Sliding mode control

Instead, it can switch from one continuous structure to another based on the current position in the state space.

Figure 1 shows an example trajectory of a system under sliding mode control.

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"[nb 1] in a tight neighborhood of the sliding surface.

Chattering can be reduced through the use of deadbands or boundary layers around the sliding surface, or other compensatory methods.

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).

(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.

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).

[4]: "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[nb 2] of applying a continuous signal to an output that can only take discrete states.

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.

In sliding-mode 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.

So the sliding-mode control acts like a stiff pressure always pushing in the direction of the sliding mode where

So the switching function is like a topographic map with a contour of constant height along which trajectories are forced to move.

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.

[7] The following theorems form the foundation of variable structure control.

, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero.

surface is reachable is given by That is, when initial conditions come entirely from this space, the Lyapunov function candidate

Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where

condition after some initial transient during the period while the system finds the sliding mode.

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.

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

Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods.

These non-linear high-gain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time.

Additionally, switched-mode observers have attractive measurement noise resilience that is similar to a Kalman filter.

[11][12] For simplicity, the example here uses a traditional sliding mode modification of a Luenberger observer for an LTI system.

The nonlinear control law v can be designed to enforce the sliding manifold so that estimate

equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them.

For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output

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

Figure 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 . The particular surface is chosen because it has desirable reduced-order dynamics when constrained to it. In this case, the surface corresponds to the first-order LTI system , which has an exponentially stable origin.