Computational Approach for Adjudging Feasibility of

Acceptable Disturbance Rejection

Vinay Kariwala and Sigurd SkogestadDepartment of Chemical Engineering

NTNU, Trondheim, Norway

skoge@chemeng.ntnu.no

2

Outline

• Problem Formulation

• Previous work

• L1 - optimal control approach (Practical)

• Case studies

• Branch and bound (Theoretical)

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

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

Mathematical FormulationLinear time-invariant systems

Skogestad and Wolff (1992), Hovd, Ma and Braatz (2003)

• Achievable with

• Minimal to have

• Largest with ,

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

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

Disturbance Rejection using Feedback

Minimax formulation• even non-causal• Scales poorly

Theoretical!

FeedbackExplicit controllerComputationally attractive

Practical!

Feedback approach also provides– Upper bound for minimax formulation

9

Optimal for rational, causal, feedback-based linear controller

Feedback

YoulaParameterization

Annn approach

a - optimalControl

10

Annn approach: Steady-stateConversion of to standard LP

• Vectorize as

• Equivalent problem (simple algebra)- Bound each element :

- Sum of rows of :

• Standard linear program

11

Annn approach : Frequency-wise

Polyhedral approximation(Hovd and Kookos, 2005)

Semi-definite programStill Convex!

Used in approach

Absolute value of complex number is non-linear

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 (NQ) until convergence– Standard LP (same as steady-state) with

constraints

variables

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

Example 1: Blown Film Extruder

• - circulant, steady-state• is parameterized by (spatial correlation)• Hovd, Ma and Braatz (2003) - bilinear formulation

aaann(Feedback)

Achievable Output ErrorBilinear

(Non-Causal)Case

0.783

0.894

0.382

0.783

0.935

0.409

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

Example 3: Dynamic system

• Interpolation constraint: – Useful for avoiding unstable pole-zero cancellation– Explicit consideration unnecessary as u is bounded

Input bound

Unstable zeroTime delay

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

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

Computational Approach for Adjudging Feasibility of

Acceptable Disturbance Rejection

Vinay Kariwala and Sigurd SkogestadDepartment of Chemical Engineering

NTNU, Trondheim, Norway

kariwala,skoge@chemeng.ntnu.no