Input-to-State Stability in Nonlinear Control Systems

Input-to-State Stability (ISS) offers a unifying framework for assessing how external inputs affect the internal state of nonlinear dynamical systems. By quantifying the maximum deviation of system trajectories in response to bounded disturbances, ISS encapsulates both robust stability against perturbations and convergence properties. Central to this concept is the construction of a Lyapunov function whose rate of decay satisfies a dissipation inequality, balancing the influence of state and input magnitudes. Over the past decade, extensions such as integral ISS (iISS) and incremental ISS have addressed systems with energy-like dissipation and trajectory convergence under varying conditions. These developments have broadened the applicability of ISS theory to large-scale interconnected networks, infinite-dimensional systems governed by partial differential equations, switched and hybrid dynamics, and data-driven control contexts. Practical applications span power system stability, biological population dynamics under environmental forcing, and robotic motion control subject to external disturbances. The ISS paradigm not only provides a systematic way to derive stability margins via small-gain theorems and comparison functions but also underpins scalable analysis and synthesis methodologies for complex nonlinear architectures.

Research from Nature Portfolio

Recent studies have introduced a data-driven ISS certification framework that constructs Lyapunov candidates directly from measured trajectories, eliminating the need for explicit model identification while preserving rigorous stability guarantees. By combining kernel-based function estimation with convex optimisation, the method yields non-quadratic Lyapunov functions suited to highly nonlinear behaviour. Another line of research has generalised ISS to stochastic networked systems by integrating graph-theoretic small-gain conditions with probabilistic performance measures. This work establishes that the input-to-state gain operator of a network of stochastic subsystems remains contractive in expectation, thereby ensuring mean-square boundedness of states under random disturbances. A third recent development extends ISS concepts to operator-theoretic descriptions of infinite-dimensional systems, deriving novel dissipation inequalities for systems modelled by Volterra integral operators. These advances push the boundaries of ISS theory towards data-centric and stochastic control applications while retaining the core analytical structure of Lyapunov-based stability analysis.

Research from all publishers

A series of nonlinear small-gain theorems has been formulated for infinite interconnections of ISS subsystems, demonstrating that network stability follows from spectral-radius conditions on the induced gain operator. This extends classical finite-dimensional results to countably infinite and heterogeneous assemblies of systems, offering scalable criteria for large-scale infrastructures. Complementing this, dissipation inequalities have been developed for partial differential equation models, leveraging sum-of-squares programming to compute certificates of ISS and induced-norm boundedness for both domain and boundary control scenarios. These methods facilitate automated verification of stability properties in distributed parameter systems such as fluid flows and elastic structures. Additionally, integral-input-to-state stability has been studied for switched nonlinear systems under dwell-time constraints, where sufficient conditions ensure preservation of iISS through mode transitions. By characterising comparison-function properties of individual modes, this work clarifies how switching frequency and subsystem dissipativity jointly determine overall robustness to persistent disturbances.

Input-to-State Stability in Nonlinear Control Systems publication trend

The graph below shows the total number of articles in input-to-state stability in nonlinear control systems across all publications each year (not limited to Nature Index journals).

Technical terms

Input-to-State Stability (ISS): A property whereby the system state remains bounded in response to bounded inputs and converges to zero as inputs vanish.

Lyapunov function: A scalar function of state used to certify stability by exhibiting decaying behaviour along system trajectories subject to input effects.

Small-gain theorem: A criterion asserting that feedback interconnections of ISS subsystems remain stable if the composition of their gain functions is contractive.

Comparison function: A class of continuous, strictly increasing functions used to relate state magnitudes to input effects in ISS and iISS analyses.

Integral input-to-state stability (iISS): A variant of ISS capturing systems whose state remains ultimately bounded by the integral of the input magnitude over time.

References

1. Dissipation inequalities for the analysis of a class of PDEs. Automatica (2016).
2. Nonlinear small-gain theorems for input-to-state stability of infinite interconnections. Mathematics of Control, Signals, and Systems (2021).
3. Integral-Input-to-State Stability of Switched Nonlinear Systems Under Slow Switching. IEEE Transactions on Automatic Control (2021).
4. Boundedness, persistence and stability for classes of forced difference equations arising in population ecology. Journal of Mathematical Biology (2019).
5. Studying State Convergence of Input-to-State Stable Systems with Applications to Power System Analysis. Energies (2019).