Sliding-Mode & Nonlinear Control — Engineering Reference

1. At a glance

Sliding-mode control (SMC) is a class of variable-structure feedback laws in which the controller is deliberately discontinuous (or, in modern variants, quasi-continuous of fractional power) so as to drive the system trajectory onto a designer-chosen manifold $s(x)=0$ — the sliding surface — and constrain it there in finite time. Once the trajectory is “on the surface,” the closed loop behaves like a reduced-order ideal dynamics that is, by construction, insensitive to matched uncertainty and matched disturbances. The same idea generalises to a family of nonlinear-control techniques — backstepping, feedback linearisation, passivity-based control, gain scheduling, computed torque — that share the use of Lyapunov stability rather than linear pole placement as their primary design tool.

SMC was born in the Soviet electromechanical-systems literature (Emelyanov 1967, Utkin 1977 IEEE TAC) and has been industrial reality for forty years: it is the mathematical engine behind Direct Torque Control (DTC) as deployed in ABB ACS800/ACS880 drives, behind InstaSPIN-FOC on TI C2000, behind the current loops of nearly every commercial buck/boost converter chipset, behind anti-lock braking (ABS) and traction-control modules in passenger cars, and behind the autopilots of several missile and UAV families. The classical bang-bang implementation suffers from chattering — high-frequency switching that excites unmodelled fast dynamics, wastes energy, and wears actuators — which kept it niche in many process industries. Levant’s super-twisting algorithm (1993, refined 1998–2003) and the broader higher-order SMC family solved chattering by making the control continuous while preserving finite-time convergence on the sliding surface; this single advance is what re-opened SMC for mainstream motor drives, power electronics, and aerospace actuators.

Where it sits in the control stack: above classical control (classical-control) and state-space methods (state-space-methods) when the plant has significant matched uncertainty (parameters drift, mass changes, friction is poorly known) and finite-time convergence is required; complementary to MPC (mpc-control) which dominates constrained / multivariable / economic problems; alternative to H∞ robust control when the disturbance bound is known but its frequency content is not.

1.1 Historical arc

The variable-structure-systems (VSS) literature began in the Soviet Union in the late 1950s with Emelyanov’s work on relay control of unstable plants. Filippov 1960 supplied the mathematical foundation — differential equations with discontinuous right-hand sides — required to make rigorous the notion of “motion on the discontinuity surface.” Utkin 1977 brought the framework to the English-language community and unified the proliferating Soviet variants into the modern SMC. The 1980s and early 1990s saw industrial uptake in motor drives (Sabanovic, Bose) and aerospace (DeCarlo, Spurgeon, Edwards). Levant’s 1993 super-twisting paper opened the second-order era, and the 2000s broadened SMC into output-feedback, adaptive, and discrete-time variants. The 2010s saw chattering finally tamed by the implicit-discretisation results of Acary–Brogliato and the strict-Lyapunov formulation of Moreno–Osorio — the last mathematical barriers to deployment in safety-critical embedded loops.

2. Why it matters

Linear feedback (PID, LQR, H∞) buys robustness against bounded parameter variation by moving poles further into the left half-plane — exponential decay with rate $|\mathrm{Re}(\lambda )|$. The trade-off is well-known: higher gain means more actuator effort, more sensor-noise amplification, less margin against unmodelled high-frequency dynamics. SMC and its nonlinear siblings reach a qualitatively different operating point:

1. Finite-time convergence. Once on the sliding surface, the error reaches zero in finite time, not merely asymptotically. Reaching the surface from arbitrary initial conditions is also finite-time under classical SMC. This matters for safety-critical envelope-keeping (ABS must clamp slip before lock-up, not “eventually”).
2. Insensitivity to matched uncertainty. Disturbances and parameter errors that enter through the same channel as the control input $u$ — the “matched” subset — are completely rejected once sliding begins, regardless of magnitude, provided the switching gain dominates the disturbance bound. No linear controller offers this.
3. Model frugality. A workable SMC requires only an upper bound on uncertainty plus the input-direction-and-rank of $B(x)$. The plant model need not be accurate.
4. Bang-bang actuator match. Where the physical actuator is naturally discontinuous — power semiconductors switching at 10–100 kHz, on/off solenoid valves, relay heaters — SMC is the native control law. Classical PID-plus-PWM works, but loses information by interposing a linear regulator between a continuous reference and a discrete actuator.

Backstepping and feedback linearisation address a different limitation: cascade or strict-feedback nonlinear plants (robot manipulator dynamics, missile body+actuator dynamics, induction-motor torque+flux dynamics) where a direct linear controller leaves modelled nonlinearities on the table. The resulting control law is model-based in the strict sense (Lyapunov + the actual nonlinear $f(x)$ enter the formula) and produces provably stable closed loops with explicit performance bounds.

3. First principles — classical sliding-mode control

3.1 The sliding surface

Consider a single-input nonlinear plant in regular form:

$\dot{x}=f(x)+g(x)u+d(t),x\in {\mathbb{R}}^n,u\in \mathbb{R}$

Choose a scalar sliding variable $s(x)$ — typically a linear combination of the tracking error and its derivatives:

$s(x)=c_{1}e+c_2\dot{e}+\cdots +c_{n-1}{e}^{(n-2)}+e^{(n-1)},e=x_1-x_{1,d}$

The polynomial ${\lambda }^{n-1}+c_{n-1}{\lambda }^{n-2}+\cdots +c_1$ must be Hurwitz — its roots define the reduced-order ideal sliding dynamics that govern the closed loop once $s=0$. The design freedom is exactly the choice of these coefficients (or, equivalently, the closed-loop time constants on the surface). A common second-order case: $s=\dot{e}+\lambda e$ with $\lambda >0$ — once on the surface, $\dot{e}=-\lambda e$, exponential decay with time constant $1/\lambda$.

