Adaptive Robust Control

Adaptive Robust Control Adaptive control: modifies the control law to cope with the fact that the system and environment are uncertain. Robust control: sacrifices performance by guaranteeing stability and performance for all ranges of parameters within bounds. Adaptive Robust Control: optimizes performance by modifying performance so that the controller is robust and stable. A good algorithm embeds elements of both ideas., CMU 1

Structure of Adaptive Control Desired Performance Objectives Control Process Measure 1. Defining clear business objectives (Performance Model) 2. Developing plans to achieve the objectives (Control) 3. Systematically monitoring progress against the plan (Gap analysis) 4. Adapt the objectives and the plans as needs and opportunities change Robert McNamara, (US Secretary of State) on Business Management (1995) 2

Structure of Adaptive Control Adapt Performance Objectives (3b) Desired Performance Objectives Adapt Control Process Measure Evaluate Adapt Control (3a) 1. Measure, evaluate and critique (Gap analysis) 2. Robust Control strategies (Model Predictive Control) 3. Adaptation a) Adapt Model and Controllers b) Adapt Performance Objectives (closed loop performance) 3

Feedback System Feedforward System CE Architecture

Inter-related Problems Model representation step/impulse, transfer function, state space, nonlinear, physics, Volterra, neural network, How to adapt the model select informative data, trade-off between seed model and data, include model constraints,.. Large scale application Centralized (state space id), decentralized (tf), internal constraints (collinearity),.. How to adapt the controller tune for robustness, interaction with identification, How to excite the process to generate informative data Include nonlinearities Input, output, process, discrete events,

Feed Forward System -- Identification 1. Gradient/Projection Algorithms 2. Recursive least squares with modifications a) Covariance Reset b) Forgetting Factors c) Deadzone d) Leakage 3. Moving Horizon Estimation 4. Non-convex optimization 5. Sub-space identification 6. + Many, many more, (check identification toolbox) System Identification, Theory for the User by Lennart Ljung. 6

Feedback System: Feedback Control 1. Pole Assignement 2. Linear Quadratic Control with Kalman filter 3. PID with Feedforward 4. Model Predictive Control 5. Model Reference Control 6. Minimum Variance Control 7. H_infinity Robust Control 8. + Many, many more, 7

Requirements The estimated model must be controllable/stabilizable with respect to the chosen control law. The control law must be robust with respect to un-modelled dynamics An internal model of the disturbances should be included (use appropriate noise filters) The estimator must retain adaptation Excitation is needed to achieve optimal controls A dead-zone is useful to stop estimation and avoid chaotic bursting

Adaptive PID Process Model: First order deadtime Identification from data Internal Model Control Max bandwidth Minimize filter constant subject to Overshoot constraint (1.2) Gain margin =2 Phase margin=60deg s t p d e s K s G 1 ) ( + = τ 1 2 2 1 2 1 2 1 + = + = + + = d d d I d c d p c t t t t K K τ τ τ τ τ τ τ τ c

$!#!" $!#!" ) ( disturbance response 1 1 ) ( systemresponse 1 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( t e t x t v q D q C t u q A q B t y + = Process Identification Problem: Find A, B, (C, D) from the data

u B( ( q ) q -t d A ( q ) x + e y xˆ ( t) = Bˆ( q Aˆ( q 1 ) u( t) ) 1 $!#!" output model ˆ 1 1 A( q ) xˆ( t) = Bˆ( q ) u( t) Problem: Need to estimate states and parameters to global optimality. N 2 T min y( t) xˆ( t), X (0) = ( x(0), x( 1),... x( n)) Xˆ (0), ˆ θ t= 1 Subject to: x( t) ( ) ˆ θ = ( a 1,..., a n, b,..., b aˆ x t aˆ x t n bˆ u t bˆ 1 ˆ( 1) +,,, + n ˆ( ) + ( 1) +,,, + mu( t m), t = 1,... N ˆ = 1 1 m ) T

Solve as Non-Convex Nonlinear (QP)Program McCormick estimators for bi-linear constraints. Exploit (banded) structure. Efficient QP solver (KKT conditions) Branch and bound with big M constraint. Consistency: (Parameters (A,B) converge to true values ) 1.G is stable 2.{A,B} co-prime 3. u(t) persistently excited 4. v(t) independent of u(t)

Control Input and Process Output Control Input u Process Output y

System and Control Parameters Process Model PID Parameter

Heat exchanger Experiment 1. Identify model with global optimization 2. Implement controller

Industrial Examples: Martin Titan IV Booster Challenges: Pole changes during flight Long set-up time Very expensive equipment G( s) = 400 ( s 1)( s + 2.25) ( s + 0.01± 8i)( s + 0.02 ± 7i)( s + 4)( s + 3.8)( s + 3.6) 0.1 Impulse Response 10 Pole-Zero Map 0 5-0.1 Amplitude -0.2 Imaginary Axis 0-0.3-5 -0.4 0 2 4 6 8 10 Time (sec) -10-4 -3-2 -1 0 1 Real Axis

Robust Control Design

System Identification 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 f i (u j,z), k =1,...,n PV i =1,...,n Distributed Control System (DCS) Control Inputs u Measured Outputs y Setpoints y* Linear equation error model Interface Layer (SCADA) LT FT e(t) =y(t) G p (q 1 )u(t) Feed TT CT FT FT Product Cooling water return

Why is it Important? 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 C CA rank CA CA 2 n 1 = 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 Markov-Laguerre Models are automatically stabi 23

