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Sliding mode control

Sliding mode control (SMC) is a variable structure control methodology for nonlinear dynamic systems that employs a discontinuous control action to constrain the system states to a prescribed sliding surface in the state space, thereby achieving desired performance while exhibiting robustness to matched uncertainties, parameter variations, and external disturbances.^[1] The core principle involves two phases: a reaching phase where the control drives the trajectory to the sliding surface, and a sliding phase where the system dynamics are confined to this surface, effectively reducing the system's order and decoupling the design process.^[1] The origins of SMC trace back to the late 1950s in the Soviet Union, with foundational work by S.V. Emelyanov on variable structure systems for delay systems in 1959, followed by significant advancements in the 1960s through collaborative efforts including Vadim I. Utkin, who formalized the theory and introduced key concepts like equivalent control for designing sliding mode dynamics.^[2] Utkin's seminal 1977 paper on variable structure systems with sliding modes marked a pivotal moment, establishing SMC as a robust alternative to linear control methods during an era dominated by optimal control paradigms.^[2] By the 1980s, research addressed practical issues such as chattering—a high-frequency switching phenomenon that can excite unmodeled dynamics—leading to innovations like continuous approximations. Higher-order sliding modes were developed in the 1990s to further mitigate chattering.^[3] SMC's advantages include its simplicity in implementation using standard power electronics, insensitivity to bounded disturbances without requiring precise plant models, and applicability to both linear and highly nonlinear systems, making it ideal for order reduction and feedback decoupling.^[1] Notable applications span diverse fields, including robotics for manipulator trajectory tracking and exoskeleton control, aerospace for spacecraft attitude stabilization and quadrotor flight, power electronics for converter regulation, and biomedical systems for blood glucose management in artificial pancreases.^[2]^[4] Recent developments integrate SMC with adaptive, fuzzy, and neural network techniques to handle unmatched uncertainties and further mitigate chattering, enhancing its utility in complex, real-time systems like underwater vehicles and nuclear reactors.^[2] Despite these strengths, ongoing challenges involve ensuring finite-time convergence and extending robustness to distributed parameter systems.^[3]

Fundamentals

Definition and Principles

Sliding mode control (SMC) is a nonlinear control strategy that drives the trajectories of a dynamic system onto a predetermined sliding surface in the state space and constrains the system's subsequent motion to remain on that surface, ensuring robust performance in the presence of uncertainties and disturbances. This method relies on discontinuous control actions that switch based on the system's state, effectively altering the system's structure to achieve the desired dynamics. As part of variable structure systems (VSS), SMC employs high-frequency switching to enforce the sliding motion, where the control input changes discontinuously across the sliding surface to counteract deviations.^[1]^[5] SMC is typically formulated for nonlinear systems of the form \dot{x} = f(x, t) + g(x, t)u, where x \in \mathbb{R}^n is the state vector, u \in \mathbb{R}^m is the control input, f(x, t) represents the known dynamics, and g(x, t) is the input distribution matrix, with the sliding surface defined as s(x) = 0, where s: \mathbb{R}^n \to \mathbb{R}^m is a smooth function. The control law is designed such that the system's Filippov solution converges to the surface and exhibits sliding motion along it, effectively reducing the system's order by one during this phase. This setup allows for the specification of reduced-order dynamics on the surface, independent of certain external influences.^[1]^[5] A fundamental principle of SMC is its insensitivity to matched uncertainties—those perturbations that act through the same input channel as the control, lying in the range space of g(x, t)—once the system enters the sliding mode on the surface. In this regime, the equivalent control automatically compensates for these uncertainties, rendering the closed-loop dynamics invariant to parameter variations and bounded disturbances without needing exact model knowledge. This robustness stems from the discontinuous nature of the control, which maintains the state on the surface through infinite switching frequency in ideal conditions.^[6]^[1] The concepts underlying SMC originated in the early 1950s in the Soviet Union, where researchers like Stanislav Emelyanov and Vadim Utkin laid the groundwork for variable structure control and identified sliding modes as a means to achieve system invariance.^[5]

