Adaptive Suboptimal Second-Order Sliding Mode

• Adaptive Suboptimal Second-Order Sliding Mode is a control framework that adapts gains online to enforce both a sliding variable and its derivative to zero or to a disturbance-dependent neighborhood.
• It integrates adaptive gain laws, disturbance compensation, and data-driven techniques to mitigate overdesign and reduce chattering compared to fixed-gain methods.
• The approach achieves finite-time stabilization and robust performance by combining theoretical Lyapunov guarantees with observer-based perturbation estimation and smooth multivariable formulations.

Adaptive Suboptimal Second-Order Sliding Mode (ASSOSM) denotes a class of second-order sliding-mode control schemes in which the sliding regime is enforced by online adaptation rather than by fixed, conservatively chosen gains. In the representative formulations considered here, ASSOSM combines second-order sliding dynamics with adaptive gain laws, disturbance compensation, and, depending on the setting, smooth multivariable control, observer-based perturbation reconstruction, or data-driven synthesis from noisy trajectories. The shared objective is to drive a sliding variable and its derivative to zero in finite time, or, under bounded perturbations, to a disturbance-dependent neighborhood, while mitigating gain overdesign and chattering relative to classical fixed-gain constructions (Wang, 2020, Xiong et al., 2018, Samari et al., 4 Aug 2025).

1. Conceptual setting and problem class

Second-order sliding mode control differs from first-order sliding design by targeting both a sliding variable and its derivative. In the multivariable smooth formulation, with tracking error $e=x_1-x_{1d}$, a standard first-order sliding variable is $s:=e$, and second-order sliding seeks to enforce $s\to 0$ and $\dot s\to 0$ in finite time. The canonical error variables may be interpreted as $x_1\equiv s$ and $x_2\equiv \dot s+\text{(known terms)}$, so the control problem becomes the finite-time regulation of $(x_1,x_2)$ (Wang, 2020).

A scalar canonical form appears in adaptive super-twisting analysis:

$$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$

where $\rho_0(t)$ is a matched perturbation and $\alpha,\beta>0$ are time-varying gains. This representation is the standard super-twisting algorithm (STA) form used for Lyapunov analysis and adaptive gain synthesis (Xiong et al., 2018).

A more recent data-driven formulation considers a continuous-time perturbed nonlinear control system of strict-feedback form

$s:=e$0

with all of $s:=e$1, $s:=e$2, $s:=e$3, and $s:=e$4 unknown. There the sliding variable is defined as

$s:=e$5

where $s:=e$6 is a virtual controller synthesized from data. This shifts the ASSOSM task from model-based gain selection to a two-stage procedure: data-driven stabilization of the upper dynamics, followed by second-order sliding enforcement on the full-order system (Samari et al., 4 Aug 2025).

2. Canonical control structures

The literature considered here contains three closely related ASSOSM-type structures.

Formulation	Core structure	Adaptive object
Adaptive STA	$s:=e$7, $s:=e$8	$s:=e$9
Smooth multivariable SOSM	Closed-loop canonical error system with fractional powers $s\to 0$0, $s\to 0$1, $s\to 0$2	Scalar $s\to 0$3 and gains $s\to 0$4
Data-driven ASSOSM	$s\to 0$5, $s\to 0$6, $s\to 0$7	$s\to 0$8

For a simplified perturbed channel $s\to 0$9, the smooth multivariable design uses a super-twisting-type law with fractional powers and an integral term. Its gains are parameterized by a single scalar adaptation variable $\dot s\to 0$0:

$\dot s\to 0$1

with $\dot s\to 0$2 and a structural condition on $\dot s\to 0$3 ensuring positive definiteness of the Lyapunov matrix used in the proof (Wang, 2020).

In the data-driven strict-feedback design, the physical input $\dot s\to 0$4 is kept continuous by artificially increasing the relative degree:

$\dot s\to 0$5

The auxiliary dynamics are written as

$\dot s\to 0$6

with $\dot s\to 0$7, $\dot s\to 0$8, and unknown $\dot s\to 0$9. The discontinuity is thus placed at the $x_1\equiv s$0-level rather than directly at the plant input (Samari et al., 4 Aug 2025).

This comparison shows that ASSOSM is not a single formula but a design pattern: second-order sliding is retained, while the gain law, plant embedding, and regularization mechanism vary with the problem class.

3. Adaptive gains and the meaning of “suboptimal”

The defining feature of ASSOSM is gain adaptation without exact disturbance bounds. In adaptive STA, the principal adaptive variable is $x_1\equiv s$1, updated by

$x_1\equiv s$2

where $x_1\equiv s$3 is the observer estimate of the perturbation magnitude, $x_1\equiv s$4 is the main user parameter, $x_1\equiv s$5 determines how closely $x_1\equiv s$6 tracks the disturbance magnitude, and $x_1\equiv s$7 enforces a minimum robustness level. The second gain $x_1\equiv s$8 is not tuned independently; it is computed from $x_1\equiv s$9 through Lyapunov parameters $x_2\equiv \dot s+\text{(known terms)}$0, so that the pair $x_2\equiv \dot s+\text{(known terms)}$1 remains in the region where a strict quadratic Lyapunov function exists (Xiong et al., 2018).

