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

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

2 Sliding Mode Control2 Outline What is this Sliding mode and how did its study start? What is this Sliding mode and how did its study start? How to design controller using this concept? How to design controller using this concept?

3 Sliding Mode Control3 Primitive Examples - Electrical

4 Sliding Mode Control4 Primitive Examples-Mechanical

5 Sliding Mode Control5 First Formal Steps The first steps of sliding mode control theory originated in the early 1950s initiated by The first steps of sliding mode control theory originated in the early 1950s initiated by S. V. Emelyanov. S. V. Emelyanov. Started as VSC – Variable Structure Control Started as VSC – Variable Structure Control Varying system structure for stabilization. Varying system structure for stabilization.

6 Sliding Mode Control6 Variable Structure Control – Constituent Systems Mode 1 Mode 2

7 Sliding Mode Control7 Piecing together …

8 Sliding Mode Control8 Properties of VSC Both constituent systems were oscillatory and were not asymptotically stable. Both constituent systems were oscillatory and were not asymptotically stable. Combined system is asymptotically stable. Combined system is asymptotically stable. Property not present in any of the constituent system is obtained by VSC Property not present in any of the constituent system is obtained by VSC

9 Sliding Mode Control9 Another Example – Unstable Constituent Systems

10 Sliding Mode Control10 Analysis … Both systems are unstable Both systems are unstable Only stable mode is one mode of system Only stable mode is one mode of system IF the following VSC is employed IF the following VSC is employed

11 Sliding Mode Control11 Combined..

12 Sliding Mode Control12 In this case,… Again, property not present in constituent systems is found in the combined system. 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. 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. However, a more interesting behaviour can be observed if we use a different switching logic.

13 Sliding Mode Control13 The regions

14 Sliding Mode Control14 Sliding Mode New trajectory that was not present in any of the two original systems

15 Sliding Mode Control15 Sliding Mode ? Defined : Motion of the system trajectory along a chosen line/plane/surface of the state space. 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. Sliding Mode Control : Control designed with the aim to achieve sliding mode. Is usually of VSC type Is usually of VSC type Eg : Previous problem can be perceived as Eg : Previous problem can be perceived as

16 Sliding Mode Control16 What is the advantage? Consider a n-th order system represented in the phase variable form Consider a n-th order system represented in the phase variable form Also consider the sliding surface defined as Also consider the sliding surface defined as

17 Sliding Mode Control17 Advantage … Thus entire dynamics of the system is governed by the sliding line/surface parameters only Thus entire dynamics of the system is governed by the sliding line/surface parameters only In sliding mode, dynamics independent of system parameters (a 1,a 2,…). In sliding mode, dynamics independent of system parameters (a 1,a 2,…). ROBUST ROBUST

18 Sliding Mode Control18 Required Properties For sliding mode to be of any use, it should have the following 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) System stability confined to sliding surface (unstable sliding mode is NOT sliding mode at all) Sliding mode should not take forever to start Sliding mode should not take forever to start

19 Sliding Mode Control19 Stable Surface Consider the system Consider the system If the sliding function is designed as If the sliding function is designed as then confined to this surface ( ), the dynamics of can be written as

20 Sliding Mode Control20 The Surface … If K is so designed that has eigenvalues on LHP only, then the dynamics of is stable. If K is so designed that has eigenvalues on LHP only, then the dynamics of is stable. Since, the dynamics of is also stable. Since, the dynamics of is also stable. Hence, if the sliding surface is designed as Hence, if the sliding surface is designed as, the system dynamics confined to, the system dynamics confined to s=0 is stable. (Requirement 1) s=0 is stable. (Requirement 1) Note : Strictly speaking, it is not necessary for s to be a linear function of x

21 Sliding Mode Control21 Convergence to s=0 The second requirement is that sliding mode should start at a finite time. The second requirement is that sliding mode should start at a finite time. Split the requirement into further bits Split the requirement into further bits Sliding mode SHOULD start. Sliding mode SHOULD start. It should do so in finite time. It should do so in finite time.

22 Sliding Mode Control22 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. 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 … Mathematics … This is called the reachability condition This is called the reachability condition

23 Sliding Mode Control23 A figure to help out … s=0 s<0 s>0

24 Sliding Mode Control24 Insufficient Consider the case, Consider the case, This gives the solution of This gives the solution of is not enough (Violates Requirement 2) is not enough (Violates Requirement 2)

25 Sliding Mode Control25 -reachability -reachability With only, s slows down too much when close to zero to have finite time convergence With only, s slows down too much when close to zero to have finite time convergence Stronger condition is needed for finite time convergence. Stronger condition is needed for finite time convergence. Defined as -reachability condition Defined as -reachability condition s has a minimum rate of convergence s has a minimum rate of convergence

