SlidingModeControl.md

Sliding Mode Control

Overview & Motivation

Variable-structure control systems change their structure depending on the current system state. Sliding Mode Control (SMC) is the most widely used instantiation: the controller switches between two or more structures to drive the state onto a designer-specified manifold — the sliding surface — and keep it there. Once constrained to the surface, the closed-loop dynamics are governed entirely by the surface geometry, independent of the plant model or matched disturbances. This structural robustness makes SMC a preferred choice for motor drives, DC-DC power converters, and any embedded plant whose parameters drift or are poorly known.

Mathematical Theory

State-Space Plant

The controller operates on a discrete-time linear plant

$$x_{k+1} = A x_k + B u_k, \quad x \in \mathbb{R}^n,; u \in \mathbb{R}^m$$

where $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$.

Sliding Surface

A linear sliding surface is defined by

$$s(x) = S x, \quad S \in \mathbb{R}^{m \times n}$$

The sliding manifold ${x : s(x) = 0}$ is an $(n-m)$-dimensional subspace. The matrix $S$ is designed so the reduced-order dynamics on the manifold are stable and meet the desired closed-loop poles.

Equivalent Control

The equivalent control $u_{eq}$ is the unique input that holds the state on $s = 0$ (i.e., $\dot{s} = 0$) for the nominal plant:

$$u_{eq} = -(S B)^{-1} S A x$$

This exists if and only if $S B$ is nonsingular, which is the relative-degree-one condition: each input channel must directly influence its corresponding sliding variable.

Switching (Reaching) Term

The switching term adds a robust push toward the surface:

$$u_{sw} = (S B)^{-1} K \mathrm{sat}(s/\phi)$$

where $K \in \mathbb{R}^{m}$ is the per-channel switching gain and $\phi > 0$ is the boundary-layer thickness. The saturation function

$$\mathrm{sat}(\sigma) = \begin{cases} \sigma & |\sigma| \le 1 \ \mathrm{sign}(\sigma) & |\sigma| > 1 \end{cases}$$

replaces the discontinuous $\mathrm{sign}(s)$ of ideal SMC with a continuous ramp inside $|s| \le \phi$, eliminating infinite-bandwidth chattering while maintaining the reaching property.

Total Control Law

$$u = u_{eq} - u_{sw} = -(S B)^{-1}\bigl[S A x + K,\mathrm{sat}(s/\phi)\bigr]$$

Reaching Condition

The Lyapunov function $V = \tfrac{1}{2} s^\top s$ satisfies $\dot{V} < 0$ outside the boundary layer when $K_i > |d_i|$ for each matched-disturbance channel $d_i$. This guarantees finite-time arrival at $|s| \le \phi$.

Sliding-Mode Dynamics

On the surface $s = 0$, the state evolves according to the $(n-m)$-dimensional reduced-order system. For a single-input system with $S = [c_1, \ldots, c_{n-1}, 1]$, the sliding pole is determined by the characteristic polynomial of the first $(n-1)$ rows under the surface constraint.

Complexity Analysis

Operation	Time	Space	Notes
Construction	$O(m^3 + n m)$	$O(nm + m^2)$	$(SB)^{-1}$ inversion once
ComputeControl	$O(n^2 + m^2)$	$O(1)$ extra	$S A x$ dominates for large $n$
Surface	$O(nm)$	$O(1)$ extra	monitoring only

All data is stored in fixed-size arrays; no heap allocation at any point.

Step-by-Step Walkthrough

Consider a double-integrator plant ($n=2$, $m=1$):

$$A = \begin{bmatrix}0 & 1\0 & 0\end{bmatrix}, \quad B = \begin{bmatrix}0\1\end{bmatrix}$$

with surface $S = [1; 1]$, gain $K = 2$, boundary layer $\phi = 0.05$.

1. Precompute $SB = 1$, so $(SB)^{-1} = 1$.
2. At state $x = [1, 0]^\top$: $s = 1 > \phi$, $\mathrm{sat}(s/\phi) = 1$.
3. Equivalent control: $u_{eq} = -(SA)x = -[0;1][1;0]^\top = 0$.
4. Switching term: $u_{sw} = 1 \cdot 2 \cdot 1 = 2$.
5. Total: $u = 0 - 2 = -2$. The large negative input decelerates the state toward the surface.
6. Once $|s| \le 0.05$, the saturation ramps linearly and the control becomes smooth.

Pitfalls & Edge Cases

• Singular $SB$: the equivalent control is undefined. This occurs when the surface does not satisfy the relative-degree-one condition. Assert non-singularity at construction.
• Chattering: ideal SMC ($\phi \to 0$) switches at infinite frequency, exciting unmodeled dynamics and wearing actuators. The boundary layer is not optional for real hardware.
• Gain too small: if $K_i \le |d_i|$, the switching term cannot overcome the disturbance and the surface is never reached. The state remains bounded but does not converge.
• Boundary-layer error: the steady-state tracking error is $O(\phi)$. Reducing $\phi$ improves accuracy at the cost of higher-frequency control activity.
• Discrete-time reaching: the Zeno-like finite-time result holds in continuous time; in discrete time the state enters an $O(\phi + \Delta t)$ neighborhood of the surface.

Variants & Generalizations

• Higher-order SMC (super-twisting): drives both $s$ and $\dot{s}$ to zero simultaneously, eliminating chattering without a boundary layer at the cost of requiring $\dot{s}$ estimates.
• Terminal SMC: uses a nonlinear surface $s = \dot{e} + \beta e^{p/q}$ to achieve finite-time convergence to the origin (not just to the surface).
• Integral SMC: augments the surface with an integral of the state to achieve zero steady-state error in the presence of constant disturbances.
• Adaptive switching gain: adjusts $K$ online to match the unknown disturbance bound, avoiding over-gain chattering.

Applications

• Brushless motor current and speed control: fast switching, large disturbance rejection.
• DC-DC converters: inherent switching structure matches SMC's variable-structure nature.
• Pneumatic and hydraulic actuators: strong friction/backlash rejection via matched-disturbance cancellation.
• Satellite attitude control: robustness to inertia uncertainty and external torques.

Connections to Other Algorithms

• LQR: smooth optimal alternative; minimises quadratic cost but has no guaranteed robustness to matched disturbances.
• Disturbance Observer (DOB): estimates and cancels the disturbance algebraically; complementary to SMC for unmatched disturbances.
• SaturationRateLimiter: the boundary-layer sat function is semantically identical to the saturation block already in the library.
• LuenbergerObserver: may be combined with SMC when the full state is not measured (output feedback SMC).

References & Further Reading

• V. Utkin, "Variable Structure Systems with Sliding Modes," IEEE Transactions on Automatic Control, 22(2), pp. 212–222, 1977.
• J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, 1991, Chapter 7.
• H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, 2002, Chapter 14.