A Survey of Some Sliding Mode Control Designs Dennis Driggers EE691 March 16, 2006

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A Survey of Some Sliding Mode Control
DesignsDennis DriggersEE691March 16, 2006
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Overview

• Most types of system control techniques
  incorporate some type of disturbance waveform
  modeling. Even if the disturbance waveform is
  completely unknown, a disturbance
  characterization of the waveform is assumed.
  This assumption is usually made on a worst case
  basis to insure stability of the targeted system.
  

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• Classical and Modern Control theory incorporates
  waveform characterization of disturbances with
  and without waveform structure. Modern control
  theory is centered around modeling the
  disturbance to either completely reject,
  minimize, or to even utilize the disturbance in
  controlling system behavior. In all of these
  circumstances it is necessary to model the
  waveform.

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Some Waveform Models used in Modern Control Design
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Introduction to Sliding Mode Control

• Sliding Mode Control does not require a
  disturbance waveform characterization to
  implement the control law. The main advantage of
  Sliding Mode Control (SMC) is the robustness to
  unknown disturbances. Required knowledge of the
  disturbance is limited to the disturbance
  boundary. Traditional SMC was, however, limited
  by a discontinuous control law. Depending on the
  plant dynamics, high frequency switching may or
  may not be an issue to contend with. There are
  techniques to limit and eliminate the
  high-frequency switching associated with
  traditional SMC. It is the intent of this paper
  to look at several SMC techniques utilizing an
  aircraft model with bounded external disturbances.

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Agenda

• Background of SMC
• Definitions
• SMC Design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Agenda

• Background of SMC
• Definitions
• SMC Design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Background

• Sliding Mode Control (SMC) theory was founded
  and advanced in the former Soviet Union as a
  variable structure control system.
• SMC is a relatively young control concept dating
  back to the 1960s.
• SMC theory first appeared outside Russia in the
  mid 1970s when a book by Itkis (1976) and a
  survey paper by Utkin (1977) were published in
  English.
• The SMC reachability condition is based on the
  Russian mathematician, Lyapunov, and his theory
  of stability of nonlinear systems.

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Agenda

• Background of SMC
• Definitions
• SMC Design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Definitions

• State Space An n-dimensional space whose
  coordinate axes consist of the x1 axes,x2
  axis,,xn axes.
• State trajectory- A graph of x(t) verses t
  through a state space.
• State variables The state variables of a system
  consist of a minimum set of parameters that
  completely summarize the systems status.
• Disturbance Completely or partially unknown
  system inputs which cannot be manipulated by the
  system designer.

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Definitions

• Sliding Surface A line or hyperplane in
  state-space which is designed to accommodate a
  sliding motion.
• Sliding Mode The behavior of a dynamical system
  while confined to the sliding surface.
• Signum function (Sign(s))
• Reaching phase The initial phase of the closed
  loop behaviour of the state variables as they are
  being driven towards the surface.

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Agenda

• Background of SMC
• Definitions
• SMC design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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SMC Design MethodologyThree Basic Steps

• Design a sliding manifold or sliding surface in
  state space.
• Design a controller to reach the sliding surface
  in finite time.
• Design a control law to confine the desired state
  variables to the sliding manifold.

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SMC Graphical Illustration

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Agenda

• Background of SMC
• Definitions
• SMC design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Aircraft Modeled Parameters

• Simplified aircraft model consist of angle of
  attack, aircraft pitch rate, and elevator
  deflection represented as a ,q, and de.
• Aircraft parameters for a particular
  airframe at a particular attitude and altitude.
• Changes in airframe due to damage
  (unknown, uncertain, and bounded)
• Horizontal tail and rudder areas.
• Flight profile filters.

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Aircraft and Disturbance Models used in
Simulations
where
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Derivations for Traditional SMC

• It is necessary to find the relative degree of
  the system in state-space. Relative degree, ,
  is determined by the number of times the output
  has to be differentiated before any control input
  appears in its expression.
• The aircraft model in scalar format is
• The relative degree of the plant is 3 as the
  control u appears as follows

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Sliding Surface Design

• The sliding manifold is formulated as
• where
• then .
• and are deigned to make the dynamic
  sliding surface stable. This is achieved by
  making the equation Hurwitz stable. The equation
  from the ITAE tables for a 2nd order system is
• and for a then C1 and C2 are 14 and
  100 respectively.

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Derivation for reaching phase

• To guarantee an ideal sliding motion the
  ?-reachability condition must be met and is
  given by

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Reaching Phase Design

• Introduce a Lyapunov function candidate.
• 
  
• The derivative of the Lyapunov function is
• The initial conditions are given as
• and
  .
• Desire seconds, then

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SMC Controller Design

• The controller can be implemented with the signum
  function as follows

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Simulink Diagram for Traditionial SMC
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Supertwist Design

• It has been shown (not in this brief) that the
  solution to the following differential equation
• 
  
• and its derivative converge to zero in finite
  time if
• , , and
  .
• On this basis u is introduced as

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Supertwist Design

• Supertwist utilizes the same sliding surface and
  values as the traditional SMC. The signum control
  function is replaced with the function
• The values for L1.5 are

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Supertwist Block Diagram
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SMC Observer Design
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SMC Observer Design
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Disturbance Observer Block Diagram
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Agenda

• Background of SMC
• Definitions
• SMC design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Disturbances
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Phase Diagramof the Sliding Surface
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Traditional SMC
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Supertwist
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SMC Observer
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Agenda

• Background of SMC
• Definitions
• SMC design Methodology
• Derivations
• Traditional SMC
• Supertwist
• SMC driven by SMC observer
• Simulation Results
• Conclusions

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Conclusion and Comments

• Traditional SMC.
• High frequency switching controller.
• Simple controller design.
• High quality control.
• Supertwist
• Continuous control function.
• Controller is more complex.
• High quality control.
• Disturbance SMC Driven by SMC Observer
• Continuous controller.
• More complex than supertwist.
• Very high quality control.
• All SMC designs provided high quality of control
  without disturbance waveform modeling.

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Summary

• Reviewed some background and definitions related
  to SMC.
• Derived three types of sliding mode controllers,
  traditional, Supertwist, and SMC Driven by a SMC
  Observer.
• Simulated each controller in Simulink using a
  partial plant model of a F-16 aircraft.
• Simulated a phase portrait of the sliding surface
  in state space.
• Compared simulation results of the error and
  control output for each design.

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References

• Shtessel, Y., Buffington, J., and Banda,
  S.Multiple Timescale Flight Control Using
  Reconfigurable Sliding Modes, Journal of
  Guidance, Control, and Dynamics, Vol. 22, No. 6,
  Nov. Dec. 1999, pp. 873-883
• Edwards, Christopher, and Surgeon, Sarah, K.
  Sliding Mode Control, Theory and Applications,
  Taylor and Frances Inc., 1900 Frost Road, Suite
  101, Bristol, PA 19007
• Brogan, William, L. Modern Control Theory,
  Third edition, Prentice Hall, Englewood Cliffs,
  New Jersey 07632
• Dorf, Richard, C., and Bishop, Robert, H, Modern
  Control Systems, Ninth edition, Prentice Hall,
  Upper Saddle River, NJ 07457