9.4 Sliding mode control

Sliding mode control fundamentals

Sliding mode control (SMC) is a robust nonlinear control technique that forces a system's state onto a predefined surface in state space, then constrains it to slide along that surface toward equilibrium. Its value in robust control comes from a key property: once on the sliding surface, the system becomes insensitive to a whole class of uncertainties and disturbances (those satisfying the matching condition). This gives SMC a significant edge over classical linear methods for uncertain nonlinear plants.

Three core advantages distinguish SMC from classical approaches:

• Matched uncertainty rejection: Disturbances and model errors that enter through the same channel as the control input are completely rejected during sliding mode.
• Finite-time convergence: The system state reaches the sliding surface in finite time, rather than the asymptotic convergence typical of linear controllers.
• Order reduction: On the sliding surface, the system dynamics reduce to a lower-order system, which simplifies both analysis and design.

Definition of sliding mode control

SMC is a variable structure control strategy that uses a discontinuous control law to drive the system state toward a sliding surface. This surface is a hyperplane in state space defined by a chosen function of the system states. Once the state reaches this surface, the discontinuous control keeps it there, and the state then evolves along the surface toward the desired equilibrium.

The "variable structure" label comes from the fact that the control law switches between different structures depending on which side of the sliding surface the state currently occupies.

Advantages vs classical control methods

• Handles uncertainties and disturbances satisfying the matching condition, providing deterministic robustness guarantees rather than relying on gain/phase margins
• Finite-time convergence to the sliding surface, giving a hard bound on the time to reach nominal behavior
• Reduces the effective system order on the sliding surface, making the remaining dynamics easier to shape
• Applicable to a broad class of nonlinear systems, including those with non-differentiable or discontinuous dynamics where linearization-based methods fail

Sliding surface design

The sliding surface is the single most important design choice in SMC. It determines how the system behaves once sliding mode is established. A poorly chosen surface can yield stable but sluggish response, or even instability in the reduced-order dynamics. The surface is typically a function of the system states, and its coefficients are selected to place the reduced-order poles where you want them.

Selection of switching function

The switching function $s(x)$ defines the sliding surface as the set $\{x : s(x) = 0\}$. Three common forms are used:

• Linear sliding surface: $s(x) = cx$, where $c$ is a constant row vector. The coefficients in $c$ determine the reduced-order dynamics. For a second-order system with states $x_1$ and $x_2$, choosing $s = x_2 + \lambda x_1$ yields first-order dynamics $\dot{x}_1 = -\lambda x_1$ on the surface, so $\lambda$ directly sets the convergence rate.
• Terminal sliding surface: $s(x) = x_2 + \lambda x_1^{p/q}$, where $p$ and $q$ are positive odd integers with $p < q$, and $\lambda > 0$. This produces finite-time convergence to the origin along the surface, not just finite-time reaching of the surface.
• Integral sliding surface: $s(x) = cx + k\int_0^t (cx)\, d\tau$, where $k > 0$. The integral term can eliminate the reaching phase entirely by designing the initial value of $s$ to be zero, so the system starts in sliding mode from $t = 0$.

Existence of sliding mode

For sliding mode to exist, the system state must actually reach the surface in finite time. This requires the reaching condition: the switching function must decrease in magnitude whenever the state is off the surface. The standard form is:

$s(x)\dot{s}(x) < 0$

This condition says that $s$ and $\dot{s}$ must have opposite signs. If $s > 0$, then $\dot{s} < 0$ (the state moves toward the surface from above), and vice versa. A stronger version, $s\dot{s} < -\eta|s|$ with $\eta > 0$, guarantees reaching in finite time bounded by $|s(0)|/\eta$.

Stability of sliding mode

Reaching the surface is not enough. The reduced-order dynamics on the surface must also be stable. Once $s(x) = 0$ is enforced, the system evolves according to lower-dimensional dynamics determined by the surface coefficients.

