1. Sliding Mode Control – An Introduction
S. Janardhanan
IIT Delhi

2. Outline What is this ‘Sliding mode’ and how did its study start?
How to design controller using this concept?
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3. Primitive Examples - Electrical
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4. Primitive Examples-Mechanical
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5. First ‘Formal’ Steps
The first steps of sliding mode control ‘theory’ originated in the early 1950’s initiated by
S. V. Emel’yanov.
Started as VSC – Variable Structure Control
Varying system structure for stabilization.
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6. Variable Structure Control – Constituent Systems
Both systems have their eigenvalues on the imaginary axis. Neither system is asymptotically stable.
Mode 2
Mode 1
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7. Piecing together …
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8. Properties of VSC
Both constituent systems were oscillatory and were not asymptotically stable.
‘Combined’ system is asymptotically stable.
Property not present in any of the constituent system is obtained by VSC
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9. Another Example – Unstable Constituent Systems
Zeta and alpha are both positive. In both cases, the system is unstable.
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10. Analysis … Both systems are unstable
Only stable mode is one mode of system
IF the following VSC is employed
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11. Combined ..
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12. In this case,…
Again, property not present in constituent systems is found in the combined system.
A stable structure can be obtain by varying between two unstable structures.
However, a more interesting behaviour can be observed if we use a different ‘switching’ logic.
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13. The regions
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14. Sliding Mode
New trajectory that was not present in any of the two original systems
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15. Sliding Mode ?
Defined : Motion of the system trajectory along a ‘chosen’ line/plane/surface of the state space.
Sliding Mode Control : Control designed with the aim to achieve sliding mode.
Is usually of VSC type
Eg : Previous problem can be perceived as
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16. What is the advantage?
Consider a n-th order system represented in the phase variable form
Also consider the sliding surface defined as
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17. Advantage …
Thus entire dynamics of the system is governed by the sliding line/surface parameters only
In sliding mode, dynamics independent of system parameters (a1,a2,…).
ROBUST
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18. Required Properties
For sliding mode to be of any use, it should have the following properties
System stability confined to sliding surface (unstable sliding mode is NOT sliding mode at all)
Sliding mode should not take ‘forever’ to start
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19. Stable Surface Consider the system
If the sliding function is designed as
then confined to this surface ( ), the dynamics of can be written as
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20. The Surface …
If K is so designed that has eigenvalues on LHP only , then the dynamics of is stable.
Since , the dynamics of is also stable.
Hence, if the sliding surface is ‘designed’ as
, the system dynamics confined to
s=0 is stable. (Requirement 1)
Note : Strictly speaking, it is not necessary for s to be a linear function of x
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21. Convergence to s=0
The second requirement is that sliding mode should start at a finite time.
Split the requirement into further bits
Sliding mode SHOULD start.
It should do so in finite time.
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22. Run towards the surface
To be sure that sliding mode starts at some time t>0, irrespective of the initial state x(0), we should be sure that the state trajectory is always moving towards s=0, whenever s is not zero.
Mathematics …
This is called the ‘reachability condition’
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23. A figure to help out …
s>0
s<0
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s=0

24. Insufficient Consider the case, This gives the solution of
is not enough (Violates Requirement 2)
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25. -reachability
With only , s slows down too much when close to zero to have finite time convergence
Stronger condition is needed for finite time convergence.
Defined as -reachability condition
s has a minimum rate of convergence
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26. Discontinuity Observe So, at , is discontinuous.  -
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27. Discontinuous Dynamics
Thus, for s>0, the system dynamics are
and for s<0
Thus, at s=0, the dynamics is not well defined.
The dynamics along the sliding surface is determined using continuation method
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28. Continuation Method
Using continuation method as proposed by Filippov*, it is said that when s=0, the state trajectory moves in a direction in between and
*A. F. Filppov, “Differential Equations with discontinuous righthand sides”Kluwer Academic Publishers,The Netherlands, 1988
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29. Diagrammatically Speaking …
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30. The reaching law approach
In reaching law approach, the dynamics of the sliding function is directly expressed. It can have the general structure
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31. Few Examples Constant rate reaching law Constant+Proportional rate
Power-rate reaching law
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32. The Control Signal Now, consider the condition Thus, Or, control is
And the system dynamics is governed by
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33. The Chattering Problem
When, s is very close to zero, the control signal switches between two structures.
Theoretically, the switching causes zero magnitude oscillations with infinite frequency in x.
Practically, actuators cannot switch at infinite frequency. So we have high frequency oscillations of non-zero magnitude.
This undesirable phenomenon is called chattering.
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34. The picture Ideal Sliding Mode Practical – With Chattering
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35. Why is chattering undesirable?
The ‘high frequency’ of chattering actuates unmodeled high frequency dynamics of the system. Controller performance deteriorates.
More seriously, high frequency oscillations can cause mechanical wear in the system.
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36. Chattering avoidance/reduction
The chattering problem is because signum function is used in control.
Control changes very abruptly near s=0.
Actuator tries to cope up leading to ‘maximum-possible-frequency’ oscillations.
Solution :
Replace signum term in control by ‘smoother’ choices’
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37. Chattering Avoidance…
Some choices of smooth functions
Saturation function
Hyperbolic tangent
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38. Disadvantage of ‘smoothing’
If saturation or tanh is used, then we can observe that near s=0
Where represents the saturation or tanh function.
The limit in both cases is zero.
So, technically the sliding mode is lost
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39. What are the actual conditions for achieving Sliding Mode
System is stable confined to
Control moves states towards this stable sliding surface
And does it in finite time.
