Nonovershooting Control Problem

• Nonovershooting control is a paradigm that designs feedback laws to keep system outputs strictly within preset safety boundaries while achieving closed-loop stability.
• Methodologies such as backstepping with extremum-seeking, homogeneous finite-time techniques, and sliding-mode controllers enable robust performance in varying dynamic and uncertain environments.
• Applications include collision avoidance in safety-critical systems, precision motion control, and networked multi-agent systems where maintaining strict output constraints is essential.

The nonovershooting control problem refers to the synthesis and analysis of feedback control laws that guarantee the controlled variable or vector never “overshoots” a prescribed target or constraint boundary—i.e., its trajectory never crosses from one side of the reference (often from below or above) for all time, while still achieving closed-loop stability, output regulation, or performance objectives. This requirement arises in safety-critical systems, collision avoidance, actuator safety, and whenever constraint-vio-lation carries unacceptable risk or cost. Modern formulations address not only SISO linear systems, but also MIMO, nonlinear, strict-feedback, uncertain, time-delay, switched, and networked control settings.

1. Problem Formulation and Fundamental Principles

The nonovershooting property is defined with respect to an output or error variable, say $h_1(t) = x_1(t) - y_r(t)$, with the requirement

$h_1(t) \leq \bar D, \quad \forall t \geq 0$

where $\bar D > 0$ is a tunable overshoot bound—typically required to be zero (“strict” nonovershoot) or arbitrarily small (“practical” nonovershoot). For safety-critical settings, the nonovershooting constraint is often formalized as a barrier, e.g.,

$\mathcal{H}(x_1, t) = y_r(t) - x_1(t) \geq 0$

so the trajectory must remain within or below the moving boundary for all time.

In strictly-feedback nonlinear systems with unknown control direction, a canonical model is: $$\dot{x}_i = x_{i+1} + \psi_i(x_1, ..., x_i), \quad i=1,...,n-1$$

$\dot{x}_n = g(x)u + \psi_n(x), \quad y = x_1$

with potentially nontrivial and unknown $g(x)$ including its sign. The nonovershooting-from-below objective is then imposed on $h_1 = x_1 - y_r$.

In broader settings (e.g., linear MIMO systems, switched or time-delay systems), the nonovershooting property is expressed for each component or output as “never changes sign compared to the target,” or as invariance of a cone or polyhedral set: $h_i^\top x(t) \geq 0, \quad \forall t \geq 0$ for suitable directions $h_i$.

2. Methodologies for Nonovershooting Control Synthesis

2.1. Backstepping and Extremum-Seeking Under Unknown Control Direction

For strict-feedback systems with unknown control direction, the construction employs a backstepping coordinate transformation: $$h_1 = x_1 - y_r,\quad h_i = x_i - \alpha_{i-1}(x_{1:i-1}, y_r^{(0:i-1)}) - y_r^{(i-1)}, \quad i=2,..,n$$ with recursively defined virtual controls $\alpha_i$, parameterized by gains $c_i$.

An extremum-seeking control law modulates the input,

$$u(t) = \sqrt{\omega} \left[ \beta\cos(\omega t) - \lambda \sin(\omega t) V(h(t)) \right]$$

where $V(h)$ is a radially unbounded function tuned to penalize the norm of $h$, and the parameters $\lambda$, $\beta$, $\omega$ are selected to guarantee effective Lie-bracket averaging. As $\omega\to\infty$, the closed-loop behavior approaches that of an averaged system for which Lyapunov analyses show the overshoot in $h_1$ can be made arbitrarily small by increasing the backstepping and extremum-seeking gains. The unknown sign of $g(x)$ is thus handled adaptively through the extremum-seeking dither, eliminating the need for explicit Nussbaum-type gains and ensuring the nonovershooting property even under control-direction uncertainty (Lu et al., 15 Jan 2026).

2.2. Nonovershooting High-Relative-Degree Safety Filters

In safety-critical applications (e.g., collision avoidance), the nonovershooting requirement typically enters as a high-relative-degree constraint: $y(t) - r(t) \geq 0$, with the input $u$ affecting $h_1$ only after $n$ differentiations. For such problems, a modular architecture consists of (i) a backstepping-style controller synthesizing virtual feedbacks for the $n$-integration chain, and (ii) a filter or override law that replaces the nominal control input $u_0$ by a safe control $\bar{u}(x, \hat{\theta}, r)$ generated to strictly maintain the barrier. The controller and parameter-adaptation modules are designed separately to ensure all safety violations are bounded by a function of the parameter error and its rate, which in turn converges as the identifier adapts (Lyu et al., 23 Sep 2025).