Historical Development

Sliding mode control originated in the Soviet Union during the 1950s, where researchers such as S.V. Emelyanov explored variable structure systems employing relay control mechanisms to manage nonlinear dynamics in first-order delay systems.^[2] Emelyanov's work laid the groundwork for discontinuous control strategies that would later define the field, focusing on systems that switch structures to achieve robust performance against uncertainties.^[2] In the 1960s and 1970s, development accelerated in the USSR, with V.I. Utkin formalizing the theory of variable structure systems featuring sliding modes, emphasizing their invariance properties for disturbance rejection.^[7] Concurrently, A.F. Filippov provided a mathematical framework for handling discontinuities through differential inclusions, enabling rigorous analysis of solutions in systems with discontinuous right-hand sides. This period solidified sliding mode control as a viable approach for complex, uncertain systems within Soviet control engineering. Western adoption gained momentum in the 1980s, facilitated by Utkin's influential survey on variable structure systems, which introduced core concepts to international audiences. By the early 1990s, English-language resources proliferated, notably Utkin's comprehensive treatment of sliding modes in control optimization and J.-J.E. Slotine and W. Li's textbook on applied nonlinear control, which popularized practical design methods and applications. Key advancements in the 1990s included the introduction of higher-order sliding modes by A. Levant, extending traditional first-order techniques to reduce chattering while preserving finite-time convergence. In 2008, J.A. Moreno and M. Osorio advanced second-order sliding modes with the super-twisting algorithm, providing a Lyapunov-based design that ensures robust, chattering-free performance through strict stability analysis.^[8] As of 2025, recent progress integrates sliding mode control with artificial intelligence, such as neural networks for adaptive parameter tuning, enhancing robustness in applications like permanent magnet synchronous motor drives for autonomous vehicles. This fusion addresses real-time uncertainties in path tracking and vehicle dynamics, with hybrid learning-based strategies demonstrating improved convergence and disturbance rejection in electric propulsion systems.

Control Mechanism

Sliding Surface Design

The sliding surface s(\mathbf{x}) = 0 serves as the core geometric constraint in sliding mode control, guiding the system states onto a lower-dimensional manifold that reduces the effective order of the controlled system by one while ensuring invariance to matched disturbances and parameter uncertainties once sliding occurs.^[9] This reduction in order simplifies the dynamics to a stable, prescribed motion independent of the original system's higher-order complexities, thereby enhancing robustness against external perturbations bounded in magnitude.^[9] For linear systems of relative degree one, the sliding surface is typically defined as a linear hyperplane s = \mathbf{c}^T \mathbf{x}, where \mathbf{c} is a coefficient vector selected such that the reduced-order dynamics on the sliding surface are asymptotically stable, with the roots of the corresponding characteristic polynomial (derived from the system's zeros) lying in the left half-plane, often verified using the Routh-Hurwitz stability criterion on the Hurwitz determinants for all eigenvalues with negative real parts. This approach allows pole placement for desired transient response, such as faster settling times, while maintaining the surface's insensitivity to modeling errors. In nonlinear systems, particularly for tracking tasks, the sliding surface is designed using Lyapunov-based methods to enforce convergence of the tracking error e = y - y_d, with a common form s = \dot{e} + \lambda e where \lambda > 0 is tuned for the desired bandwidth.^[10] This surface is derived by considering a Lyapunov function V = \frac{1}{2} s^2, ensuring \dot{V} < 0 for s \neq 0, which promotes asymptotic stability and error reduction along the manifold.^[10] For more complex nonlinearities, the surface may incorporate additional terms to handle unmodeled dynamics, prioritizing surfaces that align with the system's natural energy dissipation.^[9] Selection criteria for the sliding surface emphasize asymptotic stability of the sliding motion, finite-time convergence where possible, and inherent insensitivity to bounded disturbances up to the relative degree.^[9] Stability is confirmed via Lyapunov analysis, ensuring the surface induces negative definiteness in the time derivative of a suitable energy function, while finite-time properties require nonlinear terms that accelerate approach rates beyond exponential decay.^[11] Disturbance insensitivity is achieved by matching the surface to the control input's influence, rejecting uncertainties that do not affect the sliding dynamics.^[9] A representative example for a second-order system \ddot{x} = f(x, \dot{x}) + u is the first-order linear surface s = \dot{x} + c x with c > 0, which confines motion to \dot{x} = -c x for exponential stability when sliding. For finite-time convergence in tracking, the terminal sliding surface s = e + k e^{\frac{p}{q}} (with $1 < \frac{p}{q} < 2, k > 0, and odd integers p, q) ensures the error reaches zero in finite time, outperforming linear surfaces in speed for applications like robotic manipulators.^[11]