3.2 The reaching law

The control is split:

$u=u_{\mathrm{eq}}+u_{\mathrm{sw}}$

where $u_{\mathrm{eq}}$ is the equivalent control — the unique smooth input that holds $\dot{s}=0$ in the nominal (disturbance-free, known-parameter) case, computed by solving $\dot{s}(x)=0$. The switching term $u_{\mathrm{sw}}$ enforces $s\to 0$ in finite time even when uncertainty corrupts $u_{\mathrm{eq}}$:

$u_{\mathrm{sw}}=-K\cdot \mathrm{sign}(s),K>\sup |\Delta (x,t)|$

The classical reaching law is then $\dot{s}=-K\mathrm{sign}(s)$, which integrates to $|s(t)|=|s(0)|-Kt$ until $s=0$ is reached at $t_r=|s(0)|/K$. Finite-time — not asymptotic. Levant 1993 generalised this to the family $\dot{s}=-\alpha |s{|}^{\rho }\mathrm{sign}(s)-\beta s$ with $\rho \in (0,1)$.

3.3 Lyapunov proof of reaching

Pick $V(s)=\frac{1}{2}{s}^2$. Differentiate along trajectories:

$\dot{V}=s\dot{s}=s(L_{f}s+L_{g}s\cdot u+L_{d}s)=s(-K\mathrm{sign}(s)+\delta (x,t))=-K|s|+s\delta$

If $|\delta |\le D$ and $K>D+\eta$ for some $\eta >0$, then $\dot{V}\le -\eta |s|=-\eta \sqrt{2V}$ — the reaching condition. Standard comparison gives finite-time convergence: $V(t)=0$ for $t\ge \sqrt{2V(0)}/\eta$.

3.4 Ideal-sliding-mode dynamics and matched-uncertainty rejection

On the surface ($s=0$, $\dot{s}=0$), the equivalent control satisfies $L_{f}s+L_{g}s\cdot u_{\mathrm{eq}}+L_{d}s=0$. Matched uncertainty — disturbance $d$ enters via $g(x)$, i.e. $d=g(x){\delta }_m(t)$ — appears as an extra term in $L_{d}s$ that the equivalent control cancels exactly without any explicit knowledge of ${\delta }_m$. Unmatched uncertainty — entering orthogonal to $g(x)$ — is not rejected; this is the structural limitation that backstepping and integral SMC (ISMC) address.

3.5 Chattering

The mathematical $\mathrm{sign}(\cdot )$ is discontinuous. Any unmodelled fast dynamics between the controller output and the actuator (op-amp bandwidth, motor inductance, mechanical compliance) cause the trajectory to overshoot the surface, switch sign, overshoot again — a sustained limit cycle of small amplitude but very high frequency. Consequences: actuator wear (relay contacts, valve seats), EMI, audible whine, excess power dissipation, excitation of structural resonances. Mitigation strategies:

Strategy	Mechanism	Cost
Boundary layer	Replace $\mathrm{sign}(s)$ with $\mathrm{sat}(s/\epsilon )$ or $\tanh (s/\epsilon )$	Steady-state error proportional to $\epsilon$; loss of strict invariance
Higher-order SMC (super-twisting, twisting, sub-optimal)	Switching applied to a higher derivative of $s$	More complex tuning; needs $\dot{s}$ estimate (often via robust differentiator)
Reaching-law shaping	$\dot{s}=-K|s{|}^{\rho }\mathrm{sign}(s)-\beta s$ with $\rho \in (0,1)$	Slower reaching far from surface
Adaptive gain	$K$ shrinks when no chatter is detected	Risk of “bursting” if disturbance returns suddenly
Frequency-shaping	Insert low-pass filter on $\mathrm{sign}(s)$ output	Reintroduces matched disturbance leakage

3.6 Multi-input SMC

For $m$-input plants $\dot{x}=f(x)+G(x)u+d$ with $u\in {\mathbb{R}}^m$, the sliding surface is vector-valued $s(x)\in {\mathbb{R}}^m$. The reaching law

$\dot{s}=-K\mathrm{sign}(s),\mathrm{sign}(s)=[\mathrm{sign}(s_1),\dots ,\mathrm{sign}(s_m){]}^T$

works component-wise, but the equivalent control requires the decoupling matrix $L_{G}s=\partial s/\partial x\cdot G(x)$ to be invertible at every $x$ — the reaching condition. When $L_{G}s$ has full rank, $u_{\mathrm{eq}}=-(L_{G}s{)}^{-1}{L}_{f}s$. Loss of rank corresponds to a singularity of the input distribution; near it, SMC gain blows up. For systems with input-saturation or rank-deficient $G$, integral sliding mode (ISMC) — Utkin & Shi 1996 — augments $s$ with an integral term $z$ chosen so $s+z$ is invariant from $t=0$, eliminating the reaching phase.

3.7 Integral SMC

In classical SMC, the reaching phase suffers from matched and unmatched disturbance until $s=0$ is reached. Integral SMC (Utkin–Shi 1996) augments the surface:

$s_I(x,t)=s(x)-s(x(0))-{\int }_0^t{L}_{f}s+L_{g}s\cdot u_{0}d\tau$

where $u_0$ is the nominal controller (LQR, PID, etc.). $s_I(0)=0$ by construction, so the system starts on the surface — no reaching phase — and the discontinuous part of $u$ rejects matched disturbance from $t=0$. The nominal controller delivers its designed performance against unmatched components; the SMC layer adds robustness against matched components without re-tuning the nominal loop. This is the dominant pattern in modern aerospace SMC.

4. The super-twisting algorithm (Levant 1993)

4.1 Definition

The super-twisting algorithm (STA) is a second-order sliding-mode controller — it drives both $s$ and $\dot{s}$ to zero in finite time — yet the control signal $u$ is continuous. This is the property that revolutionised industrial SMC:

$u=-\alpha |s{|}^{1/2}\mathrm{sign}(s)+v,\dot{v}=-\beta \mathrm{sign}(s)$

The first term is a continuous Hölder-power feedback; the second is the integral of a discontinuous signal — which is itself continuous. The “twisting” name refers to the spiral trajectory in the $(s,\dot{s})$ phase plane.

