An Introduction to Model Predictive Control TEQIP Workshop, IIT Kanpur 22 nd Sept., 2016

Transcription

1 An Introduction to Model Predictive Control EQIP Workshop, II Kanpur 22 nd Sept., 216 Sachin C. Patwardhan Dept. of Chemical Engineering I.I.. Bombay Outline Motivation Development of MPC Relevant Linear Models Review of Linear Quadratic Optimal Control Linear Model Predictive Control Formulation Adaptive MPC Nonlinear Model Predictive Control Formulation Summary and Research Directions 2 Plant Wide Control Framework Hierarchy of control system functions Setpoints Long erm Scheduling and Planning On-line Optimization Multivariable / Nonlinear Control Regulatory (PID) Control PV, MV Market Demands / Raw material availability Slow Parameter drifts Advanced Control MV Plant PV Fast Load Disturbances 3 4 1

2 Why Multi-Variable Control? Why Model Predictive Control? Most rear systems have multiple inputs and Need to control over wide operating range multiple controlled outputs Process nonlinearities Systems exhibit complex and multi-variable Changing process parameters / conditions interactions between inputs and outputs variables Conventional approach : Multi-loop PI - difficult to tune Need to operate a system within operating Ad-hoc constraint handling using logic programming constraints (PLCs): lack of coordination Safety limits MPC deals with multivariable interactions, Input saturation constraints operating constraints, and process nonlinearity Product quality constraints 5 systematically 6 Model Predictive Control Most widely used multivariable control scheme in the process industries over last 35 years Used for controlling critical unit operations (such as reactors) in refineries world over Development of MPC Relevant Linear Perturbation Models With increasing computing power, MPC is increasingly being applied in diverse application areas: robotics, fuel cells, internet search engines, planning and scheduling, control of drives, biomedical applications 7 8 2

3 MPC Relevant Linear Model Local Linearization Critical Step: Development of a control relevant linear perturbation model for developing MPC scheme Approach 1: If a reliable mechanistic or grey-box dynamic model is available, then a linear perturbation model can be developed using local linearization Approach 2: Alternatively, a linear perturbation model Given a lumped parameter model dx State Dynamics : dt Measurement Model : n X R, U R m, Y R F(X, U,D) Y G(X), D R and steady state operating point ( X, U, D), we apply aylor series expansion in neighborhood of ( X, U, D) to develop a linear perturbation model r d can be developed using input-output data generated by deliberately exciting the system for a short period and using the system identification tools Given steady state operating inputs ( U, D) Corresponding Steady State X can be found by solving for dx F( X, U, D) dt 9 1 Local Linearization Local Linearization aylor series expansion in the neighborhood of ( X, U, D) dx F F( X, U, D) dt X F U ( X,U,D) F U( t) U D( t) D At Steady State dx F( X, U, D) (2) dt Subsract Equation (2) from Equation (1) to derive perturbation model Measurement Model Y( t) G X ( X,U,D) D X( t) X ( X,U,D) G G X( t) X Y X( t) X X ( X) G y( t) Y( t) Y X ( X) X ( X) X( t) X (1) A F X Define F ; B U computed Continuous ime Linear Perturbation Model dx Ax Bu Hd dt y Cx Perturbation variables x(t) X(t) - X u(t) U(t) - U ; at ( X, U, D ) ; matrices ; H F D y(t) Y(t) - Y d(t) D(t) - D ; C G X

4 Example: Quadruple ank System Model Parameters Pump1 V 1 ank 3 ank 1 ank 4 ank 2 Pump 2 V 2 dh1 a1 a3 2gh1 dt A A 1 dh2 a2 dt A 2 3 dh4 a4 dt A 4 Manipulated Inputs : v Measured Outputs : h and h 1 a4 2gh2 A 2 1k1 2gh3 v1 A (1 1) k1 2gh4 v1 A 1 2k2 2gh4 v 2 A dh3 a3 (1 2) k2 2gh3 v2 dt A A and v A 1, A 3 [cm 2 ] 28 A 2, A 4 [cm 2 ] 32 a 1, a 3 [cm 2 ].71 a 2, a 4 [cm 2 ].57 k c [V/cm].5 g [cm/s 2 ] 981 Model Parameters Steady state Operating Conditions P- : Minimum Phase P+ : Non-minimum Phase P - P + h 1, h 2 [cm] (12.4,12.7) (12.6,13) h 3, h 4 [cm] (1.8,1.4) (4.8,4.9) v 1,,v 2 [V] (3,3) (3.15,3.15) k 1,k 2, [cm 3 /V] (3.33,3.35) (3.14,3.29) γ 1,γ 2 (.7,.6) (.43,.34) Linearization of Quadruple ank Model Linearization of Quadruple ank Model 1 A3 1 A dx 2 dt 1 3 kc y x kc 1k1 A1 A4 A 24 x 1 (1 1) k 4 A4 Ai i a i 1 2hi g P - P + ( 1, 2 ) (62,9) (63,91) ( 3, 4 ) (23,3) (39,56) 2k2 A2 (1 2) k A 3 u for i 1,2,3, Quadruple ank System Continuous ime State Space Model Matrices P - h 1, h 2 [cm] (12.4,12.7) h 3, h 4 [cm] (1.8,1.4) v 1,,v 2 [V] (3,3) k 1,k 2, [cm 3 /V] (3.33,3.35) A γ 1,γ 2 (.7,.6) Developed at ( X, U).8325 Steady State Operating Point B C

