Parallel Compensator for Control of Multivariable Systems with Difficult Plants

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Parallel Compensator for Control of Multivariable
Systems with Difficult Plants

• Ryszard Gessing
• Silesian University of Technology Gliwice, Poland

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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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Introduction

• Smith predictor (compensator), 1958
• Deng, Iwai and Mizumoto 1999 for minimum phase
  plants
• Gessing ACC 2004 for nonminimum phase plants

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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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The case of SISO plant
is stable
Regulator
Uklad zamkniety
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Replacement plant
n-the plant order
where
Characteristic equation of the closed loop system
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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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The case of MIMO plant
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The multivariable system with PC
Decoupling of steady states
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Parallel compensator
-- replacement plant
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The choice of the replacement plant
In steady state we have
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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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Regulator
for
we obtain
or
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Closed loop (CL) system
Multiplying from the left by this matrix
we obtain
and finally
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Transfer function of the CL system
Oznaczmy
(common denominator of the i-th row of the matrix
)
Assuming
we obtain
The transients of the i-th output depend on zeros
of
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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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Design algorithm
1) Assume
2) Choose possibly small so that
where
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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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Example
The plant with 2-inputs and 2-outputs
The plant is nonminimum phase since
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Decoupling matrix
Decoupled plant
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Replacement plant
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Results of simulation for
For
the system is stable and the responses are
acceptable
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Control saturation
dla
a dla
For the assumed constraints
we obtain
for
and for
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Outline of Presentation

• Introduction
• Parallel Compensator (PC) for SISO Plants
• Parallel Compensator for MIMO Plants
• Closed Loop System Description
• Design Algorithm
• Example
• Final Conclusions

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Final conclusions

• By using PC the plant is decoupled
• The choice of the replacement plant for
  particular channels is performed independently
  and is simple as for SISO plant
• May be applied for difficult plants
• From examples it results that the system is
  robust, but it is to early to say about that
• The plant must be stable
• The regulator has a high order