Sliding Mode Control – An Introduction S. Janardhanan IIT Delhi

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

Primitive Examples - Electrical Sliding Mode Control

Primitive Examples-Mechanical Sliding Mode Control

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. Sliding Mode Control

Variable Structure Control – Constituent Systems Both systems have their eigenvalues on the imaginary axis. Neither system is asymptotically stable. Mode 2 Mode 1 Sliding Mode Control

Piecing together … Sliding Mode Control

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 Sliding Mode Control

Another Example – Unstable Constituent Systems Zeta and alpha are both positive. In both cases, the system is unstable. Sliding Mode Control

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

Combined .. Sliding Mode Control

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. Sliding Mode Control

The regions Sliding Mode Control

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

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 Sliding Mode Control

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

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 Sliding Mode Control

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 Sliding Mode Control

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

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 Sliding Mode Control

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. Sliding Mode Control

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’ Sliding Mode Control

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

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

-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 Sliding Mode Control

Discontinuity Observe So, at , is discontinuous.  - Sliding Mode Control

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 Sliding Mode Control

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 Sliding Mode Control

Diagrammatically Speaking … Sliding Mode Control

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

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

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

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. Sliding Mode Control

The picture Ideal Sliding Mode Practical – With Chattering Sliding Mode Control

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. Sliding Mode Control

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’ Sliding Mode Control

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

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 Sliding Mode Control

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. Sliding Mode Control

Some aspects of Continuous Sliding Mode Control Robustness Multivariable Sliding Mode ‘Almost’ Sliding Mode Sliding Mode Control

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. Sliding Mode Control

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

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. Sliding Mode Control

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 Sliding Mode Control

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) Sliding Mode Control

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. Sliding Mode Control

Eventual Sliding Mode Sliding Mode Control

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 Sliding Mode Control

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 Sliding Mode Control

Ordered Sliding Mode Sliding Mode Control

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 Sliding Mode Control

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) Sliding Mode Control

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 Sliding Mode Control

The motion S= S=- S=0 Sliding Mode Control

Disadvantage ‘Almost’ is NOT exact Sliding Mode Control

‘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 Sliding Mode Control

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. Sliding Mode Control

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

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. Sliding Mode Control

Typical Twisting Trajectory Sliding Mode Control

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. Sliding Mode Control

Typical Super-twisting Trajectory Sliding Mode Control

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

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. Sliding Mode Control

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

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

References 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. 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. Sliding Mode Control

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. 53-101. 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 Initial paper on TSM Sliding Mode Control

Thank You Sliding Mode Control