26 Sliding Mode Control26 Discontinuity Observe Observe So, at, is discontinuous. So, at, is discontinuous. -

27 Sliding Mode Control27 Discontinuous Dynamics Thus, for s>0, the system dynamics are Thus, for s>0, the system dynamics are and for s<0 Thus, at s=0, the dynamics is not well defined. Thus, at s=0, the dynamics is not well defined. The dynamics along the sliding surface is determined using continuation method The dynamics along the sliding surface is determined using continuation method

28 Sliding Mode Control28 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 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 sidesKluwer Academic Publishers,The Netherlands, 1988

29 Sliding Mode Control29 Diagrammatically Speaking …

30 Sliding Mode Control30 The reaching law approach In reaching law approach, the dynamics of the sliding function is directly expressed. It can have the general structure In reaching law approach, the dynamics of the sliding function is directly expressed. It can have the general structure

31 Sliding Mode Control31 Few Examples Constant rate reaching law Constant rate reaching law Constant+Proportional rate Constant+Proportional rate Power-rate reaching law Power-rate reaching law

32 Sliding Mode Control32 The Control Signal Now, consider the condition Now, consider the condition Thus, Thus, Or, control is Or, control is And the system dynamics is governed by And the system dynamics is governed by

33 Sliding Mode Control33 The Chattering Problem When, s is very close to zero, the control signal switches between two structures. 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. 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. Practically, actuators cannot switch at infinite frequency. So we have high frequency oscillations of non-zero magnitude. This undesirable phenomenon is called chattering. This undesirable phenomenon is called chattering.

34 Sliding Mode Control34 The picture Ideal Sliding Mode Practical – With Chattering

35 Sliding Mode Control35 Why is chattering undesirable? The high frequency of chattering actuates unmodeled high frequency dynamics of the system. Controller performance deteriorates. 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. More seriously, high frequency oscillations can cause mechanical wear in the system.

36 Sliding Mode Control36 Chattering avoidance/reduction The chattering problem is because signum function is used in control. The chattering problem is because signum function is used in control. Control changes very abruptly near s=0. Control changes very abruptly near s=0. Actuator tries to cope up leading to maximum- possible-frequency oscillations. Actuator tries to cope up leading to maximum- possible-frequency oscillations. Solution : Solution : Replace signum term in control by smoother choices Replace signum term in control by smoother choices

37 Sliding Mode Control37 Chattering Avoidance… Some choices of smooth functions Some choices of smooth functions Saturation function Hyperbolic tangent

38 Sliding Mode Control38 Disadvantage of smoothing If saturation or tanh is used, then we can observe that near s=0 If saturation or tanh is used, then we can observe that near s=0 Where represents the saturation or tanh function. Where represents the saturation or tanh function. The limit in both cases is zero. The limit in both cases is zero. So, technically the sliding mode is lost So, technically the sliding mode is lost

39 Sliding Mode Control39 What are the actual conditions for achieving Sliding Mode System is stable confined to System is stable confined to Control moves states towards this stable sliding surface Control moves states towards this stable sliding surface And does it in finite time. And does it in finite time.

40 Sliding Mode Control40 Some aspects of Continuous Sliding Mode Control Robustness Robustness Multivariable Sliding Mode Multivariable Sliding Mode Almost Sliding Mode Almost Sliding Mode

41 Sliding Mode Control41 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.

42 Sliding Mode Control42 Disturbance Consider the system with disturbance Consider the system with disturbance Disturbance comes through input channel Disturbance comes through input channel How does sliding mode behave in such a situation. How does sliding mode behave in such a situation.

43 Sliding Mode Control43 Disturbance Rejection The control law is designed so as to bring the system to the sliding surface. The control law is designed so as to bring the system to the sliding surface. Let us see dynamics confined to the sliding surface Let us see dynamics confined to the sliding surface Thus, Thus, Therefore, Therefore, And And Again, dynamics independent of disturbance. Again, dynamics independent of disturbance. Hence disturbance rejection.

44 Sliding Mode Control44 What if more than one input ? If system has more than one input, then the system can be transformed to the form If system has more than one input, then the system can be transformed to the form With having more than one elements. With having more than one elements. Thus, will also have multiple rows. Hence, the system can have more than one sliding surface Thus, will also have multiple rows. Hence, the system can have more than one sliding surface

45 Sliding Mode Control45 Approach to sliding surface Sliding mode will start when all sliding functions are zero. I.e, intersection of all sliding surfaces. Sliding mode will start when all sliding functions are zero. I.e, intersection of all sliding surfaces. Approach to the intersection Approach to the intersection Direct to intersection (Eventual) Direct to intersection (Eventual) Surface by surface Surface by surface In particular order (Fixed Order) In particular order (Fixed Order) First approach (Free order) First approach (Free order)

46 Sliding Mode Control46 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. 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. It does not necessary stay on any one of the sliding surfaces on approaching it.

