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Sliding Mode ControlLab Assignment Report

Submitted To:

Dr. Chris Edwards

Submitted By:SALMAN SALEEM (ss672)

Module:

Nonlinear Control (EG7040/EG4011)

Submission Date:

31st May, 2011.


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INTRODUCTION TO SMC

Variable structure control with sliding mode control was first proposed and elaborated by

several researchers from the former Russia, starting from the sixties The ideas did not appearoutside of Russia until the seventies when a book by Itkis (Itkis, 1976) and a survey paper by

Utkin (Utkin, 1977) were published in English. Since then, sliding mode control hasdeveloped into a general design control method applicable to a wide range of system typesincluding nonlinear systems, MIMO systems, discrete time models,large scale and in infinite-

dimensional systems.Essentially, sliding mode control utilizes discontinuous feedback control laws

to force the system state to reach, and subsequently to remain on, a specified surface within

the state space (the so-called sliding or switching surface). The system dynamic whenconcerned to the sliding surface is described as an ideal sliding motion and represent the

controlled system behaviour.

The advantages of obtaining such a motion are twofold: Firstly the system behaves as a system of reduced order with respect to the original plant; and secondly the

movement on the sliding surface of the system is insensitive to a particular kind ofperturbation and model uncertainties.

This latter property of invariance towards so-called matched uncertainties is the

most distinguish feature of sliding mode control and makes this methodology particularsuitable to deal with uncertain nonlinear system.

The design of a sliding mode controller (SMC) involves designing of a sliding surface thatrepresents the desired stable dynamics and a control law that makes the designed sliding

surface attractive. The phase trajectory of a SMC can be investigated in two parts,

representing two modes of the system [1]. The trajectories starting from a given initial

condition off the sliding surface tend towards the sliding surface. This is known as reachingor hitting phase and the system is sensitive to parameter variations in this part of the phase

trajectory. When the convergence to the sliding surface occurs, the sliding phase starts and

the trajectories are insensitive to parameter variations and disturbances in this phase [2].

LAB SHEETS

Consider the second order integrator regulator system

=bu, b> .(1)

Where the value of the system parameter b is not precisely known but lies within the knownrange 1


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where ,>0.(3)

Where

s(x)=c1x1+x2, c1>0....(4)

is the switchi functi n. The set of points in the st te space satisfying s(x)=0 is called theswitching or sliding line.

Part 1

In this partI plotted the control action u which is gi en in equation 3 as a function of s over

the interval [-1 1] assuming that =1 and =0.01. I wrote the code for equaton 3 on matlab

editor which is given below.

When I run this m file I had a plot which is given below.

Figure 1:Controller action u

In figure 1 I have gotthe non linear behavior ofthe controller u forthe given integratorregulator system.

Now IfI compare this controller action u with sign(s) function which is discontinues andnon linear function I wrote the following code.


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After compiling this code on Matlab I gotthe graph which is shown below.

Figure 2: (Comparison b/w controller action u with sign(s) function)In figure 2 the comparison clearly shows thatthere is nonlinear relationship with controlleraction u which is a red line and sign(s) function which is green in color.

Part 2

In this partI downloaded the runl.m and vsim.mdl and open them in Matlab. After run this Ihad a graph which is given below. The parameters thatI gave as follows:

b=2; x0T

[1 0] ; c1 =1 ; =1 ; =0.01 ;time interval=10msec

Figure 3: Phase plane trajectory x1on x-axis x2 on y-axis


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The (x1-x2) phase plane trajectories for the SMC are given in Figure 3. As it can be

seen from the figure, after reaching the sliding mode which is s(x) = 0, the phase planetrajectory is nonlinear and it smoothly enters the sliding mode region. In this region the

control action wouldnt effect. Motion of the trajectories now will depend on the parametersof the sliding surface.

Part 3

In this part I changed the value of controller smoothing coefficient delta to very closeto zero around 0.00001 then I had the chattering effect which is given below in the figure 4.

Figure 4: Chattering Effect (control action u with respect to time)

ExplanationofChatteringeffect:

We have noticed that the controller is discontinuous at s(x) !0 . Due to the effects ofsampling, switching and delays in the devices used to implement the controller, respectively

in the simulation engines used when modelling the controlled system, sliding mode controlsuffers from chattering. The figure 5 shows how delays can cause chattering. It depicts atrajectory in the region s > 0 heading toward the sliding manifold s = 0 .It first hits themanifold at a point a. In ideal sliding mode control, the trajectory should start sliding on themanifold from a point a .In reality, there will be a delay between the time the sign of schanges and the time the control switches. During this delay period, the trajectory crosses themanifold into the region s < 0 .Chattering results in low control accuracy, high heat losses inelectrical power circuits and high wear of moving mechanical parts. It may also exciteunmodeled high frequency dynamics, which degrades the performance of the system and mayeven lead to instability. There are many strategies used to avoid chattering, e.g. you canintroduce a boundary layer. Here, the sign function is made continuous by using a piecewiselinear approximation within the boundary layer you have exponentially convergence to the

sliding mode.


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Figure 5

To see the chattering effects onto the controller action u I plotted some graphs which isshown below.

Figure 6: Plot of controller action u againsttime.=0.001



From figure 6, 7 and 8, I can clearly see when I am changing the value of delta towards zerothe chattering effect is going to increase which clearly shows that when would be slightlysmall, controller will display chatter motion.


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Part 4:

Switching function is defined as

s(x)= c1x1+x2 =0, c1>0

In sliding mode s(x) = 0, i.e

c1x1+x2 =0

In state-space design

For again I need to take switching function

c1x1+x2 =0

x2 = - c1x1

After getting the state values and I wrote the following code in matlab for plotting x1(t)

and x2(t) and I got a graph regardless of the value of b which is given below in figure 9.


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Figure 9: System Dynamics in terms of phase portrait(x1 on x axis and x2 on y-axis)

In Figure 9 I have the same system dynamics in sliding mode as I had for full order system inPart 2.To see the controller output against time I changed the function in the vsim simulinkfile for first order system which is given below.

F(u)= -(1*u[1])/(abs(u[1])+delta).After simulating this model I got a graph which is given below.

Figure 10: Plot of Controller action u against time

Figure 10 clearly indicates that the we have the function which is same as we got in part 1

and as compared to sign(x1) shown in figure 10b we have again the same nonlinearrelationship.


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Figure 10b:Plot of sign(x1)

Part 5:

Figure 11:Phase portrait(x1 on x-axis and x2

on y-axis) for b=0.5


on y-axis)for b=0.8


on y-axis) for b=18


on y-axis) for b=22.

From all figures 11,12,13 and 14, The effect of the uncertainity in the model can be easily

seen from the phase plane trajectory. I can easily find the range of b that will keep system on

sliding mode after touching the sliding surface. In other words I have to calculate the values


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for 1 and 2 so that it would be 1


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of low order. System uncertainty parameter b has a certain range in which we have the good

performance and good phase plane trajectory.

References:

[1] J. Y. Hung, W. Gao, J. C. Hung, Variable

Structure Control: A survey, IEEE Transactions

on Industrial Electronics, 40(1), pp.2-22,

(1993).

[2] J. J. Slotine, S. S. Sastry, Tracking control of

nonlinear systems using sliding surfaces with

application to robotic manipulators ,

International Journal of Control, 38(2), pp.

465-492, (1983).

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