5 Discrete Dynamic Models Digital Control: Measured Outputs Development of computer oriented discrete dynamic models Assumptions 1. Measurement Sampling : Measurements are sampled at a constant rate and uniform sampling interval of sec hus, measurements, t k :k,1,2,... k y (, are available at instant Measured Output Output measurements are available only at discrete sampling instant t k k : k, 1, 2,... Where represents sampling interval ADC Measured Output Input Reconstruction with zero order hold : Manipulated inputs are piecewise constant during the sampling interval u(t) u( for t k t ( k 1) Sampling Instant Continuous Measurement from process Sampling Instant Sampled measurement sequence to computer Digital Control: Manipulated Inputs Linear Discreteime Model Manipulated Input Sequence In computer controlled (digital) systems Manipulated inputs implemented through DAC are piecewise constant u( t) u( t ) u( for t k t t 1 ( k Sampling Instant Input Sequence Generated by computer k k k ) DAC Manipulated Input Sampling Instant Continuous input profile generated by DAC 19 Computer Controlled Systems Manipulated inputs are piecewise constant u( t) u( for t k t ( k 1) Difficulty Disturbance inputs d(t) are NO piecewise constant functions! How to develop a discrete time model? Simplifying Assumption 1 : Sampling interval () is small enough so that the disturbance inputs can be approximated as piecewise constant functions during the sampling interval d( t ) d( for t k t ( k 1) 2 5

6 Unmeasured Disturbances Linear Discrete ime Model d( samples ypical piecewise constant unmeasured disturbance generated using zero mean Gaussian random process with unit variance 21 Notation x( x(t k), y( y(t k) and u( u(t k) Discrete time linear model under Assumption 1: Φ exp( A ) ; x( k 1) Φx( Γu( Γ d( y( Cx( Γ exp( A ) B d ; Γd exp( A ) H d Simplifying Assumption 2 : d( : zero mean white noise process with known covariance Cov d( Ed ( d( Qd d 22 Linear Discrete ime Model Linear Discrete ime Model Cov Now, define E w ( Γ Ed ( w ( Γ Ed ( d( Γ Γ Q Γ d w( Γ d( d d hus, w( is a zero mean stochastic process with covariance matrix Q Γ Q Γ d d d d d d d Combining all the simplifying assumptions, we arrive at a linear discrete time computer control relevant dynamic model of the form x( k 1) x( k ) u( k ) w( k ) y( k ) Cx( k ) v( k ) where w( and v( are assumed to be uncorrelated random sequences with zero mean and know variances Simplifying Assumption 3: Measurements are corrupted with zero mean white noise process with known covariance v( with Covv( Ev( v( R 23 w ( k ) w( k ) Q ; E v( k ) v( k R E ) Q quantify uncertainties in state dynamics and/or modeling errors R quantifies variability of measurement errors 24 6

7 Quadruple ank System On-line State Estimator Discrete ime State Space Model Matrices Sampling ime = 5 sec C Q.5I I 44 R A suitable state estimator can be developed using the linear perturbation model For example, steady state Kalman predictor Recursive Observer Gain Matrix Computed using Algebraic Riccati Equation P P Q LCP L P C R CP C 1 Prediction Estimator e( y( Cxˆ( k k 1) xˆ( k 1 xˆ( k k 1) u( Le( 25 9/18/216 State Estimation 26 Data Driven Model Development 4 ank Experimental Setup Man. Input Perturbations Unmeasured Disturbances System Response 3.2 Manipulated Input Physical System Measured Output Sampling Instant ARX/ARMAX/ State Space Model Identification Sampling Instant Data Driven Linear State Space Model Quadruple ank Experimental Setup at the,