Reaching and Sliding Phases

Sliding mode control operates in two distinct phases: the reaching phase and the sliding phase. In the reaching phase, the system states are driven from their initial conditions toward the sliding surface defined by s = 0, where s is the sliding variable. This transient phase ensures finite-time convergence to the surface through a discontinuous control law of the form u = u_{eq} + u_{disc}, where u_{eq} is the equivalent control component that nominally maintains motion on the surface, and u_{disc} is the discontinuous term, often implemented as K \operatorname{sign}(s) with K > 0, responsible for attracting the trajectories. The reaching condition, typically expressed as s \dot{s} \leq -\eta |s| with \eta > 0, guarantees that the system arrives at the sliding surface in finite time, bounded by t_r \leq |s(0)| / \eta, providing robustness against bounded disturbances and model uncertainties during this phase.^[12]^[9] Once the sliding surface is reached, the system enters the sliding phase, where the states are constrained to remain on s = 0, resulting in ideal sliding motion. During this phase, the discontinuous control induces high-frequency switching that enforces \dot{s} = 0, effectively reducing the system order and rendering the dynamics independent of matched uncertainties. The equivalent control u_{eq}, obtained by solving \dot{s} = 0, governs the motion along the surface, yielding reduced-order dynamics that are stable and converge to the desired equilibrium. For instance, in a second-order system such as \dot{x}_1 = x_2, \dot{x}_2 = f(x,t) + u, with sliding surface s = \dot{e} + \lambda e where e = x_1 - x_d and \lambda > 0, the sliding phase dynamics simplify to \dot{e} = -\lambda e, ensuring exponential convergence to the reference x_d. Phase portraits for this system illustrate trajectories approaching the sliding line in the reaching phase and then sliding along it toward the origin in the sliding phase.^[12]^[9] The transition from the reaching to the sliding phase occurs when the reaching condition is satisfied and the eta condition \eta > |\dot{s}| holds, ensuring the surface becomes attractive and invariant. However, ideal sliding can lead to chattering due to perfect switching, which is undesirable in physical systems. To mitigate this, a boundary layer approximation is often employed, replacing the discontinuous \operatorname{sign}(s) with a continuous saturation function \operatorname{sat}(s / \epsilon) within a thin layer of thickness $2\epsilon around s = 0, trading off some robustness for smoother control while approximately maintaining sliding motion. This approach preserves finite-time reaching outside the layer and quasi-sliding within it.^[12]^[9]

Theoretical Foundations

Existence of Sliding Mode

The existence of a sliding mode in variable structure systems is formally established by ensuring that the sliding surface s = 0 is both attractive and invariant under the discontinuous control law. Consider a nonlinear system described by \dot{x} = f(x) + g(x) u, where x \in \mathbb{R}^n is the state vector, u \in \mathbb{R} is the scalar control input, and f, g are sufficiently smooth functions. The sliding surface is defined as s(x) = 0, with its time derivative given by \dot{s} = \frac{\partial s}{\partial x} (f + g u). A fundamental theorem states that sliding motion exists on s = 0 if the control is designed such that s \dot{s} < 0 for all s \neq 0, which directs system trajectories toward the surface and prevents escape once reached, thereby ensuring invariance. This condition s \dot{s} < 0 is derived from Lyapunov stability analysis, where a Lyapunov function candidate V = \frac{1}{2} s^2 yields \dot{V} = s \dot{s} < 0, confirming asymptotic stability of the origin in the sliding variable space and finite-time convergence to the surface under bounded controls.^[13] For systems with discontinuous right-hand sides due to the switching control u, classical Carathéodory solutions may not exist at the discontinuity surface. Instead, solutions are defined in the Filippov sense, where the differential inclusion \dot{x} \in F(x) uses the convex hull F(x) = \mathrm{co} \{ f(x,t,\lambda) : \lambda \in \mathrm{arg\,disc} \} of limiting values of the vector field across the discontinuity, ensuring measurable and absolutely continuous solutions.^[14] Sliding occurs when the vector fields on either side of s = 0 have normal components pointing toward the surface, i.e., \nabla s \cdot f^+ < 0 and \nabla s \cdot f^- > 0 (or vice versa), leading to a Filippov set containing a tangential velocity that confines motion to the surface.^[13] Local existence of Filippov solutions near the sliding surface is guaranteed by Nagumo's theorem, which applies to differential inclusions with upper semicontinuous, convex, and compact-valued right-hand sides, ensuring that trajectories remain within a bounded set if initial conditions satisfy the invariance condition.^[15] This theorem confirms the well-posedness of the dynamics during the onset of sliding, provided the control magnitude suffices to overcome perturbations. Key assumptions underpinning these results include the continuity of f and g away from discontinuities, boundedness of the control input to prevent infinite switching, and a matched uncertainty structure where disturbances enter through the same channel as the control, i.e., \dot{x} = f(x,t) + g(x,t) (u + d(t,x)) with |d| \leq D for some bound D.^[16] These conditions ensure robustness while maintaining the theoretical framework for sliding mode attainment.

