Input-to-State Stability: Theory & Applications

• Input-to-state stability (ISS) is a property ensuring that system states remain bounded and eventually contract to a disturbance-dependent neighborhood determined by a gain function.
• ISS theory unifies classical stability concepts with robustness analysis and extends to hybrid, switched, and infinite-dimensional systems.
• Lyapunov function methods and small-gain theorems provide practical tools for controller design and robustness analysis in various ISS applications.

Input-to-state stability (ISS) formalizes the interplay between stability and disturbance robustness for controlled nonlinear systems by providing quantitative bounds linking the magnitude of external disturbances (“inputs”) to the evolution of system states. ISS theory unifies classical stability concepts for ordinary differential equations (ODEs) with robustness analysis and extends to hybrid, impulsive, switched, and infinite-dimensional systems including partial differential equations (PDEs). An ISS property ensures that, for any initial state and any essentially bounded input, the system’s state trajectory remains bounded and eventually contracts to a disturbance-dependent neighborhood determined by an explicit gain function.

1. Mathematical Definition and Classes of ISS

Let $x(t)\in \mathbb{R}^n$ be the state of a controlled dynamical system and $u(\cdot)$ an exogenous input. The standard ISS definition asserts that there exist comparison functions $\beta\in\mathcal{KL}$, $\gamma\in\mathcal{K}_\infty$ such that

$$|x(t)| \le \beta(|x(0)|,t) + \gamma(\|u\|_\infty) \qquad \forall t\ge 0,$$

where $\|u\|_\infty = \esssup_{s\geq 0} |u(s)|$ (Mironchenko, 25 Jun 2024, Mironchenko et al., 4 Jun 2024). The function $\beta$ quantifies the state decay from initial conditions, while $\gamma$ encodes the input-to-state gain—i.e., the maximum steady-state deviation due to bounded disturbance.

For infinite-dimensional systems on a Banach space $X$, with input space $U$ and admissible inputs $\mathcal{U}$, the ISS property is analogously defined as

$$\|\phi(t,x_0,u)\|_X \le \beta(\|x_0\|_X, t) + \gamma(\|u\|_\infty),$$

where $\phi$ is the flow map and the comparison functions satisfy the same structural properties (Heni, 11 Nov 2024, Mironchenko et al., 4 Jun 2024, Mironchenko et al., 2019). The choice of $\mathcal{K}$, $\mathcal{KL}$ and related classes is standard: $\gamma\in\mathcal{K}$ means continuous, strictly increasing, zero at zero; $\beta\in\mathcal{KL}$ is of class $\mathcal{K}$ in the first argument and decreasing to zero in the second.

Variants such as integral ISS (iISS), local/practical ISS (LISS/ISpS), and set-ISS adapt the structure of the gain and decay estimates to other robustness regimes, using more general input norms or allowing for residual inaccuracies (Ito, 2020, Sinha et al., 2022).

2. Lyapunov Characterizations

The Lyapunov-based approach is central and provides both necessary and sufficient conditions for ISS in broad classes of systems (Mironchenko, 25 Jun 2024, Heni, 11 Nov 2024, Mironchenko et al., 4 Jun 2024, Mironchenko et al., 2017). The canonical dissipative-form ISS Lyapunov function $V:X\to\mathbb{R}_+$ satisfies: $$\alpha_1(|x|) \le V(x) \le \alpha_2(|x|), \qquad \dot{V}(x,u) \le -\alpha_3(|x|) + \sigma(|u|),$$ for class $\mathcal{K}_\infty$ functions $\alpha_i$ and $\sigma$ (Mironchenko, 25 Jun 2024, Heni, 11 Nov 2024). This dissipation inequality ensures an explicit trade-off: away from the equilibrium and in the presence of small input, $V$ decays, guaranteeing stability; large inputs feed through $\sigma$ into a bounded dissipation rate, making the ultimate bound explicit.

For infinite-dimensional or hybrid systems, non-coercive Lyapunov functions (which may lack a global lower bound) are often employed, provided one establishes uniform local stability and bounded reachability sets (Mironchenko et al., 2017). Conversely, in Banach or Hilbert spaces, the existence of a coercive (or Lipschitz continuous on bounded sets) ISS Lyapunov function is equivalent to the ISS property (Mironchenko et al., 4 Jun 2024).

In discrete-time and stochastic systems, a stochastic ISS Lyapunov function mediates decay in expectation (see (Culbertson et al., 2023)): $$\mathbb{E}[V(f(x,w)) - V(x)] \le -\kappa_3(V(x)) + \kappa_4(\|w\|_{L^p}),$$ enabling analysis in the probabilistic context (ISSp, input-to-state stability in probability).

3. ISS for Hybrid, Switched, and Impulsive Systems

Hybrid Systems, Impacts, and Periodic Orbits

In systems with impulse effects, such as robotic locomotion with impact events, both continuous and discrete dynamics (flow and jumps) may contribute to the system’s evolution (Veer et al., 2017). For systems that generate periodic orbits (limit cycles), ISS of the orbit is defined via a point-to-set distance to the periodic solution. A forced Poincaré map $P$ is used to encode the effect of continuous and impulsive disturbances: $x_{k+1} = P(x_k, u_k, v_k),$ where ISS of the orbit is shown to be equivalent to ISS of the fixed point of $P$. The bounding estimate takes the form

$$|x(t)|_\mathcal{O} \le \beta( |x(0)|_\mathcal{O}, t ) + \gamma( \|u\|_\infty ) + \gamma( \|v\|_\infty ),$$

linking disturbance magnitudes and initial condition offset to the deviation from the target orbit.

