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Modelling and Control of Nonlinear Processes

Modelling and Control of Nonlinear Processes. Jianying (Meg) Gao and Hector Budman Department of Chemical Engineering University of Waterloo. Outline. Motivation Modelling Volterra series State-affine Contro l G-S PI RS&RP. Motivation Nonlinear process examples

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Modelling and Control of Nonlinear Processes

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  1. Modelling and Control of Nonlinear Processes Jianying (Meg) Gao and Hector Budman Department of Chemical Engineering University of Waterloo

  2. Outline • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Motivation • Nonlinear process examples • Two major difficulties: modelling and control! • Empirical Modelling • Volterra series, state-affine • Robust Control • Robust Stability (RS) and Robust Performance (RP) • Proportional-Integral (PI) control • Gain-scheduling PI (G-S PI) • Results and Conclusions • Continuous stirred tank reactor (CSTR) • Future application

  3. Substrate Nonlinear Process Example 1 • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Fed-batch Bioreactor Mass Balance • Linear process: constant • Nonlinear process: Monod Input Output

  4. A A & B Cooling Heat Output Input Nonlinear Process Example 2 • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Continuous Stirred Tank Reactor: CSTR • Mass Balance: • Linear process: constant • Nonlinear process: Arrhenius

  5. Linear model nonlinear model uncertainty Nonlinear Control • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • 1st Difficulty: simple & accurate model • Accurate: the model gives a good data fit • Simple: the model structure is simple to apply for control purpose • 2nd Difficulty: model is never perfect! • Uncertainty: model/plant mismatch • Controllers are desired to be ROBUST to model uncertainty! • Robust control: takes into account uncertainty!

  6. 1g/l 1g/l Empirical Modelling • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP (inlet concentration) model v y e u process controller -y measurement soft sensor

  7. at different time t t-2 t t-1 at current time 1st Order Volterra Series • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • No priori knowledge of the process is required! • Black box model between input and output

  8. A Cooling Tc A & B 0 t 1 n 1 1st Order Volterra Series • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Impulse response • 1st order Volterra kernels: Output Input t

  9. + = + = 1st Order Volterra Series • Motivation • Modelling • Volterra series • State-affine • Control • Robust control • Gain-scheduling • PI • MPC Cooling Temperature Reactor concentration

  10. Volterra Series Model • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • No priori knowledge of the process is required! • Black box model between input and output • More terms, better data fit • 2nd order Volterra kernels:

  11. Identify Volterra Kernels • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Identification of Volterra kernels • Linear least squares

  12. Linear model uncertainty Volterra Series Model • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Advantages • From input/output data • Straightforward generalization of the linear system description • Linear least squares algorithm: • Disadvantages • The output depends on past inputs raised to different powers and in different product combinations, e.g. • Not suitable for robust control approach

  13. Least squares Sontag’s algorithm Volterra series model I/O Data State-affine model Intermediate step State-affine Model • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • State-affine Model (Sontag, 1978) • State-affine system, i.e. systems that are affine in the state variables but are nonlinear with respect to the inputs • It can cover a wide range of nonlinear processes • Identified from Volterra series model kernels • Suitable for robust control

  14. State-affine Model • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Model structure • Where • Identification of matrix coefficients • Iterative matrix manipulation of Volterra kernels • Sontag (1978), Budman and Knapp (2000,2001)

  15. Nonlinearity Nominal point Linear model uncertainty State-affine Model • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • A simple example • How to treat the nonlinearity as Uncertainty?

  16. 50 5 t Nonlinearity Uncertainty • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Uncertainty is function of input: Key advantage! • Uncertainty bounds

  17. State-affine model True process output 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 50 100 150 200 250 300 350 400 450 Results 1 (modelling) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP

  18. State-affine model True process output 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 50 100 150 200 250 300 350 400 450 Results 1 (modelling) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP

  19. Results 1 (modelling) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • State-affine model for CSTR

  20. Conclusions 1 (modelling) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • A general modelling approach is proposed! • For a Nonlinear Process: • Obtained an empirical model from I/O data! • No priori knowledge required! So it can be applied to processes with unknowndynamics! • Nonlinearityis dealt with as uncertainty! • Methods for quantifying the model uncertainty from experimental data are studied. 