Reachability and Region of Attraction

In sliding mode control, reachability refers to the property that system trajectories converge to the sliding surface in finite time from initial states not on the surface. The fundamental reachability condition, known as the η-reachability condition, is given by s \dot{s} \leq -\eta |s|, where s is the sliding variable, \dot{s} its time derivative, and \eta > 0 is a positive constant representing the convergence rate. This inequality ensures that the sliding surface attracts trajectories robustly, even in the presence of bounded matched disturbances. The finite-time nature of reachability follows directly from integrating the η-reachability condition. Starting from an initial sliding variable s(0), the time t_r to reach the sliding surface satisfies t_r \leq \frac{|s(0)|}{\eta}. This bound arises because the condition implies \frac{ds}{dt} \leq -\eta \operatorname{sgn}(s), leading to exponential-like decay that terminates at s = 0 in finite duration rather than asymptotically. Reachability is commonly analyzed using Lyapunov theory, with the quadratic function V = \frac{1}{2} s^2 serving as a candidate Lyapunov function for the approach dynamics. Differentiating along trajectories yields \dot{V} = s \dot{s} \leq -\eta |s| = -\eta \sqrt{2V}, which confirms negative definiteness outside the sliding surface and finite-time convergence to V = 0. The solution to this differential inequality provides an explicit bound on the reaching time, t_r \leq \frac{\sqrt{2 V(0)}}{\eta}, reinforcing the robustness of the analysis for systems where the control law enforces the η-condition. The region of attraction, or basin of attraction, is the set of initial states from which the sliding surface is reachable in finite time under the given control law. For many sliding mode designs, this region encompasses the entire state space, achieving global reachability, particularly when the system has relative degree one with respect to the control input and disturbances are bounded. In such cases, an equivalent theorem establishes global reachability by ensuring the control magnitude exceeds the bound on uncertainties, making the η-condition hold uniformly. This global property holds provided the sliding surface is properly defined and the system satisfies matching conditions for disturbances.

Sliding Motion Stability

Once the system trajectories reach the sliding surface s(x) = 0, the sliding motion is governed by the dynamics restricted to this manifold, effectively reducing the order of the system. In standard sliding mode control for an n-th order system with a scalar sliding surface, the motion on the surface is equivalent to that of an (n-1)-th order autonomous system \dot{x}_r = f_r(x_r), where x_r represents the coordinates tangent to the surface, obtained by solving the constraint s(x_r, x_n) = 0 for the normal component x_n and substituting into the system equations.^[12] This reduction arises because the sliding surface acts as a constraint, eliminating one degree of freedom and simplifying the analysis to the dynamics within the surface.^[17] The stability of this sliding motion is analyzed by considering the reduced-order system independently. For linear systems, where the surface is defined such that s = C x = 0 with C chosen to make the reduced dynamics \dot{x}_r = A_r x_r (and A_r the corresponding submatrix), stability requires A_r to be Hurwitz, ensuring all eigenvalues have negative real parts.^[12] In the nonlinear case, stability is established using Lyapunov methods: select a positive definite function V(x_r) > 0 for x_r \neq 0 with V(0) = 0, such that its time derivative along the reduced dynamics satisfies \dot{V}(x_r) < 0 for x_r \neq 0. For instance, in tracking problems, a common choice is V = \frac{1}{2} q^T P q where q is the tracking error projected onto the surface, P > 0 solves a Lyapunov equation, and the surface design ensures exponential convergence \dot{q} = -\Lambda q with \Lambda > 0.^[18] This approach guarantees asymptotic stability of the origin in the reduced coordinates, confining trajectories to a bounded region on the surface.^[12] To sustain the sliding motion, the equivalent control u_{eq} is the continuous input that keeps \dot{s} = 0 on the surface. For a system \dot{x} = f(x) + g(x) u, with s = h(x), the equivalent control is derived from \dot{s} = \frac{\partial h}{\partial x} (f + g u_{eq}) = 0, yielding u_{eq} = - \left( \frac{\partial h}{\partial x} g \right)^{-1} \frac{\partial h}{\partial x} f, assuming g is invertible and matched to the surface.^[17] In practice, u_{eq} is approximated or estimated, as the actual control combines it with a discontinuous term to enforce reaching; however, on the surface, the discontinuous component averages to zero, leaving u_{eq} to dictate the motion.^[18] Sliding mode control exhibits robustness to matched uncertainties—those entering through the same channel as the control g(x)—due to the invariance of the surface dynamics under such perturbations. However, unmatched uncertainties, which affect the system through channels orthogonal to g(x), can destabilize the reduced-order dynamics unless additional conditions are met, such as bounded perturbations and surface design that incorporates robustness margins. Limitations arise because the equivalent control cannot compensate for unmatched terms, potentially leading to deviations from the surface or degraded stability; robust variants, like adaptive laws, are often required to mitigate this, though they may compromise the order reduction.^[12]^[18]

