Modelling and Control of Nonlinear Processes

1
Modelling and Control of Nonlinear Processes

• Jianying (Meg) Gao and Hector Budman
• Department of Chemical Engineering
• University of Waterloo

2
Outline

• 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

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

3
Nonlinear Process Example 1

• Fed-batch Bioreactor Mass Balance
• Linear process constant

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

• Nonlinear process Monod

Input
Output
4
Nonlinear Process Example 2

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

• Continuous Stirred Tank Reactor CSTR
• Mass Balance
• Linear process constant
• Nonlinear process Arrhenius

5
Nonlinear Control

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

• 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
Empirical Modelling

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

Type First principles model Empirical model
How? Mass, energy balance Input/output data
Choose? Difficult, complex Easy
(inlet concentration)
model
v
y
e
u
process
controller
-y
measurement
soft sensor
7
1st Order Volterra Series

• No priori knowledge of the process is required!
• Black box model between input and output

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

8
1st Order Volterra Series

• Impulse response
• 1st order Volterra kernels

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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

• No priori knowledge of the process is required!
• Black box model between input and output
• More terms, better data fit
• 2nd order Volterra kernels

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

11
Identify Volterra Kernels

• Identification of Volterra kernels
• Linear least squares

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

12
Volterra Series Model

• 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

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

13
State-affine Model

• 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

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

14
State-affine Model

• Model structure
• Where
• Identification of matrix coefficients
• Iterative matrix manipulation of Volterra kernels
• Sontag (1978), Budman and Knapp (2000,2001)

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

15
State-affine Model

• A simple example
• How to treat the nonlinearity as Uncertainty?

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

16
Nonlinearity Uncertainty

• Uncertainty is function of input Key advantage!
• Uncertainty bounds

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

17
Results 1 (modelling)

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

18
Results 1 (modelling)

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

19
Results 1 (modelling)

• State-affine model for CSTR

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Conclusions 1 (modelling)

• 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 unknown dynamics!
• Nonlinearity is dealt with as uncertainty!
• Methods for quantifying the model uncertainty
  from experimental data are studied. 

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

21
G-S PI Design

• PI controller
• Proportional gain
• Integration time
• Gain-Scheduling PI controller

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

22
Traditional Gain-Scheduling

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

5
50
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G-S PI Design

• Continuous G-S PI controller state-space
• Design parameters

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

Math manipulation
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Closed-loop System APS

• Affine Parameter-dependent System the
  closed-loop
• Assumption1Affine dependence on the uncertain
  parameters

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Uncertain Parameter

• Assumption 2 Each uncertain parameter is bounded
• Convexity Parameter vector
• is valued in a hyper-rectangle
• called the parameter box

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Robust Stability

• Lyapunov function
• Energy 0
• Stable position zero energy
• Path

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

27
Robust Stability

• CSTR avoid overheating, maintain target
• General RS condition

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

Max
stable
unstable
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Robust Performance

• Disturbance Rejection performance index
• Smaller , better performance
• Larger , worse performance

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

? Disturbance in A
A B
Output
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Robust Performance

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

• Fed-batch bioreactor product quality!
• RP Solve for and controller parameters

Good
Bad
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Robust Control Design

• Empirical model of the nonlinear process
• State-affine model
• Controller structure
• PI
• G-S PI
• Closed-loop system
• RS and RP conditions are checked

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Results 2 (Linear PI)

• Linear PI RS and RP regions

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

Good
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Results 2 (G-S PI RS)

• Improve over linear PI

Kc2.42,taui1.1545

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Results 2 (G-S PI RP)

• Improve over linear PI

Kc2,taui1.1545

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Results 2 (PI)

• Simulation G-S PI is much BETTER!

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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 PIbetter
-0.5
-1
2
4
6
8
10
12
14
16
18
20

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Conclusions 2 (Control)

• 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!

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

PI
Linear 2 1.15 0 0 0.96 0.38
G-S 1.37 2.95 -0.004 0.001 0.39 0.22
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Conclusions

• 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!

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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Application

• 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!

• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

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• Motivation
• Modelling
• Volterra series
• State-affine
• Control
• G-S PI
• RSRP

Thank You!
Any Questions?
39
Unconstrained MPC

• p prediction horizonm control horizon
• k sampling point, same as t

Km
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Unconstrained MPC

• Quadratic Design Objective
• Solution

1
2
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Unconstrained MPC

• Least Squares Solution
• MPC Solution Best Sequence of m Control Moves
• Present Control Move is Implemented

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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

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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

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Robust G-S MPC
Operation Range
1(u-1)
2(u0)
3(u1)
Step 1
State Affine 23
State Affine 12
Step 2
RS RP
MPC 2-3
MPC 1-2
Step 3
Step 4
Switching
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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

• 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

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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

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Results (2)

• CSTR Simulation Figure 1

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Conclusions (2)

• is a Good performance index
• Consistence between analysis and simulation
  (Table 2)
• Robust G-S MPC Design
• Global G-S MPC designed with guaranteed RS and RP
  
• Analysis is the worst case
• Future Directions
• Separation of operation range, of separations
• Reducing conservatism of the design
• Process with more nonlinearity

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Thank You!
Any Questions?
53
Traditional G-S Design

• A typical G-S design procedure for nonlinear
  plants
• Step1 select n operating points which cover the
  range of the plants dynamics
• Step 2 Linearize a first principle model or
  identify linear models around each operating
  point
• Step 3 Design local linear controller for each
  local linear model
• Step 4 In between operating points, the gains of
  the local controllers are interpolated, or
  scheduled, resulting in a global controller.