You verify stability of these reduced-order dynamics using Lyapunov theory. A common candidate is the quadratic Lyapunov function:

$V(s) = \frac{1}{2}s^2$

If you can show $\dot{V} < 0$ for all $s \neq 0$, the sliding mode is stable. For multi-input systems, you'd use $V = \frac{1}{2}s^T s$ and verify $\dot{V} < 0$.

Reaching phase analysis

The reaching phase is the interval between the initial time and the moment the state first hits the sliding surface. During this phase, the system is not on the surface, so the robustness guarantees of sliding mode do not yet apply. This makes the reaching phase a vulnerability: the system is exposed to uncertainties and disturbances before it reaches the surface.

Minimizing the duration of the reaching phase is therefore a practical design priority.

Reaching condition for sliding mode

The reaching condition ensures the state moves toward the sliding surface during this phase. A commonly used strengthened form is:

$s(x)\dot{s}(x) < -\eta |s(x)|$

where $\eta > 0$ is the reaching rate. This guarantees that the distance to the surface decreases at a rate of at least $\eta$, giving an upper bound on reaching time of $|s(0)|/\eta$.

Techniques for reducing reaching phase

• High-gain control: Increasing the discontinuous gain $k$ speeds up reaching but increases control effort and worsens chattering.
• Disturbance observers: Estimating and feedforward-compensating disturbances reduces the burden on the discontinuous term, allowing faster and smoother reaching.
• Reaching law approaches: These explicitly shape the trajectory during the reaching phase. Common reaching laws include:
  • Constant rate: $\dot{s} = -q\,\text{sign}(s)$, which drives $s$ to zero at a constant rate $q$
  • Power rate: $\dot{s} = -k|s|^\alpha \text{sign}(s)$ with $0 < \alpha < 1$, which accelerates convergence near the surface
  • Exponential: $\dot{s} = -q\,\text{sign}(s) - \varepsilon s$, which combines constant-rate reaching far from the surface with exponential decay near it
• Integral sliding surfaces: As mentioned above, these can eliminate the reaching phase altogether by ensuring $s(0) = 0$.

Equivalent control method

The equivalent control method provides a way to analyze what happens during sliding mode. The idea is to find the continuous control input that would be needed to keep the state exactly on the sliding surface. This "equivalent control" captures the average behavior of the actual discontinuous controller during sliding.

Derivation of equivalent control

For a system of the form:

$\dot{x} = f(x) + b(x)u$

with sliding surface $s(x) = 0$, the sliding condition requires $\dot{s}(x) = 0$ during sliding mode. Expanding using the chain rule:

$\dot{s} = \frac{\partial s}{\partial x}\dot{x} = \frac{\partial s}{\partial x}[f(x) + b(x)u] = 0$

Solving for $u$ gives the equivalent control:

$u_{eq} = -\left(\frac{\partial s}{\partial x}b(x)\right)^{-1}\frac{\partial s}{\partial x}f(x)$

This requires that $\frac{\partial s}{\partial x}b(x) \neq 0$, which is the condition that the control input actually influences the sliding variable (a controllability-like requirement for the surface).

Role in sliding mode control design

The equivalent control represents the average continuous effort needed to maintain sliding. Substituting $u_{eq}$ back into the system equation gives the reduced-order dynamics on the surface, which you then analyze for stability.

The full SMC law is typically composed as:

$u = u_{eq} - k\,\text{sign}(s(x))$

where $k > 0$ is chosen large enough to satisfy the reaching condition despite bounded uncertainties. The $u_{eq}$ term handles the nominal dynamics, while the discontinuous $-k\,\text{sign}(s)$ term provides robustness.

Chattering phenomenon

Chattering is the most significant practical limitation of standard SMC. It manifests as high-frequency oscillation of the state around the sliding surface, caused by the discontinuous switching of the control law. In real systems, chattering leads to excessive actuator wear, high energy consumption, and excitation of unmodeled high-frequency dynamics (flexible modes, parasitic dynamics) that can destabilize the system.

