In control theory, sliding mode control, or SMC, is a form of variable structure control (VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system by application of a highfrequency switching control. The statefeedback 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 multiple control structures are designed so that trajectories always move toward a switching condition, and so the ultimate trajectory will not exist entirely within one control structure. Instead, the ultimate trajectory will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode^{[1]} and the geometrical locus consisting of the boundaries is called the sliding (hyper)surface. Figure 1 shows an example trajectory of a system under sliding mode control. The sliding surface is described by s = 0, and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the context of modern control theory, any variable structure system, like a system under SMC, may be viewed as a special case of a hybrid dynamical system.
Intuitively, sliding mode control uses practically infinite gain to force the trajectories of a dynamic system to slide along the restricted sliding mode subspace. Trajectories from this reducedorder sliding mode have desirable properties (e.g., the system naturally slides along it until it comes to rest at a desired equilibrium). The main strength of sliding mode control is its 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, optimality requires the use of bang–bang control; hence, sliding mode control describes the optimal controller for a broad set of dynamic systems.
One application of sliding mode controllers is the control of electric drives operated by switching power converters.^{[2]}^{:"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 pulsewidth modulation or a similar technique^{[nb 1]} of applying a continuous signal to an output that can only take discrete states.
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 slidingmodecontrol action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics.^{[3]}^{:554–556} Continuous control design methods are not as susceptible to these problems and can be made to mimic slidingmode controllers.^{[3]}^{:556–563}
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Consider a nonlinear dynamical system described by

where
is an ndimensional state vector and
is an mdimensional input vector that will be used for state feedback. The functions and are assumed to be continuous and sufficiently smooth so that the Picard–Lindelöf theorem can be used to guarantee that solution to Equation (1) exists and is unique.
A common task is to design a statefeedback control law (i.e., a mapping from current state at time t to the input ) to stabilize the dynamical system in Equation (1) around the origin . That is, under the control law, whenever the system is started away from the origin, it will return to it. For example, the component x_{1} of the state vector may represent the difference some output is away from a known signal (e.g., a desirable sinusoidal signal); if the control can ensure that x_{1} quickly returns to x_{1} = 0, then the output will track the desired sinusoid. In slidingmode control, the designer knows that the system behaves desirably (e.g., it has a stable equilibrium) provided that it is constrained to a subspace of its configuration space. Sliding mode control forces the system trajectories into this subspace and then holds them there so that they slide along it. This reducedorder subspace is referred to as a sliding (hyper)surface, and when closedloop feedback forces trajectories to slide along it, it is referred to as a sliding mode of the closedloop system. Trajectories along this subspace can be likened to trajectories along eigenvectors (i.e., modes) of LTI systems; however, the sliding mode is enforced by creasing the vector field with highgain feedback. Like a marble rolling along a crack, trajectories are confined to the sliding mode.
The slidingmode control scheme involves
Because sliding mode control laws are not continuous, it has the ability to drive trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better than asymptotic). However, once the trajectories reach the sliding surface, the system takes on the character of the sliding mode (e.g., the origin may only have asymptotic stability on this surface).
The slidingmode designer picks a switching function that represents a kind of "distance" that the states are away from a sliding surface.
The slidingmodecontrol law switches from one state to another based on the sign of this distance. So the slidingmode control acts like a stiff pressure always pushing in the direction of the sliding mode where . Desirable trajectories will approach the sliding surface, and because the control law is not continuous (i.e., it switches from one state to another as trajectories move across this surface), the surface is reached in finite time. Once a trajectory reaches the surface, it will slide along it and may, for example, move toward the origin. So the switching function is like a topographic map with a contour of constant height along which trajectories are forced to move.
The sliding (hyper)surface is of dimension where n is the number of states in and m is the number of input signals (i.e., control signals) in . For each control index , there is an sliding surface given by

The vital part of VSC design is to choose a control law so that the sliding mode (i.e., this surface given by ) exists and is reachable along system trajectories. The principle of sliding mode control is to forcibly constrain the system, by suitable control strategy, to stay on the sliding surface on which the system will exhibit desirable features. When the system is constrained by the sliding control to stay on the sliding surface, the system dynamics are governed by reducedorder system obtained from Equation (2).
