sliding-mode-control

name	sliding-mode-control
description	Sliding mode control (SMC) — sliding surface design (relative degree, manifold s=0), equivalent control, reaching law (constant rate, power rate, exponential), chattering reduction (boundary layer, super-twisting algorithm), higher-order SMC (HOSM), Lyapunov stability analysis, disturbance rejection, SMC for mechanical systems (position, velocity, force control), adaptive SMC, terminal SMC, and applications in robotics, power electronics, and automotive.
metadata	{"priority":7,"promptSignals":{"phrases":["sliding mode control","SMC","sliding surface","chattering","variable structure control","super-twisting"],"minScore":3}}

Sliding Mode Control (SMC) — Complete Skill

Fundamentals of Sliding Mode Control

Variable Structure Systems

Concept: control law switches discontinuously based on system state; creates a "sliding surface" in state space that the system is forced to reach and then slide along

Key property: once on the sliding surface, system dynamics are determined by the surface design (reduced-order system, independent of plant uncertainties and disturbances within bounds)

Two phases:

1. Reaching phase: system moves from initial state to the sliding surface s = 0
2. Sliding phase: system constrained to surface; converges to equilibrium

Sliding Surface Design

First-Order SMC

System: SISO nonlinear: ẋ = f(x) + g(x)×u + d(t) [d = disturbance; |d| ≤ D] Error: e = x - x_d (position error); ė, ë = derivatives

Sliding surface (relative degree r): For r=1 (relative degree 1): s = e For r=2: s = ė + c × e [c > 0; sliding surface slope] For r=n: s = e^(n-1) + c_(n-1) × e^(n-2) + ... + c₁ × e [λ polynomial Hurwitz]

Hurwitz condition for coefficients: s = ė + c_e × e; for convergence on surface: s = 0 → ė = -c × e → e(t) = e(0)×exp(-c×t) Choose c to set convergence time on surface: τ_surface = 1/c

Relative Degree 2 Example (Mechanical System)

Second-order system: mẍ = F_u + F_dist [position control; relative degree 2] Error: e = x - x_d; ė = v - v_d; ë = a - a_d = (F_u + F_dist)/m - ẍ_d

Sliding surface: s = ė + c × e

Dynamics on surface (s=0): ė = -c × e → exponential convergence of tracking error

Time derivative of s: ṡ = ë + c × ė = (F_u + F_dist)/m - ẍ_d + c × ė

Control Law and Equivalent Control

Equivalent Control

Setting ṡ = 0 (to stay on surface): 0 = (F_u_eq + F_dist)/m - ẍ_d + c × ė → F_u_eq = m × (ẍ_d - c×ė) - F_dist [requires knowledge of disturbance — not available]

Nominal equivalent control (without disturbance): F_u_eq_nom = m × (ẍ_d - c × ė) [model-based feedforward]

Switching control (to handle disturbance): F_u = F_u_eq_nom - K_s × sign(s) [sign(s) = +1 if s>0; -1 if s<0] K_s > |F_dist|_max = D × m [gain must exceed maximum disturbance amplitude]

Reaching Condition — Lyapunov Analysis

Stability Proof

Lyapunov function: V = (1/2) × s² Time derivative: V̇ = s × ṡ

For reaching condition: V̇ = s × [(F_u_nom - K_s×sign(s) + F_dist)/m - ẍ_d + c×ė] = s × [F_u_nom/m - ẍ_d + c×ė] + s × [F_dist/m] - K_s × |s|/m × m [s×sign(s) = |s|] = s × 0 (equivalent control cancels) + s × F_dist/m - K_s × |s| V̇ ≤ |s| × D - K_s × |s| = |s| × (D - K_s) < 0 when K_s > D

Finite time reaching: V̇ ≤ -η × |s| for some η > 0 → surface reached in t_reach ≤ |s(0)|/η

Reaching Laws

Common Reaching Laws

Constant rate reaching: ṡ = -K × sign(s) [constant speed approach to surface; discontinuous] Reaching time: t_r = |s(0)| / K

Exponential (proportional) reaching: ṡ = -ε × s - K × sign(s) [s(t) = s(0)×exp(-εt) - K/ε×(1-exp(-εt))×sign(s)] Approaches surface exponentially; additional constant term ensures finite reaching ε > 0; K > 0 (K handles disturbances; ε speeds up reaching)

Power rate reaching: ṡ = -K × |s|^α × sign(s) [α ∈ (0,1); faster reaching near surface than constant rate] Useful for reducing chattering while maintaining finite reaching time