8 Splitting Data for Identification and Validation Identification Data: Inputs y1 Input and output signals u 1 (, volts.5 Identification Data: Manipulated Inputs u Identification Data 5 1 Samples Validation data u 2 (, volts ime (sec) 29 3 Identification Data: Outputs Data Driven State Space Model y 1 (, cm y 2 (, cm 1 Identification Data: Output Perturbations ime (sec) 31 Innovation form of State Space Model (or observer) Sampling ime = 5 sec x( k 1) Φx( Γu( Le( y( Cx( e( L 1 C ; (Model developed using System Identification oolbox of MALAB) 9/18/

9 Model Validation: Inputs Model Validation: Outputs u 1 (, volts u 2 (, volts.5 Validation Data: Manipulated Input Perturbations ime (sec) 33 Perturbation Level 1 (cm) Model Validation: Output 1 Model Simulation Measured Output ime Comparison of simulated model output with the measured outputs in the validation data set Perturbation Level 2 (cm) Identified models have reasonably accurate predictions Model Validation: Output 2 Model Simulation Measured Output ime 34 Linear Quadratic Regulator Brief Review of Linear Quadratic Optimal Control Model: x( k 1) Ax( Bu( y( Cx( Objective Regulate the process at origin of the state space in the face of sudden impulse like, disturbances, which result in non-zero initial conditions Determine input sequence u(),..., u( N 1) such that Final State x( N) WN x( N) min N 1 u(),..., u( N 1) x( W x x( u( Wu u( k Square of Subject to Penalize distance Large x(k 1) x( u( from Origin manipulated y( Cx( inputs W W,W : Symmetric PositiveDefinite seighting matrices x, u N

10 Summary: Quadratic Optimal Control Algebraic Riccati Equation ime varying state feedback controllaw G( W u u( -G( x( Gain matrix computed by solving Discrete time Riccati Equation S ( k 1) S ( k ) [ G ( k )] S ( k 1)[ G ( k )] W G ( k ) W G ( k ) Equation solved backward in time starting from N, N -1,...1 with S(N) W s( : Symmetric and ve definite for each k which ensures optimality of solution at each stage 1 N S ( k 1) N should be known a-priori and gain matrices have to be saved : not quite practical in many situations x u 37 When N becomes large, S( S ; G( G which can be computed by solving Algebraic Riccati Equation (ARE) as 1 G W S u S S [ G ] S [ G ] W G W G u( -G x( W x u and control law assumes form ARE has many solutions. However, if (, ) is controllable and if (, ) is observable where u then there exists a unique symmetric and non negative definite solution to ARE. 38 Nominal Stability Analysis Nominal Stability Analysis heorem 1: Consider the time invariant dynamic model together with the LQ loss function. Assume that a positive-definite steady state solution exists for the algebraic Riccati equations. hen the steady state optimal strategy gives an asymptotically stable closed-loop system Proof: Define Lyapunov function Note : S is a ve definite matrix hus, the closed loop system is asymptotically stable for any choice of positive definite Wx and positive semi-definite Wu matrices Simultaneously guarantees closed loop stability and good closed loop performance By selecting Wx and Wu appropriately, it is easy to compromise between speed of recovery and magnitude of control signals

11 Closed Loop Poles Linear Quadratic Optimal Output Regulator he poles of the closed loop system obtained by solving the characteristic equation In many situations we are only interested in controlling certain outputs of a system It can be shown that the poles are the n stable eigenvalues of the generalized eigenvalue problem he above modified objective function can be rearranged as follows his equation is called the Euler equation of the LQ problem. heorem 1 shows that LQ controller gives a stable closed loop system, i.e. all poles of - G are inside the unit circle. 41 and by setting we can use the Riccati equations derived above for controller design. 42 Linear Quadratic Gaussian Regulator Nominal Closed Loop Stability Linear Quadratic Gaussian (LQG) Regulator Design optimal state estimator (Kalman Predictor / Kalman Filter) Implement control law using estimated states Process Dynamics x( k 1) x( k ) u( k ) w( k ) y( k ) Cx( k ) v( k ) Controller implementation using Kalman Predictor e( k 1) y( k 1) Cxˆ( k 1 k 2) xˆ( k k 1) xˆ( k 1 k 2) u( k 1) L e( k 1) u( k ) G xˆ( k k 1) Is the closed loop stable under the nominal conditions? 43 x( k 1) Φ ΓG ε( k 1 k ) [ ] I Φ ΓG det [ ] Combined Closed LoopDynamics ΓG L C ε( k k 1) I L v( k ) 9/18/216 State Feedback Control 44 ΓG det I L C x( k ) Closed Loop Characteristic Equation Φ ΓG 1 and LC I [] w( k ) Note : hrough Lyapunov stability arguments, we have established that 1 Eigenvalues of the closedloop equation inside the unit circle det I Φ ΓG I L C hus, even though the observer and controller are designed separately to be a-stable, the nominal closed loop system, implemented using the observer based feedback controller, is asymptotically stable 11