47 Sliding Mode Control47 Eventual Sliding Mode

48 Sliding Mode Control48 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 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

49 Sliding Mode Control49 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 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

50 Sliding Mode Control50 Ordered Sliding Mode

51 Sliding Mode Control51 Chattering Refreshed A conventional sliding mode behaviour would have a sliding surface dynamics of the form 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 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 Chattering – Finite frequency, finite amplitude oscillations about the sliding surface

52 Sliding Mode Control52 Almost Sliding Mode To remedy chattering, the strict requirement of movement on sliding surface is relaxed. To remedy chattering, the strict requirement of movement on sliding surface is relaxed. We try to get Almost – sliding mode (Quasi sliding mode) We try to get Almost – sliding mode (Quasi sliding mode)

53 Sliding Mode Control53 Saturation function based Sliding Mode Control Instead of Instead of Inside the band |s|<, the reaching law is linear as This is also called boundary layer technique

54 Sliding Mode Control54 The motion S=0 S=- S=

55 Sliding Mode Control55 Disadvantage Almost is NOT exact Almost is NOT exact

56 Sliding Mode Control56 Newer Avenues Two phases in sliding motion : Reaching Phase and Sliding Phase Two phases in sliding motion : Reaching Phase and Sliding Phase Improvements Improvements Reaching Phase - Higher Order Sliding Mode Control Reaching Phase - Higher Order Sliding Mode Control Sliding Phase - Terminal Sliding Mode Control Sliding Phase - Terminal Sliding Mode Control

57 Sliding Mode Control57 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. Basic Definition of Sliding Mode : s(x)=0 in finite time. Sliding surface reached in finite time and stays on it. Problem : Chattering results Problem : Chattering results Solution : Try to get ds/dt = 0, additionally in finite time. Second order sliding mode. 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. Get the first n-1 derivatives of s(x) to zero in finite time. n-th order sliding mode.

58 Sliding Mode Control58 Advantage Smooth control results. No Chattering. Smooth control results. No Chattering. Disadvantage : Not very straight forward. Disadvantage : Not very straight forward.

59 Sliding Mode Control59 HOSM : Twisting Algorithm Applicable to systems of relative degree 2. Applicable to systems of relative degree 2. Input appears in 2 nd derivative of sliding function. Input appears in 2 nd derivative of sliding function. Input is still discontinous. However, there is no chattering in states. Input is still discontinous. However, there is no chattering in states.

60 Sliding Mode Control60 Typical Twisting Trajectory

61 Sliding Mode Control61 Super Twisting Algorithm For systems with relative degree 1. For systems with relative degree 1. Switching shifted to derivative of input. Switching shifted to derivative of input. Input continuous and so is derivative of sliding function. Input continuous and so is derivative of sliding function. No chattering here too. No chattering here too.

62 Sliding Mode Control62 Typical Super-twisting Trajectory

63 Sliding Mode Control63 New Idea If we are concerned with getting an output to zero, why not set s=y!! If we are concerned with getting an output to zero, why not set s=y!! Are there any extra conditions? Are there any extra conditions? Zero Dynamics Zero Dynamics

64 Sliding Mode Control64 Terminal Sliding Mode Higher order sliding mode is about reaching the sliding surface smoothly. 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. 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.

65 Sliding Mode Control65 The sliding surface Terminal Sliding Mode Terminal Sliding Mode Fast close to origin. Finite time convergence. Fast-Terminal Sliding Mode Fast-Terminal Sliding Mode

66 Sliding Mode Control66 Terminal Sliding Surface … In case of systems with more than 2 states, For the system in phase variable form, For the system in phase variable form,

67 Sliding Mode Control67 References V. Utkin, Variable Structure Systems with Sliding Mode, IEEE Trans. Automat. Contr., AC-12, No. 2, pp. 212-222, 1977 V. Utkin, Variable Structure Systems with Sliding Mode, IEEE Trans. Automat. Contr., AC-12, No. 2, pp. 212-222, 1977 An introductory paper on VSC and sliding mode control. 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 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. 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 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. 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.

68 Sliding Mode Control68 C. Edwards and S. Spurgeon, Sliding Mode control: Theory and Applications, Taylor and Francis, London, 1998 C. Edwards and S. Spurgeon, Sliding Mode control: Theory and Applications, Taylor and Francis, London, 1998 A good book on the subject. 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. 53-101. 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. 53-101. An initial paper on HOSM An initial paper on HOSM X. Yu and Z. Man, On finite time convergence: Terminal sliding modes, in Proc. 1996 Int. Workshop on Variable Structure Systems, Kobe, Japan, 1996. pp. 164–168 X. Yu and Z. Man, On finite time convergence: Terminal sliding modes, in Proc. 1996 Int. Workshop on Variable Structure Systems, Kobe, Japan, 1996. pp. 164–168 Initial paper on TSM Initial paper on TSM

69 Sliding Mode Control69


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