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40. Some aspects of Continuous Sliding Mode Control
Robustness
Multivariable Sliding Mode
‘Almost’ Sliding Mode
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41. Robustness of CSMC
When in sliding mode, entire system dynamics is governed by sliding surface parameters and not original system parameters.
Hence, sliding mode is robust.
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42. Disturbance Consider the system with disturbance
Disturbance comes through input channel
How does sliding mode behave in such a situation.
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43. Disturbance Rejection
The control law is designed so as to bring the system to the sliding surface.
Let us see dynamics confined to the sliding surface
Thus,
Therefore,
And
Again, dynamics independent of disturbance.
Hence disturbance rejection.
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44. What if more than one input ?
If system has more than one input, then the system can be transformed to the form
With having more than one elements.
Thus, will also have multiple rows. Hence, the system can have more than one sliding surface
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45. Approach to sliding surface
Sliding mode will start when all sliding functions are zero. I.e, intersection of all sliding surfaces.
Approach to the intersection
Direct to intersection (Eventual)
Surface by surface
In particular order (Fixed Order)
First approach (Free order)
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46. Eventual Sliding Mode
In this type of sliding mode, the state trajectory moves to the intersection of the sliding surfaces through a connected subset in the state space.
It does not necessary stay on any one of the sliding surfaces on approaching it.
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47. Eventual Sliding Mode
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48. Fixed order Sliding Mode
In fixed order sliding mode, the state trajectory moves to one pre-specified sliding surface and staying on it moves to the intersection of the first surface with the next pre-specified sliding surface
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49. Free order sliding mode
In free order sliding mode, the state trajectory remains on a sliding surface once the state approaches it. However, there is no particular order in which the surfaces are reached
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50. Ordered Sliding Mode
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51. Chattering Refreshed
A conventional sliding mode behaviour would have a sliding surface dynamics of the form
However, due to finite bandwidth of the actuator, the input cannot switch fast enough near the sliding surface
Chattering – Finite frequency, finite amplitude oscillations about the sliding surface
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52. Almost Sliding Mode
To remedy chattering, the strict requirement of “movement on sliding surface” is relaxed.
We try to get ‘Almost’ – sliding mode (Quasi sliding mode)
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53. Saturation function based Sliding Mode Control
Instead of
Inside the band |s|<, the reaching law is linear as
This is also called ‘boundary layer technique
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54. The motion
S=
S=-
S=0
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55. Disadvantage
‘Almost’ is NOT exact
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56. ‘Newer’ Avenues
Two phases in sliding motion : Reaching Phase and Sliding Phase
Improvements
Reaching Phase - Higher Order Sliding Mode Control
Sliding Phase - Terminal Sliding Mode Control
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57. Higher order Sliding Mode
Basic Definition of Sliding Mode : s(x)=0 in finite time. Sliding surface reached in finite time and stays on it.
Problem : Chattering results
Solution : Try to get ds/dt = 0, additionally in finite time.  Second order sliding mode.
Get the first n-1 derivatives of s(x) to zero in finite time.  n-th order sliding mode.
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58. Advantage Smooth control results. No Chattering.
Disadvantage : Not very straight forward.
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59. HOSM : Twisting Algorithm
Applicable to systems of relative degree 2.
Input appears in 2nd derivative of sliding function.
Input is still discontinous. However, there is no chattering in states.
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60. Typical Twisting Trajectory
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61. Super Twisting Algorithm
For systems with relative degree 1.
Switching shifted to derivative of input.
Input continuous and so is derivative of sliding function.
No chattering here too.
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62. Typical Super-twisting Trajectory
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63. New Idea
If we are concerned with getting an output to zero, why not set s=y!!
Are there any extra conditions?
Zero Dynamics
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64. Terminal Sliding Mode
Higher order sliding mode is about reaching the sliding surface smoothly.
Terminal sliding mode deals with design of the sliding function such that the system reaches origin in FINITE TIME one the sliding surface is reached.
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65. The sliding surface Terminal Sliding Mode
Fast close to origin. Finite time convergence.
Fast-Terminal Sliding Mode
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66. Terminal Sliding Surface …
In case of systems with more than 2 states,
For the system in phase variable form,
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67. References
V. Utkin, “Variable Structure Systems with Sliding Mode”, IEEE Trans. Automat. Contr., AC-12, No. 2, pp , 1977
An introductory paper on VSC and sliding mode control.
J.Y.Hung, W.Gao, J.C.Hung, “Variable Structure Control – A Survey”, IEEE Trans. Ind. Electron., Vol. 20, No. 1, pp. 2-22,Feb. 1993
A survey paper on VSC and sliding mode control concepts.
B. Draženović , "The invariance conditions in variable structure systems", Automatica, vol. 5, pp. 287, 1969
The paper proving that in case of matched disturbance, one can eliminate disturbance effect using appropriate control. Cited more than 250 times ‘officially’. Work done in one night.
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68. C. Edwards and S. Spurgeon, Sliding Mode control: Theory and Applications, Taylor and Francis, London, 1998
A good book on the subject.
L. Fridman and A. Levant, "Higher order sliding modes," in Sliding Mode Control in Engineering, Eds. W. Perruquetti and J. P. Barbot, Marcel Dekker Inc., 2002, pp
An initial paper on HOSM
X. Yu and Z. Man, On finite time convergence: Terminal sliding modes,” in Proc Int. Workshop on Variable Structure Systems, Kobe, Japan, pp. 164–168
Initial paper on TSM
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69. Thank You
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