Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
Chattering Problem in Sliding Mode Control Systems
Vadim Utkin and Hoon Lee
The Ohio State University
Abstract
In practical applications of sliding mode
control,
engineers
may
experience
undesirable phenomenon of oscillations
having finite frequency and amplitude,
which is known as ‘chattering’. At the first
stage of sliding mode control theory
development the chattering was the main
obstacle for its implementation. The major
attention was paid to the systems with
motion equations in canonical space – space
of a system output and its high order
derivatives. Small time constants of real
differentiators could not be disregarded, if
control actions were discontinuous state
functions, and they led to oscillations in the
vicinity of discontinuity surfaces in the
system state space. Chattering is a harmful
phenomenon because it leads to low control
accuracy, high wear of moving mechanical
parts, and high heat losses in power circuits.
There are two reasons which can lead to
chattering.
Similarly to the systems in canonical space,
the chattering can be caused by fast
dynamics which were neglected in the ideal
model. These ‘unmodeled’ dynamics with
small time constants are usually disregarded
in models of servomechanisms, sensors and
data processors.
The second reason of chattering is utilization
of digital controllers with finite sampling
rate, which causes so called ‘discretization
chatter’. Theoretically the ideal sliding
mode implies infinite switching frequency.
Since the control is constant within a
sampling interval, switching frequency can
not exceed that of sampling, which lead to
chattering as well.
caused by unmodeled dynamics, the second
order sliding mode control system is studied
x&1 = x2
x&2 = ax1 + bx2 + c sin x1 + du (t )
where a and b are negative constant while c and
d are positive constant values. Those equations
govern a simple unstable ‘inverted pendulum’
system with x1 as an angular displacement. It is
assumed that there exist certain dynamics of the
actuator, which are not taken into account in the
ideal model:
x&1 = x 2
x& 2 = ax1 + bx 2 + c sin x1 + dw(t )
1
w( p ) =
u ( p ),
( μp + 1) 2
wherev u ( p ) and w( p ) is are Laplace
transforms of u (t ) and w(t ) , the constant μ is
regarded to be a sufficiently small value. At the
presence of the actuator unmodeled dynamics,
the actual input to the system will be w(t )
instead of u (t ) . The control input and the
sliding surface are chosen as
u = − Msign( s )
s = λx1 + x2
where λ and M are positive constants. The ideal
sliding mode cannot be expected to occur in this
case since x& and s& become continuous time
functions.
u (t )
s
w (t )
Actuator
(Unmodeled dynamics)
s = λx + x&
To demonstrate why the chattering can be
1-4244-0208-5/06/$20 ©2006 IEEE
1
( μ p + 1) 2
Figure 1
346
&x& = ax+bx& +csinx + dw
Plant
x , x&
Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
In accordance with singular perturbation
theory, in systems with continuous control, a
fast component of the motion decays rapidly
while a slow component depends on the
small time constants continuously. In
discontinuous control systems the solution
depends
on
the small parameters
continuously as well. But unlike continuous
systems, the switching in control excites the
unmodeled dynamics, which leads to
oscillations of the state vector at a high
frequency. Based on the Lyapunov function’
methodology it is shown that the system
motion is unstable in some finite vicinity of
the discontinuity surface s = 0, while for
high deviations the trajectories are
converging to this surface. It explains
qualitatively why undamped oscillations, or
chattering, are excited. Next figures
demonstrate simulation results for the
inverted pendulum with the second order
unmodeled dynamics.
Figure 4
The efficient recipe for chattering suppression is
use of asymptotic observers. The main idea of
using an asymptotic observer to prevent
chattering is to generate ideal sliding mode in
the auxiliary loop including the observer. In the
observer loop, sliding mode is generated from
the control software; therefore, any unmodeled
dynamics which cause chattering can be
excluded. As can be seen in the following figure,
controller uses estimated states x̂ instead of
measured states x directly from the plant so the
observer is free of any unmodeled dynamics of
actuators or sensors.
s
s
Observer
Loop
μ z& = ...
x& = ...
Actuator
(Unmodeled dynamics)
Plant
x&ˆ = ...
x (t )
xˆ ( t )
Observer
Controller
Figure 2
Figure 5
ε (μ )
Although x& becomes continuous, x&ˆ in the
observer is a discontinuous time function,
therefore, sliding mode may be enforced.