4.2 Stability gains

For a plant with bounded uncertainty derivative $|\dot{\Delta }|\le L$, classical sufficient conditions (Levant 1993; refined Moreno & Osorio 2008 via strict Lyapunov function):

$\beta >L,{\alpha }^2\ge 4L\cdot \frac{\beta +L}{\beta -L}$

A common practical choice: $\alpha =1.5\sqrt{L}$, $\beta =1.1L$. The Moreno–Osorio Lyapunov function

$V(s,v)=2\beta |s|+\frac{1}{2}{v}^2+\frac{1}{2}(\alpha |s{|}^{1/2}\mathrm{sign}(s)-v{)}^2$

establishes finite-time convergence and explicit settling-time bounds.

4.3 Robust exact differentiator (Levant 1998)

A by-product of the super-twisting algorithm is a finite-time exact differentiator:

${\dot{z}}_0=-{\lambda }_0|z_0-f(t){|}^{1/2}\mathrm{sign}(z_0-f)+z_1,{\dot{z}}_1=-{\lambda }_1\mathrm{sign}(z_0-f)$

with $z_1\to \dot{f}$ in finite time, exactly, for any Lipschitz $f$. This is widely used as a clean alternative to high-gain observers when noise levels permit.

5. Higher-order sliding modes

A $k$-th order sliding mode drives $s,\dot{s},\dots ,s^{(k-1)}$ all to zero in finite time. Beyond super-twisting (2-SMC), the dominant families:

• Twisting algorithm (2-SMC, discontinuous control): $u=-r_1\mathrm{sign}(s)-r_2\mathrm{sign}(\dot{s})$ with $r_1>r_2>0$. Requires both $s$ and $\dot{s}$.
• Sub-optimal algorithm (Bartolini–Ferrara–Usai 1998): geometric switching on $s-\frac{1}{2}{s}_{\max }$.
• Quasi-continuous (Levant 2005): control depends only on $s$ and $\dot{s}$, but is continuous except at $s=\dot{s}=0$.
• $k$-SMC for arbitrary $k$ (Levant 2003): nested $\mathrm{sign}$ structure; algebraic, no derivative estimation past order $k-1$.

Adaptive gain SMC (Plestan et al. 2010, Shtessel et al. 2012) addresses the practical pain point that disturbance bound $L$ is rarely known a priori — the switching gain is grown when chattering metrics rise and shrunk when they fall:

$\dot{K}=\{\begin{array}{ll}\gamma |s|& \text{if }|s|>\mu \\ -{\gamma }_{1}K& \text{if }|s|\le \mu \end{array}$

with safeguards to keep $K\ge K_{\min }$.

5.1 Comparison of SMC variants

Algorithm	Order	Control continuity	Sliding accuracy	Best fit
Classical SMC	1	Discontinuous	$\mathcal{O}(T_s)$ — first-order	On/off actuators, simple plants
Boundary-layer SMC	1	Continuous (saturated)	$\mathcal{O}(\epsilon )$	Quick fix where chatter unacceptable
Super-twisting	2	Continuous	$\mathcal{O}(T_s^2)$ — second-order	Motor drives, power electronics
Twisting	2	Discontinuous	$\mathcal{O}(T_s^2)$	When derivative $\dot{s}$ available
Sub-optimal (Bartolini)	2	Continuous	$\mathcal{O}(T_s^2)$	Chatter-critical, simple tuning
$k$-SMC nested	$k$	Discontinuous	$\mathcal{O}(T_s^k)$	Output-feedback with $k-1$ unmeasured derivatives
Adaptive SMC	1–2	Either	Same as base	Unknown disturbance bound
Integral SMC	1	Discontinuous in $u_{\mathrm{sw}}$	Eliminates reaching phase	Nominal controller + robustness layer
Terminal SMC	1	Discontinuous	Finite-time to zero (vs exponential)	Aerospace, fast tracking

Terminal SMC (Man, Paplinski & Wu 1994; Yu & Man 1996) deserves mention — the sliding surface $s=\dot{e}+\beta |e{|}^{\alpha }\mathrm{sign}(e)$ with $\alpha \in (0,1)$ replaces the linear $\lambda e$ term with a fractional-power one. On the surface, $\dot{e}=-\beta |e{|}^{\alpha }\mathrm{sign}(e)$ converges to zero in finite time rather than exponentially — useful when the post-sliding settling time matters. Non-singular terminal SMC (Feng, Yu & Man 2002) avoids the singularity at $e=0$ that plagues the original formulation.

6. Backstepping (Krstic, Kanellakopoulos, Kokotovic 1995)

6.1 The strict-feedback structure

Backstepping designs a stabilising control law recursively for plants in strict-feedback form:

