Sliding Mode Control - Control Scheme

Control Scheme

Consider a nonlinear dynamical system described by

\dot{\mathbf{x}}(t)=f(\mathbf{x},t) + B(\mathbf{x},t)\,\mathbf{u}(t)

where

is an -dimensional state vector and

is an -dimensional 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 state-feedback control law (i.e., a mapping from current state at time 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 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 quickly returns to, then the output will track the desired sinusoid. 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. Sliding mode control forces the system trajectories into this subspace and then holds them there so that they slide along it. This reduced-order subspace is referred to as a sliding (hyper)surface, and when closed-loop feedback forces trajectories to slide along it, it is referred to as a sliding mode of the closed-loop 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 high-gain feedback. Like a marble rolling along a crack, trajectories are confined to the sliding mode.

The sliding-mode control scheme involves

  1. Selection of a hypersurface or a manifold (i.e., the sliding surface) such that the system trajectory exhibits desirable behavior when confined to this manifold.
  2. Finding feedback gains so that the system trajectory intersects and stays on the manifold.

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 sliding-mode designer picks a switching function that represents a kind of "distance" that the states are away from a sliding surface.

  • A state that is outside of this sliding surface has .
  • A state that is on this sliding surface has .

The sliding-mode-control law switches from one state to another based on the sign of this distance. So the sliding-mode 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 is the number of states in and is the number of input signals (i.e., control signals) in . For each control index, there is an sliding surface given by

\left\{ \mathbf{x} \in \mathbb{R}^n : \sigma_k(\mathbf{x}) = 0 \right\}

The vital part of SMC 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 reduced-order system obtained from Equation (2).

To force the system states to satisfy, one must:

  1. Ensure that the system is capable of reaching from any initial condition
  2. Having reached, the control action is capable of maintaining the system at

Read more about this topic:  Sliding Mode Control

Famous quotes containing the words control and/or scheme:

    “The child knows only that he engages in play because it is enjoyable. He isn’t aware of his need to play—a need which has its source in the pressure of unsolved problems. Nor does he know that his pleasure in playing comes from a deep sense of well-being that is the direct result of feeling in control of things, in contrast to the rest of his life, which is managed by his parents or other adults.”
    —Bruno Bettelheim (20th century)

    “In the scheme of our national government, the presidency is preeminently the people’s office.”
    —Grover Cleveland (1837–1908)