Design Methods

Standard Control Laws

In sliding mode control, the standard control law is synthesized as a combination of an equivalent control component and a discontinuous switching component to drive the system states onto the predefined sliding surface and maintain them there despite uncertainties. The overall control input takes the form u = u_{eq} + u_{sw}, where u_{eq} is a continuous term that enforces the desired dynamics on the sliding surface by satisfying \dot{s} = 0, and u_{sw} is the discontinuous term responsible for robustness against matched disturbances and model inaccuracies.^[1] The switching component in the basic signum-based law is typically u_{sw} = -k \operatorname{sgn}(s), with k > 0 a design gain, ensuring that the sliding surface dynamics satisfy \dot{s} = -\eta \operatorname{sgn}(s) for some \eta > 0; this reaching law guarantees finite-time convergence to the surface while providing insensitivity to bounded uncertainties in the system model.^[19] To approximate the ideal signum function and mitigate high-frequency chattering induced by the discontinuity, a smooth switching term can be employed, such as u_{sw} = -k \frac{s}{|s| + \epsilon} for a small \epsilon > 0, which behaves like the signum function away from the surface but transitions continuously near s = 0.^[19] A common practical implementation involves boundary layer control, where the signum is replaced by the saturation function \operatorname{sat}(s / \epsilon) = \begin{cases} \operatorname{sgn}(s) & |s| \geq \epsilon \\ s / \epsilon & |s| < \epsilon \end{cases}, confining the switching to outside a thin layer around the surface and applying linear feedback within it to reduce oscillations while preserving quasi-sliding behavior.^[19] The design process begins by computing u_{eq} based on the chosen sliding surface equation, typically solving G B u_{eq} = -G f for the system \dot{x} = f(x,t) + B(x,t) u with output s = G x, followed by selecting the gain k sufficiently larger than the bound on the uncertainty, such as k > |f + g u_{nom}| + \delta for some margin \delta > 0, ensuring the reaching condition holds.^[1] This approach, while introducing a small tracking error proportional to \epsilon, balances robustness with implementation feasibility in physical systems.^[19]

Higher-Order Sliding Modes

First-order sliding mode control (SMC) suffers from the chattering phenomenon, primarily caused by the discontinuous sign function \operatorname{sgn}(s) in the control law, which leads to high-frequency oscillations in the control input and system states when implemented with real actuators. Higher-order sliding modes (HOSMs) address this limitation by enforcing sliding motion on higher-order time derivatives of the sliding variable s, thereby allowing continuous control inputs while preserving the robustness and finite-time convergence properties of standard SMC.^[20] A prominent example of second-order sliding modes is the super-twisting algorithm, which achieves finite-time convergence to the sliding surface for systems with relative degree one. The control law is given by

u = -k_1 |s|^{1/2} \operatorname{sgn}(s) + v, \quad \dot{v} = -k_2 \operatorname{sgn}(s),

where k_1 > 0 and k_2 > 0 are gains tuned to ensure stability, and the resulting control u is continuous despite the discontinuous nature of the algorithm. This approach reduces chattering significantly compared to first-order methods by compensating for uncertainties through the integrated dynamics of v. In general, higher-order sliding modes of order r enforce finite-time convergence such that \dot{s}_i = 0 for i = 1, \dots, r, where s_1 = s and s_{i+1} = \dot{s}_i. These are realized using homogeneous control laws of negative degree, which scale appropriately under time rescaling and ensure robustness to bounded perturbations.^[20] Such controllers generalize the sliding mode concept to higher relative degrees, enabling precise tracking in nonlinear systems with matched and unmatched uncertainties. A key application of HOSMs is Levant's robust exact differentiator, which serves as a special case for real-time estimation of derivatives up to order r. This differentiator uses a homogeneous HOSM structure to provide exact differentiation in the presence of Lipschitz perturbations, with the error converging to zero in finite time.^[21] Recent developments in the 2020s have focused on practical real-time implementations, including refinements to the twisting algorithm—a second-order method with control u = -k_1 \operatorname{sgn}(s) - k_2 \operatorname{sgn}(\dot{s})—and sub-optimal algorithms that approximate optimal convergence while minimizing computational load. The sub-optimal second-order controller, originally proposed for uncertain systems, has seen extensions for energy-efficient and discrete-time applications, enhancing suitability for embedded systems in robotics and power electronics.