\dot{x}_1 &= f_1(x_1) + g_1(x_1) x_2 \\ \dot{x}_2 &= f_2(x_1, x_2) + g_2(x_1, x_2) x_3 \\ &\vdots \\ \dot{x}_n &= f_n(x_1, \ldots, x_n) + g_n(x_1, \ldots, x_n) u \end{aligned}$$ Each $x_{i+1}$ acts as a **virtual control input** for the $x_i$ subsystem. Start at $x_1$: treat $x_2$ as a controller you can choose, pick $x_2 = \alpha_1(x_1)$ to stabilise $x_1$ via a Lyapunov function $V_1$. Now $x_2$ is *not* equal to $\alpha_1$ — define the error $z_2 = x_2 - \alpha_1(x_1)$ and build $V_2 = V_1 + \tfrac{1}{2} z_2^2$. Iterate: at step $i$, treat $x_{i+1}$ as virtual control to drive $z_i \to 0$, define $z_{i+1} = x_{i+1} - \alpha_i$, continue. At step $n$, $u$ is the *real* input — pick it to drive $z_n \to 0$. ### 6.2 Properties and limits - **Uses nonlinearities, doesn't cancel them.** Unlike feedback linearisation, backstepping leaves stabilising nonlinear terms in place and only cancels destabilising ones — yielding *better* transients and lower gains. - **Provably stable.** The composite Lyapunov $V_n = \sum \tfrac{1}{2} z_i^2$ certifies asymptotic stability of the full state. - **Adaptive variant** handles unknown parameters with parameter estimation laws. - **Explosion of complexity** — at step $i$, $\dot{\alpha}_{i-1}$ must be computed analytically, and contains derivatives of all earlier $\alpha_j$. For $n > 3$ this becomes intractable by hand. Two practical fixes: **dynamic surface control (DSC)** (Swaroop–Hedrick 2000) introduces a first-order filter at each step to approximate $\dot{\alpha}$; **command-filtered backstepping** (Farrell–Polycarpou 2008) replaces analytic differentiation with a tracking-differentiator. ### 6.3 When to use it - Cascade plants with measurable intermediate states (manipulator joint+link, motor flux+torque, missile body-rate+attitude). - Plants where matched-only assumptions of SMC are too restrictive — unmatched uncertainty can be handled term by term. - Adaptive control with unknown but constant parameters. ## 7. Other nonlinear control approaches | Method | Core idea | Strength | Weakness | |---|---|---|---| | **Feedback linearisation** (Isidori 1985) | Change of coordinates + state feedback transforms nonlinear plant into linear one; design linear controller | Provably exact when model is known | Requires accurate model; cancels useful nonlinearities; fails at unmatched zeros | | **Computed torque** (robotics) | Subset of feedback linearisation for Euler–Lagrange systems: $u = M(q)\ddot{q}_d + C(q,\dot{q})\dot{q} + g(q) + \tau_{\rm fb}$ | Standard in industrial manipulator control; tight tracking | Sensitive to model parameters (especially friction, payload) | | **Passivity-based control (PBC)** | Exploit energy / port-Hamiltonian structure; design controller as energy shaper | Physical insight; large region of attraction | Limited to systems with well-defined energy functions | | **Lyapunov-based design (LBD)** | Construct $V$, choose $u$ to make $\dot{V} < 0$ | General, intuitive | Finding $V$ is hard; conservative | | **Gain scheduling** | Interpolate linear controllers over operating envelope | Industrially familiar (aerospace, automotive) | No global stability guarantee; "hidden coupling" between schedule and plant | | **Internal model control (nonlinear)** | Embed reference / disturbance generator in controller | Clean structure | Inverse must exist | | **Adaptive control** | Online parameter estimation + certainty-equivalence control | Handles slow drift | Bursting; persistent-excitation requirement | ### 7.1 SMC vs other modern controllers | Property | SMC | LQR | $H_\infty$ | MPC | Backstepping | |---|---|---|---|---|---| | **Matched-disturbance rejection** | Exact (in sliding) | Bounded | Bounded | Bounded | Bounded | | **Unmatched-disturbance rejection** | None (classical) | Bounded | Bounded | Bounded | Bounded | | **Convergence** | Finite-time | Asymptotic | Asymptotic | Asymptotic | Asymptotic | | **Constraints** | None native | None | None | Native | None | | **Model accuracy needed** | Low (bounds only) | High | Medium | High | High | | **Compute cost** | Very low | Low | Low | High (online QP) | Medium | | **Closed-loop margins** | Robust by design | $\ge 60°$ PM, $\ge 6$ dB GM | Designed via $\gamma$ | Implicit | Lyapunov-certified | | **Multivariable** | Yes | Yes | Yes | Yes | Limited to strict feedback | | **Industrial maturity** | High (drives, ABS) | Very high | Medium (aerospace) | Very high (refining) | Medium | ### 7.2 Lyapunov primer for nonlinear control The unifying tool across SMC, backstepping, and adaptive control is the **Lyapunov function**. Given an equilibrium $x^* = 0$ of $\dot{x} = f(x)$, a candidate $V: \mathbb{R}^n \to \mathbb{R}$ certifies stability if: 1. $V(0) = 0$ and $V(x) > 0$ for $x \ne 0$ (positive definite); 2. $V \to \infty$ as $\|x\| \to \infty$ (radially unbounded — only needed for global results); 3. $\dot{V}(x) = \nabla V \cdot f(x) \le 0$ (negative semi-definite — stability) or $< 0$ (negative definite — asymptotic stability). **LaSalle's invariance principle** strengthens the result: if $\dot{V} \le 0$ and the largest invariant set contained in $\{x : \dot{V}(x) = 0\}$ is $\{0\}$, the equilibrium is asymptotically stable. **Finite-time Lyapunov** (Bhat & Bernstein 2000) requires $\dot{V} \le -c V^\alpha$ with $\alpha \in (0, 1)$ — the bound separating SMC from linear feedback. The art of nonlinear control is *constructing* $V$. For SMC, $V = \tfrac{1}{2} s^2$ works automatically. For backstepping, $V$ is built incrementally as the recursion proceeds. For passivity-based control, $V$ is the *physical energy*. For adaptive control, $V$ includes parameter-error terms. The fact that this construction is non-algorithmic is what keeps nonlinear control a craft rather than a button-press. ## 8. Worked examples ### Example A — Classical SMC for a DC servo positioning loop **Plant.