212 Ketter Hall, North Campus, Buffalo, NY 14260

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• 212 Ketter Hall, North Campus, Buffalo, NY 14260
  
• www.civil.buffalo.edu
• Fax [phone] Tel [phone] x 2400
• Control of Structural Vibrations
• Lecture 7_3
• Sliding Mode Control
• Instructor
• Andrei M. Reinhorn P.Eng. D.Sc.
• Professor of Structural Engineering

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Variable Structure Control

• Controller Structure around the plant is changed
  by some external influence to obtain desired
  plant response.
• High-speed Switching Control.
• For example, the gain in the feedback path
  switches between two values according to some
  rule.
• Switching occurs with respect to a switching
  surface in State Space.
• Individual Structures may be unstable, but the
  Variable Structure is asymptotically stable.

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Illustration

• Consider the system
• where and k(x1) can be 3 (Structure
  1) or 2 (Structure 2)
• Structure 1 Structure 2
• Both the structures are unstable.

k(x1) -3
k(x1) 2
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Illustration (Contd.)

• Now consider the
• Variable Structure System
• with the switching surface given by
• and

Switching
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Illustration (Contd.)
Example s1 0.1 Initial Conditions (0.5,0.5)
For s1 lt 1
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Sliding Mode

• In the above example, when the system trajectory
  intersects the switching surface, it remains on
  there for all subsequent time. This property is
  called a sliding mode.
• A sliding mode will exist for a system, if in the
  vicinity of the switching surface, the state
  velocity vector is always directed towards the
  switching surface.
• In the above example,
• s1 lt 1 (sliding mode exists) s1 gt 1 (sliding
  mode does not exist).

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System Model

• The State Equations of the system are given by
• where x(t) ?R n and u(t) ?R m.

• Each component of the switched control is given
  by

• Where ?i(x) 0 is the ith switching (or
  discontinuity) surface associated with the (n-m)
  dimensional switching surface,
• If a sliding mode exists on ?(x) 0, it is
  called a sliding surface.

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Design of Sliding Mode Control

• Phase 1 (Sliding Surface Design) Constructing
  Switching Surfaces so that the system restricted
  to the switching surface produces a desired
  behavior.
• For convenience only linear switching surfaces of
  the form Sx(t)0 are considered in practice.
• Phase 2 (Controller Design) Constructing
  switched feedback gains which drive the plant
  state trajectory to the sliding surface and
  maintain it there.
• For the existence of a sliding mode on the
  switching surface, the state velocity vectors
  should be directed towards the surface, i..e.,
  the system must be stable to the switching
  surface. Therefore there must exist a Lyapunov
  function in the neighborhood of the switching
  surface. The feedback gains are determined to
  ensure the time derivative of a suitable Lyapunov
  function to remain negative definite.

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Sliding Surface Design

• The dynamics of the system on the switching
  surface can be determined using the Method of
  Equivalent Control.
• The existence of the sliding mode implies
• because the trajectory does not
  leave the sliding surface
• Therefore by the chain rule,
• or
• where ueq is the so-called equivalent control
  force.

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Sliding Surface Design (Contd.)

• The behavior of the system on the sliding surface
  is given
• by and constrained by
• Note that due to the m constraints , the
  dynamics of the system on the sliding surface is
  of reduced order and is governed by only (n-m)
  state variables.
• This behavior may now be designed using a
  technique such as LQR or pole placement

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Sliding Mode Control of Linear Structures (Yang
et. al., 1994)

• Equations of Motion in Second order Form
• Equations of Motion in State Space Form
• Sliding Surface (Linear) where r no.
  of controllers
• Method of Equivalent Control gives
• The matrix P has to be determined so that this
  reduced order system has desirable properties

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SMC of Linear Structures (Contd.)

• The design of the sliding surface is obtained by
  minimizing the performance index,
• or in transformed variables,
• with
• Solving the resulting Ricatti equation, one can
  obtain
• where, , and
• is the solution of the Ricatti equation

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SMC of Linear Structures (Coordinate
Transformations)

• In order to do this, the above equation is first
  converted to a regular form using the following
  transformation
• where
• We then obtain and
• where
• Partitioning the system into r and (2n-r) as
• and setting for
  convenience, we obtain

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Controller Design for SMC of Linear Structures

• Design of Controllers using Lyapunov Function

Positive Definite Lyapunov Function
Derivative must be Negative Definite
Taking the derivative and using the state
equations,
where
This condition could be satisfied using different
controllers, for example
Discontinuous ControllerH() is the step function
and a boundary layer ?0 is introduced to reduce
chattering
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Example (Yang et. al., 1994)
Properties m1 m2 m3 1 ton c1 c2 c3
1.407 kNs/m k1 k2 k3 980 kN/m
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SMC of Nonlinear Structures

• Equation of motion of Inelastic Structure
• Inelastic Restoring Forces
• Equations in State Space
• Control condition from Lyapunov Function
• Complete Compensation It can also be shown that
  if there are as many controllers as there are
  degrees of freedom, them the earthquake forces as
  well as the inelastic forces can be completely
  nullified using SMC.

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Advantages of SMC

• Robustness The behavior on the sliding mode
  depends only on the switching surface and is
  independent of the structural properties.
  Therefore the effectiveness of control is
  insensitive to parametric uncertainties of the
  model.
• Nonlinear Structures The control algorithm is
  applicable to nonlinear and inelastic structures
  unlike linear control and other frequency domain
  algorithms.
• Uses information about input as well as output
  feedback in control determination. It is
  therefore an open-closed-loop type control.