The Step/impulse Response Model used by DMC u(t)-u(t-1) q -1 q -1 q -1 q -1 g 1 g 2 g 3 g 4 q -1 g N n y(t) = g(k)δu(t k) k=1 Summing junction Step Response 2 1.5 1 0.5 0 Step Response 0.5 0 5 10 15 20 25 Time (sec) G(s) = Model order n typically 70-200 (slow convergence) 2s +2 2s 2 +5s +1 e 5s Transfer function (6 numbers) approximated with step-response with 25 numbers g T = (0 0 0 0 0 0-0.09 0.26 0.60 0.87 1.09 1.27 1.41 1.53 1.62 1.69 1.75 1.80 1.84 1.87,.,1.99) 24

Markov-Laguerre Model Delays and inverse response First order response u(t)-u(t-1) 1 q g 1 1 q g 2 1 q g 3 1! aq q! a g N N 1 1 y(t) = g(k)δu(t k)+ g(n) q a u (t k) k=1 Summing junction 0 a <1 2 1.5 Step Response Fixed pole(s) has to be chosen (Fast convergence) Amplitude 1 0.5 0 0.5 0 5 10 15 20 25 Time (sec) Extensive and very strong theory exists for system identification. Easy to convert to state space model formulation. g T = (0 0 0 0 0 0-0.09 0.8) 25

Adaptive Model Matching Trade off between new data and current model min 2C X n data i=0 C = constraint set e(t i) T Q(t i)e(t i)+( seed ) T F seed ( seed ) Q(t) = Data selection and weighting seed = seed model F seed = Fisher information matrix/inverse covariance Constraint set (applied to the Markov Laguerre mod 1. Box constraints 2. Upper/lower bounds for delays 3. Inverse response bounds 4. Time constants 5. Robust controllability 6. Mass/Energy balance (collinearity/uncollinearity)

Adaptive with Selective Memory How to stop here - Rather than here Poor control Good control Poor control Good control Poor control Good control Issue: How to turn estimation on and off?

Desired Trajectory Σ Controller Input Plant Unmeasured Disturbances Σ Output Feedback Loop Adaptation Off Controller Model Supervisor Model Σ Σ

Desired Trajectory Σ Controller Input Plant Unmeasured Disturbances Σ Output Feedback Loop Adaptation On Controller Model Supervisor Model Σ Σ

Desired Trajectory Σ Controller Input Plant Unmeasured Disturbances Σ Output Feedback Loop Adaptation Off Controller Model Supervisor Model Σ Σ

Heat Exchanger Experiments SAPC T T T HEX T Control goal: Regulate hot water outlet temperature using cold water flow AIChE Annual Meeting, Salt Lake City 2007 Dozal-Mejorada & Ydstie 31

Experiments Shell and Tube Heat Exchanger Algorithm parameters: Polynomial orders: RLS parameters: Parameter leakage: Control goal: Reduce bursting in adaptive control of the shell and tube HEX Use Extended Horizon Control RLS with selective memory

Heat Exchanger Experiments NO Supervision WITH Supervision AIChE Annual Meeting, Salt Lake City 2007 Dozal-Mejorada & Ydstie 33

Application to Power Plant Control

Model estimated using output error identification (global optimality)

CV vs MV Refinery Data

Finding Transfer Functions to Update Current Example: 68 CVs, 22 MVs, 3 DVs = 1700 TFs Update after 331 samples Calculate difference in L 2 norms between G seed = G ML Rank order importance Finds: G(34,9) > G(34,1)>G(68,22)>G(35,1) Differences: 4.2 3.7 0.6, 0.4 Steady-State Gains: TF Seed New (34,9) 9 4.55 (34,1) 9.9 5.085 (35,1) 54 18.45 (68,22) -0.5154-0.6851

Impulse Responses for identified Transfer Functions

Which Parameters to Update in Models (Multivariate Statistical Analysis - Combinatorial) Ø For reduced data set (5-25 transfer functions) Check when sufficient data is available for update Identify the most influential parameters in given model structure Only updating important parameters instead of all of them v and model selection criteria can answer these questions effectively They are MSE-based parameter ranking and selection technique They can detect if the excitation is enough to update a model They can determine which parameters we should update give a set of new observations 2 MSEP = Bias + Variance T ( True True ) T ( E( Yˆ) Y ) ( ( ˆ) ) ( ( ˆ True E Y YTrue Tr Cov Y )) MSEP( Yˆ) = E ( Yˆ Y ) ( Yˆ Y ) = +

Parameter Ranking and Selection Estimate subset of parameters which lead to lowest MSE of model predictions by using a parameter subset selection technique: r CW Estimated squared bias introduced by removing β2 terms = Estimated variance reduction when SM is used instead of EM ˆT β2e 1 1 2 1 1 2 β2e = 2 T s Tr W A W W A W E T ( WA W ) ( WA W ) ˆ ( ) Ω( ) ( ) 1 1 2 1 1 2 r CCW T ( ( P1 )) Tr M M P = ( rcw 1) n The best SM for making predictions at key operating points should have the lowest value for r ""# Eghtesadi, Z, S Wu, KB McAuley (2013) Ind. Eng. Chem. Res., 52, 12297 12308. Eghtesadi, Z, KB McAuley (2014) Ind. Eng. Chem. Res., 53 (14), 6033 6046

Collinearity/uncollinearity Constraints Steady state mass/energy balance (linear) Y = AKU A collinearity structure n PV (n MV + n DV ) K gain matrix (n MV + n DV ) (n MV + n DV ) RankK =(n MV + n DV ) y = K p G p (s) Modified constraint (one-shot-method) for steady state gain adaptation: min 2C X n data i=0 K = G(1) Y = AKU, e(t i) T Q(t i)e(t i)+( seed ) T F seed ( seed ) A defines collinearity structure Cond K apple 0 Included as an additional ridge Paper by +++++ ILS Proprietary Shared with Exxon-Mobil under N

4756 lines of assembler code 15 lines of MATLAB code Pitt CMU