Wathanyoo Khaisongkram David Banjerdpongchai

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MATLAB Based GUIs for MIMO Linear Controller
Design
Wathanyoo Khaisongkram David
Banjerdpongchai Dept. of Electrical Engineering,
Chulalongkorn Univ., THAILAND
2002 Asian Control Conference (ASCC
2002) September 25-27, 2002
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Outline

• Introduction

• Research Objective

• Convex Design Method

• MATLAB Based GUIs

• Numerical example on distillation column

• Conclusion

MATLAB Based GUIs for Linear Controller Design
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Introduction

• Typical purposes in designing controllers
• Stability Performance
• Reference tracking
• Maximum overshoot, settling time, rise time
• Maximum magnitude of control signal

MATLAB Based GUIs for Linear Controller Design
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• Basic controller design methods
• Usually not guarantee time domain performances
• Require experiences in design (trial and error)
• Even more difficult for MIMO plant

Control design via convex optimization (Boyd et
al., 1988)

• Achieve global solution or indicate if no
  solution
• Provide information on limits of performance
• Applicable to MIMO systems

MATLAB Based GUIs for Linear Controller Design
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• But,
• object-oriented CAD tool were not
• available for common users
• resulting controllers usually have
• high order

MATLAB Based GUIs for Linear Controller Design
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Research Objective

• To develop MIMO control design tools with
• graphical user interfaces for common
  users
• (control engineers)

• Benefits
• A simple way of specifying control design
• More accessibility for general users

MATLAB Based GUIs for Linear Controller Design
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Standard form of control system
Hzw Pzw PzuK(IPyuK)-1Pyw
MATLAB Based GUIs for Linear Controller Design
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Convex function of Hzw
s(t), h(t) are linear functions of Hzw Convex on
s(t), h(t) Convex on Hzw
MATLAB Based GUIs for Linear Controller Design
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Convex Design Method
MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Design Specifications
Step response constraints

• Maximum overshoot and undershoot
• Settling time and rise time
• Upper and lower bound

Other constraints

• Peak magnitude of control signal
• Peak magnitude of transfer function (H -norm)
• RMS response

MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Control system reformation
Compute Pzw, Pzu, Pyw, Pyu
MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Youla Parameterization (Youla, 1976)
Hzw Pzw PzuK (IPyuK)-1Pyw looks complicated
Then Hzw T1 T2QT3

• Hzw affine in Q
• Convex function of Hzw is a convex function
  of Q

MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Ritz Approximation (Courant Hilbert, 1953)
Hzw T1 T2QT3 Infinite dimension
Let QN be a finite dimensional estimator of Q
QN x1Q1 x2Q2 . . . xNQN
then
HN R0 x1R1 x2R2 . . . xNRN
MATLAB Based GUIs for Linear Controller Design
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Control system reformation
Youla parameterization
Ritz approximation
MATLAB Based GUIs for Linear Controller Design
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Solving the Design Problem
MATLAB Based GUIs for Linear Controller Design
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Ellipsoid Algorithm

• Simple implementation
• Computational complexity

• Initial ellipsoid must contain a minimizer

MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Minimal realization

• Resulting controllers
• usually have rather
• high order
• Minimal realization
• yields reduced order
• with same performance

MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs
Objective provide more accessibility to common
users (control engineers)

• Basic knowledge required
• Easy to operate

• Advantages
• Familiar object-oriented system
• MATLAB is widely used in control

MATLAB Based GUIs for Linear Controller Design
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Program Structure
MATLAB Based GUIs for Linear Controller Design
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Numerical example on distillation column

• 14th order 2 inputs
• 2 outputs linearized
• plant
• augmenting integrator
• results in 16th order
• plant
• using 2nd-order pade
• approximation

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Design specifications

• Design objective
• Minimize settling time from r2 to y2

• Design constraints (Partial)
• Maximum signal peak from r2 to y1 lt 0.5
• Maximum signal peak from r1 to y2 lt 0.5
• Peak magnitude of each control signals lt 0.5
• Lower bound-upper bound constraints
• on step responses from r1 to y1 and r2 to y2

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Design Results
76 iterations,
Without min. real. 21th order controller
With min. real. 15th order controller
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MATLAB Based GUIs for Linear Controller Design
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Conclusion

• MIMO Linear controller design can be cast as
  convex
• optimization problem

• MATLAB GUIs provide a simpler way in specifying
  
• MIMO linear control design problem

• Including minimal realization in the CAD tools
• enhances the usefulness of the design
  procedure

• Numerical example demonstrated the satisfactory
  MIMO
• linear controller for approximated
  distillation column

MATLAB Based GUIs for Linear Controller Design