  21. Slope= u e PI 0 t 0 t G-S PI Design • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • PI controller: • Proportional gain: • Integration time: • Gain-Scheduling PI controller

  22. A A & B Output Linear Model 2 Linear Model 1 Linear Model 3 ? ? switch switch Traditional Gain-Scheduling • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP 5 50

  23. G-S PI Design • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Continuous G-S PI controller: state-space • Design parameters: Math manipulation

  24. Closed-loop System: APS • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Affine Parameter-dependent System: the closed-loop • Assumption1:Affine dependence on the uncertain parameters

  25. 100 5 t W Uncertain Parameter • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Assumption 2: Each uncertain parameter is bounded • Convexity: Parameter vector is valued in a hyper-rectangle called the parameter box

  26. Robust Stability • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Lyapunov function • Energy 0 • Stable position: zero energy • Path:

  27. Min W Robust Stability • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • CSTR: avoid overheating, maintain target • General RS condition: Max stable unstable

  28. e 1g/l Robust Performance • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Disturbance Rejection; performance index • Smaller , better performance • Larger , worse performance  =Disturbance in A A & B Output

  29. Max Min Robust Performance • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Fed-batch bioreactor: product quality! • RP: Solve for and controller parameters Good Bad

  30. uncertainty linear model controller closed-loop RS ? RP ? Robust Control Design • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Empirical model of the nonlinear process • State-affine model • Controller structure • PI: • G-S PI: • Closed-loop system • RS and RP conditions are checked

  31. 20 18 16 RS region 14 12 RP region 10 8 6 4 2 0.5 1 1.5 2 2.5 3 3.5 Results 2 (Linear PI) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Linear PI: RS and RP regions Good

  32. 0.5 Good 0 -0.5 -1 -1 -0.5 0 0.5 1 1.5 Results 2 (G-S PI RS) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Improve over linear PI Kc=2.42,taui=1.1545

  33. 0.5 Good 0 -0.5 -1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Results 2 (G-S PI RP) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Improve over linear PI Kc=2,taui=1.1545

  34. Results 2 (PI) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Simulation: G-S PI is much BETTER! 1 Disturbance 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 1 Linear PI 0.5 0 G-S PI:better -0.5 -1 2 4 6 8 10 12 14 16 18 20

  35. Conclusions 2 (Control) • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Design and simulation results • is a good performance index • Consistence between analysis and simulation • A general robust design approach is proposed! • Based on empirical model from I/O data! • G-S PI! Much better performance for wide operation range!

  36. Conclusions • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Two difficulties are solved efficiently! • Modelling: state-affine • Control: robust control • Our contributions! • Quantify uncertainty from I/O data! • Develop global RP conditions! • Propose Continuous G-S PI structure!

  37. Application • Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP • Empirical Modelling • Models: nonlinear chemical and biochemical processes • Robust Control Design • Nonlinear processes when nonlinearity is treated as uncertainty! • Uncertain processes with real and time-varying uncertainty!

  38. Motivation • Modelling • Volterra series • State-affine • Control • G-S PI • RS&RP Thank You! Any Questions?

  39. Unconstrained MPC • p: prediction horizon;m: control horizon • k: sampling point, same as t K+m

  40. Unconstrained MPC 0 0 • Quadratic Design Objective • Solution [1] [2]

  41. Unconstrained MPC • Least Squares Solution • MPC Solution: Best Sequence of m Control Moves • Present Control Move is Implemented

  42. MPC Design Parameters • No general guide on design parameters! • Control horizon: m • Small: a robust controller that is relatively insensitive to model errors • Large: computational effort increases; excessive control action • Prediction horizon: p • Large: more conservative control action which has a stabilizing effect but also increase the computational effort • weighting matrix for outputs : • Usually set

  43. State-space MPC • Weighting matrix for inputs: • More important than other parameters • Small more aggressive control, less stable • Large less aggressive control, more stable • State-space MPC (Zanovello and Budman,1999) • Closed-loop System: APS

  44. Robust G-S MPC Operation Range 1(u=-1) 2(u=0) 3(u=1) Step 1 State Affine 1&2 State Affine 2&3 Step 2 RS & RP MPC 1-2 MPC 2-3 Step 3 Step 4 Switching

  45. Outline • Traditional G-S Design • Design procedure and disadvantages • Robust G-S Design • Affine parameter-dependent systems (APS) • RS and its LMI formulation • RP and its LMI formulation • Robust G-S MPC design • State-affine model and uncertainty quantification • State-space formulation of MPC • Results and Conclusions • Case study: nonlinear CSTR

  46. Nonlinear CSTR Pure A Mix of A and B Q , C , T f f f Q , C , T f r V , , T u(t)=Tc • 1st –order exothermal reaction 1st –order exothermal reaction

  47. Results (1) • Optimization Design Results: Table 1 • Evenly Separated Ranges

  48. Conclusions (1) • Efficient Robust G-S MPC Design • Simulation test with disturbances, e.g. IMA, and etc. • Global G-S MPC designed with guaranteed RS and RP • Analysis is the worst case which covers all the simulations • Observations of the Robust G-S MPC Design • Performance index close to each other • Even separations may not capture the process nonlinearity • Conservatism of the design • Robust G-S MPC Performance depends on • # of separations • Separation point locations: evenly or not • Nonlinear dynamics

  49. Results (2) • Comparison of two controllers: Analysis & Simulation • G-S MPC 5-1: designed based on optimization • G-S MPC 5-2: chosen randomly • Table 2

  50. Results (2) • CSTR Simulation: Figure 1

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