Sliding Mode Controller

Sliding Mode Control
Dr.R.Subasri
Professor, Kongu Engineering College,
Perundurai, Erode, Tamilnadu, INDIA
Dr.R.Subasri, KEC,INDIA

Variable-Structure Systems: Discontinuous feedback
control strategies.
-Theory of the control inputs typically take values
from a discrete set, such as the extreme limits of a relay,
or from a limited collection of pre-specified feedback
control functions.
Sliding mode control: one of the most popular techniques
under VSS by Utkin
• choice of a switching surface of the state space
according to the desired dynamical specifications of the
closed-loop system.
• The switching logic, and thus the control law, are
designed so that the state trajectories reach the surface
and remain on it.

The main advantages of this method are:
• its robustness against a large class of perturbations or
model uncertainties
• the need for a reduced amount of information in
comparison to classical control techniques
• the possibility of stabilizing some nonlinear systems
which are not stabilizable by continuous state feedback
laws
Drawback:
The actuators had to cope with the high
frequency bang-bang type of control actions that could
produce premature wear, or even breaking.

However, this main disadvantage, called chattering,
could be reduced, or even suppressed, using techniques
such as nonlinear gains, dynamic extensions, or by using
more recent strategies, such as higher-order sliding
mode control
Applications
•Control of electrical motors, DTC
•Observers and signal reconstruction
• Mechanical systems
•Control of robots and manipulators
• Magnetic bearings

single-dimensional motion of a unit mass is considered.
A state-variable description is easily obtained by
introducing variables for the position and the velocity
x1=x and x2 =
1 2 1 10
2 1 2 2 20
x x x (0) x
x u f (x ,x ,t) x (0) x
 
  
&
&
x&
where u is the control force, and the disturbance term f(x1,
x2, t) which may comprise dry and viscous friction as well
as any other unknown resistance forces, is assumed to be
bounded, i.e., 1 2f(x ,x ,t) L 0 
The problem is to design a feedback control law u
=u(x1,x2)that drives the mass to the origin asymptotically.
In other words, the control u is supposed to drive the state
variables to zero 1 2
t
lim x ,x 0

Dr.R.Subasri, KEC,INDIA

This apparently simple control problem is a challenging
one, since asymptotic convergence is to be achieved in the
presence of the unknown bounded disturbance f(x1, x2, t).
For instance, a linear state-feedback control law
1 1 2 2 1 2u k x k x , k 0 , k 0    
provides asymptotic stability of the origin only for
f(x1, x2, t )=0 and typically only drives the states to a
bounded domain 1 2 1 2Ω(k ,k ,L) for f(x ,x ,t) L 0 
Single-dimensional motion of a unit mass
Asymptotic convergence for f(x1, x2, t )=0

The results of the simulation of the system with
x1(0) =1 , x2(0) =-2, k1=3 , k2=4 and f(x1,x2,t)=sin(2t)
Asymptotic convergence for f(x1, x2, t )= sin 2t

Main Concepts of Sliding Mode Control
Let us introduce desired compensated dynamics for the example
system. A good candidate for these dynamics is the homogeneous
linear time-invariant differential equation
1 1x cx 0, c>0 &
A general solution of above eqn and its derivative is given by
1 1
2 1 1
x (t) x (0)exp( ct)
x (t) x (t) cx (0)exp( ct)
 
   &
Both x1(t)and x2(t) converge to zero asymptotically. Note, no
effect of the disturbance f(x1, x2,t) on the state compensated
dynamics is observed. How could these compensated dynamics
be achieved?
First, we introduce a new variable in the state space of the system
in 1 2 2 1σ σ(x (t),x (t)) x cx , c>0  

In order to achieve asymptotic convergence of the state variables x1,
x2 to zero with a given convergence rate in the presence of the
bounded disturbance f(x1, x2, t), we have to drive the variable to
zero in finite time by means of the control u.
The variable  is called a sliding variable
1 2 2 1σ σ(x ,x ) x cx , c>0  The equation corresponds to a
straight line in the state space of the system and is referred
to as a sliding surface.
A condition to make the trajectory of the system driven
towards the sliding surface and remains on it thereafter is
known as Reachability Condition
α
σσ σ
2
&
1 2 2 1σ σ(x (t),x (t)) x cx , c>0  

The control u(x1 ,x2) that drives the state variables x1, x2 to the
sliding surface in finite time , and keeps them on the surface
thereafter in the presence of the bounded disturbance f(x1, x2, t) is
called a sliding mode controller
The control law consists of two parts , one as feedback linearisation
and other as switching function depends on  which is sgn( ).
Where  is control gain.
1 σ 0
sgn(σ) 0 σ 0
1 σ 0


 
 