12 LQOC Formulation Limitations of LQOC LQ Optimal Controller: Linear quadratic regulator can be further modified to Difficult to incorporate or handle operating constraints explicitly Reject non-stationary (drifting) unmeasured Limits/constraints on the manipulated inputs disturbances Constraints on process outputs (arising from racking arbitrarily changing setpoints for the product quality, safety considerations) controlled outputs Algebraic Riccati Equations: AREs is notoriously Achieve robustness in the face of mismatch difficult to solve for large dimensional systems between the plant and the model Model Predictive Control Linear Model Predictive Control Multivariable control based on on-line use of dynamic model and constrained optimization Developed by industrial researchers Dynamic Matrix Control (DMC) developed by Shell in U.S.A. (Cutler and Ramaker, 1979) Model Algorithmic Control developed by Richalet et. al. (1978) in France Can be used for controlling complex, large dimensional and non-square systems

13 Advantages of MPC MPC: Basic Idea Can be viewed as a modified version of the classical optimal control problem Can systematically and optimally handle Multivariable interactions Operating input and output constraints Basic Idea: Given a reasonably accurate model for plant dynamics, possible consequences of the current and future input moves on the future plant behavior can be forecasted on-line and used while deciding the input moves Finite Horizon formulation: Optimization problem is formulated over a finite window of time starting from current instant, i.e. over [k, k+p] (unlike over [k, ) in the classical optimal control) Pro-active constraint management: Using the dynamic model, on-line forecasting is carried out foresee and avoid any possible constraint violations over the time window [k,k+p] 49 5 MPC: Basic Idea Moving Horizon Formulation On-line Constrained Optimization: At each sampling instant, a constrained optimization problem is formulated over the window and solved online to determine the current input u( Output Constraints Moving horizon implementation: he time window for control keeps moving or receding From [k, k+p] o [k+1, k+p+1]. and so on (Kothare et al, (2), IEEE Control Systems echnology)

14 MPC: Schematic Diagram Components of MPC Optimization Dynamic Prediction Model MPC Set point rajectory Inputs Disturbances Process Dynamic Model Plant-model mismatch Dynamic Model: used for on-line forecasting over a moving time horizon (window) Outputs 53 Internal model and state estimator Discrete Linear State Space Model developed from mechanistic approach or time series modeling (FIR or Finite Step Response models were used initially) State Estimator: Open loop observer / Kalman Predictor/ Kalman Filter / Luenberger Observer / Innovation form of state observer developed from ARX / ARMAX / BJ model Prediction of Future Plant Behavior Key issue: Handling unmeasured drifting disturbances and plant model mismatch On-line constrained optimization strategy Quadratic programming Linear programming 54 MPC with State Estimation State Estimation and Prediction Consider state estimation and prediction using prediction form of observer Such a observer can be developed using any of the following approaches Kalman predictor Luenberger predictor State realization of ARX / ARMAX / BJ model On-line Optimizer Dynamic Model Prediction estimate of the current state and innovation (Kothare et al, (2), IEEE Control Systems echnology)

15 Future Prediction Future rajectory Prediction In absence of model plant mismatch, the innivations e( y( yˆ( k k 1) y( Cxˆ( k k 1) is a zero mean white noise, i.e. when the steady state Kalman predictor is used as observer in absence of model plant mismatch (MPM), model predictions over future time window [k + 1; k+p] can be generated as follows E e( zˆ( k j 1) Φzˆ( k j) Γu( k j yˆ ( k j) Czˆ( k j) with zˆ( xˆ( k k 1) for j 1,2,..., p 57 In practice, Model Plant Mismatch (MPM) arises due to Changes in the steady state operating conditions Abrupt step changes/drifts in the unmeasured disturbances In the presence of MPM, the innovation sequence is no longer a zero mean white noise. he mean starts drifting and the sequence becomes a colored noise. hus, the model predictions have to be corrected to account for for MPM 58 Compensation for MPM State Estimation and Prediction he innovation signal contains a signature of MPM, which is typically a low frequency signal However, the innovation signal also contains the measurement noise, which is typically in the high frequency range hus, a filtered version of the innovation sequence can be used as a proxy for MPM and can be used for correcting the future predictions Innovation Bias Approach: Effect of model plant mismatch and /or unmeasured disturbance signal is extracted by filtering the innovation through a unity gain low pass filter model predictions over future time window [k + 1; k + p] with compensation for MPM are generated as follows