** Permanent-magnet DC servo motor driving a $0.5$ kg precision stage through a $5\,{\rm mm/rev}$ ballscrew. Nominal parameters: $$J\ddot{\theta} + b\dot{\theta} = K_t i, \quad J = 1.2 \times 10^{-4}\,{\rm kg\,m^2},\ b = 5 \times 10^{-4}\,{\rm N\,m\,s/rad},\ K_t = 0.085\,{\rm N\,m/A}$$ Inertia uncertain ±30 % due to payload variation; viscous friction uncertain factor of 2; load disturbance torque bounded by $\tau_d \le 0.1\,{\rm N\,m}$. **Sliding surface.** $s = \dot{e} + \lambda e$ with $\lambda = 50\,{\rm s^{-1}}$ — closed-loop time constant on the surface $1/\lambda = 20\,{\rm ms}$. **Equivalent control** (nominal): $$i_{\rm eq} = \frac{1}{K_t}\left( J(\ddot{\theta}_d - \lambda \dot{e}) + b \dot{\theta} \right)$$ **Switching term** with boundary layer $\varepsilon = 0.05\,{\rm rad/s}$: $$i_{\rm sw} = -K_{\rm sw} \, \mathrm{sat}(s/\varepsilon), \qquad K_{\rm sw} = 5\,{\rm A}$$ **Reaching condition.** Worst-case disturbance + parameter mismatch on $s$-dynamics: $$|\delta| \le \frac{0.3 J |\ddot{\theta}_d - \lambda \dot{e}| + b|\dot{\theta}| + |\tau_d|}{K_t} \approx 2.8\,{\rm A}$$ so $K_{\rm sw} = 5\,{\rm A}$ leaves $\eta \approx 2.2\,{\rm A}$ margin. Reaching time from $s(0) = 5\,{\rm rad/s}$: $t_r \le 5/2.2 = 2.3\,{\rm s}$ (worst case); in practice $\sim 50\,{\rm ms}$. **Performance.** A $5\,{\rm mm}$ step command (one revolution) tracks in $80\,{\rm ms}$ with $\pm 5\,\mu{\rm m}$ steady-state error (set by the $\varepsilon = 0.05$ boundary layer mapping back through $\lambda$). Disturbance torque doubling causes no visible response. Pure $\mathrm{sign}$ implementation produces 8 kHz chatter measurable as 40 mA RMS ripple on the current sensor — boundary-layer version reduces this to 2 mA. ### Example B — Super-twisting current loop for buck converter **Plant.** Synchronous-buck converter, $L = 22\,\mu{\rm H}$ (±20 %), $R_{DCR} = 8\,{\rm m\Omega}$, $V_{\rm in} = 24\,{\rm V}$ (varies 18–36 V), switching frequency 200 kHz. Current loop bandwidth target: 10 kHz. **State equation.** $L \dot{i}_L = d \cdot V_{\rm in} - R i_L - v_{\rm out}$, with duty cycle $d \in [0, 1]$ the control input. **Sliding variable.** $s = i_L - i_{\rm ref}$ (first-order; current is the primary state). **Super-twisting control.** $$d = d_0 - \frac{1}{V_{\rm in,nom}}\left[ \alpha |s|^{1/2} \mathrm{sign}(s) + v \right], \qquad \dot{v} = \beta \, \mathrm{sign}(s)$$ with $d_0 = (v_{\rm out} + R i_{\rm ref})/V_{\rm in,nom}$ (feedforward). Disturbance derivative bound from worst-case $\dot{V}_{\rm in}$ and load-step transients: $L \approx 5 \times 10^3\,{\rm A/s}$. Gains $\alpha = 1.5 \sqrt{L} = 105$, $\beta = 1.1 L = 5500$. **Result.** Current loop reaches reference in $< 50\,\mu{\rm s}$ for a $5\,{\rm A}$ step. Current ripple < 1 % of mean — versus 10 % observed with a classical $\mathrm{sign}$-based SMC at the same gain. Continuous duty cycle eliminates audible switching modulation. This is the topology used in TI's InstaSPIN-FOC current regulator (TMS320F28x family) for permanent-magnet motor drives. **Robustness check.** Sweep $L$ over $[17.6, 26.4]\,\mu{\rm H}$ (±20 %), $V_{\rm in}$ over $[18, 36]\,{\rm V}$, load current over $[0, 20]\,{\rm A}$. Closed-loop bandwidth varies $9.2$–$11.5\,{\rm kHz}$; settling time $40$–$55\,\mu{\rm s}$; no oscillation or instability observed in any corner. A PI controller tuned at nominal would have lost $\sim 6\,{\rm dB}$ of gain margin at the worst-case corner — the super-twisting design holds margin across the entire envelope without re-tuning. ### Example C — Backstepping altitude controller for quadrotor **Plant** (altitude channel, decoupled from attitude): $$\dot{z} = w, \qquad \dot{w} = \frac{T \cos\phi \cos\theta}{m} - g$$ with $m = 1.4\,{\rm kg}$, $g = 9.81\,{\rm m/s^2}$, total thrust $T$ as the control input. Attitude $(\phi, \theta)$ assumed small ($\cos\phi\cos\theta \approx 1$). **Step 1 — virtual control on $w$.** Define $z_1 = z - z_d$. Pick $w_d = \dot{z}_d - k_1 z_1$ with $k_1 = 2.5\,{\rm s^{-1}}$. Then $\dot{z}_1 = -k_1 z_1 + (w - w_d)$. Lyapunov $V_1 = \tfrac{1}{2} z_1^2$ satisfies $\dot{V}_1 = -k_1 z_1^2 + z_1 z_2$ where $z_2 = w - w_d$. **Step 2 — real control on $T$.** $\dot{z}_2 = T/m - g - \dot{w}_d$. Pick $$T = m (\dot{w}_d + g - k_2 z_2 - z_1)$$ with $k_2 = 3.0\,{\rm s^{-1}}$. Composite Lyapunov $V_2 = V_1 + \tfrac{1}{2} z_2^2$ gives $\dot{V}_2 = -k_1 z_1^2 - k_2 z_2^2 < 0$ — global asymptotic stability. **Numbers for a $1\,{\rm m}$ step command.** $\dot{w}_d = -k_1 \dot{z}_1 = +k_1^2 z_1$ at $t = 0^+$; thrust pulses to $T = m(g + k_1^2 \cdot 1) = 1.4 \cdot (9.81 + 6.25) = 22.5\,{\rm N}$, settles to hover $m g = 13.7\,{\rm N}$. Rise time $\sim 0.3\,{\rm s}$, no overshoot, zero steady-state error. ## 9. Design heuristics ### When SMC is the right tool - **Matched uncertainty dominates.** Plant gain known to factor of two; disturbance bound known; no need for tight asymptotic behaviour. - **Bang-bang or switching actuator is native.** Power converters, on/off valves, relay heaters, ABS modulators — anything where the actuator inherently switches. - **Finite-time convergence is a hard requirement.** Safety-envelope clamping, fault-detection deadlines, aerospace mission timing. - **Sample rate is high relative to system bandwidth.