Observers and Extensions

Sliding Mode Observers

Sliding mode observers (SMOs) are robust estimation techniques that apply sliding mode principles to state observation in dynamic systems, particularly those affected by uncertainties, disturbances, or nonlinearities. Unlike traditional observers such as Luenberger or Kalman filters, SMOs enforce a sliding regime on the output estimation error, ensuring finite-time convergence to the true states under bounded uncertainties. This approach originated in the 1970s and 1980s, with foundational work by Utkin on linear systems in canonical form and extensions by Slotine to nonlinear systems, providing robustness without requiring precise model knowledge.^[22]^[23] The core structure of an SMO for a nonlinear system \dot{x} = f(x) + g(x)u + d(t), y = h(x), where d(t) represents bounded disturbances, is given by \dot{\hat{x}} = f(\hat{x}) + g(\hat{x})u + l \cdot \text{sgn}(e), with the estimation error e = y - \hat{y} and \hat{y} = h(\hat{x}). Here, l is a sufficiently large gain matrix chosen to dominate uncertainties, driving the error to zero in finite time. For linear systems, this often employs a canonical coordinate transformation to separate observable and output subspaces, yielding \dot{\hat{x}}_1 = A_{11}\hat{x}_1 + A_{12}\hat{y} + B_1 u + L \nu and \dot{\hat{y}} = A_{21}\hat{x}_1 + A_{22}\hat{y} + B_2 u - \nu, where \nu = M \text{sgn}(e_y) and e_y = \hat{y} - y. The gain L ensures error stability, while M enforces the sliding condition.^[22]^[24]^[25] The sliding surface for observation is defined as s_o = e = 0, where the output error e is driven to the origin, analogous to the reaching phase in control but focused on estimation. Once on the surface, the observer error dynamics become \dot{\tilde{e}}_1 = (A_{11} - L A_{21}) \tilde{e}_1, which is asymptotically stable if A_{11} - L A_{21} has negative eigenvalues, ensuring finite-time attraction and asymptotic state convergence. This surface design leverages the discontinuous sign function to counteract matched uncertainties, providing insensitivity to parameter variations.^[24]^[26] A key feature of SMOs is the equivalent output injection, obtained by low-pass filtering the discontinuous term \nu to extract \nu_{eq}, the average value needed to maintain sliding. This signal approximates unknown inputs or model mismatches, such as \nu_{eq} \approx D f(t) for disturbances f(t) entering through matrix D, enabling reconstruction without direct measurement. The filtering, often via \tau \dot{\nu}_{eq} + \nu_{eq} = \nu with small \tau > 0, yields continuous estimates while preserving robustness.^[26]^[27] In fault detection, SMOs generate residuals from the equivalent injection or error signals to identify actuator and sensor faults. For instance, actuator faults f_i(t) manifest in \nu_{eq} as \nu_{eq} \approx -D_2 f_i(t), allowing reconstruction via f_i(t) \approx (D_2^T D_2)^{-1} D_2^T \nu_{eq}, while sensor faults alter the output directly for threshold-based detection. This method excels in robust fault isolation for linear and nonlinear systems, outperforming linear observers in noisy environments.^[28]^[29]^[30] Extensions to unknown input observers (UIOs) address systems with disturbances entering non-matching channels, requiring conditions like \text{rank}(C D) = q for disturbance dimension q and stable invariant zeros. Designs transform the system into a form where sliding enforces insensitivity to unknowns, as in \dot{\hat{z}} = F \hat{z} + T B u + G y + K \text{sgn}(e), yielding simultaneous state and input estimation. These UIO-SMOs, building on Walcott-Zak frameworks, ensure bounded errors under Lyapunov stability.^[31]^[32]^[25]