16 Future rajectory Prediction Future rajectory Prediction Future instant (k+1) Future instant (k+2) Future instant (k+p) Future rajectory Prediction Future rajectory Prediction Interpretation of p step output prediction equation Note: he predictions generated using the innovation bias approach is equivalent to carrying out predictions using the observer augmented with an artificially introduced integrated white noise model, i.e. prediction Generated using the following dynamic system Future output Effect of the past prediction state on future outputs Effect of future inputs on future outputs Effect Disturbances on future outputs of Plant Model Mismatch and Unmeasured p is called as Prediction Horizon Introduction of integrated white noise in predictions helps in achieving offset free closed loop behavior

17 Constraints on Inputs Constraints on Inputs o reduce the degree dimension of the on-line optimization problem, degrees of freedom available for shaping the future trajectory are often restricted to first q moves Bounds on the manipulated inputs Bounds on rate of change of manipulated inputs by imposing input constraints of the form q is called the Control Horizon In a practical implementation control horizon (q) << prediction horizon (p) Since predictions are carried out online at each control instant, it is possible to choose future inputs moves such that the above constraints are respected Control Horizon Input Blocking Constraints Schematic Representation of Control Horizon and Input Bounds Alternatively, degree of freedom for shaping the future trajectory can be used through input blocking constraints q is called the Control Horizon In a practical implementation control horizon (q) << prediction horizon (p) 67 9/18/216 State Feedback Control 68 17

18 Input Blocking Constraints Input Blocking and Bounds Bounds on the manipulated inputs Schematic Representation of Input Blocking and Input Bounds Bounds on rate of change of manipulated inputs Since predictions are carried out online at each control instant, it is possible to choose future inputs moves such hat the above constraints are respected 9/18/216 State Feedback Control 69 7 Future Setpoint rajectory Future Setpoint rajectory In addition to predicting the future output trajectory, at each instant, a filtered future setpoint trajectory is generated using a reference system of the form

19 Steady State arget Computation Constrained MPC formulation Given the prediction model, input constraints and desired set point trajectory, the MPC problem at sampling instant k is formulated as follows Case: Number of manipulated inputs equals the number of controlled outputs and unconstrained solution exists 9/18/216 State Feedback Control Constrained MPC formulation Subject to following constraints (a) Model Prediction Equations and Constrained MPC formulation W is a symmetric positive definite error weighting matrix E W U hese matrices are treated as tunig parameters, which are used to shape the closedloop input and output behavior is symmetric positive semidefinite input weighting matrix (b) Bounds on future inputs and predicted outputs he terminal state weighting matrix discrete Lyapunov equation. When poles of are inside the unit circle, W W can be found by solving can be found by solving discrete Lyapunov equation When some poles of Φ are outside unit circle, the procedure for computing the terminal weighting matrix is given in Muske and Rawlings (1993)

20 Moving Horizon Implementation Moving Horizon Formulation he resulting constrained optimization problem is solved online each sampling instant using any standard constrained optimization method. he controller is implemented in a moving horizon framework. hus, after solving the optimization problem over window [k,k+p], only the first optimal move is implemented on the plant, i.e. he optimization problem is reformulated at the next sampling instant over time windows [k+1, k+p+1] based on the updated information from the plant and resolved. Optimization problem transformed to Quadratic Programming (QP) problem for improving computing efficiency on-line and solved using efficient QP solvers available commercially. MPC formulation can control Non-square multi-variable systems i.e. systems with number of controlled outputs not equal to the number of manipulated inputs. In many practical situations, not all outputs have to be controlled at fixed setpoints but need to be maintained in some zone. Such zones can be easily defined using constraints on predicted outputs Quadratic Programming (QP) QP Formulation A constrained optimization problem is called as Quadratic Programming (QP) formulation if it Has following standard form Min 1 U HU F U U 2 Subject to AU b A large dimensional QP formulation can be solved very quickly using an efficient search method hrough a series of algebraic manipulations, the Constrained MPC formulation can be transformed to a Quadratic Programming (QP) Problem 79 o understand how the MPC optimization problem can be transformed to a quadratic programming problem, consider MPC formulation without terminal state weighting the prediction model can be expressed as follows (Note: QP formulation can be carried out with terminal state weighting also. It has been neglected here to keep the expressions relatively simple) 8 2