Causes of chattering

Three main sources contribute to chattering:

• Discontinuous control law: The signum function switches instantaneously between $+k$ and $-k$. In an ideal system with infinite switching frequency, this produces perfect sliding. In practice, finite switching rates cause oscillation.
• Unmodeled dynamics: Actuator delays, sensor lags, and neglected fast dynamics create a gap between the commanded and actual control, causing the state to overshoot the surface repeatedly.
• Digital implementation: Finite sampling rates and quantization in sensors and actuators prevent the controller from switching at the exact moment the state crosses the surface.

Techniques for chattering reduction

• Boundary layer approach: Replace the discontinuous control with a continuous approximation inside a thin layer around the surface (detailed in the next section).
• Higher-order sliding modes: Use higher derivatives of $s$ to produce a continuous control signal while still achieving finite-time convergence of $s$ and its derivatives to zero.
• Disturbance observers: Estimate and compensate for the disturbances that force the discontinuous gain to be large. A smaller required gain means less chattering.
• Adaptive gain tuning: Adjust $k$ online so it's only as large as needed to maintain the reaching condition, rather than using a fixed conservative upper bound.

Boundary layer approach

The boundary layer approach is the simplest and most widely used chattering mitigation technique. Instead of enforcing $s = 0$ exactly (which requires discontinuous control), you accept $|s| \leq \phi$ for some small boundary layer thickness $\phi > 0$. Inside this layer, the control transitions smoothly rather than switching abruptly.

Continuous approximation of discontinuous control

The signum function is replaced with a continuous function inside the boundary layer. Two common choices:

• Saturation function: $\text{sat}(s/\phi) = \begin{cases} s/\phi & \text{if } |s| \leq \phi \\ \text{sign}(s) & \text{if } |s| > \phi \end{cases}$
• Hyperbolic tangent: $\tanh(s/\phi)$, which provides a smooth transition everywhere

The modified control law becomes:

$u = u_{eq} - k\,\text{sat}(s/\phi)$

Outside the boundary layer ($|s| > \phi$), the controller behaves identically to standard SMC. Inside, it acts as a high-gain linear controller.

Tradeoff between robustness and performance

This approach introduces a fundamental tradeoff:

• Larger $\phi$: Less chattering, smoother control signals, but the state only converges to a neighborhood of the surface (steady-state error proportional to $\phi$). Robustness to matched disturbances is degraded.
• Smaller $\phi$: Better tracking accuracy and robustness, but chattering returns as $\phi \to 0$.

Adaptive boundary layer methods address this by adjusting $\phi$ online. For example, $\phi$ can be reduced when the state is near the surface and disturbances are small, and increased when chattering is detected.

Higher-order sliding modes

Higher-order sliding modes (HOSM) offer a more principled solution to chattering than the boundary layer approach. Instead of approximating the discontinuous control, HOSM pushes the discontinuity into higher derivatives of the control signal, producing a continuous (or even smooth) actual control input while still achieving finite-time exact convergence to $s = 0$.

The order of a sliding mode refers to the number of consecutive time derivatives of $s$ that are driven to zero. Standard SMC is first-order (only $s = 0$). Second-order drives both $s = 0$ and $\dot{s} = 0$.

Second-order sliding mode control

Second-order sliding mode (SOSM) algorithms drive $s$ and $\dot{s}$ to zero in finite time. The control input itself is continuous (the discontinuity appears in $\dot{u}$), which directly addresses chattering. Three well-known SOSM algorithms:

• Twisting algorithm: Applies different control magnitudes depending on the signs of $s$ and $\dot{s}$. Converges in finite time but requires measurement of $\dot{s}$.
• Super-twisting algorithm: Only requires measurement of $s$ (not $\dot{s}$), making it particularly practical. The control law has the form $u = -k_1|s|^{1/2}\text{sign}(s) + v$, $\dot{v} = -k_2\,\text{sign}(s)$. The integral term $v$ provides the continuous control component.
• Sub-optimal algorithm: Uses only the value of $s$ and its maximum, requiring minimal information about the system.