2.3. Homogeneous Finite-Time Invariant-Set Stabilization

In the linear multi-input or input-output-linearizable setting, the nonovershooting control problem is addressed by constructing invariant cones or polyhedral sets (e.g., $\Sigma = \{x: h_i^\top x \geq 0\}$). A linear asymptotic nonovershooting stabilizer is first synthesized by finding a $K$ such that the cone is invariant and the closed-loop is Hurwitz. This is then extended to a homogeneous finite-time stabilizer: $$u(x) = K_0 x + \|x\|_d^{1+\mu}(K - K_0)d(-\ln\|x\|_d)x$$ with negative degree $\mu < 0$, homogeneous norm $\|x\|_d$, and dilation $d(s)$, so that finite-time convergence and cone-invariance (hence nonovershoot) are achieved. The homogeneity framework supports robust invariance under bounded disturbances (input-to-state safety, ISSf) (Polyakov et al., 2023).

2.4. Sliding-Mode and Quasi-Continuous Laws

For (perturbed) double-integrator systems, robust nonovershooting convergence is achieved by quasi-continuous sliding-mode controllers of the form: $u(x) = -|x_1|^{-1} (\gamma x_1 + |x_2| x_2)$ possibly with maximum or additive regularization to suppress chattering at $x_1=0$. Such laws guarantee for all initial conditions that the output never crosses zero (sign is preserved), and admit uniform input-to-state stability bounds for matched and unmatched disturbances (Ruderman et al., 5 Jun 2025, Ruderman et al., 5 Dec 2025). Extensions via smoothed (tanh-based) sliding modes enable chattering-free nonovershooting stabilization in practical UAV applications (Wang et al., 2024).

2.5. Delay-Adaptive Nonovershooting Backstepping

For strict-feedback chains with unknown actuator delays (ODE–transport-PDE–ODE “sandwich” systems), nonovershooting is synthesized via predictor-based backstepping combined with high-order barrier function design. Adaptive delay-identifier modules (finite-time, BaLSI-type) estimate and compensate unknown delay, and a one-sided QP safety filter enforces the tracking constraint $y_1 \geq s$ even before delay convergence, ensuring that the output never violates the “safe” region regardless of initial conditions or unknown delays (Zhao et al., 2024).

3. Theoretical Guarantees and Representative Results

3.1. Lyapunov and Averaging Analyses

For extremum-seeking-based backstepping controllers, Lyapunov functions $W = \frac{1}{2}\bar{h}^T\bar{h}$ for the averaged system yield practical nonovershooting bounds: $\dot{W} \leq -2(c_m-1)W + \frac{1}{\kappa_n}$ which provide uniform ultimate boundedness of $h_1$, with the bound rendered arbitrarily small by increasing the feedback and extremum-seeking gains (Lu et al., 15 Jan 2026).

3.2. Fixed- and Finite-Time Stabilization

Homogeneous feedback can yield either finite-time or fixed-time nonovershooting convergence, with explicit settling-time estimates and Lyapunov certificates. Barrier functions (homogeneous or high-order classical) are constructed to guarantee statewise invariance for all initial data in the “safe region,” with precise handling of boundary behavior during the override phase (Polyakov et al., 2023, Polyakov et al., 2022).

3.3. Safety under Uncertainty and Adaptive Robustness

In presence of parametric or delay uncertainty, modular adaptive override control guarantees that any violation of the nonovershooting barrier is bounded by a known function of the estimator’s parameter error and adaptation rate—this bound vanishes as estimation converges, guaranteeing asymptotic preservation of the constraint (Lyu et al., 23 Sep 2025, Zhao et al., 2024). Such architectures are robust to model uncertainties and intrinsic observation delays.

4. Representative Applications and Implementation

4.1. Safety-Critical Systems and Collision Avoidance

Nonovershooting controllers are natively suited to collision-avoidance, inter-vehicle spacing, and airspace management, where output variables must be rigorously constrained for all time. For example, in delay-coupled vehicle platoons, nonovershooting delay-adaptive backstepping ensures that inter-vehicle distances remain above the safe threshold despite large, unknown transport delays (Zhao et al., 2024).