21 QP Formulation QP Formulation Matrix relating the effect of past states to future predictions Matrix relating the effect of past unmeasured disturbances and model plant mismatch on the future predictions Matrix relating the effect of future manipulated inputs On future predictions Consists of impulse response coefficients of the model Referred to as Dynamic Matrix in MPC literature Unconstrained QP Formulation Unconstrained QP Formulation Using these notations, unconstrained version of MPC problem can be stated as follows

22 Unconstrained QP Formulation Unconstrained QP Formulation he unconstrained optimization problem can be reformulated as a Quadratic Programming problem as follows Since only the first input move is implemented on the process With some algebraic manipulations, the above control law can be rearranged as follows From the above expression, it is easy to see that unconstrained MPC is a form of state feedback control law he optimum solution to above minimization problem is Advantage of unconstrained formulation: closed form control law can be obtained and, as a consequence, on-line computation time is small Constrained QP Formulation Constrained QP Formulation he constrained MPC formulation at k th sampling instant can be re-cast as

23 Alternate Formulations Nominal Stability o achieve offset free control, it is possible to develop MPC formulation based on the augmented state space model (see Muske and Rawlings, 1993; Yu et al., 1994). Early formulations of MPC, such as Dynamic Matrix Control (DMC), were based on open loop observer and were meant for open loop stable systems. hese formulations can be derived by setting L = [] in the innovation bias formulation. MPC formulation in this presentation has been developed using Kalman predictor. It is straightforward to develop a similar formulation based on the Kalman filter. 89 Proved for the deterministic version of MPC under certain simplifying assumptions Assumption 1: here is no model plant mismatch or unmeasured disturbances are absent and both internal model (i.e. observer) and plant evolve according to Assumption 2: he true states are perfectly measurable Assumption 3: It is desired to control the system at the origin 9 Nominal Stability Nominal Stability Let us formulate MPC in terms of a generalized loss function Let us denote the optimal solution of the resulting constrained optimization problem at instant k as

24 Nominal Stability Nominal Stability Let optimum solution of the MPC problem over the window [k + 1, k + p + 1] be denoted as hus, it follows that We want to examine A non-optimal but feasible solution for the optimization problem over window [k + 1, k + p + 1] is For this feasible solution, the following inequality holds Nominal Stability Nominal Stability hus, it follows that and the nominal closed loop system is globally asymptotically stable. It is remarkable that we are able to construct a Lyapunov function using the MPC objective function. hus, under the nominal conditions, MPC guarantees global asymptotic stability and optimal performance

25 uning of MPC uning of MPC Set Point Set Point Filter MPC Robustness Filter Facilitates Performance specification Inputs Unknown Disturbances Process Dynamic Model Plant-model mismatch Guard again plant-model mismatch Outputs Prediction Horizon: ypically chosen close to open loop settling time (6 to 1 samples) Control Horizon: ypically chosen small (5 to 1) to avoid model inversion problems Input rate constraints Zone / Range Control: Not necessary to specify set points on each output. Instead, high and low limits can be defined within which output should be maintained Example: Shell Control Problem Shell Control Problem (SCP) Controlled Outputs : (y 1 ) op End Point (y 2 ) Side Endpoint (y 3 ) Bottom Reflux emperature Manipulated Inputs : (u 1 ) op Draw (u 2 ) Side Draw (u 3 ) Bottom Reflux Duty Unmeasured Disturbances: (d 1 ) Upper reflux (d 2 ) Intermediate reflux 99 y( s) G ( s) u( s) G ( s) d( s) 27s 4.5e 28s 1.77e 5s 1 6s 1 18s 14s 5.39e 5.72e G u ( s) 5s 1 6s 1 2s 22s 4.38e 4.42e 33s 1 44s 1 27s 1.2e 27s 1.44e 45s 1 4s 1 15s 15s 1.52e 1.83e G ( ) d s ` 25s 1 2s s 1 32s 1 u d 27s 5.88e 5s 1 15s 6.9e 4s 1 19s 7.2e 19s 1 Characteristics Large time delays High degree of multivariable interactions 1 25