Arbitrary-order sliding mode control

Arbitrary-order sliding mode (AOSM) generalizes HOSM to drive $s, \dot{s}, \ddot{s}, \ldots, s^{(r-1)}$ all to zero in finite time, where $r$ is the chosen order. Higher orders produce smoother control signals, since the discontinuity is pushed into the $r$-th derivative of $u$.

The tradeoff is design complexity: AOSM controllers require knowledge of (or bounds on) higher-order derivatives of the sliding variable, and parameter tuning becomes more involved as the order increases. In practice, second-order methods (especially the super-twisting algorithm) cover most applications.

Sliding mode observers

Sliding mode observers (SMOs) apply SMC principles to state estimation rather than control. They use a discontinuous injection term to drive the output estimation error to zero in finite time, after which the observer "slides" and the remaining state estimates converge. The finite-time convergence and robustness to certain disturbances make SMOs attractive compared to standard Luenberger observers for uncertain systems.

Design of sliding mode observers

For a system:

$\dot{x} = Ax + Bu + f(x, u, t), \quad y = Cx$

where $f(x, u, t)$ represents uncertainties, the SMO takes the form:

$\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x}) + K\,\text{sign}(y - C\hat{x})$

The design involves two gain matrices:

• $L$ (linear observer gain): Shapes the estimation error dynamics, analogous to a Luenberger observer gain
• $K$ (sliding mode gain): Provides robustness by injecting a discontinuous correction based on the output error $e_y = y - C\hat{x}$

The sliding surface for the observer is $s_o = y - C\hat{x} = 0$. The gains $L$ and $K$ must be selected so that: (1) the reaching condition is satisfied for $s_o$, and (2) the estimation error dynamics are stable once sliding occurs.

Applications in fault detection and estimation

SMOs are particularly useful for fault detection and isolation (FDI):

• During normal operation, the observer tracks the system output and the sliding variable $s_o$ remains near zero.
• When a fault occurs, it drives the output away from the observer's prediction, causing $s_o$ to deviate from zero. Monitoring $s_o$ provides a residual signal for fault detection.
• The equivalent output error injection term (the average value of the discontinuous injection during sliding) contains information about the fault magnitude, enabling fault estimation, not just detection.

For example, in a sensor fault scenario, the SMO estimates what the sensor output should be. A persistent discrepancy between the estimated and measured outputs indicates a sensor fault, and the equivalent injection term quantifies the fault magnitude.

Sliding mode control applications

SMC's robustness and ability to handle nonlinearities have led to widespread use across engineering domains. The following sections highlight three major application areas.

Robotic manipulators

Robot arms are a natural fit for SMC because they involve highly nonlinear dynamics (coupled inertias, Coriolis and centrifugal terms, gravity), and the model parameters change with payload and configuration. SMC can provide accurate trajectory tracking despite these uncertainties.

A typical design defines a sliding surface for each joint as $s_i = \dot{e}_i + \lambda_i e_i$, where $e_i$ is the tracking error for joint $i$. The controller then drives each $s_i$ to zero, achieving exponential tracking error convergence with rate $\lambda_i$ on the surface. Force control and impedance control variants also exist for tasks requiring contact with the environment.

Electric motors and power systems

SMC has been applied to DC motors, induction motors, and permanent magnet synchronous motors (PMSMs) for speed, torque, and position control. The robustness to parameter variations (resistance changes with temperature, inductance varies with saturation) is particularly valuable here.

In power systems, SMC-based controllers have been used for voltage regulation in power converters, frequency control in microgrids, and power flow management. The fast finite-time response helps maintain grid stability during transient events.

