Integration of Design and Control A Robust Control

Integration of Design and Control: A Robust Control Approach by Nongluk Chawankul Peter L. Douglas Hector M. Budman University of Waterloo, Ontario, Canada

Background Control performance depends on the controller and the design of the process. Traditional design procedure: Step 1: Process design (sizing + nominal operating conditions) Step 2: Control design Idea of integrating design and control: Process design + Process control = Integrated approach

Background Traditional design and control design Two steps design Step 1: Process design Cost = Capital cost(x) + operating cost(x) where Min x is design variable h(x) = 0 g(x) 0 Optimum design Step 2: Control design Objective Function (Cost) Cost = capital cost(x) + operating cost(x) + cost related to closed loop system(x, y) x is design variable y is control tuning parameter Process constraints Equality constraints, h(x) = 0 Inequality constraints, g(x, y) 0 Min Design controller Optimum design Only one step design where Cost(x) x s. t. Integrated design and control design Cost(x, y) x, y Closed loop system s. t. h(x) = 0 g(x, y) 0

Integrated Design and Control Design Previous studies • Nonlinear Dynamic Model (difficult optimization problem) • Variability cost not into cost function: Multi-objective optimization • Decentralized Control : PI /PID Our study • Linear Nominal Model + Model Uncertainty (Simple optimization problem) • Variability cost into cost function : One objective function • extended to Centralized Control : MPC

Case study • Case study II: SISO MPC • Case study III: MIMO MPC SISO system MIMO system XD RR Feed Ethane Propane Isobutane N-Butane N-Pentane N-Hexane XD RR A + IMC or MPC A 1 + - XD* Feed Ethane Propane Isobutane N-Butane N-Pentane N-Hexane Q XB - XD* MPC Q + - XB * A 2 XB

Case Study III: MIMO MPC MIMO case study: Rad. Frac model in ASPEN PLUS was used. Different column designs, 19 – 59 stages were studied. Product specifications Mole fraction of propane in distillate product = 0. 783 Mole fraction of isobutane in bottom product = 0. 1 Design variables are functions of nominal RR at specific product compositions.

Optimization Objective Function Minimize Cost(U, C) = CC(U) + OC(U) + max VC(U, C) U, C Such that Lm h(U) = 0 g(U, C) 0 (equality constraints) (inequality constraints) U is a vector of design variables. C is a vector of control variables. Lm is a set of uncertainty.

Objective Function

Capital Cost (CC) and Operating Cost (OC) • Capital Cost, CC – Cost of sizing, e. g. number of stages N and column diameter D – Capital cost for distillation column from Luyben and Floudas, 1994 ($/day) • Operating Cost, OC – Operating cost from Luyben and Floudas, 1994 ($/day) where tax = tax factor HD = reboiler duty (GJ/hr) OP = operating period (hrs) UC = Utility cost ($/GJ)

Variability Cost (VC) Variability Cost, VC - Variability cost, VC = inventory cost - sinusoid disturbance induces process variability - consider holding tank to attenuate the product variation t V 1 RR t A 1 + t Feed Ethane Propane Isobutane N-Butane N-Pentane N-Hexane - XD* MPC Q + - XB * A 2 V 2 t t

Calculation of Variability Cost (VC) - 1 Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids) With phase lag Consider worst case variability : Related to maximum VC

Calculation of Variability Cost (VC) - 2 Objective Function (-cont-) From column To customer Cin Cout V Q in Q out Apply Laplace transform The product volume in the holding tank VC 1 = W 1 P 1 V 1 (A/P, i, N) VC 2 = W 2 P 2 V 2 (A/P, i, N) VC = VC 1+ VC 2

Equality Constraints

Equality Constraints: Process models -1 Process Models: ASPEN PLUS simulations at specific product compositions Q(RR) N(RR) D(RR) HD(RR)

Equality Constraints: Process models - 2 Process Models: ASPEN PLUS simulations BF(RR) DF(RR)

Equality Constraints: Process models - 3 Process Models: Input/Output Model for 2 2 system yi Sn First Order Model S 1 S 2 S 3 t Process gains y and -35% +35% y 1 -1% +1% 0 1% y 2 time In a similar fashion, time constants and dead time p(RR) and p(Q) (RR) Kp 1(RR) for paring x. D-RR Kp 2(RR) for paring x. B-RR Kp 3(Q) for paring x. D-Q Kp 4(Q) for paring x. B-Q

Equality Constraints: Process models - 4 Process gains for 2 2 system

Equality Constraints: Process models - 5 Process time constants: p(RR) and p(Q) Process dead time: (RR)

