Editing Sliding mode control (section)

==Introduction==
[[File:First order sliding mode control.svg|thumb|Figure&nbsp;1: [[Phase plane]] trajectory of a system being stabilized by a sliding mode controller. After the initial reaching phase, the system states "slides" along the line <math>s=0</math>. The particular <math>s=0</math> surface is chosen because it has desirable reduced-order dynamics when constrained to it. In this case, the <math>s=x_1 +\dot{x}_1 = 0</math> surface corresponds to the first-order [[LTI system]] <math>\dot{x}_1 = -x_1</math>, which has an [[exponentially stable]] origin.]]

Figure&nbsp;1 shows an example trajectory of a system under sliding mode control. The sliding surface is described by <math>s=0</math>, and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the theoretical description of sliding modes, the system stays confined to the sliding surface and need only be viewed as sliding along the surface. However, real implementations of sliding mode control approximate this theoretical behavior with a high-frequency and generally non-deterministic switching control signal that causes the system to "chatter"<ref group="nb">'Chatter' or 'chattering' is the undesirable phenomenon of [[oscillation]]s having finite frequency and amplitude. Chattering is a harmful phenomenon because it leads to low control accuracy, high wear of moving mechanical parts, and high heat losses in power circuits.

For more details, see Utkin, Vadim; Lee, Jason Hoon (July 2006), ''Chattering Problem in Sliding Mode Control Systems'', vol. 10.1109/VSS.2006.1644542., pp. 346–350</ref> in a tight neighborhood of the sliding surface. Chattering can be reduced through the use of [[deadband]]s or boundary layers around the sliding surface, or other compensatory methods. Although the system is nonlinear in general, the idealized (i.e., non-chattering) behavior of the system in Figure&nbsp;1 when confined to the <math>s=0</math> surface is an [[LTI system]] with an [[exponentially stable]] origin.
One of the compensatory methods is the adaptive sliding mode control method proposed in 
<ref name="Zeinali2010">{{cite journal
 | author1 = Zeinali M.
 | author2 = Notash L.
 | year = 2010
 | title = Adaptive sliding mode control with uncertainty estimator for robot manipulators
 | journal = International Journal of Mechanism and Machine Theory
 | volume = 45
 | number = 1
 | pages = 80–90
 | doi = 10.1016/j.mechmachtheory.2009.08.003
}}</ref>
<ref name="Zeinali2018">{{cite conference |url= http://www.vss-graz.com/index.php/en/|title= First-Order Continuous Adaptive Sliding Mode Control for Robot Manipulators with Finite-Time Convergence of Trajectories to Real sliding Mode|last= Zeinali|first=Meysar |location=Graz University of Technology, Austria |date= 2018 |conference= 15th International Workshop on Variable Structure Systems and Sliding Mode Control }}</ref> which uses estimated uncertainty to construct continuous control law. In this method chattering is eliminated while preserving accuracy (for more details see references [2] and [3]). The three distinguished features of the proposed adaptive sliding mode controller are as follows: (i) The structured (or parametric) uncertainties and unstructured uncertainties (un-modeled dynamics, unknown external disturbances) are synthesized into a single type uncertainty term called lumped uncertainty. Therefore, a linearly parameterized dynamic model of the system is not required, and the simple structure and computationally efficient properties of this approach make it suitable for the real-time control applications. (ii) The adaptive sliding mode control scheme design relies on the online estimated uncertainty vector rather than relying on the worst-case scenario (i.e., bounds of uncertainties). Therefore, a-priory knowledge of the bounds of uncertainties is not required, and at each time instant, the control input compensates for the uncertainty that exists. (iii) The developed continuous control law using fundamentals of the sliding mode control theory eliminates the chattering phenomena without trade-off between performance and robustness, which is prevalent in boundary-layer approach.

Intuitively, sliding mode control uses practically infinite [[Gain (electronics)|gain]] to force the trajectories of a [[dynamic system]] to slide along the restricted sliding mode subspace. Trajectories from this reduced-order sliding mode have desirable properties (e.g., the system naturally slides along it until it comes to rest at a desired [[stationary point|equilibrium]]). The main strength of sliding mode control is its [[robust control|robustness]]. Because the control can be as simple as a switching between two states (e.g., "on"/"off" or "forward"/"reverse"), it need not be precise and will not be sensitive to parameter variations that enter into the control channel. Additionally, because the control law is not a [[continuous function]], the sliding mode can be reached in ''finite'' time (i.e., better than asymptotic behavior). Under certain common conditions, [[optimal control|optimality]] requires the use of [[bang–bang control]]; hence, sliding mode control describes the [[optimal control]]ler for a broad set of dynamic systems.

One application of sliding mode controller is the control of electric drives operated by switching power converters.<ref name="Utkin93">{{Cite journal|doi=10.1109/41.184818|last=Utkin|first= Vadim I.|year=1993|title=Sliding Mode Control Design Principles and Applications to Electric Drives|journal=IEEE Transactions on Industrial Electronics|volume=40|issue=1|pages=23–36|citeseerx=10.1.1.477.77}}</ref>{{rp|"Introduction"}} Because of the discontinuous operating mode of those converters, a discontinuous sliding mode controller is a natural implementation choice over continuous controllers that may need to be applied by means of [[pulse-width modulation]] or a similar technique<ref group="nb">Other pulse-type modulation techniques include [[delta-sigma modulation]].</ref> of applying a continuous signal to an output that can only take discrete states. Sliding mode control has many applications in robotics. In particular, this control algorithm has been used for tracking control of unmanned surface vessels in simulated rough seas with high degree of success.<ref>[https://www.youtube.com/watch?v=hD1hxwupECs "Autonomous Navigation and Obstacle Avoidance of Unmanned Vessels in Simulated Rough Sea States - Villanova University"]</ref><ref name="Mahini2013">{{Cite journal|last=Mahini|year=2013|title=An experimental setup for autonomous operation of surface vessels in rough seas
|journal=Robotica|volume=31|issue=5|pages=703–715|doi=10.1017/s0263574712000720|s2cid=31903795 |display-authors=etal}}</ref>

Sliding mode control must be applied with more care than other forms of [[nonlinear control]] that have more moderate control action. In particular, because actuators have delays and other imperfections, the hard sliding-mode-control action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics.<ref name="Khalil02"/>{{rp|554&ndash;556}} Continuous control design methods are not as susceptible to these problems and can be made to mimic sliding-mode controllers.<ref name="Khalil02"/>{{rp|556&ndash;563}}