Firstly, it drives the nonlinear plant’s state trajectory onto a specified
and user chosen surface in the state space which is called the sliding
or switching surface.
This surface is called the switching surface because a control path
has one gain if the state trajectory of the plant is “above” the
surface and a different gain if the trajectory drops “below” the
surface.
Secondly, it maintains the plant’s state trajectory on this surface for
all subsequent times. During the process, the control system’s
structure varies from one to another and thereby earning the name
variable structure control. The control is also called as the sliding
mode control

The reachability condition is : σσ 0&
 
1 2
1 1 2
σ(t) cx(t) x(t) cx (t) x (t)
σ(t) cx (t) u f x ,x ,t
   
  
& & && & &
& &
  1 1 2σσ(t) σ (cx (t) u f x ,x ,t  & &
1u(t) cx (t) ηsgn(σ)  &
Secondly, to satisfy the condition, we design the sliding mode
controller as
Let us design the sliding mode control law for the previous
tsystemwith the initial conditions x1(0)= 1; x2(0)=- 2, the control
gain = 2, the parameter c=1.5, and the disturbance f(x1, x2, t)
=sin 2t
First, design the sliding mode surface as
1 2σ(t) cx(t) x(t) cx (t) x (t)   &
Where c has to satisfy the Hurwitz condition, c > 0

The reachability condition is : σσ 0&
 
d
d 1 2
σ(t) ce(t) e(t) ce(t) x(t) x (t)
σ(t) ce(t) u x (t) f x ,x ,t
    
   
& && && && &&
&& &&
  d 1 2σσ(t) σ (ce(t) u x (t) f x ,x ,t   && &&
du(t) ce(t) x (t) ηsgn(σ)   & &&
Secondly, to satisfy the condition, we design the sliding mode
controller as
Let us design the sliding mode control law for tracking the
desired position as xd = sin t, with the initial conditions x1(0)=
0.5; x2(0)=-1.0, the control gain = 0.5, the parameter c=0.5,
and the disturbance f(x1, x2, t) =sin 2t
First, design the sliding mode surface as σ(t) ce(t) e(t)  &
Where c has to satisfy the Hurwitz condition, c > 0 and e is
tracking error e = x-xd

System without disturbances
• function
[sys,x0,str,ts]=mdlInitializeSizes
• sizes = simsizes;
• sizes.NumContStates = 2;
• sizes.NumDiscStates = 0;
• sizes.NumOutputs = 2;
• sizes.NumInputs = 1;
• sizes.DirFeedthrough = 0;
• sizes.NumSampleTimes = 0;
• sys=simsizes(sizes);
• x0=[1 -2];
• str=[];
• ts=[];
• function sys=mdlDerivatives(t,x,u)
• sys(1)=x(2);
• sys(2)=u;
• %sys(2)=u+sin(2*t);
• function sys=mdlOutputs(t,x,u)
• sys(1)=x(1);
• sys(2)=x(2);
•

Controller design without disturbances
• function
• sys = simsizes(sizes);
• x0 = [];
• str = [];
• ts = [];
• x1=u(1);
• x2=u(2);
• k=1.5*x1+x2;
• ut=-k1*x1-k2*x2
• %ut=-1.5*x2-2*sign(k);
• sys(1)=ut;

System with disturbances
• function
• sys=simsizes(sizes);
• x0=[1 -2];
• str=[];
• ts=[];
• function sys=mdlDerivatives(t,x,u)
• sys(1)=x(2);
• sys(2)=u+sin(2*t);
• sys(1)=x(1);
• sys(2)=x(2);
•

Controller design with disturbances
• function
• x0 = [];
• str = [];
• ts = [];
• x1=u(1);
• x2=u(2);
• %si=1.5*x1+x2;
• ut=-k1*x1-k2*x2
• %ut=-1.5*x2-2*sign(si);
• sys(1)=ut;

Sliding mode Controller design with
disturbances
• function
• x0 = [];
• str = [];
• ts = [];
• x1=u(1);
• x2=u(2);
• xd=sin(t);
• si=1.5*x1+x2;
• %si=0.5*x1+x2;
• e
• ut=-1.5*x2-2*sign(si);
• sys(1)=ut;

Sliding mode control is a nonlinear control method that alters the
dynamics of a nonlinear system by the multiple control structures
are designed so as to ensure that trajectories always move
towards a switching condition. Therefore, the ultimate trajectory
will not exist entirely within one control structure.
The state-feedback control law is not a continuous function of
time. Instead, it switches from one continuous structure to
another based on the current position in the state space.
Hence, sliding mode control is a variable structure control method.
The motion of the system as it slides along these boundaries is
called a sliding mode[3] and the geometrical locus consisting of
the boundaries is called the sliding (hyper) surface.

There are two steps in the SMC design. The first step is
designing a sliding surface so that the plant restricted to
the sliding surface has a desired system response. This
means the state variables of the plant dynamics are
constrained to satisfy another set of equations which
define the so-called switching surface.
The second step is constructing a switched feedback
gains necessary to drive the plant’s state trajectory to the
sliding surface. These constructions are built on the
generalized Lyapunov stability theory.

Thank you

Sliding Mode Controller

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