Variable-Structure & Sliding-Mode Control

A variable-structure controller switches among continuous laws according to state. In sliding-mode control (SMC), the switching is chosen so trajectories reach a manifold $s(x)=0$ and then evolve along it with desirable reduced-order dynamics.

For a second-order tracking error, one common surface is

$$s=\dot e+\lambda e,\qquad \lambda>0.$$

If control makes $s$ reach zero, motion on the surface satisfies $\dot e=-\lambda e$, so tracking error decays exponentially.

flowchart TD
  X1["state: s > 0"] -->|"switching law drives s downward"| M["sliding manifold s = 0"]
  X2["state: s < 0"] -->|"switching law drives s upward"| M
  M -->|"reduced dynamics"| O["tracking objective"]

Reaching condition #

A standard Lyapunov candidate is $V=s^2/2$. A sufficient reaching condition is

$$\dot V=s\dot s\le-\eta|s|,\qquad \eta>0.$$

An ideal law contains a discontinuous term such as

$$u=u_{eq}-k\,\mathop{\mathrm{sign}}(s),$$

where $u_{eq}$ would keep $\dot s=0$ on the manifold and $k$ must dominate bounded uncertainty entering through the same channel as the control. The switching term points toward the surface from both sides.

The phrase matched uncertainty is crucial. If an unknown disturbance enters in the input direction, enough switching authority may reject it. Unmatched disturbances that push state through another channel generally affect the sliding dynamics and are not magically canceled.

Ideal sliding versus physical switching #

Mathematical sliding assumes arbitrarily fast switching. Real sensors, processors and actuators have finite bandwidth, delay, hysteresis and unmodeled modes. The result is chattering: high-frequency motion around the surface. Chattering can heat actuators, excite structural resonance, amplify noise and invalidate the model used to justify robustness.

Common mitigations include:

• replace $\mathop{\mathrm{sign}}(s)$ with a saturation function inside a boundary layer $|s|<\phi$;
• add hysteresis or enforce a switching period;
• use higher-order sliding algorithms when their assumptions hold;
• design actuator dynamics as part of the plant;
• estimate unmeasured state with bandwidth consistent with noise.

A boundary layer trades exact invariance for bounded tracking error. State that bound; do not keep the ideal robustness claim after smoothing the controller.

Design procedure #

1. Choose outputs and define tracking error.
2. Select a surface whose internal/sliding dynamics are stable.
3. Derive $\dot s$ including disturbances and actuator dynamics.
4. Prove a reaching condition over a stated uncertainty set.
5. Check control magnitude, rate and switching-frequency limits.
6. Replace ideal switching with an implementable law and re-derive the error bound.
7. Simulate and test noise, sampling, delay, saturation and unmodeled resonances.

The surface must also be reachable from allowed initial states without violating constraints. A proof that ignores saturation may ask for unbounded control exactly where robustness is claimed.

What SMC is and is not #

SMC is attractive when the plant admits a useful switching surface, uncertainty bounds are credible, and fast actuation is physically appropriate. It is not synonymous with any relay controller: an ordinary thermostat has switching and hysteresis but no designed invariant manifold. It is not a universal remedy for model uncertainty, and it does not remove the need to model sensing and actuation.

The robust object is the closed-loop motion near a particular manifold under a particular uncertainty structure. Preserve those qualifiers whenever applying the method.