Computational Approach for
Adjudging Feasibility of
Acceptable Disturbance Rejection
Vinay Kariwala and Sigurd Skogestad
Department of Chemical Engineering
NTNU, Trondheim, Norway
[email protected]
Outline
• Problem Formulation
• Previous work
• L1 - optimal control approach (Practical)
• Case studies
• Branch and bound (Theoretical)
2
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
3
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)
4
Mathematical Formulation
Linear time-invariant systems
Skogestad and Wolff (1992), Hovd, Ma and Braatz (2003)
• Achievable
• Minimal
• Largest
with
to have
with
,
5
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
6
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
7
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
8
Annn approach
Feedback
Youla
Parameterization
a - optimal
Control
Optimal for rational, causal, feedback-based linear controller
9
Annn approach: Steady-state
Conversion of
• Vectorize
to standard LP
as
• Equivalent problem (simple algebra)
- Bound each element :
- Sum of rows of
:
• Standard linear program
10
Annn approach : Frequency-wise
Absolute value of complex number is non-linear
Polyhedral approximation
(Hovd and Kookos, 2005)
Semi-definite program
Still Convex!
Used in
approach
11
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
12
Summary
approach:
• Exact solutions for practical (feedback) cases
– Steady-state
– Frequency-wise
– Dynamic Case (discrete time)
• Upper bound for minimax (non-causal)
formulation
13
Example 1: Blown Film Extruder
•
- circulant, steady-state
•
is parameterized by
(spatial correlation)
• Hovd, Ma and Braatz (2003) - bilinear formulation
Case
Achievable Output Error
Bilinear
aaann
(Non-Causal) (Feedback)
0.783
0.783
0.894
0.935
0.382
0.409
14
Example 2: Fluid Catalytic Cracker
• Process:
transfer matrices
• Steady-state: Perfect control possible
• Frequency-wise computation
Non-Causal (Hovd and Kookos, 2005)
Upper bound
Lower bound
(upper bound on solution using minimax formulation)
15
Example 3: Dynamic system
• Interpolation constraint:
– Useful for avoiding unstable pole-zero cancellation
– Explicit consideration unnecessary as u is bounded
Input bound
Time delay
Unstable zero
16
Branch and Bound
•
approach provides practical solution
• Minimax formulation – theoretical interest
• 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
• Blown film extruder (16384 options for d)
– Optimal solution by resolving 6, 45 and 47 nodes
17
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
18
Computational Approach for
Adjudging Feasibility of
Acceptable Disturbance Rejection
Vinay Kariwala and Sigurd Skogestad
Department of Chemical Engineering
NTNU, Trondheim, Norway
kariwala,[email protected]