** $f_s \ge 20 \times f_{\rm bw}$ to keep chatter tolerable. ### When SMC is the wrong tool - **Smooth continuous output is mandatory** (acoustic loudspeakers, fragile optical paths) and super-twisting cannot achieve the residual chatter target → use H∞ or LQG. - **Plant is accurately LTI** with little uncertainty — pole placement, LQR, or $H_2$ design are simpler and tighter. - **Unmatched uncertainty is the dominant problem** — use integral SMC, backstepping, or robust MPC. - **Sample rate is low** ($< 10 \times$ bandwidth) — discretisation amplifies chatter and can destabilise. ### Application matrix — industry deployments | Industry | Application | Variant typically deployed | Reference | |---|---|---|---| | Industrial drives | PMSM / IM torque & speed control | DTC (hysteresis-SMC), super-twisting current loop | ABB ACS880, Siemens SINAMICS | | Power electronics | Buck / boost / flyback current control | Super-twisting, hysteretic SMC | TI InstaSPIN, LTC controllers | | Automotive | ABS slip control, traction control | Classical SMC + boundary layer | Bosch ESP9, Continental MK100 | | Automotive | EV traction motor flux observer | Sliding-mode observer (SMO) | ST X-CUBE-MCSDK, TI motorControlSDK | | Aerospace | Missile pitch-yaw autopilot | Backstepping + integral SMC | RAFAEL, MBDA published references | | Aerospace | UAV attitude control | Super-twisting attitude controller | PX4 advanced controllers | | Robotics | Manipulator joint impedance | Computed torque + sliding-mode robustness | KUKA LWR-IV, FRANKA EMIKA | | Robotics | Quadruped balance control | Backstepping for cascade dynamics | Boston Dynamics Spot (published) | | Process | pH neutralisation | Adaptive backstepping | Honeywell adaptive control suite | | Energy | Grid-tied inverter current loop | Super-twisting + repetitive | Schneider Conext, ABB PVS | | Biomedical | Continuous glucose pump | Integral SMC | Medtronic 770G (closed loop) | ### Boundary-layer tuning The boundary layer $\varepsilon$ exchanges **strict invariance** for **practical chatter reduction**. Steady-state error on the surface is approximately $\varepsilon / \lambda$ (in units of $e$). Rule of thumb: pick $\varepsilon$ so chatter amplitude on the actuator falls below the actuator dead-band; if steady-state error then violates spec, raise the surface gain $\lambda$ or switch to super-twisting. ### Super-twisting gain selection Start with the Levant-recommended $\alpha = 1.5\sqrt{L}$, $\beta = 1.1 L$ where $L$ is the bound on $\dot{\Delta}$. If you cannot compute $L$ analytically, identify it from worst-case open-loop simulation: drive the plant with extreme inputs and parameter values, measure $\max |\dot{\Delta}|$. **Add 50 % safety margin.** Use adaptive gain (Plestan 2010) if $L$ varies widely. ### Backstepping pitfalls - **Differentiate, do not approximate.** Numerical differentiation of $\alpha_i$ injects noise that propagates through the recursion. Use command filters (DSC) when $n \ge 3$. - **Watch initial conditions.** The composite Lyapunov function is positive but the *transient* can be large because each $z_i$ starts at $x_i(0) - \alpha_{i-1}(0)$, which may be far from zero. - **Adaptive backstepping** needs persistent excitation for parameter convergence — guaranteed only when the reference is rich enough. ### Real-world implementation Sample at $\ge 10 \times$ the closed-loop bandwidth, ideally $\ge 20 \times$. ADC quantisation sets a hard floor on achievable chatter — a 12-bit ADC on a $\pm 10\,{\rm V}$ range has $5\,{\rm mV}$ LSB, and the controller cannot distinguish $|s|$ below that. Anti-alias the surface variable derivative if it is computed numerically; the robust differentiator of §4.3 is the cleanest option. ### Sliding-mode observers (SMO) The same machinery used for control applies in *reverse* to state estimation. A sliding-mode observer for the system $\dot{x} = A x + B u + \phi(x, u)$, $y = C x$ takes the form $$\dot{\hat{x}} = A \hat{x} + B u + \phi(\hat{x}, u) + L_{\rm eq} (y - C \hat{x}) + L_{\rm sw} \mathrm{sign}(y - C\hat{x})$$ The discontinuous correction term enforces $y - C\hat{x} \to 0$ in finite time, after which the estimate is *exact* in the observable subspace despite bounded unknown nonlinearity $\phi$. SMOs are the textbook example for **sensorless motor flux estimation**: the back-EMF, treated as a disturbance, is reconstructed from the equivalent injection. Walcott–Zak 1987 gave the canonical fault-tolerant SMO; Edwards–Spurgeon 1994 the practical step-by-step design. Production: ST X-CUBE-MCSDK and TI InstaSPIN both ship SMO-based sensorless drivers for PMSM and induction machines. ### Combining SMC with linear outer loops A widely used architectural pattern is **cascaded linear-outer / SMC-inner**: the outer loop is a PI or LQR for tracking + setpoint shaping, the inner loop is super-twisting for the fast actuator dynamics. Each layer is designed independently — the outer loop assumes the inner loop tracks perfectly (legitimate because finite-time convergence makes the inner-loop transient negligible at outer-loop timescales). This pattern is what makes SMC compatible with existing PID-tuned plants: the operator still sees a familiar setpoint dial. ## 10. Edge cases & gotchas ### Chattering excites unmodelled fast dynamics Parasitic inductance, capacitor ESR, mechanical compliance, sensor bandwidth — all can be excited by the kHz-class switching of classical SMC. The result is sustained limit cycles that masquerade as "noise" but waste real power and damage hardware. **Always characterise the high-frequency plant** at 10–100× the design bandwidth before deploying SMC, even with super-twisting. ### Reaching-phase transients Far from the surface, the switching gain $K$ drives the actuator to its limit. Without input saturation handling, the integrator-like states (e.g. position error in a velocity-error sliding surface) can overshoot dramatically. Mitigation: explicit saturation in