie Feebac Error Correction Feeforwar term: new measrement Assme 1.p-1

25 Define: 1. i i 2. Step Response Moel State Definition assming vj0 for 2. Effect of past inpts on n-step ftre preictions is the crrent state: 0 1 n 1 x T T T T Sstem otpt: 0 [I 0 0] x

26 2. Step Response Moel State Upate x1 M 1 x

27 2. Step Response Moel State Upate n n n n 1 1 v S x M x step response shift rop S 1 S 2 S n-1 S n State Upate Eqation

28 1. Finite Implse Response Moel Moel Preictions Moel preiction of Moel preiction error Cx 0 e m Moel preiction error: effect of nmeasre istrbances an moel ncertainties v Maniplate Variables Measre Distrbances i S1 v i 1 Si v i Effect of crrent/ftre inpts e i 1 e Bias correction of moel error Effect of past inpts

29 2. Step Response Moel Mlti-Step Preiction x Effect of ftre inpt moves to be etermine

30 Feeforwar term: new measrement Assme 1 p-10 Ω Ω e e p p p p 2. Step Response Moel Mlti-Step Preiction Ftre inpt moves to be ecie Feebac Error Correction The state store in memor Preicte ftre otpt samples Dnamic Matrices mae of step response coefficients

31 3. State-Space Moel Obtaining Discrete-Time State-Space Moel 1 Cz B B Az z Cz B B Az z ~ ~ ~ ~,, z g z f z Fnamental Moel Linearization Discretization { } N i i i 1,,, or s s G s s G s z z G z z G z Test Data I/O moel Ientification State-Space Realization State-Space Ientification

32 3. State-Space Moel State Upate for Difference Moel Cz B B Az z Cz B B Az z B B z A C z C B B z A z [ ] z I CB B CB B z I CA A z 1 x x x Ξ Γ Γ Φ State Upate

33 3. State-Space Moel Mlti-Step Preiction Ω Ω ΞΦ ΞΦ e e p p x p p x Ξ e m Ftre inpt moves to be ecie Feeforwar term: new measrement Assme 1 p-10 Feebac Error Correction The state store in memor Preicte ftre otpt samples Moel preiction of Moel preiction error Dnamic Matrix mae of step response coefficients

34 Smmar Regarless of moel form, one gets the preiction eqation in the form of Assmptions Measre DV remains constant at the crrent vale of Moel preiction error e remains constant at the crrent vale of e 1 1 U b nown e x Y p p L L e L x L

35 Moel Preictive Control Lectre VIb: Sstem Ientification Niet S. Kaisare Department of Chemical Engineering Inian Institte of Technolog Maras

36 Aim Overview Use inpt/otpt ata to obtain empirical moel eg., step or implse response coefficients Major steps Plant testing: Var inpts an measre otpt response Data conitioning an pre-treating Moel strctre selection an parameterization Parameter estimation Moel valiation

37 Sstem Ientification for DMC Direct estimation of step or implse response coefficients Estimation of alternate moel form e.g., transfer fnction or state space an conversion to FIR moel Can we se step / implse tests to obtain FIR coefficients? What are the isses involve? Is this the best metho to obtain a moel?

38 Step Tests

39 Plse Tests

40 Ke Isses 1. Choice of sampling perio h Sampling perio is sall chosen sch that N 30~40 Consier 1 e 7s 1 4s Settling time: 4~5 * time constant time ela secons h 1 secon, N 32 is a reasonable choice Ma be ajste base on control objectives

41 Ke Isses 2. Choice of step / implse size Too small Small otpt change Low signal to noise ratio Too large Shift process to nesirable conitions Introce non-linearit Trial an error reqire Size of plse reqire is mch larger than step size

42 Ke Isses 3. Mltiple experiments nee to be performe Averaging reces impact of noise on S i or H i Steps / implses of ifferent sizes ma be se Set of experiments are chosen as training ata an another set as test ata. Moel is valiate sing the test ata. 4. Detection of stea state in presence of noise! 5. Inpt excitation Step tests accratel ientif low freqenc stea state characteristics bt not the high freqenc characteristics Plse tests are theoreticall perfect nbiase excitation signals In practice, plse tests are not attractive options

43 Least Sqares Ientification Disavantages of Step / Plse testing Long test times Ma be impossible for some inpts More sstem excitation is neee to generate more information Alternative: Ranom inpt signals that excite all freqencies with same energ are se instea. Eg., Ranom Noise Signal Ranom Binar Signal RBS Pseo-Ranom Binar Signal PRBS Least sqares ientification Fit the FIR coefficients to the ata b minimizing qaratic norm of resials

44 Tpes of Inpts Ranom RBS PRBS

45 Least Sqares Fitting M 1 M Process Aim: Given inpt otpt ata, obtain the implse response coefficients Since n i 1 H i i H H 2 M M 1 M 2 M n H n Y U H

46 Least Sqares Fitting Note that the moel is not a perfect escription of the sstem. Aim is to minimize the moeling errors resials min H Y UH e e Y UH Least Sqares Fitting: Fin H sch that: T T ee min Y UH Y UH H T 1 T H U U U Y

47 Ke Points

48 Ke Points