Automotive and aerospace systems

Automotive applications include:

• Anti-lock braking systems (ABS): SMC regulates wheel slip ratio despite varying road surface conditions
• Traction control and electronic stability control (ESC): SMC handles the nonlinear tire-road friction characteristics
• Engine and transmission control: Robust to manufacturing tolerances and aging effects

Aerospace applications include attitude control of satellites and aircraft, where SMC rejects external disturbances (aerodynamic forces, gravity gradient torques) and compensates for actuator faults. Fault-tolerant control is a particularly active area: an SMC-based attitude controller can maintain desired orientation even with partial actuator failures, provided sufficient control authority remains.

Adaptive sliding mode control

Adaptive sliding mode control (ASMC) addresses a limitation of standard SMC: the discontinuous gain $k$ must be chosen based on known bounds of the uncertainties. If these bounds are conservative, the gain is unnecessarily large, worsening chattering. If the bounds are wrong, robustness is lost. ASMC removes this requirement by estimating uncertain parameters online.

Combining sliding mode with adaptive control

In ASMC, the control law is augmented with adaptive terms that estimate unknown parameters. For a system:

$\dot{x} = (A + \Delta A)x + (B + \Delta B)u$

where $\Delta A$ and $\Delta B$ are unknown, the ASMC law takes the form:

$u = u_{eq} - K\,\text{sign}(s) - \hat{\Delta A}\,x - \hat{\Delta B}\,u$

The estimates $\hat{\Delta A}$ and $\hat{\Delta B}$ are updated by adaptation laws derived from Lyapunov stability analysis. As the estimates improve, the discontinuous gain $K$ can be reduced, directly decreasing chattering.

Handling parametric uncertainties

ASMC can handle parametric uncertainties satisfying the matching condition (uncertainties entering through the same channel as the control input) or, in some formulations, the extended matching condition.

The adaptation laws are typically derived by choosing a composite Lyapunov function that includes both the sliding variable and the parameter estimation errors. For example:

$\dot{\hat{\Delta A}} = \Gamma_1 x s^T, \quad \dot{\hat{\Delta B}} = \Gamma_2 u s^T$

where $\Gamma_1$ and $\Gamma_2$ are positive definite gain matrices that set the adaptation speed. Larger gains give faster adaptation but can introduce oscillations in the estimates. The Lyapunov analysis guarantees that the parameter errors remain bounded and that the sliding variable converges to zero.

Discrete-time sliding mode control

Discrete-time sliding mode control (DSMC) adapts SMC for digital implementation, where control inputs are updated only at sampling instants. This is the form that actually runs on microcontrollers and DSPs, so understanding the differences from continuous-time SMC is essential for practical implementation.

Sliding mode control for sampled-data systems

In DSMC, the sliding surface is defined at discrete time steps:

$s(k) = cx(k)$

The key difference from continuous SMC is that you cannot achieve true sliding mode in discrete time. Instead, the state reaches a quasi-sliding mode band around the surface, with width proportional to the sampling period $T_s$.

A common discrete reaching law is:

$s(k+1) - s(k) = -qT_s\,\text{sign}(s(k)) - \varepsilon T_s\, s(k)$

where $q > 0$ and $0 < \varepsilon < 1$. The first term provides constant-rate reaching, and the second term adds exponential decay. The system reaches the quasi-sliding mode band in a finite number of steps.

Digital implementation considerations

Several practical factors affect DSMC performance:

• Sampling period: Must be small enough that the quasi-sliding mode band is acceptably narrow (band width scales with $T_s$). Too small a $T_s$ wastes computational resources; too large degrades performance and can cause instability.
• Quantization: Finite sensor and actuator resolution introduces additional error. If the quantization level is comparable to the quasi-sliding mode band width, it becomes the dominant source of steady-state error.
• Computation delay: The time needed to compute $u(k)$ means the control is actually applied slightly late. For fast systems, this delay can be significant and may require predictive compensation (computing $u(k)$ based on a one-step-ahead prediction of $x(k+1)$).
• Multirate schemes: Using different sampling rates for the sliding variable computation and the control update can improve performance in systems where sensor and actuator bandwidths differ.