Chattering Problem and Remedies

Chattering Mechanism

Chattering: high-frequency oscillation around s=0 due to discontinuous sign(s) in control; caused by:

• Finite switching frequency of actuators
• Unmodeled dynamics (parasitic poles)
• Sensor noise amplified by sign function

Practical consequence: mechanical wear, heat, vibration, structural damage; unacceptable for most applications

Boundary Layer Method

Replace sign(s) with saturation function: u = -K × sat(s/Φ) where sat(s/Φ) = {s/Φ if |s| ≤ Φ; sign(s) if |s| > Φ} [Φ = boundary layer thickness]

Effect: smooth control within boundary layer → no chattering; approximate sliding Cost: steady-state tracking error ≤ Φ/c (position error ≤ boundary layer / surface gain) Selection: Φ = trade-off between chattering and tracking accuracy

Equivalently: replace sign with hyperbolic tangent: u = -K × tanh(s/Φ) [infinitely differentiable; avoids discontinuity entirely]

Super-Twisting Algorithm (STA) — Second-Order SMC

STA (Levant 1993): achieves finite-time convergence WITHOUT chattering for second-order s dynamics u = -λ × |s|^(1/2) × sign(s) + u₁ [fractional power of s; smooth control] u₁̇ = -W × sign(s) [integral action handles disturbance]

Conditions for stability: W > C (disturbance Lipschitz constant); λ > 0 Convergence: s → 0 and ṡ → 0 in finite time simultaneously (second-order sliding)

Advantage: output u is Lipschitz (no chattering); suitable for mechanical systems Application: used in motor drives, hydraulic actuators, automotive ABS

Higher-Order SMC (HOSM)

Arbitrary-order algorithm (Levant 2001): For r-th order sliding mode: design control so that s, ṡ, ..., s^(r-1) → 0 in finite time r=1: standard SMC; r=2: STA or twisting; r=3: quasi-continuous algorithms

Nested sliding modes: each inner loop handles one integration For r=3: u₃_dot = -α × sign(s); u₂_dot = -β × |s|^(1/2)×sign(ṡ+...) → complex

Quasi-continuous control: u = -α × (ṡ + |s|^(1/2)sign(s)) / (|ṡ| + |s|^(1/2)) [bounded; continuous except on surface]

Application — Robot Joint Position Control

System: τ = M(q)q̈ + C(q,q̇)q̇ + G(q) + d(t) [torque equation; d = friction+unmodeled] Error: e = q - q_d; ė = q̇ - q̇_d

Sliding surface: s = ė + Λ × e [Λ = diagonal gain matrix]

Control law: τ = M_nom × (q̈_d - Λ×ė) + C_nom×q̇ + G_nom - K × sign(s) K ≥ ||ΔM||×||q̈_d - Λ×ė|| + ||ΔC||×||q̇|| + ||ΔG|| + D [gain must exceed all uncertainties]

With STA: replace K×sign(s) with super-twisting → no chattering

Boundary layer tracking error: ||e_ss|| ≤ Φ_max / λ_min(Λ) [steady-state error bounded by Φ/Λ]

Adaptive Sliding Mode Control

Adaptive gain to avoid over-conservative K: K̇ = γ × |s| [adapt gain upward when error increases] K̈ = -γ × |s| only when system on surface [or use modified adaptation law to avoid over-adaptation]

Advantage: K automatically tunes to match actual disturbance level; minimal chattering

Standards and References

Standard	Scope
Utkin "Sliding Modes in Control and Optimization" (1992)	Foundational reference
Shtessel et al. "Sliding Mode Control and Observation" (2014)	HOSM and STA reference
Levant "Higher-Order Sliding Modes" (2001) IJOC	HOSM theory
IEEE Transactions on Industrial Electronics	SMC application papers

Output

Provide: system description (order; state equations; relative degree r; disturbance bounds D; uncertainties), sliding surface design (s = ė + c×e or higher order; c selection for convergence τ_surface [s]; Hurwitz check of characteristic polynomial), control law (equivalent control F_u_eq; switching gain K_s > D; total u), reaching law (type: constant/exponential/power; ε, K values; reaching time estimate t_reach [s]), Lyapunov stability proof (V̇ ≤ -η|s|; K > D: verified?), chattering mitigation (boundary layer Φ; sat() or tanh(); resulting steady-state error bound [mm or rad]; OR STA: λ, W parameters), simulation results recommendation (ode45/Simulink; compare sign vs. sat vs. STA; chattering amplitude), closed-loop performance (tracking error norm; convergence time on surface; disturbance rejection: response to step d [N or N·m]), and reference (Utkin 1992; Shtessel 2014; Levant 2001 for HOSM).