System Identification for MPC

Transcription

1 System Identification for MPC Conflict of Conflux? B. Erik Ydstie, Carnegie Mellon University Course Objectives: 1. The McNamara Program for MPC 2. The Feldbaum Program for MPC 3. From Optimal Control to MPC to SYSID for Control and Back 4.Towards Tractable Optimization Formulations 5. State of Art (Challenging Problem of Integration and Software) B Erik Ydstie, CMU 1

2 System Identification (SYSID) Review Mass and Energy Balance Constraints (nonlinear) dz i dt =p i(z)+ y k =h k (z), n MV X+n DV j=1 Linear (output) error model e(t) =y(t) f i (u j,z), k =1,...,n PV G p (q 1 )u(t) i =1,...,n Model Predictive Controller Distributed Control System (DCS) Control Inputs u Interface Layer (SCADA) Measured Outputs y Setpoints y* Capture Flowsheet structure Energy and material balances - Collinearity - Uncollinearity Feed FT LT TT CT FT Used for very large systems 50 + MV/DVs 100+ CVs FT Product Cooling water return

3 Data Flow Prior Information Step-response State Space Laguerre, Tuning Parameters,.XML.TXT MPC Control ABB Honeywell Aspen Emerson Process B. Erik Ydstie, ILS Inc. 3

4 MPC SYSID Boiler Master Turbine Master Controls (Emerson/Ovation) Turbine Controls for Siemens SYSID Data from Model estimated using output error identification (global optimality) B. Erik Ydstie

5 B. Erik Ydstie

6 Prior Information: System Strucure Prior structure of G Digraph (edges in the process network) Parametric representation for each G ij (nodes) Information of collinearity structure Process Data Semi-closed loop Experiment Design N(n, m) = 400 G 25 Collinearity: SVD RGA Angles G 38 (n 1)n(m 1)m 4 = 2970, 7! 0.12 deg separation Bilinear constraints Digraph Network I/O Data 10 MV/DVs 12 CVs Problem: Define the Operator G that best matches the prior information and process data Bayes Estimation Problem with Constraints

7 Prior Information: Model Structures Polynomials used B A,B A,B,C B,F Name FIR, SR ARX, Equation error, Instrumental variables,.. ARMAX, AML OE (Output Error, Markov-Laguerre, Kautz,..) State space representations have become popular for multivariable systems after the introduction of sub-space identification. Halfway Conclusion: The components are in place for systematic SYSID Software is lacking Quite difficult to do due to non-stationary disturbances Theory not that easy to understand completely Comprehensive Software solutions not available yet. B. Erik Ydstie, ILS Inc. 7

8 The Admissibility Problem rank [ B AB A B A B] 2 n-1 = n Reachability: Any state can be reached in a finite amount of time B( q) y ( t) = u( t) A( q) Observability: Any state can be determined in a finite amount of time éc ê ê CA rankêca ê ê ê ëca 2 n-1 ù ú ú ú ú ú ú û = n A(q) and B(q) no common factors = Observable+Controllable (Co-prime) A(q) and B(q) no common unstable factors = Detectable+Stabilizable Detectable: Stabilizable: Any unstable state is observable Any unstable state is reachable The FIR / Markov-Laguerre Models are automatically stabilizable 8

9 MISO Identification NX u(k)u(k i) N k=1 MV7 is most excited MV2 is least excited Cond(F) = O(1) Excitation MV 1 D MV2 A MV3 A MV 4 A MV 5 A MV 6 A CV 3 Prior (blue) Update (red) Data (yellow) Data is persistently excited from a SISO case. We get (Ljung, Wahlberg, Forsell) Bias and Variance: Bias Variance B. Erik Ydstie, ILS Inc. 9

10 Generating Multivariable Input Signals u 1 y 1 System u 2 Same results hold as long as PE and independent noise and disturbance sequences. Results based law of large numbers, difficult to achieve using PRBS type excitation. y 2 Orthogonal inputs: 1 N NX u(k)u(k i) =V (N) T V (N i) =R(N) = k=1 R>0, i =0 0, i 6= 0 Input sequences must be independent in time wrt to the network [0.2019] Mass balance constraints in the process apple B. Erik Ydstie, ILS Inc. 10

