Bilinear Control Systems and Stabilization Techniques

Bilinear control systems occupy a distinctive position between linear and fully nonlinear dynamical models, characterised by control inputs that multiply state variables. This structural form captures a wide array of processes including chemical reactors, thermal systems and certain classes of quantum dynamics. The central challenge lies in designing controllers that ensure stability and performance in the face of intrinsic nonlinear interaction terms. Stabilization techniques draw on Lyapunov theory, feedback linearization, integral action and spectral assignment methods to guarantee that trajectories converge to desired equilibria while respecting constraints on inputs and states. Recent advances have extended classical results to systems with positivity constraints, time-varying delays and distributed parameters, thereby broadening the applicability of bilinear control to networked processes, spatially extended systems and energy conversion devices. These developments underscore the global significance of bilinear stabilisation, offering both theoretical insight and practical algorithms for robust regulation across disciplines.

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Recent studies have unified controller design for a broad class of bilinear systems by integrating integral actions on inputs and outputs within a single framework. A composite controller was shown to encompass static, input-integral and output-integral variants. Stability analysis employed a custom-crafted Lyapunov function whose level sets yield estimates of admissible initial conditions under state and input constraints. Application to a multi-cell heat exchanger demonstrated effective temperature regulation and flow-rate management within preset bounds.

Another line of work addressed positive bilinear systems, where states and inputs remain nonnegative. By exploiting a max-separable Lyapunov candidate, researchers derived simple box-shaped regions of attraction that facilitate real-time implementation. Two dynamic controllers were proposed: one based on summing max-separable functions of state and input, the other coupling quadratic input penalisation with integral action on a performance output. Both schemes achieved asymptotic stability and output regulation, highlighting their suitability for processes such as chemical networks and compartmental thermal systems.

Bilinear Control Systems and Stabilization Techniques publication trend

The graph below shows the total number of articles in bilinear control systems and stabilization techniques across all publications each year (not limited to Nature Index journals).

Technical terms

Bilinear control system: A dynamical model in which control inputs enter multiplicatively with state variables, yielding a system that is linear in each argument separately but not jointly linear.

Lyapunov function: A scalar function of the system state used to assess stability by verifying that it decreases along trajectories of the closed-loop system.

Integral action: A control strategy that incorporates the time integral of an error or output signal to eliminate steady-state errors and improve disturbance rejection.

Region of attraction: The set of initial states from which the closed-loop system trajectories converge to a desired equilibrium under a given controller.

Positive system: A system whose states and outputs remain nonnegative whenever driven by nonnegative inputs, important in modelling physical quantities such as concentrations and temperatures.

References

1. Stabilization for a Class of Bilinear Systems: A Unified Approach. IEEE Control Systems Letters (2023).
2. Stabilization Criteria for Bilinear Systems with Time-varying Delay. Universal Journal of Electrical and Electronic Engineering (2014).
3. Stabilization for a Class of Positive Bilinear Systems. IEEE Control Systems Letters (2023).
4. Constrained Feedback Stabilization for Bilinear Parabolic Systems. Intelligent Control and Automation (2015).