26 SCP: MPC uning Parameters SCP: PID uning Parameters Operating Constraints Input Limits.5 ui.5 for i 1,2,3.5 di.5 for i 1,2 Rate Limits.5 u i.5 for i 1,2,3 Output Constraints W diag 1 e.5 y 1.5 y 1 W 1.5diag 1 u 3.5 Prediction Horizon : 4 Control Horizon : Sampling interval () = 2 min Multi-loop PID control: hree independent PID controllers with no coordination among them PID Pairing and uning (y1) op End Point - (u1) op Draw Kc =.3, i = 13 min, d = (y2) Side Endpoint - (u2) Side Draw Kc =.23, i = 3 min, d = (y3) Bottom Reflux - (u3) Bottom Reflux Duty Kc =.28, i = 9 min, d = Comparison of Servo Responses Comparison of Servo Responses Controlled Outputs Manipulated Inputs

27 Comparison of Regulatory Responses Comparison of Regulatory Responses Controlled Outputs Manipulated Inputs (Open Loop Observer Based MPC Formulation) Comparison of Regulatory Responses SCP: Sequential Servo Changes Unmeasured Disturbances Measured Outputs Shell Control Problem y(1) y(2) y(3) Sampling instant With drifting Unmeasured disturbances Note: Decoupled Servo Response. Change in one Setpoint Does not affect the other outputs

28 SCP: Sequential Servo Changes Commercial Products.5 Shell Control Problem Manipulated Inputs u(2) u(3) u(1) Sampling instant (Ref.: Qin and Badgwell, 23) 19 9/18/216 State Feedback Control 11 Linear MPC Applications (23) Industrial Application: Ammonia Plant (Ref.: Qin and Badgwell, 23) 9/18/216 State Feedback Control 111 (Ref.: Qin and Badgwell, 23) 9/18/216 State Feedback Control

29 Dealing with Model-Plant Mismatch Adaptive and Non-Linear Model Predictive Control Adaptive Model Predictive Control: Active approach On-line Model Maintenance: Identify model parameters on-line, either intermittently using a batch of data, or, on-line using recursive parameter estimation Robust Model Predictive Control: Passive approach Incorporate robustness at the design stage Adaptive MPC 4 ank Experimental Setup Set point Disturbances Faults Model Predictive Controller Identified Model Parameters Inputs Process System Identification Outputs Online model parameter estimation: using Recursive Least Squares/ Pseudo-linear Regression Quadruple ank Experimental Setup at the,

30 AMPC of Quadruple ank System racking Performance AMPC uning Parameters Parameter Value Prediction Horizon(p) 1 Control Horizon(q) 5 1 Forgetting Factor Filter Coefficient (in recursive least squares) u.95m / hr 3% u( k k j) 3% yˆ( k k j) 4cm ACODS Relative Sensitivity Index racking Performance Model Sensitivity Matrix G( C( [I - ] Relative Sensitvity Index Gij ( Gij () % Sij ( 1 G () ij 1 u 119 ACODS

31 Example: Control of ennessee Eastman Problem E Problem: Objective Function Primary controlled variables: Product concentration of G Product Flow rate 9/18/216 State Feedback Control 121 9/18/216 State Feedback Control 122 E Problem: Operating Constraints E Problem: ransition Control Primary Controlled Outputs Managing large setpoint transitions needs either on-line model adaptations or use of nonlinear prediction models 9/18/216 State Feedback Control 123 9/18/216 State Feedback Control

32 E Problem: ransition Control E Problem: ransition Control Secondary Controlled Outputs Manipulated Inputs Adaptive Model Predictive Control is still an open research area. No commercial adaptive MPC is available yet E Problem: ransition Control Need for Nonlinear Control Manipulated Inputs Linear prediction model based MPC: limits applicability to small regions around operating point Real systems are nonlinear: use of linear controllers can generate sub-optimal performance Nonlinear MPC Need to achieve tight control of highly nonlinear systems Control of time varying (batch / semi-batch) systems Grade transition problems in polymer processing 9/18/216 State Feedback Control

33 Models for Nonlinear MPC (NMPC) Models for Nonlinear MPC First Principles / Phenomenological / Mechanistic / Grey Box Based on physics of the problem Energy and material balances hermodynamic models Conservation laws: conservation of charge Valid over wide operating range Provide insight in the internal working of systems Development and validation process: difficult and time consuming, requires a domain expert for development Data Driven / Black Box Models Dynamic models developed directly from inputoutput data Model Forms Nonlinear Difference Equations (NARX, NARMAX etc.) Artificial Neural Networks Limitations Valid over limited operating range Provide no insight into internal working of systems Development process: much less time consuming and comparatively easy Nonlinear MPC: Vendors NMPC: Applications (23) (Ref.: Qin and Badgwell, 23) (Ref.: Qin and Badgwell, 23)