11 Closed Loop System: Closed Loop Identification y(t) =G c 0(q 1 )u(t)+h0(q c 1 )v(t) G c 0(q 1 )=S0(q c 1 )G 0 (q 1 ) H0(q c 1 )=S0(q c 1 )H 0 (q 1 ) Excitation Process Issues for closed loop identification: MPC S0(q c 1 1 Model parameterization )= 1+G 0 (q 1 )K(q 1 ) Algorithm and mathematical approach Filters to shape bias and variance Excitation (complete theory for SISO, Lacking for MIMO, some progress for Networks Extension to multivariable case (treated very superficially in most books and papers) Methods that may fail: Regression type models (equation error, instrumental variables) Subspace methods Compensation methods (direct and indirect) Correlation/spectral methods Use output error methods for identification (open and closed loop) B. Erik Ydstie, ILS Inc. 11

12 Integration of SYSID with MPC: The Decision Problem Model and Desired Performance Objectives Robert McNamara,1960, (CEO Ford, US Secretary of State) Control/Plan Process Measure 1. Defining clear business objectives (Stable/Robust Performance) 2. Developing plans to achieve the objectives (Predictive Control) 3. Systematically monitoring progress against the plan (Feedback, Filter) 4. Adapt objectives/plans as new needs and opportunities arise (Identification) Repeated Identification -> Iterative Learning -> Adaptive Control 12

13 The McNamara Program for MPC Performance Objectives Model and Desired Performance Objectives MPC Design/Identify/Adapt MPC Process Measure Evaluate Critic Current Practice Predictive Model 1. Measure, evaluate and critique (Gap analysis) 2. Control strategies (Optimal Control/Model Predictive Control/H infinity ) 3. Identification, Learning, Adaptation a) Adapt Controllers (directly or indirectly) b) Adapt Performance Objectives (closed loop, Q,R/move suppression) The Decision problem is driven by Uncertainty (more than accurate models) Numerous Practical and Theoretical Challenges Remain MPC provides a fruitful Paradigm to Study these Challenges 13

14 min u2u,y2y The Feldbaum (1961) Program 1X (y(t + i + 1) y(t + i + 1) ) 2 + ru(t + i) 2 i=1 Optimal (Certainty Equivalent, LQ Optimal Control, 1980 to MPC) Caution (Robust H 1 Control, 1980 ) * Probing (System ID / Adaptation, 1980 ) Each field well advanced, but poorly integrated (Especially on the software side) * min u(t+i) TX ˆx(t + i) T Qˆx(t + i)+u(t + i) T Ru(t + i) i=0 {z } Finite Horizon Cost Subject to: +ˆx(t + i) T P ˆx(t + i) {z } Terminal Cost ˆx(t + 1) = Aˆx(t) + Bu(t) ˆx(t + i) min apple ˆx(t + i) apple ˆx(t + i) max u(t + i) min apple u(t + i) apple u(t + i) max B Erik Ydstie, CMU 14

15 From LQ to MPC and Back Again Step 1: Formulate a (linear) model x(t + 1) = Ax(t)+Bu f (t)+ke(t) ŷ(t) = T x(t)+du f (t)+e(t) Step 2: Split Objective in Two and use Predictions from Model X T 1 min u2u,ŷ2y i=0 (ŷ(t + i + 1) y(t + i + 1) ) 2 + ru(t + i) 2 {z } Model Predictive Control + 1X i=t (ŷ(t + i + 1) y(t + i + 1) ) 2 + ru(t + i) 2 {z } LQ Control Step 3: Ignore last part Step 4: Solve QP and use first control. Step 3: Repeat Step 4 (and hope for the best) x(t + T ) T P T x(t + T ) Theory for robust stability and performance B Erik Ydstie, CMU 15

16 MPC and SYSID: Learn from the Past and Control Into the Future Step 2B: Split Objective in Three and use Past Information min 2 i=0 min u2u,ŷ2y NX (ŷ(t i) y(t i) ) 2 +( 0 ) T F 0 ( 0 ) {z } SYSID Bayes 1X (ŷ(t + i + 1) y(t + i + 1) ) 2 + ru(t + i) 2 {z } i=1 Robust MPC Model Identified from Past Data Control Into the Future Adaptive Control Iterative Control Closed Loop Identification Identification for Control +++ Basic Idea: Controller works while data is collected (0) 7! (t 1 ) 7! (t 2 ),... B Erik Ydstie, CMU 16

