Integration of Design and Control : Robust approach using MPC and PI controllers N. Chawankul, H. M. Budman and P. L. Douglas Department of Chemical Engineering University of Waterloo

Outline  Introduction  Objectives  Methodology  Case study  Results  Conclusions

Introduction Traditional Approach 1- Design: min (capital + operating costs) 2- Control for designed plant -stability -actuator constraints -performance specs: Small Overshoot Short Settling Time Large closed loop bandwidth Integrated Approach 1- Design+Control min (capital +operating +variability costs) st stability actuator constraints

Variability Cost Variability Cost = Cost of Imperfect Control For a disturbance d (green): What is the cost due to off- spec product (blue)? disturbance output

Robust Control Approach To test stability and calculate variability cost we need a model. Nonlinear model: stability (Lyapunov-difficult) variability (numerically- difficult). “Robust” linear model: nonlinear model= family of linear models family of linear models= nominal model +model uncertainty (error)

Nonlinear Dynamic Model (difficult optimization problem) Variability cost not into cost function: Multi-objective optimization Decentralized Control : PI /PID Linear Dynamic Nominal Model + Model Uncertainty (Simple optimization problem) Variability cost into cost function : One objective function Centralized Control : MPC Previous Approaches Our approach Introduction

Objectives of the current work  Variability using MPC based on a nominal model and model error.  Cost of variability in one objective function together with the design cost.  Model uncertainty (as a function of design variables) into the objective function.  The robust stability criteria as a process constraint.  Compare the traditional method to integrated method.  Preliminary study on SISO system (distillation column) with MPC.

Methodology Model Predictive Control (MPC) Nominal Step Response and Uncertainty Process Variability Optimization - Objective Function - Constraints

MPC Controller SdSd Process MPC d W y r u(k) k k+1 k+n k-1        k+2 k+3   target past future y(k) y(k+1/k) Simplified MPC block diagram u 

Nominal Step Response and Uncertainty t y 1 t S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 SnSn Step Response Model, S n Nominal step response model, S n,nom Uncertainty, Actual S n Upper bound Nominal step response Lower bound t S n -S n,nom u t u 1 0 0

u t +1 y t  u = +1  u = -1

Process variability-1 y = f(W) Process MPC W (Sinusoid unmeasured disturbance) y r= u Substitute  (k),  u(k-1) into the first equation and apply Z-transform

Process variability-2 Assume, W is sinusoidal disturbance with specific  d. (alternatively, superposition of sinusoids) With phase lag Consider worst case variability : disturbance output Amplitude of disturbance,W Amplitude of output,y

u is a vector of design variables. c is a vector of control variables. MinimizeCost(u,c) = Capital Cost + Operating Cost + Variability Cost u,c Such thath(u,c) = 0(equality constraints) g(u,c)  0(inequality constraints) Optimization

Constraints  h(u,c) = 0 (equality constraints) -steady state empirical correlations  g(u,c)  0 (inequality constraints) -manipulated variable constraint -robust stability

1.Manipulated variable constraint Inequality Constraints- 1 Consider the MPC controller gain, K MPC : where is a manipulated variable weight The infinity norm of A is the amplitude of the disturbance.

Inequality Constraints Robust stability constraint (Zanovello and Budman, 1999) LiLi Mp K mp c T1T1 T2T2 H N1 W1  W2 N2 Z -1 I N1 - M  (k+1/k)  u(k) U(k) U(k-1) Z(k)w(k) H H Block diagram of the MPC and the interconnection M-   Z -1 I U(k) U(k+1) M w z

Case study- Distillation Column Preliminary study: SISO system Feed = (propane) RR  Depropanizer column from Lee, 1994  adjust reflux ratio to control the mole fraction of propane in distillate Ethane Propane Isobutane N-Butane N-Pentane N-Hexane A MPCMPC XD*XD* + - Q

Process Model  RadFrac model in ASPEN PLUS  different column design, 19 – 59 stages  design variables (number of stages and column diameter) are functions of nominal RR Number of stages VS. RR The mathematic expressions of the process variables (N, D, Q) as functions of RR (Equality Constraints)

Input/Output Model  First Order Model Nominal step response Upper bound Lower bound Step change on RR by  10 % ( stages) y t S1S1 S2S2 S3S3 SnSn Dynamic simulation using ASPEN DYNAMICS 63.2 % (Equality Constraints)

Cost = CC(u) + OC(u) + VC(u,c)  Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day)  Operating cost, OC ($/day) where Q = reboiler duty (GJ/hr) OP = operating period (hrs) UC = Utility cost ($/GJ) Objective Function

 Variability cost, VC ($/day) - assume sinusoid unmeasured disturbance, W - disturbance induces process variability - consider a holding tank to attenuate the product variation - calculate the volume of the holding tank - calculate the loss due to the product held in the tank Variability Cost (Inventory cost) - 1 where P = product price, N = payoff period (10 years), i = interest rate (10%) and V = volume of the holding tank V1 V2

The required volume of the holding tank F in F out  C in  C out  The worst case variability: Distillation Column Feed disturbance A simple mass balance Holding V Variability Cost (Inventory cost) - 2 spec

Objective Function (-cont-) From the simple mass balance Assume Apply Laplace transform The volume of the holding tank  and define V F in F out  C in  C out  The worst case variability: Where B is the bound of the output.

Two different approaches Integrated Method Traditional Method Robust Performance (Morari, 1989)

Traditional method Step 1 Design Step 2 Control design Robust Performance test Problem Formulation-1 CC+OC Choose RR yes no RR * Choose , RR* yes no  *, * M M’  Robust Performance (Morari, 1989)

Integrated method Problem Formulation-2 TC = CC(RR) + OC(RR) + VC(RR,, ) Choose RR, No Choose 1. Maximum VC (RR,, ) ? 2. ? Calculate CC (RR) + OC(RR) Yes Calculate VC (RR,,) RR *,, 1. Minimum TC ? 2. and ? No Yes RR,

Results-1 Results from Integrated Method: W is a product price multiplier. WRR D (m)NCapital cost ($/day) Operating cost ($/day) Variability cost ($/day) Total cost ($/day)

Results-2 Comparison using Traditional and Integrated methods WTotal Cost of integrated method ($/day) Total Cost of traditional method ($/day) Saving ($/day)% Saving

Disturbance Process variability

IMC Control Controllerkpkckpkc II DD FF PID  Internal Model Control, IMC C(s) G(s) F(s) d r y’ y  IMC-based PID parameters for (Morari and Zafiriou, 1989) is used.

IMC Control  Internal Model Control, IMC C(s)G p (s) F(s) d r y’ y  IMC-based PID parameters for (Morari and Zafiriou, 1989) is used. G d (s) G p (s)

Results Comparison using both methods P Integrated MethodTraditional Method Saving ($/day) % Saving RR  c (sec) Total cost ($/day) RR  c (sec) Total cost ($/day)

Conclusions single objective function linear dynamic model + model uncertainty MPC variability cost is explicitly incorporated in the objective function integrated approach results in lower costs - savings can be significant; >13% for high value products On-going work: Formulate the MIMO problem with MPC

Acknowledgement Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)