34 Summary Current Research Directions Model Predictive Control Developing reliable nonlinear models capturing provides a coordinated approach to handling of multivariable interactions and operating constraints deal with control problems of non-square systems effects of unmeasured disturbances Incorporating robustness at design stage transparent way of tuning controller through objective Integrating fault diagnosis with MPC/NMPC function weights and rate limits to achieve desirable formulations closed loop performance Development of improved state estimation can handle nonlinear systems effectively Very flexible control tool for addressing wide variety of control problems schemes Embedding MPC / NMPC on a chip Current Research Directions References Fast NMPC for robotic and other fast applications like automobiles Improving MPC relevant optimization schemes: guaranteed convergence Coordinated MPC: Developing multiple MPC that cooperate and control a large system Stochastic MPC: Handling uncertainty in unmeasured disturbances and parameters Books with excellent material on LQOC and MPC Astrom, K. J. and B. Wittenmark, Computer Controlled Systems, Prentice Hall, 199. Camacho, E. C. and C. Bourdons, 1999, "Model Predictive Control", Springer Verlag, London. Franklin, G. F. and J. D. Powell, Digital Control of Dynamic Systems, Addison-Wesley, Goodwin, G., Graebe, S. F., Salgado, M. E., Control System Design, Phi Learning, 29. Glad,., Ljung, L. Control heory: Multivariable and Nonlinear Methods, aylor and Francis, 2. Sodderstrom,. Discrete ime Stochstic Systems. Springer, 23. Rawlings, J. B., Mayne, D. Q., Model Predictive Control: heory and Design, Nob Hill Publishing,

35 References References MPC and Related Important Review Articles Garcia, C. E., Prett, D. M., Morari, M. Model predictive control: heory and practice - A survey. Automatica, 25 (1989), Morari, M., Lee, J.H., Model Predictive Control: Past, Present and Future, Comp. Chem. Engg., 23 (1999), Henson, M.A. (1998). Nonlinear Model Predictive Control : Current status and future directions. Computers and Chemical Engg,23, Lee, J.H. (1998). Modeling and Identification for Nonlinear Model Predictive control:requirements present status and future needs. International Symposium on Nonlinear Model Predictive control,ascona, Switzerland. Meadows, E.S., Rawlings, J. B. Nonlinear Process Control, ( M.A. Henson and D.E. Seborg (eds.), New Jersey: Prentice Hall, Chapter 5.(1997). Qin, S.J., Badgwell,.A. A servey of industrial model predictive control technology, Control Engineering Practice 11 (23) Useful / Important Papers Muske, K. R., Rawlings, J. B. ; Model Predictive control with linear models, AIChE J., 39 (1993), Muske, K. R. ;Badgwell,. A. Disturbance modeling for offset-free linear model predictive control. Journal of Process Control, 12 (22), Ricker, N. L., Model Predictive Control with State Estimation, Ind. Eng. Chem. Res., 29 (199), Yu, Z. H., Li, W., Lee, J.H., Morari, M. State Estimation Based Model Predictive Control applied to Shell Control Problem: A Case Study, Chem. Eng. Sci., (1994), Patwardhan S.C. and S.L. Shah (25) From data to diagnosis and control using generalized orthonormal basis filters. Part I: Development of state observers, Journal of Process Control,15,7, References References Srinivas, K., Shaw, R., Patwardhan, S. C., Noronha, S. Adaptive model predictive control of multivariable timevarying systems. Ind. Eng. Chem. Res., 28, 47, Badwe, A., Singh, A., Patwardhan, S. C., Gudi. R. D., A Constrained Recursive Pseudo-linear Regression Scheme for On-line Parameter Estimation in Adaptive Control. Journal of Process Control, 2, , 21. Srinivasarao,M.; Patwardhan,S. C.; Gudi, R. D. Nonlinear predictive control of irregularly sampled multi-rate systems using nonlinear black box observers. Journal of Process Control, 27, 17, Srinivasrao, M.; Patwardhan, S.C. ; Gudi, R. D. From data to nonlinear predictive control. 2.. Improving regulatory performance using identified observers. Ind. Eng. Chem. Res., 26, 45, Prakash, J.; Patwardhan, S. C.;Narasimhan, S. Integrating model based fault diagnosis with model predictive control. Ind. Eng. Chem. Res., 25, 44, Patwardhan, S.C. ; Manuja, S.; Narasimhan, S.; Shah, S. L From data to diagnosis and control using generalized orthonormal basis filters. Part II: Model predictive and fault tolerant control. Journal of Process Control, 26, 16, Deshpande, A., Patwardhan, S. C., Narasimhan, S. Intelligent State Estimation for Fault olerant Nonlinear Model Predictive Control, Journal of Process Control, 19, ,