the reaching-law amplitude, plus a more conservative $\lambda$ for the surface coefficients. ### Adaptive gain "bursting" If the disturbance temporarily vanishes, an adaptive switching gain $K$ shrinks; if the disturbance returns suddenly, $K$ cannot grow fast enough and $s$ blows up before adaptation catches up. The Ioannou σ-modification adds a leakage term to bound $K$ from below; the dead-zone modification freezes adaptation when $|s|$ is small. ### Discretisation effects Continuous-time analysis is the design layer; the implementation runs at finite $f_s$. The discretised sign function on a sampled signal exhibits **discretization chatter** independent of unmodelled dynamics — switching at every sample. The effective steady-state band scales as $K/f_s$ in the worst case. Use Euler-discretised super-twisting (Brogliato 2020) or implicit schemes for chatter-free discrete implementation. ### Sliding-surface design is an art The Hurwitz polynomial coefficients $c_i$ set the *speed* of the ideal dynamics but also the *aggressiveness* of the equivalent control. Faster surfaces demand larger $u_{\rm eq}$ — and saturate sooner. Practical guideline: surface time-constant 3–5× faster than the desired closed-loop response; never faster than $1/(5 T_s)$ where $T_s$ is the sample period. ### Quasi-continuous and industrial compromises Many commercial SMC products are not pure mathematical SMC: they use **proportional switching** ($\propto s$ inside a small band, saturated outside), **PWM averaging** of the switching signal, or **hysteresis** with deadband. These are pragmatic compromises that buy chatter reduction at the cost of strict invariance — the engineer must verify performance under worst-case disturbance, not just nominal. ### Hybrid and multi-input plants Multi-input SMC ($m > 1$) requires care with the **reaching-condition decoupling matrix** $L_g s$ being invertible everywhere. Hybrid plants (continuous dynamics with discrete switches — gear changes, mode transitions) need switched Lyapunov analysis (Bartolini 1998) and minimum-dwell-time guarantees. ### Discrete-time implementation choices The continuous-time $\mathrm{sign}(s)$ has no faithful direct digital implementation. Three common discretisations: 1. **Explicit Euler with sign:** $u_k = -K \mathrm{sign}(s_k)$ — simplest, but exhibits sample-driven chattering of amplitude $\propto K T_s$ regardless of plant. 2. **Implicit (Brogliato 2020):** solve $s_{k+1} + K T_s \mathrm{sign}(s_{k+1}) = s_k$ for $s_{k+1}$ — gives chatter-free finite-time discrete sliding mode at the cost of a scalar nonlinear solve per step. 3. **Euler discretised super-twisting:** straightforward forward-Euler on $u$ and $v$ in §4.1 — for super-twisting the continuous control makes the discretisation well-posed; chatter scales as $T_s^2$, not $T_s$. Sample-rate selection: $f_s \ge 20\, f_{\rm bw}$ for super-twisting acceptable; $f_s \ge 50\, f_{\rm bw}$ for classical SMC if smooth output is needed. Many PMSM drives run inner-loop SMC at $f_s = 10$–$20\,{\rm kHz}$ for a closed-loop bandwidth of $500$–$1000\,{\rm Hz}$. ## 11. Tools & software ### Simulation and design environments - **MATLAB Simulink** + Control System Toolbox — `slidemode` examples in Aerospace Blockset and Powertrain Blockset; custom S-functions for arbitrary switching laws. The MathWorks Variable-Structure Control example library covers Levant differentiator, super-twisting, twisting algorithm. - **MathWorks Simscape Electrical** — switching-converter SMC implementations directly on physical-network models. - **Python `python-control` + `scipy`** — for prototyping; no built-in SMC blocks, but the underlying numerical ODE solvers handle discontinuous right-hand sides via event-detection in `scipy.integrate.solve_ivp`. - **Julia `ControlSystems.jl`** + `DifferentialEquations.jl` — event-driven simulation of discontinuous controllers is a first-class capability. - **PLECS** (Plexim) — power-electronics-oriented; the de-facto simulator for switching-converter SMC. - **OpenModelica** + Modelica Standard Library — physical-modelling-first; supports hybrid event-driven simulation. ### Code generation and deployment - **MATLAB Embedded Coder** + Simulink — generates ANSI C for STM32, NXP S32K, TI C2000 from Simulink SMC blocks. - **TI C2000 InstaSPIN-FOC and InstaSPIN-MOTION** — production-ready FOC current loops use an internal sliding-mode observer (SMO) for sensorless flux/angle estimation; documented in TI app note SPRABQ3. - **TI motorControlSDK** — open-source reference for super-twisting current control on TMS320F280049. - **STM32 X-CUBE-MCSDK** (ST) — motor-control firmware package; includes sliding-mode observer for PMSM sensorless. - **dSPACE Real-Time Interface** — Simulink-to-MicroLabBox / MicroAutoBox; standard aerospace and automotive HIL platform for SMC validation. - **Speedgoat real-time** — MathWorks-native real-time target; supports custom switching frequencies up to 1 MHz on FPGA I/O. ### Industrial products embedding SMC variants - **ABB ACS800 / ACS880 drives** — Direct Torque Control (DTC) is a quasi-SMC: stator flux and torque each have a hysteresis band, switching law selects voltage vector that drives both toward setpoint. Effectively a 2-input, hysteresis-modulated SMC. - **Siemens SINAMICS S120** — implements DTC and a "Servo Control" mode with sliding-mode-derived flux observer. - **Bosch / Continental ABS modules** — wheel-slip control is a classical SMC implementation, well-documented in SAE 2002-01-1583. - **Honeywell HG1700 / HG4930 IMUs** — internal calibration loops use sliding-mode observers for bias estimation. ### Research codebases - **levant-toolbox** (GitHub, Levant