Adaptive and Robust Variants

Adaptive sliding mode control (ASMC) extends standard sliding mode control by dynamically adjusting the control gain to accommodate unknown or time-varying bounds on system disturbances and uncertainties, thereby ensuring bounded tracking errors without requiring prior knowledge of the bound magnitude. In ASMC, the switching gain \hat{\phi} is updated using the adaptation law \dot{\hat{\phi}} = \gamma |s|, where \gamma > 0 is the adaptation rate and s is the sliding variable; this law increases the gain proportionally to the sliding surface deviation, driving the system states toward the sliding surface while preventing excessive gain growth that could amplify chattering. This approach guarantees ultimate boundedness of the tracking error, with the error norm converging to a ball whose radius depends on the adaptation parameters and disturbance characteristics.^[33] Integral sliding mode (ISM), introduced by Utkin and Shi in 1996,^[34] incorporates an integral term in the sliding surface to enhance robustness against unmatched uncertainties—those not directly counteracted by the control input—ensuring that the sliding motion commences immediately at t=0 without a reaching phase, thus eliminating transient responses vulnerable to unmatched perturbations. The ISM surface is typically defined as s(t) = [G](/page/G) \left( x(t) - x(0) - \int_0^t \dot{x}_{\mathrm{eq}}(\tau) \, d\tau \right), where G is the output matrix and \dot{x}_{\mathrm{eq}} is the equivalent dynamics, allowing the controller to confine motion to the sliding manifold from the initial instant and reject both matched and unmatched uncertainties robustly. This variant is particularly effective in systems like aerospace vehicles where unmatched terms, such as aerodynamic disturbances, persist throughout operation. In the 2020s, hybrid approaches integrating fuzzy logic or neural networks with ASMC have emerged to dynamically tune sliding surfaces and gains in real-time, leveraging artificial intelligence for handling complex, nonlinear uncertainties without explicit modeling. These methods employ fuzzy systems to approximate unknown functions or neural networks (e.g., radial basis function networks) to estimate disturbance bounds online, adapting the sliding surface parameters based on error feedback and achieving superior performance in applications like robotic manipulators and autonomous vehicles compared to purely adaptive schemes. For instance, neural-fuzzy ASMC adjusts surface slopes via network weights trained on instantaneous tracking errors, reducing chattering while maintaining finite-time convergence.^[35] Stability of ASMC is established via Lyapunov analysis, where a candidate function V = \frac{1}{2} s^2 + \frac{1}{2\gamma} \tilde{\phi}^2 (with \tilde{\phi} = \hat{\phi} - \phi the estimation error) yields the inequality \dot{V} \leq -\beta \sqrt{V} for some \beta > 0 under the control law u = u_{eq} - \hat{\phi} \text{sign}(s); this demonstrates finite-time convergence of s to zero and bounded \tilde{\phi}, confirming asymptotic stability and robustness.^[33] Compared to fixed-gain sliding mode control, ASMC offers improved performance for systems with slowly varying parameters, as the adaptive mechanism tracks parameter drifts to maintain smaller steady-state errors and reduced control effort, whereas fixed gains may lead to over-conservatism or instability under parameter changes exceeding the preset bound.

Applications and Implementations

Engineering Examples

One prominent engineering example of sliding mode control (SMC) is the stabilization of an inverted pendulum, a benchmark underactuated system prone to instability due to gravity. The dynamics are modeled in state-space form as \dot{x}_1 = x_2, \dot{x}_2 = f(x) + g(x)u + d, where x_1 represents the pendulum angle from the vertical, x_2 is the angular velocity, u is the control input, and d denotes an external disturbance such as wind or friction variations.^[36] To achieve stabilization at the upright position (x_1 = 0, x_2 = 0), a linear sliding surface is selected as s = \lambda x_1 + x_2, with \lambda > 0 chosen to ensure desirable error dynamics (typically \lambda = 1 for balanced response). The control law compensates for the nonlinear dynamics via u = u_{eq} - k \operatorname{sgn}(s), where u_{eq} is the equivalent control approximating the known model, and k > |d| (e.g., k = 10 to $100) enforces robustness against bounded disturbances.^[36] Simulations of this design, implemented in MATLAB/Simulink with parameters such as pivot mass M = 1 kg, pendulum mass m = 0.010 kg, and length L = 0.5 m, demonstrate effective performance. State trajectories x_1(t) and x_2(t) converge to zero rapidly from initial conditions like x_1(0) = 0.1 rad and x_2(0) = 0, with higher k (e.g., k=100) yielding near-instantaneous settling and reduced overshoot. The control signal u(t) exhibits chattering due to the discontinuous \operatorname{sgn}(s) term, manifesting as high-frequency oscillations around the equilibrium, though bounded within actuator limits. Compared to traditional PID controllers, SMC demonstrates superior stability and faster recovery under disturbances.^[36] Another illustrative example is speed control of a DC motor subject to load torque disturbances, common in robotics and electric vehicles. The motor dynamics, derived from armature voltage control, are typically \dot{\omega} = -\frac{B}{J} \omega + \frac{K_t}{J} i_a - \frac{T_L}{J}, coupled with electrical equation \dot{i_a} = -\frac{R_a}{L_a} i_a - \frac{K_e}{L_a} \omega + \frac{1}{L_a} v_a, where \omega is angular speed, i_a armature current, T_L load torque disturbance, and parameters include inertia J, viscous friction B, torque constant K_t, back-EMF constant K_e, resistance R_a, and inductance L_a. For a separately excited motor (e.g., nominal 12 V, J = 8.25 \times 10^{-6} kg·m²), a first-order sliding surface \sigma = \lambda_1 e + \lambda_0 \int e \, dt + \dot{e} is used, with error e = \omega_{ref} - \omega and gains \lambda_1 = 1/\tau_e, \lambda_0 \leq 4/\tau_e for \tau_e = 1.52 \times 10^{-4} s. The control voltage v_a = v_{eq} + v_{sw} includes an equivalent component for nominal tracking and switching v_{sw} = K_d \sigma + \delta \operatorname{sgn}(\sigma) (e.g., K_d = 0.83, \delta = 1.29) to drive finite-time convergence to \sigma = 0.^[37] MATLAB simulations for a reference speed \omega_{ref} = 1200 rpm reveal finite-time response, with \omega(t) reaching setpoint quickly and robust settling. Under an output disturbance at t = 0.1 s, SMC rejects the disturbance effectively with minimal deviation and fast recovery, showcasing finite-time attractivity. Chattering appears in v_a(t) as ripples, mitigated partially by continuous approximations like hyperbolic tangent. In contrast, PID (gains K_p = 1381.34, K_i = 11.327, K_d = 0.0232) exhibits higher control effort and poorer disturbance rejection under the same conditions, highlighting SMC's superior robustness.^[37] An example from aerospace is the application of SMC to quadrotor flight control for attitude stabilization and trajectory tracking. In such systems, SMC handles nonlinear dynamics and uncertainties like wind gusts by defining sliding surfaces for roll, pitch, and yaw errors, ensuring robust performance. Simulations and real-time tests show reduced tracking errors (under 5°) compared to linear methods.^[2] Post-2010 advancements include automated design tools leveraging sum-of-squares (SOS) programming for Lyapunov-based SMC synthesis, particularly for polynomial systems. The SOSTOOLS MATLAB toolbox facilitates this by solving semidefinite programs to construct SOS Lyapunov functions V(x) satisfying \dot{V} < -\eta V^\gamma (with $0 < \gamma < 1 for finite-time stability), enabling gain selection and surface design without manual tuning. For instance, in the inverted pendulum case, SOS decompositions verify positive definiteness of V = x^T P x and negativity of \dot{V} along s = 0, reducing design iteration time. These tools integrate with YALMIP for optimization and have been used in robust SMC design.^[38]