To force the system states to satisfy , one must:
Note that because the control law is not continuous, it is certainly not locally Lipschitz continuous, and so existence and uniqueness of solutions to the closedloop system is not guaranteed by the Picard–Lindelöf theorem. Thus the solutions are to be understood in the Filippov sense^{[4]}^{[1]}. Roughly speaking, the resulting closedloop system moving along is approximated by the smooth dynamics ; however, this smooth behavior may not be truly realizable. Similarly, highspeed pulsewidth modulation or deltasigma modulation produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different nonlinear control design method that produces a continuous controller. In some cases, slidingmode control designs can be approximated by other continuous control designs.^{[3]}
The following theorems form the foundation of variable structure control.
Consider a Lyapunov function candidate

where is the Euclidean norm (i.e, is the distance away from the sliding manifold where ). For the system given by Equation (1) and the sliding surface given by Equation (2), a sufficient condition for the existence of a sliding mode is that
in a neighborhood of the surface given by .
Roughly speaking (i.e., for the scalar control case when m = 1), to achieve , the feedback control law is picked so that σ and have opposite signs. That is,
Note that
and so the feedback control law has a direct impact on .
To ensure that the sliding mode is attained in finite time, must be more strongly bounded away from zero. That is, if it vanishes too quickly, the attraction to the sliding mode will only be asymptotic. To ensure that the sliding mode is entered in finite time,^{[5]}
where μ > 0 and are constants.
This condition ensures that for the neighborhood of the sliding mode ,
So, for ,
which, by the chain rule (i.e., with ), means
where D ^{+} is the upper righthand derivative of and the symbol denotes proportionality. So, by comparison to the curve z(t) = z_{0} − μt which is represented by differential equation with initial condition z(0) = z_{0}, it must be the case that for all t. Moreover, because , must reach in finite time, which means that V must reach V = 0 (i.e., the system enters the sliding mode) in finite time.^{[3]} Because is proportional to the Euclidean norm of the switching function σ, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero.
In the context of sliding mode control, this condition means that
where is the Euclidean norm. For the case when switching function σ is scalar valued, the sufficient condition becomes
Taking α = 1, the scalar sufficient condition becomes
which is equivalent to the condition that
That is, the system should always be moving toward the switching surface σ = 0, and its speed toward the switching surface should have a nonzero lower bound. So, even though σ may become vanishingly small as approaches the surface, must always be bounded firmly away from zero. To ensure this condition, sliding mode controllers are discontinuous across the σ = 0 manifold; they switch from one nonzero value to another as trajectories cross the manifold.
For the system given by Equation (1) and sliding surface given by Equation (2), the subspace for which the surface is reachable is given by
That is, when initial conditions come entirely from this space, the Lyapunov function candidate V(σ) is a Lyapunov function and trajectories are sure to move toward the sliding mode surface where . Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where is more strongly bounded away from zero in finite time. Hence, the sliding mode σ = 0 will be attained in finite time.
Let
be nonsingular. That is, the system has a kind of controllability that ensures that there is always a control that can move a trajectory to move closer to the sliding mode. Then, once the sliding mode where is achieved, the system will stay on that sliding mode. Along sliding mode trajectories, is constant, and so sliding mode trajectories are described by the differential equation
If an equilibrium is stable with respect to this differential equation, then the system will slide along the sliding mode surface toward the equilibrium.
The equivalent control law on the sliding mode can be found by solving
for the equivalent control law . That is,
and so the equivalent control
That is, even though the actual control is not continuous, the rapid switching across the sliding mode where forces the system to act as if it were driven by this continuous control.
Likewise, the system trajectories on the sliding mode behave as if
The resulting system matches the sliding mode differential equation
and so as long as the sliding mode surface where is stable (in the sense of Lyapunov), the system can be assumed to follow the simpler condition after some initial transient during the period while the system finds the sliding mode. The same motion is approximately maintained provided the equality only approximately holds.