Equality Constraints: Process models - 6 Model uncertainty y Sn, upper x. D-RR x. B-RR Sn, nom Sn, lower x. D-Q x. B-Q Time

Equality Constraints: Process models - 7 Model Uncertainty for 2 2 system

Inequality Constraints

Inequality Constraints- 1 1. Manipulated variable constraint is a tuning parameter. Large less aggressive control Two manipulated variables Calculate RR and Q and

Inequality Constraints- 2 2. Robust stability constraint (Zanovello and Budman, 1999) Block diagram of the MPC and the connection matrix M U(k( (k+1/k( + - Li Kmp u(k( ++ T 1 W 1 Z-1 I T 2 U(k-1( H Z(k( w(k( N 1 M W 2 ++ c ++ Mp H H N 1 -+ N 2 Z-1 I U(k+1( U(k( M z w

Two different approaches Traditional Method Robust Performance (Morari, 1989) Integrated Method Where U is manipulated variables

Results

Results 1 Method RR* l* CC ($/day) OC ($/day) VC ($/day) TC ($/day) Integrated 1. 91 0. 23 196. 1 586. 2 23. 34 805. 5 Traditional 1. 92 0. 5 195. 5 586. 6 65. 6 847. 8 Smaller RR* smaller delay smaller interaction easier to control

Results - 2 Results from Integrated design and control design approach w 1 w 2 RR* * 11 N D (m) 1 1 1. 913 0. 2350 3. 65 26 3. 392 5 1 1. 911 0. 2341 3. 63 26 3. 392 10 1 1. 908 0. 2338 3. 62 27 3. 391 15 1 1. 753 0. 2331 3. 03 38 3. 370 20 1 1. 753 0. 2332 3. 03 38 3. 370 1 5 1. 912 0. 1886 3. 64 26 3. 392 1 10 1. 909 0. 1848 3. 62 26 3. 391 1 15 1. 906 0. 1836 3. 61 27 3. 391 1 20 1. 904 0. 1830 3. 60 27 3. 390 w 1 or w 2 increases; -RR* decreases smaller dead time - 11 decreases interaction decreases as RR decreases - * decreases RS constraint is easy to satisfy as 11 decreases

Results - 3 Compare Results from Traditional and Integrated design and control design approaches. Why? In the traditional method the RR is determined only once by the minimization of CC and OC and does not change with product price as in the integrated approach!

Results - 4 Effect of RRmax on Total Cost (TC) RR max RR* * N D (m) CC ($/day) OC ($/day) VC ($/day) TC ($/day) 2. 633 1. 913 0. 2350 26 3. 392 195. 98 586. 21 23. 34 805. 53 2 1. 913 0. 2349 26 3. 392 195. 98 586. 21 23. 34 805. 53 1. 9 1. 854 0. 2024 38 3. 370 259. 76 570. 48 72. 62 902. 86 1. 8 1. 761 0. 2021 39 3. 370 262. 60 570. 20 93. 04 925. 84

Conclusions 1 - For the case ≠ 0, using the integrated method, the optimization tends to select smaller RR values which correspond to smaller dead time and smaller interaction. 2 - The optimal design obtained using the integrated method resulted in a lower total cost as compared to the traditional method. 3 - Limit on manipulated variable affects the closed loop performance and leads to more cost.

Calculation of Variability Cost (VC) -1 Process variability W (Sinusoid unmeasured disturbance) r=0 +- MPC u Process + + y Substitute (k), u(k-1) into the first equation and apply z-transform

Results - 1 Results from Integrated design and control design approach for = 0 w 1 w 2 RR* * 11 N D (m) 1 1 1. 921 0. 2503 3. 68 26 3. 393 5 1 1. 955 0. 2509 3. 83 26 3. 399 10 1 1. 980 0. 2518 3. 95 25 3. 404 15 1 2. 012 0. 2521 4. 11 25 3. 410 20 1 2. 103 0. 2527 4. 63 24 3. 431 1 5 1. 960 0. 2511 3. 86 26 3. 400 1 10 1. 992 0. 2516 4. 01 25 3. 406 1 15 2. 102 0. 2522 4. 63 24 3. 431 1 20 2. 234 0. 2540 5. 57 23 3. 465 w 1 or w 2 increases; -RR* increases uncertainty decreases as RR increases - 11 increases interaction increases as RR increases - * increases RS constraint is more difficult to satisfy as 11 increases

Results - 4 Compare savings when = 0 and 0 =0 0