17 SYSID and MPC - Conflict or Conflux? Adapted from Polderman (1986) Definition: An Identification Based Control is said to be Self-Tuning if SYSID gives the correct control Set of Identified Models : Set of Parameters with correct controls : Control and Estimation are Self Tuningif : G H H G Exampe: LTI System: u y(t) = ay(t 1) + bu(t 1) e(t) =y(t) ŷ(t) y System Model - + Linear feedback: u(t) =K(â, ˆb)y(t) Question: Will system satisfy performance specifications when the same control is applied to both systems? (The question of (Roust Lagrange) stability for closed loop identification and control was addressed by 1995)

18 Analysis: Assume model output matches plant output G = {â, ˆb : ay(t 1) bk( )x(t) {z } y(t) =ây(t 1) ˆbK( )x(t) } {z } ŷ(t) An infinite number of solutions. These depend on K. Example 1: One step ahead predictive control b The admissible set Solve for u(t) : y(t + 1) = ay(t)+bu(t) a u(t) = â ˆb y(t) G = Get correct control even if parameter estimates are off. â, ˆb : a b = â ˆb Thm: Any identifier that minimizes prediction error is self tuning when used with minimum variance control. Admissibility Problem (close to singularity gives large, oscillatory controls) (Problem of direction ) B Erik Ydstie, CMU 18

19 The set that gives correct controls H = â, ˆb : a b y(t)) = â ˆb y(t) H G b Example 2: Pole placement control (Vogel and Edgar, Find gains so that : y(t) =a 0 y(t 1), 0 <a 0 < 1 u(t) = â a 0 ˆb Admissible set y(t) a H = â, ˆb : H G a a 0 b y(t)) = â a 0 ˆb y(t) Thm: Any identifier that minimizes prediction error is self tuning when used with pole-placement. In this case Admissibility is more Complex as we require: Observability and Controllability Can be expressed as Bilinear Constraints in SYSID problem. B Erik Ydstie, CMU It is going well so far!! 19

20 Example 3: Model Predictive Control min (y (t + 1)2 + ru(t)2 ) + py(t + 1)2 u u(t) = ba r + b2 5 G= H= H ( a, b : a a, b : a = (b a a0 b b ) y(t)) = r + b 2 ) a a0 ˆ ba b y(t) H 6 G unless : 0 r = 0 and/or G a = a, b = b H Thm: MPC does NOT satisfy the self-tuning principle. B Erik Ydstie, CMU 20

21 Problem: Information in the Past is Not Connected to Future Information Additional means are needed to get optimal controls for MPC. Persistent Excitation to Converge Controls More Complex Controls to Align sets G and H? Intelligent Excitation (SYSID for Control, Dual Control) B Erik Ydstie, CMU 21

22 From Feldbaum to MPC and Back min u2u,y2y TX (y(t + i) y(t + i) ) 2 + ru(t + i) 2 + i=1 {z } Model Predictive Control 1X i=t +1 (y(t + i) y(t + i) ) 2 + ru(t + i) 2 {z } LQ Control min u2u,ŷ2y TX (ŷ(t + i) y(t + i) ) 2 + ru(t + i) 2 + x(t + i) T F (t + i) 1 x(t + i) +x(t)p T x(t) {z } {z } Robust CE AMPC Information Gathering i=1 ŷ(t + i) F (t + i) Maximum Likelihood Estimate Fisher Information Matrix Challenges (identification using past data is the easiest): Solve a robust control problem on line (structured and unstructured uncertainty) Back out control signals from forward propagation of Fisher matrix What to do with the arrival cost B Erik Ydstie, CMU 22

23 Special Case (TA Heirung/ J Morinelly) Fix the transition matrix A (step-response/kautz model) Solve CE (rather than robust H infinity ) control problem (caution related to parameter uncertainty only) Ignore arrival cost Computationally Expensive and Untested

24 So What are the Issues? Data Rich But Information Poor Systems (Nature is not a kind adversary) MPC and SYSID - Conflict or Conflux? How to represent/parametrize the system How to excite the system How to manage changing models Directionality Complexity Software 24

25 Conclusions MPC is not self tuning There is a strong" inter-action between control and identification Different ways to solve the problem More complex control External Excitation (setpoints/inputs) Identification for control Need to Retune Controller when model changes Collinearity issue is not well understood Very Large scale Applications are challenging MPC Maintenance is still challenging 25

26 4756 lines of assembly code (1983) 15 lines of MATLAB code (2014) Golden Opportunities Pitt CMU