In the smooth multivariable design, the adaptive mechanism is a scalar $x_2\equiv \dot s+\text{(known terms)}$2 that grows only when the sliding variable is outside a prescribed neighborhood:

$x_2\equiv \dot s+\text{(known terms)}$3

with $x_2\equiv \dot s+\text{(known terms)}$4 and $x_2\equiv \dot s+\text{(known terms)}$5. When the error is large, all gains rise as powers of $x_2\equiv \dot s+\text{(known terms)}$6; once $x_2\equiv \dot s+\text{(known terms)}$7, the gains are frozen, avoiding unbounded growth and excessive control effort (Wang, 2020).

In the data-driven strict-feedback construction, the adaptive suboptimal SOSM gain is

$x_2\equiv \dot s+\text{(known terms)}$8

where $x_2\equiv \dot s+\text{(known terms)}$9 denotes the maximum of the sequence $(x_1,x_2)$0, and $(x_1,x_2)$1 are design parameters. The role of this law is to increase $(x_1,x_2)$2 until a trajectory-wise analogue of the unknown non-adaptive bound is satisfied, thereby removing the need to compute $(x_1,x_2)$3, $(x_1,x_2)$4, and $(x_1,x_2)$5 explicitly (Samari et al., 4 Aug 2025).

The qualifier “suboptimal” is used in two related senses. First, the adaptation laws do not search for the mathematically smallest admissible gains; they seek gains that become sufficiently large in finite time and then remain bounded. Second, in the smooth multivariable construction, smooth fractional-power terms replace discontinuous sign injections, trading minimality and simplicity for reduced chattering and smoother signals. This suggests that “suboptimal” refers to gain selection and structural regularization, not to abandonment of robustness guarantees under the stated assumptions (Wang, 2020, Samari et al., 4 Aug 2025).

4. Disturbance observers, differentiators, and internal estimation

Observer design is central to several ASSOSM realizations. In adaptive STA, the perturbation $(x_1,x_2)$6 is reconstructed by a third-order sliding mode observer:

$(x_1,x_2)$7

with $(x_1,x_2)$8, gains

$(x_1,x_2)$9

and $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$0. The resulting observer error system is the standard third-order HOSM differentiator structure, and for $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$1 the estimate satisfies $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$2 after a finite time $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$3 (Xiong et al., 2018).

The smooth multivariable line employs an adaptive multivariable smooth disturbance observer (AMSDO) based on the same structural template as the controller. For $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$4, an auxiliary observer state $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$5 is introduced, the observation error is $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$6, and the disturbance estimate $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$7 is generated by the same smooth second-order sliding mechanism with adaptive gains parameterized by $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$8. Because the observer error dynamics can be cast into the same canonical second-order sliding form as the controller error dynamics, the same Lyapunov framework applies (Wang, 2020).

The data-driven strict-feedback scheme does not reconstruct the matched disturbance itself. Instead, it estimates $$\dot z_1=-\alpha |z_1|^{1/2}\operatorname{sgn}(z_1)+z_2,\qquad \dot z_2\in -\beta \operatorname{sgn}(z_1)+\rho_0(t),$$9 using Levant’s robust exact differentiator:

$\rho_0(t)$0

with $\rho_0(t)$1, $\rho_0(t)$2, and sufficiently large $\rho_0(t)$3. The same differentiator is also used to detect extremal values $\rho_0(t)$4 by monitoring zero crossings of $\rho_0(t)$5, which is essential for the suboptimal switching rule (Samari et al., 4 Aug 2025).

A common misconception is that ASSOSM always relies on crude low-pass estimates of uncertainty. The observer-based STA design explicitly contrasts exact perturbation reconstruction via a third-order sliding mode observer with the conventional approximation obtained from a first-order low-pass filter, while the strict-feedback design relies on exact differentiation of the sliding variable rather than model identification of the disturbance (Xiong et al., 2018, Samari et al., 4 Aug 2025).

5. Stability statements and convergence regimes

The stability guarantees in ASSOSM are finite-time but not identical across formulations. In the smooth multivariable setting, a transformed state $\rho_0(t)$6 and quadratic Lyapunov function $\rho_0(t)$7 are used to derive a scalar differential inequality of the form

$\rho_0(t)$8

with $\rho_0(t)$9. When $\alpha,\beta>0$0, this reduces to a fast finite-time convergence result: $\alpha,\beta>0$1 and $\alpha,\beta>0$2 in finite time. When $\alpha,\beta>0$3, the corresponding theorem yields fast finite-time uniformly ultimately boundedness (UUB), so the states converge to a compact disturbance-dependent neighborhood (Wang, 2020).