Practical Challenges and Solutions

One of the primary practical challenges in implementing sliding mode control (SMC) is the chattering phenomenon, which arises from the high-frequency switching inherent in the discontinuous control law, particularly near the sliding surface. This switching generates undesirable oscillations in the control signal and system states, leading to mechanical wear on actuators, increased heat losses in power circuits, and reduced control accuracy over time.^[39] Chattering can be quantified and analyzed using Fourier analysis or describing function methods, which decompose the oscillatory behavior into frequency components to assess amplitude and dominant frequencies, enabling performance evaluation and gain tuning for mitigation.^[40] Several strategies have been developed to mitigate chattering while preserving the robustness of SMC. The boundary layer approach replaces the discontinuous sign function with a continuous saturation function within a thin layer around the sliding surface, smoothing the control action and reducing high-frequency components at the cost of a small steady-state error.^[41] Higher-order sliding modes extend this by enforcing smoothness up to higher derivatives of the sliding variable, effectively attenuating chattering through finite-time convergence without direct discontinuity in the control input.^[42] Additionally, time-delay estimation integrates a delay-based observer to approximate uncertainties and disturbances, allowing a reduced switching gain that minimizes chattering in nonlinear systems like manipulators.^[43] In discrete-time implementations, sampling introduces further challenges, as the finite sampling period can cause instability or exacerbate chattering due to approximation errors in the continuous-time design, potentially leading to loss of robustness against matched disturbances.^[44] Solutions include zero-order hold (ZOH) discretizations, which model the digital actuator as holding the control value constant between samples, enabling stability analysis and controller redesign to ensure quasi-sliding mode behavior within a bounded error band.^[45] Numerical simulation of SMC poses difficulties due to the discontinuities in the dynamics, which standard solvers struggle to handle accurately, often resulting in artificial chattering or divergence. Event-based methods, particularly for Filippov systems that convexify discontinuities along sliding surfaces, address this by advancing simulation only at event times (e.g., surface crossings), providing efficient and precise trajectories without excessive computational load.^[46] As of 2025, perspectives on practical deployment emphasize hardware-in-the-loop (HIL) testing for embedded systems, where real-time simulations integrate physical controllers with virtual plants to validate SMC performance under realistic constraints like sensor noise and computational delays, as demonstrated in applications to flexible robotic arms.^[47]

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