For the above inverted pendulum
system, an asymptotic observer is designed,
assuming that the state x1 is measured only:
xˆ&1 = xˆ2 − L1 ( x1 − xˆ1 )
xˆ&2 = axˆ1 + bxˆ2 + c sin xˆ1 + du (t ) − L2 ( x1 − xˆ1 )
Figure 3
where L1 and L2 are constant values determined
347
Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
such that the error e = x1 − x̂1 reduces to
zero. As can be seen in simulation results,
the system with the observer is free from
chattering.
positive value and A and B are m × m and
m × n matrices, respectively. The matrix A is
assumed to have eigenvalues with negative real
parts. Instead of the system state vector x, the
controller becomes a function of
vector
x * ( x* ∈ ℜ n )
⎧ M ( s* < 0)
,
u=⎨
⎩− M ( s* > 0)
s* = cx* ,
x* = Hz ,
− HA −1 B = I (an identity matrix).
The control is assumed to be a periodic function
represented by the first term of the Furie series
u = u 0 + u1 sin ωt.
As follows from the describing function method,
u 0 corresponds to the control in the ideal sliding
Figure 6
The describing function method is applied to
estimate the parameters of chattering –
amplitude and frequency of oscillations.
mode, u1 is proportional to the magnitude of
control M, the chattering frequency ω is inverse
proportional to the time constant of the
unmodeled dynamics μ.
It means that the chattering amplitude can be
reduced for discontinuous control with state
dependent gain:
u 2 = − M 0 ( x1 + δ ) sign( s )
The behavior of an arbitrary order
system with scalar discontinuous control at
presence of unmodeled dynamics is studied:
x& = f ( x) + b( x)u
(x ∈ ℜn )
⎧ M ( s < 0)
, s = cx ,
u=⎨
⎩− M ( s > 0)
M = M ( x1 ) = M 0 ( x1 + δ ),
s = cx1 + x 2
instead of
u1 = − M 0 sign( s )
in the second order system
x&1 = x 2
x& 2 = a1 x1 + a 2 x 2 +u.
c is a -vector 1 × n , M is constant value,
The unmodeled dynamics is governed by
μz& = Az + Bx
As can be seen in the next figure 7, the
chattering amplitude is reduced considerably.
with small time constant μ , whose state is
characterized by an intermediate state vector
z ( z ∈ ℜ m ) and the state vector x is
regarded as an input of this subsystem.
μz& = Az + Bx , (A and B are
m × m and m × n matrices, respectively )
The time constant μ is a sufficiently small,
348
Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
Figure 7
Any methods would be helpful to suppress
chattering if it can decrease the gain M
properly holding the establishment of sliding
mode. In previous part M was reduced along
with the system states. The gain M can be
adjusted in the other ways; e.g. M becomes a
function of an equivalent control ueq . This
methodology also looks promising since ueq
converges to zero as sliding mode arises
along the discontinuity surface s.
The control law may be selected as
s=x
u = − M 0 ( σ + δ ) sign( s )
where M 0 , δ are positive constants. σ is
the average value of sign( s ) . By using a
low-pass filter, the average of sign( s ) can be
approximated. Similarly to the systems with
state dependent control gain the chattering
can be reduced (see the next figure). The
method can be applied for the plants
subjected to unknown disturbances.
Figure 8
Chattering in discrete-time systems is caused by
discontinuities in control, since switching
frequency cannot exceed that of sampling.
Increasing a sampling frequency to decrease the
chattering amplitude seems unjustified. We
believe that using a computer is adequate to
control system dynamics if a sampling frequency
corresponds to average, slow system motion
rather than to a high frequency component.
The state trajectories in discrete-time systems
with discontinuous control are not confined to
the switching manifold but to some domain
around it. So the problem is to design control for
the discrete-time system
xk +1 = F ( xk , uk ), uk = u ( xk ), xi ∈ ℜ n , u ∈ ℜ m
such that qualitatively this motion has the same
properties as its continuous-time counterpart.
The newly introduced Definition embraces both
discrete- and continuous-time systems:
Definition .
The set S ( x) in the manifold
s( x) = 0, s ∈ ℜm is the domain of sliding
mode if there exists vicinity ε of S such that
s ( xk +1 ) = s[ F ( xk , uk )] = 0 for xk ∈ ε .
349
Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
The solution to the equation in Definition is
called equivalent control ukeq as well, since,
similar to continuous systems, it results in
motions with state trajectories in the
manifold s ( x) = 0 and finite time is needed
to reach the manifold in discrete-time
system as well. The fundamental difference
is that the control should be a continuous
function of the state.
Experimental results for sliding mode
control of inductions motors with observers are
discussed.
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