Key consequences include:

• Controller design for periodic gaits reduces to ensuring the linearization of the Poincaré map is stable, i.e., eigenvalues inside the unit circle. Robustness under bounded continuous/discrete disturbances follows from the established LISS of the Poincaré map.
• This framework generalizes classic Poincaré return map arguments for asymptotic stability to the input-robust setting.

Switched and Impulsive Systems

For switched systems, ISS can be enforced even when some individual modes are not ISS, provided the switching signal meets certain constraints, such as average dwell time or (in more general frameworks) growth-constrained activation and transition counts governed by class $\mathcal{FK}_\infty$ functions (Kundu et al., 2015, Kundu et al., 2015). The ISS property under such switching rules is verified using multiple Lyapunov functions $V_i$ and associating switching to walks on a weighted digraph, where contractivity conditions on cycles imply exponential-like decay and overall ISS.

In impulsive and hybrid systems, dwell-time conditions are critical (Dashkovskiy et al., 2012, Dashkovskiy et al., 2016, Liu et al., 2022):

• When continuous dynamics are ISS but impulses are destabilizing, a minimum dwell-time (spacing between jumps) is needed.
• Conversely, if impulses are stabilizing and flows possibly destabilizing, the necessity is for a maximal dwell-time.
• If both contributions are stabilizing, ISS is achieved unconditionally.

Time-delay and functional differential systems admit Razumikhin- and Krasovskii-style ISS Lyapunov functionals, yielding criteria that unify delay-dependent and delay-independent settings (Liu et al., 2022, Sinha et al., 2022).

4. ISS in Infinite-Dimensional Systems

For evolution equations in Banach or Hilbert spaces (including semilinear PDEs, reaction-diffusion, parabolic or hyperbolic equations), ISS generalizes with suitable adjustments for infinite-dimensional analysis (Mironchenko et al., 4 Jun 2024, Dashkovskiy et al., 2012, Mironchenko et al., 2019):

• State and input spaces are Banach spaces, admissible input spaces must be shift-invariant and support concatenation.
• Lyapunov theory (both coercive and non-coercive) provides necessary and sufficient conditions for ISS.
• For linear systems, ISS is equivalent to exponential stability of the generating semigroup plus $L^p$-admissibility of the control operator.
• Applications extend to boundary-controlled systems, delay equations, and coupled network PDEs.

Small-gain theorems ensure that interconnections of ISS subsystems (with nonlinear gains) remain ISS under appropriate gain operator spectral conditions (Mironchenko et al., 4 Jun 2024, Dashkovskiy et al., 2012). Modularity of ISS via max-type Lyapunov constructions and vector Lyapunov approaches is crucial for large-scale or networked infinite-dimensional control problems.

5. Applications and Extensions

• Robotics: Limit-cycle walking/running robot control employs ISS theory for hybrid gaits under external and impulsive disturbances (Veer et al., 2017). Controller synthesis operates via Poincaré map eigenvalue assignment and gain tuning.
• Thermal Phase Change: Stefan problem stabilization with interface heat loss, moving interfaces, and PDE-ODE coupling is analyzed using backstepping and Lyapunov functionals to exhibit explicit ISS bounds (Koga et al., 2019).
• Switched/Networked Systems: Power electronics, networked control, and traffic flow models exploit ISS under quantifiable switching, boundary, or internal disturbances (Kundu et al., 2015, Mironchenko et al., 2019).
• Stochastic Systems: ISS in probability (ISSp) handles stochastic disturbances, leveraging martingale inequalities and probabilistic Lyapunov conditions to ensure high-probability state bounds (Culbertson et al., 2023).

Further extensions (supported by Razumikhin-type approaches, iISS concepts, practical/partial ISS) address robustness to delays, set stability for invariant sets (beyond equilibrium), and partial input/output-to-state stability properties (Sinha et al., 2022, Ito, 2020).

6. Key Theoretical and Practical Implications

• Superposition and Equivalence Theorems: ISS is equivalent to the combination of uniform local stability and a strong limit property (ULIM), generalizing Sontag–Wang theory to Banach spaces and generalizing stability characterizations beyond finite dimensions (Mironchenko et al., 4 Jun 2024, Mironchenko et al., 2017).
• Lyapunov Construction: Both direct (via functional analysis/sobolev methods for PDEs) and converse (using reachability and solution continuity) results guarantee the depth of the ISS paradigm.
• Robustness-Centric Design: ISS provides a systematic method to relate disturbance attenuation, residual bounds, and attractivity rates; modular stability and controller/observer synthesis become tractable for complex, nonlinear, hybrid, or infinite-dimensional systems.

7. Open Issues and Research Directions

Notable open questions relate to:

• Non-Lyapunov characterizations and more general superposition theorems for abstract infinite-dimensional or hybrid systems (Mironchenko et al., 2019).
• ISS for time-varying systems (including delay systems, hybrid and sampled-data systems) (Heni, 11 Nov 2024).
• Computational tools, such as sum-of-squares methods for constructing ISS Lyapunov functionals for PDEs and networked systems.
• Extensions to nonlinear small-gain theorems and robustness notions such as input-to-output or practical ISS.

The ongoing development of ISS theory, supported by recent advances in Lyapunov function methods, small-gain interconnection results, and extensions to hybrid, stochastic, and infinite-dimensional settings, maintains its doctrinal centrality for rigorous analysis and control design in contemporary nonlinear and large-scale control systems.