SLIDING MODE CONTROL
Sliding mode control is a particular type of variable structure control systems. Variable structure control systems are characterized by a suite of feedback control laws and a decision rule. The decision rule, named the switching function selects a particular feedback control in accordance with the systems behavior. In sliding mode control, VSCS are designed to drive the system states to a particular surface in the state space, named sliding surface. Once the sliding surface is reached, sliding mode control keeps the states on the close neighborhood of the sliding surface. Hence the sliding mode control is a two part controller design. The first part involves the design of a switching function so that the sliding motion satisfies design specifications. The second is concerned with the selection of a control law that will make the switching surface attractive to the system state [1].
There are two main advantages of sliding mode control. First is that the dynamic behavior of the system may be tailored by the particular choice of the sliding function. Secondly, the closed loop response becomes totally insensitive to some particular uncertainties. This principle extends to model parameter uncertainties, disturbance and non-linearities that are bounded.
Sliding mode control due to its nature is discontinuous. This creates some problems in application such as chattering; infinite frequency of the control effort is required at sliding surface to keep the system states on the sliding surface. Therefore it is not usually advised to implement sliding mode control to mechanical systems due to fast wearing of the parts. Filtering and continuous approximation of the control law can be used to prevent that issue. However robustness property of the sliding mode control is lost.
Variable structure control systems are systems where the control law is deliberately changed during the process according to predefined rules which depend on the state of the system [1]. This yields to a switching surface in the state space in the sliding mode control. In sliding mode control full state feedback control structure is used with an addition of a switching term that is aimed to cancel the effects of uncertainties. To better visualize the control law change process and switching surface, an illustration from [1] is used. The second order system,
………(1)
is used for the illustration of variable control structure systems, u is the control input and x is the angular position. First consider the effect of using only a negative position feedback for the control law (k>0).
………(2)
Substituting (2) in (1) and multiplying both sides gives
………(3)
Integrating (3) gives the following equation, where c is constant of integration.
………(4)
Equation (4) represents an ellipse in the phase portrait, special case being a circle when k=1. In Figure 1 two ellipses are shown with two different k values, k1<1 in one and k2>1.
Figure 1. Phase portrait of the system with k1<1 and k2>1 [1].
By changing the value of k during the system process it can be achieved to move the system states from an initial point on the phase portrait to the origin. Note that this is equivalent to decreasing the distance of an initial point on state space to zero. Therefore the control law may be changed to the following, where 0<k1<1<k2.
if
………(5)
Figure 2.2. Phase portrait of the system under VSCS [1].
Sliding mode control uses the same
principle described above to take the states to the origin. In the above
described controller approach,
was
used for switching the controller input. In sliding mode control, the switching
function is chosen as
.
Therefore sliding mode control would switch with the sign of
,
which is called the sliding surface.
represents
a line that passes through origin in state space. Therefore if one can find a
controller that first reaches
and
that consequently keeps the states so that
,
the system would reach to origin of the state space (regulator). If a
control structure can be established so that whenever
is
greater than zero
is
less than zero and vice versa then it is possible to keep the states on the
sliding surface. Once is
satisfied then the system would behave as a first order system,
(
),
and slide to the origin on the sliding surface. However this is only possible if
the switching occurs at zero time, the frequency of the controller is infinity.
This is called ideal sliding motion.
Following is the representation of an nth order uncertain or non-linear time invariant system with m control inputs.
………(6)
The function f(t,x,u) is assumed to be unknown but bounded by some known functions of the states. Different restrictions can be placed on this function, which represents parameter uncertainties, or nonlinearities in the system, or even disturbances.
Sliding mode control can be divided into two terms, one being the linear control for the linear part of (6) and the other being the discontinuous term to overcome the term of (6).
Remaining of the chapter is divided into two subsections. First section explains the linear control law part of the sliding mode control supposing that the sliding surface is reached. Second section explains the reachability of the sliding surface by using a second order system and the application of the sliding mode control to a second order system, where the system has bounded non-linearities, or uncertainties.
The system given with (6) is first assumed to be linear. Hence dropping from equation (6) gives the following linear system.
………(7)
Recall that the aim is to reach
and
keep the states on the sliding surface. Sliding surface can be represented
as
,
where
.
Multiplying both sides of equation (7) with S gives
………(8).
Once the sliding surface is
reached,
would be satisfied and hence,
equation (8) is equal to zero. Therefore linear control can be obtained as
………(9),
which is a linear feedback control law with
,
designed such that the states would remain on.
Note that SB must be non-singular. This is not difficult for S is a design
parameter and B has a rank of m. Because the rank of B is m, the system
equations can be partitioned as to give
.
Hence equation (7) can be written in the following form:
………(10-a)
………(10-b),
where
and
.
The compatible transformation of the switching function
would result in
.
Therefore during ideal sliding, the motion is
………(11)
Dividing equation (11) by S1 and replacing S2-1S1 by M, equation (10-a) can be written as follows
………(12)
Equations (11) & (12) represent the ideal sliding
motion. Note that the order of the system is reduced to m or to equation
(10-b) as soon as S is designed so that
, in
equation (12), has stable eigenvalues.
Consequently, if of (6) is only in the input channel, i.e., can be completely rejected whilst ideal sliding is established via (11).
In this section the second order system given below will be considered. For the reachability of multi input multi output systems refer [1].
………(13)
Sliding mode control for this system may be selected as
………(14)
The linear part of this control input is state feedback
law, which is in agreement with the structure of equation (9 and the
discontinuous part is to be arranged so that the system is insensitive to.
The sliding surface,,
can be written as
,
by selecting S2 as 1 and using S2-1S1
for m (for one of S1 and S2 is enough to define the sliding
surface). Once sliding surface is reached the system would behave as the first
order system
………(15),
which is insensitive to
.
Then next step is to reach the sliding surface, which is possible if
………(16)
Equations (13) and (14) are used to replace
.
………(17)
The convenient selection of k1 = Φm and k2
= Φ+m (all parameters are positive) gives;
then replacing
with
s, and
with
results
.
Since
,
then replacing
(η
is a small positive design parameter) would give
………(18)
Equation (18) is negative by the definition of the
parameters. Therefore sliding surface can be reached in finite time. It can be
shown that
if
small parameter, η, in equation (18) is neglected.
k1 and k2, state feedback law parameters, can be designed by pole placement. Φ and m are the poles of the 2nd order system.
For questions about this tutorial please feel free to contact vefa@drexel.edu.
