1. Computational Approach for Adjudging Feasibility of Acceptable Disturbance Rejection Vinay Kariwala and Sigurd Skogestad Department of Chemical Engineering NTNU, Trondheim, Norway skoge@chemeng.ntnu.no

2. 2 Outline Problem Formulation Previous work L 1 - optimal control approach (Practical) Case studies Branch and bound (Theoretical)

3. 3 Process Controllability Analysis Ability to achieve acceptable control performance Limited by plant itself, Independent of controller Useful for finding How well the plant can be controlled? What control structure should be selected? –Sensors, Actuators, Pairing selection What process modifications will improve control? –Equipment sizing, Buffer tanks, Additional sensors and actuators

4. 4 Disturbance Rejection Measure Is it possible to keep outputs within allowable bounds for the worst possible combination of disturbances, while keeping the manipulated variables within their physical bounds? Flexibility (e.g. Swaney and Grossman, 1985) Disturbance rejection measure (Skogestad and Wolff, 1992) Operability (e.g. Georgakis et al., 2004)

5. 5 Mathematical Formulation Linear time-invariant systems Skogestad and Wolff (1992), Hovd, Ma and Braatz (2003) Achievable with Minimal to have Largest with,

6. 6 Previous Work Steady-state: Hovd, Ma and Braatz (2003) –Conversion to bilinear program using duality –Solved using Baron Kookos and Perkins (2003) –Inner minimization replaced by KKT conditions –Integer variables to handle complementarity conditions

7. 7 Previous Work Frequency-wise solution: Skogestad and Postlethwaite (1996) –SVD based necessary conditions Hovd and Kookos (2005) –Absolute value of complex number is non-linear –Bounds by polyhedral approximations

8. 8 Disturbance Rejection using Feedback Minimax formulation even non-causal Scales poorly Theoretical! Feedback Explicit controller Computationally attractive Practical! Feedback approach also provides –Upper bound for minimax formulation

9. 9 Optimal for rational, causal, feedback-based linear controller Feedback Youla Parameterization Annn approach a - optimal Control

10. 10 Annn approach: Steady-state Conversion of to standard LP Vectorize as Equivalent problem (simple algebra) - Bound each element : - Sum of rows of : Standard linear program

11. 11 Annn approach : Frequency-wise Polyhedral approximation (Hovd and Kookos, 2005) Semi-definite program Still Convex! Used in approach Absolute value of complex number is non-linear

12. 12 Annn approach : Dynamic System Continuous-time formulation –Difficult to compute -norm –Formulation using bounds - highly conservative Discrete-time formulation –Finite impulse response models of order N –Increase order of Q (N Q ) until convergence –Standard LP (same as steady-state) with constraints variables

13. 13 Summary approach: Exact solutions for practical (feedback) cases –Steady-state –Frequency-wise –Dynamic Case (discrete time) Upper bound for minimax (non-causal) formulation

14. 14 Example 1: Blown Film Extruder - circulant, steady-state is parameterized by (spatial correlation) Hovd, Ma and Braatz (2003) - bilinear formulation aaann (Feedback) Achievable Output Error Bilinear (Non-Causal) Case 0.783 0.894 0.382 0.783 0.935 0.409

15. 15 Example 2: Fluid Catalytic Cracker Process: transfer matrices Steady-state: Perfect control possible Frequency-wise computation Upper boundNon-Causal (Hovd and Kookos, 2005) Lower bound (upper bound on solution using minimax formulation)

16. 16 Example 3: Dynamic system Interpolation constraint: –Useful for avoiding unstable pole-zero cancellation –Explicit consideration unnecessary as u is bounded Input bound Unstable zero Time delay

17. 17 Branch and Bound Blown film extruder (16384 options for d) –Optimal solution by resolving 6, 45 and 47 nodes Exact solution using branch and bound –Branch on –Upper bound using approach –Tightening of upper bound using divide and conquer –Lower bound using worst-case d for approach approach provides practical solution Minimax formulation – theoretical interest

18. 18 Conclusions Disturbance rejection measure Minimax formulation –Theoretical interest – can be non-causal –Previous work – scales poorly approach –Practical controllability analysis –Exact solutions for steady-state, frequency-wise and dynamic (discrete time) cases –Computationally efficient Efficient theoretical solution using Branch and bound

19. Computational Approach for Adjudging Feasibility of Acceptable Disturbance Rejection Vinay Kariwala and Sigurd Skogestad Department of Chemical Engineering NTNU, Trondheim, Norway kariwala,skoge@chemeng.ntnu.no