Wathanyoo Khaisongkram David Banjerdpongchai

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MATLAB Based GUIs for Linear Controller
Design via Convex Optimization
Wathanyoo Khaisongkram David
Banjerdpongchai Department of Electrical
Engineering, Chulalongkorn University
2002 American Control Conference (ACC02) May 8,
2002
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Main Topics

• Introduction

• Research Objective

• Convex Design Method

• MATLAB Based GUIs

• Numerical example on flexible robot arm

• 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)

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

• Achieve global solution or indicate if no
  solution exists
• Provide information on limits of performance

But, object-oriented CAD tools 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 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 flexible arm

• 4th order model by Finite Element Method (FEM)
• Load mass 0.1 kg.
• Load position 0.45 m.

MATLAB Based GUIs for Linear Controller Design
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Design specifications

• Design objective
• Minimize settling time

• Design constraints (Partial)
• Maximum overshoot less than 10
• No undershoot
• Peak of TF. magnitude less than 1 dB
• Peak of control signal less than 5 Volts

MATLAB Based GUIs for Linear Controller Design
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Design Results
48 iterations,
Without min. real. 24th order controller
With min. real. 5th order controller
MATLAB Based GUIs for Linear Controller Design
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MATLAB Based GUIs for Linear Controller Design
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Conclusion

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

• MATLAB GUIs provide a simpler way in specifying
  
• linear controller design

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

• Numerical example demonstrated the satisfactory
• linear controller for flexible arm

MATLAB Based GUIs for Linear Controller Design