4.2. High-Performance Tracking and Precision Motion

By combining continuous-reset elements with loop-shaping and phase compensation, it is possible to stack multiple integrators for precision tracking without incurring overshoot or windup, breaking the classical waterbed tradeoff (Karbasizadeh et al., 2021). Reset-based and sliding-mode nonovershooting controllers allow high-gain performance while maintaining safety constraints in high-speed, high-precision mechanical systems.

4.3. Networked, Multi-Agent, and Switched Systems

Distributed output-regulation schemes extend nonovershooting guarantees to cooperative multi-agent networks, ensuring every agent’s regulated variable always tracks the leader from one side with strictly monotonic convergence (Schmid et al., 2018). In switched systems, eigenstructure-assignment techniques formally address the feasibility and design of nonovershooting transitions across multiple subsystems and switching scenarios (Wulff et al., 2024).

5. Relationship to Performance Limitations and Fundamental Constraints

Absolute structural limitations preclude zero overshoot for certain plant classes. Specifically, continuous-time SISO plants with right-half-plane (RHP) zeros or poles, or nonminimum-phase exosystem zeros, are subject to explicit lower bounds on achievable overshoot—the minimum possible overshoot is determined by the slowest unstable mode, and duality frameworks (via Borel measures) provide tight optimality certificates (wenczel et al., 2011). For stable minimum-phase plants, zero overshoot is theoretically achievable, but for RHP zero plants, nonovershooting control is only possible up to the hard limit imposed by the system’s zero-pole configuration.

Moreover, nonovershooting/monotonicity requirements may restrict achievable performance (e.g., convergence rate), especially in sparse-actuation, non-minimum-phase, or high-relative-degree settings. Necessary and sufficient solvability conditions for monotonic tracking in general LTI MIMO systems are characterized in terms of geometric output-nulling subspaces and Radó-type combinatorial rank conditions (Ntogramatzidis et al., 2014, Ntogramatzidis et al., 2014).

6. Synthesis Procedures and Tuning Guidelines

Across the methodologies, typical synthesis involves:

• For backstepping/extremum-seeking: recursive construction of virtual controls, extremum-seeking law parameter selection ($\omega$, $\beta$, $\lambda$), and Lyapunov bounding/averaging arguments.
• For homogeneous finite-time design: computation of invariant cone constraints, solution of structured LMIs for both the Hurwitz and cone-invariance criteria, and explicit construction of $d$-homogeneous norms.
• For sliding-mode/quasi-continuous control: definition of the sliding manifold tailored for monotonic convergence, choice of gains based on disturbance bounds and initial conditions, and regularization of discontinuous switching via “tanh” or boundary-layer design to suppress chattering (Ruderman et al., 5 Jun 2025, Wang et al., 2024).
• For adaptive architectures: modular decomposition into controller and identification/filter modules, with explicit Lyapunov or ISSf-type bounds on safety violations and practical guidelines for gain and initialization selection to minimize “practical” overshoot (Lyu et al., 23 Sep 2025).

A summary of principal techniques is given below.

Methodology	Key Ingredients	Guarantees
Backstepping + Extremum Seeking	Backstepping coordinates, dither law, averaging	Arbitrarily small overshoot
Homogeneous Feedback	Cone invariance, negative-degree homogeneity	Finite/fixed-time, ISSf
Sliding/Quasi-Continuous Mode	Sub-optimal damping, phase-plane partition	Robust, chattering-free
Delay-Adaptive Nonovershooting	Predictor-based transforms, QP safety filter	Safe tracking under delay
Modular Adaptive Override	Separate controller/identifier, ISSf analysis	Bounded violation under $\tilde{\theta}$

7. Perspectives and Extensions

The nonovershooting control problem admits mature formulations for a wide variety of system classes and is under active extension in directions including output-regulation for nonlinear/nonminimum-phase and underactuated systems, fixed-time and prescribed-time safety filter design for safety-critical robotics, and distributed nonovershooting consensus on networks with heterogeneous delays and uncertainties. Open research questions remain regarding systematic design under severe model uncertainty, scalability to high-dimensional/constrained MIMO plants, and precise fundamental tradeoffs between nonovershoot and convergence rate in multi-objective optimal control (Lu et al., 15 Jan 2026, Polyakov et al., 2023, Lyu et al., 23 Sep 2025).

The integration of nonovershooting architectures with learning-based or data-driven adaptive schemes, as well as practical implementation with quantized or sampled-data measurements, continues to be a significant theme in advanced safety-critical control design.