Adaptive STA is based on a strict Lyapunov function $\alpha,\beta>0$4, where

$\alpha,\beta>0$5

With the Lyapunov-based mapping from $\alpha,\beta>0$6 to $\alpha,\beta>0$7, one obtains $\alpha,\beta>0$8 and finite-time convergence $\alpha,\beta>0$9 for all $s:=e$00. After the observer transient, the gain tracking error $s:=e$01 also converges to zero in finite time, implying

$s:=e$02

after a finite time. Thus the gain converges to the smallest value compatible with the chosen robustness margin $s:=e$03 and floor $s:=e$04 (Xiong et al., 2018).

The data-driven strict-feedback design has a layered guarantee. First, if the data-dependent semidefinite program is feasible, then the upper subsystem

$s:=e$05

is globally asymptotically stable with quadratic Lyapunov function $s:=e$06. Second, the adaptive suboptimal SOSM layer enforces

$s:=e$07

in finite time on any prescribed bounded set. The resulting overall property is semi-global asymptotic stability (S-GAS): for any given bounded set of initial conditions, design parameters can be chosen so that the origin is asymptotically stable for all initial conditions in that set (Samari et al., 4 Aug 2025).

These guarantees should not be conflated. Fast finite-time convergence, fast finite-time UUB, exact perturbation reconstruction, and S-GAS are distinct statements, each tied to a particular disturbance model, observer condition, and plant class.

6. Multivariable, data-driven, and implementation-oriented extensions

A major extension of ASSOSM is direct multivariable design. The smooth second-order sliding formulation is explicitly intended for multi-input and multi-output systems and treats the MIMO case directly rather than through a simple aggregation of decoupled SISO laws. The paper states that classical ASSOSM methods are often SISO and require decoupling for MIMO systems, whereas the proposed approach uses a direct multivariable structure and can provide better coordination among channels. In simulation, choosing $s:=e$08, for example $s:=e$09, yields significantly smoother state and control trajectories and disturbance estimates than adaptive multivariable super-twisting sliding mode control with $s:=e$10, with reduced control amplitude and high-frequency content (Wang, 2020).

Another extension is exact discrete-time implementation of gain adaptation. In adaptive STA, forward Euler discretization of the inclusion for $s:=e$11 can create chattering and numerical issues for large $s:=e$12, so the gain update is implemented by an implicit backward Euler step. The resulting mixed variational inclusion is solved exactly at each step by geometrical enumeration, and the paper states that this produces monotone, non-chattering evolution of $s:=e$13 consistent with the continuous-time inclusion. In the electromechanical emulator example, observer errors converge quickly, within $s:=e$14 s, and chattering in $s:=e$15 is greatly reduced compared to using fixed large STA gains (Xiong et al., 2018).

The most substantial recent development is data-driven ASSOSM with noisy data. The method begins with a single finite-time experiment, forms data matrices $s:=e$16, $s:=e$17, and $s:=e$18, and imposes the noise bound $s:=e$19. The virtual controller is then obtained from an SDP in the decision variables $s:=e$20, $s:=e$21, $s:=e$22, and $s:=e$23. A notable point is the rank condition

$s:=e$24

which is stated to be strictly weaker than the rank condition usually required in Willems-type data-driven methods. The input used during data collection may be a feedback law or even identically zero, provided this rank condition holds. The reported case studies include an inverted pendulum, an unstable linear benchmark used to contrast the method with De Persis–Tesi-type requirements, and a highly nonlinear four-state system in which increasing $s:=e$25, $s:=e$26, and $s:=e$27 enlarges the region of attraction, including experiments with initial conditions of order $s:=e$28 (Samari et al., 4 Aug 2025).

Several recurring misconceptions can be resolved directly from these formulations. “Suboptimal” does not mean loss of rigor; the cited schemes retain strict Lyapunov proofs, finite-time convergence results, or UUB/S-GAS statements under explicit assumptions. Second-order sliding does not require a discontinuous plant input; super-twisting uses an internal discontinuous channel, the strict-feedback ASSOSM integrates a discontinuous $s:=e$29 to obtain continuous $s:=e$30, and the smooth multivariable construction replaces sign terms by fractional powers. At the same time, the limitations are equally explicit: exact perturbation reconstruction in adaptive STA requires $s:=e$31; data-driven feasibility can become conservative or fail as $s:=e$32 grows; and semi-globality in the strict-feedback setting requires larger $s:=e$33, $s:=e$34, and $s:=e$35 as the prescribed initial set expands (Xiong et al., 2018, Wang, 2020, Samari et al., 4 Aug 2025).