lab) — MATLAB / Octave implementations of the robust differentiator, super-twisting, and quasi-continuous algorithms with verified gain-selection scripts. - **smc-utils** (Plestan group, ECN) — Adaptive-gain SMC reference implementations. - **HOSM Toolbox** (Shtessel et al.) — higher-order sliding-mode control + observer Simulink library. - **CasADi** + **acados** — when SMC needs to coexist with predictive layers; same QP solver supports both worlds. ## 12. Cross-references - `[[Engineering/classical-control]]` — PID, lead-lag, frequency-domain design; the linear layer below SMC. - `[[Engineering/state-space-methods]]` — Lyapunov stability, controllability, observability; mathematical foundation shared by SMC and backstepping. - `[[Engineering/mpc-control]]` — constrained optimal control; the complementary modern controller; SMC and MPC are routinely combined (MPC outer loop, SMC inner loop). - planned `[[Engineering/h-infinity-robust]]` — alternative frequency-domain robust controller; same companion batch as this note. - planned `[[Engineering/state-space-methods]]` — Luenberger and Kalman observers; sliding-mode observers (SMO) are the nonlinear-robust complement. - planned `[[Engineering/digital-control]]` — z-domain, discretisation, sample-rate selection for SMC implementation. - `[[Engineering/electric-motors]]` — PMSM and induction-motor control; DTC and FOC are SMC-flavoured. - `[[Engineering/power-electronics]]` — buck, boost, and three-phase inverter control; classical SMC application domain. - planned `[[Engineering/adaptive-control]]` — parameter-adaptive variants; deep overlap with adaptive SMC and adaptive backstepping. - planned `[[Robotics/pid-control]]` — manipulator dynamics, computed torque, impedance control; backstepping origin domain. - planned `[[Robotics/pid-control]]` — robot-joint SMC and chattering mitigation at the actuator level. - `[[Languages/Tier3/scientific]]` — language reference for the canonical SMC tool. ## 13. Citations - Utkin, V. I. (1977). "Variable Structure Systems with Sliding Modes." *IEEE Trans. Automat. Contr.* 22(2), 212–222. The canonical SMC paper that introduced the framework to the English-language literature. - Utkin, V. I., Guldner, J. & Shi, J. (2009). *Sliding Mode Control in Electromechanical Systems* (2nd ed.). CRC Press. The industrial reference, especially for motor drives. - Edwards, C. & Spurgeon, S. K. (1998). *Sliding Mode Control: Theory and Applications*. Taylor & Francis. The British school's textbook treatment. - Levant, A. (1993). "Sliding order and sliding accuracy in sliding mode control." *Int. J. Control* 58(6), 1247–1263. The super-twisting algorithm paper. - Levant, A. (1998). "Robust exact differentiation via sliding mode technique." *Automatica* 34(3), 379–384. The robust differentiator. - Levant, A. (2003). "Higher-order sliding modes, differentiation and output-feedback control." *Int. J. Control* 76(9–10), 924–941. The $k$-SMC family. - Levant, A. (2005). "Quasi-continuous high-order sliding-mode controllers." *IEEE Trans. Automat. Contr.* 50(11), 1812–1816. - Moreno, J. A. & Osorio, M. (2008). "A Lyapunov approach to second-order sliding mode controllers and observers." *Proc. 47th IEEE CDC*, 2856–2861. The strict Lyapunov function for super-twisting. - Krstic, M., Kanellakopoulos, I. & Kokotovic, P. V. (1995). *Nonlinear and Adaptive Control Design*. Wiley. The canonical backstepping textbook. - Kanellakopoulos, I., Kokotovic, P. V. & Morse, A. S. (1991). "Systematic design of adaptive controllers for feedback linearizable systems." *IEEE Trans. Automat. Contr.* 36(11), 1241–1253. Origin of backstepping. - Khalil, H. K. (2002). *Nonlinear Systems* (3rd ed.). Prentice Hall. The canonical nonlinear-control textbook; Chapter 14 covers SMC, Chapter 13 backstepping. - Slotine, J.-J. E. & Li, W. (1991). *Applied Nonlinear Control*. Prentice Hall. The accessible practitioner's text; popularised the boundary-layer approach. - Isidori, A. (1995). *Nonlinear Control Systems* (3rd ed.). Springer. Differential-geometric approach to feedback linearisation. - Plestan, F., Shtessel, Y., Bregeault, V. & Poznyak, A. (2010). "New methodologies for adaptive sliding mode control." *Int. J. Control* 83(9), 1907–1919. The adaptive-gain SMC paper. - Shtessel, Y., Edwards, C., Fridman, L. & Levant, A. (2014). *Sliding Mode Control and Observation*. Birkhäuser. Comprehensive modern reference covering observers and HOSM. - Bartolini, G., Ferrara, A. & Usai, E. (1998). "Chattering avoidance by second-order sliding mode control." *IEEE Trans. Automat. Contr.* 43(2), 241–246. The sub-optimal algorithm and chattering analysis. - Swaroop, D., Hedrick, J. K., Yip, P. P. & Gerdes, J. C. (2000). "Dynamic surface control for a class of nonlinear systems." *IEEE Trans. Automat. Contr.* 45(10), 1893–1899. Dynamic surface control (DSC). - Farrell, J. A., Polycarpou, M., Sharma, M. & Dong, W. (2009). "Command filtered backstepping." *IEEE Trans. Automat. Contr.* 54(6), 1391–1395. - Brogliato, B., Polyakov, A. & Efimov, D. (2020). "The implicit discretization of the super-twisting sliding-mode control algorithm." *IEEE Trans. Automat. Contr.* 65(8), 3707–3713. Discretisation analysis. - Anderson, B. D. O. & Moore, J. B. (1971). *Linear Optimal Control*. Prentice Hall. The LQR margin results that backstepping and SMC must beat to justify the added complexity. - SAE 2002-01-1583. *Sliding-Mode Control for ABS Wheel-Slip Regulation.* Automotive industry standard reference. - Texas Instruments SPRABQ3 (2013). *Sensorless-FOC With Flux-Weakening and MTPA for IPMSM Motor Drives Using InstaSPIN-FOC.* The TI C2000 super-twisting / sliding-mode observer reference.