It follows from these theorems that the sliding motion is invariant (i.e., insensitive) to sufficiently small disturbances entering the system through the control channel. That is, as long as the control is large enough to ensure that and is uniformly bounded away from zero, the sliding mode will be maintained as if there was no disturbance. The invariance property of sliding mode control to certain disturbances and model uncertainties is its most attractive feature; it is strongly robust.
As discussed in an example below, a sliding mode control law can keep the constraint
in order to asymptotically stabilize any system of the form
when has a finite upper bound. In this case, the sliding mode is where
(i.e., where ). That is, when the system is constrained this way, it behaves like a simple stable linear system, and so it has a globally exponentially stable equilibrium at the origin.


Sliding mode control can be used in the design of state observers. These nonlinear highgain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time. Additionally, switchedmode observers have attractive measurement noise resilience that is similar to a Kalman filter.^{[6]}^{[7]} For simplicity, the example here uses a traditional sliding mode modification of a Luenberger observer for an LTI system. In these sliding mode observers, the order of the observer dynamics are reduced by one when the system enters the sliding mode. In this particular example, the estimator error for a single estimated state is brought to zero in finite time, and after that time the other estimator errors decay exponentially to zero. However, as first described by Drakunov,^{[8]} a sliding mode observer for nonlinear systems can be built that brings the estimation error for all estimated states to zero in an finite (and arbitrarily small) time.
Here, consider the LTI system
where state vector , is a vector of inputs, and output y is a scalar equal to the first state of the state vector. Let
where
The goal is to design a highgain state observer that estimates the state vector using only information from the measurement y = x_{1}. Hence, let the vector be the estimates of the n states. The observer takes the form
where is a nonlinear function of the error between estimated state and the output y = x_{1}, and is an observer gain vector that serves a similar purpose as in the typical linear Luenberger observer. Likewise, let
where is a column vector. Additionally, let be the state estimator error. That is, . The error dynamics are then
where is the estimator error for the first state estimate. The nonlinear control law v can be designed to enforce the sliding manifold
so that estimate tracks the real state x_{1} after some finite time (i.e., ). Hence, the sliding mode control switching function
To attain the sliding manifold, and σ must always have opposite signs (i.e., for essentially all ). However,
where is the collection of the estimator errors for all of the unmeasured states. To ensure that , let
where
That is, positive constant M must be greater that a scaled version of the maximum possible estimator errors for the system (i.e., the initial errors, which are assumed to be bounded so that M can be picked large enough; al). If M is sufficiently large, it can be assumed that the system achieves e_{1} = 0 (i.e., ). Because e_{1} is constant (i.e., 0) along this manifold, as well. Hence, the discontinuous control v(σ) may be replaced with the equivalent continuous control v_{eq} where
So
This equivalent control v_{eq} represents the contribution from the other (n − 1) states to the trajectory of the output state x_{1}. In particular, the row A_{12} acts like an output vector for the error subsystem
So, to ensure the estimator error for the unmeasured states converges to zero, the vector L_{2} must be chosen so that the matrix (A_{2} + L_{2}A_{12}) is Hurwitz (i.e., the real part of each of its eigenvalues must be negative). Hence, provided that it is observable, this system can be stabilized in exactly the same way as a typical linear state observer when A_{12} is viewed as the output matrix (i.e, "C"). That is, the v_{eq} equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control forces the estimate of the measured state to have zero error in finite time. Additionally, white zeromean symmetric measurement noise (e.g., Gaussian noise) only affects the switching frequency of the control v, and hence the noise will have little effect on the equivalent sliding mode control v_{eq}. Hence, the sliding mode observer has Kalman filter–like features.^{[7]}
The final version of the observer is thus
where
That is, by augmenting the control vector with the switching function , the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal is viewed as a control input to the 2input LTI system.
For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output y = x_{1}). However, a similar procedure can be used to design a sliding mode observer for a vector of weighted combinations of states (i.e., when output uses a generic matrix C). In each case, the sliding mode will be the manifold where the estimated output follows the measured output with zero error (i.e., the manifold where ).
