Control Theory for Linear and Nonlinear Systems

Control theory provides a mathematical and conceptual framework for influencing the behaviour of dynamic systems so that they perform in a desired manner. In the linear domain, systems are described by linear differential or difference equations, enabling analysis by superposition and spectral methods. Key notions such as controllability and observability determine whether a system’s state can be steered or fully inferred by inputs and outputs. Classical design techniques include pole placement, frequency-domain methods and optimal control, notably the linear quadratic regulator. These methods yield elegant solutions based on eigenvalue assignments or algebraic Riccati equations, with applications ranging from aerospace autopilots to power-grid stabilisers. Nonlinear control theory extends these ideas to systems whose behaviour cannot be approximated by a single linear model. Lyapunov stability theory provides a systematic way to verify convergence to equilibrium points by constructing energy-like functions. Feedback linearisation transforms certain classes of nonlinear plants into equivalent linear systems over limited operating regions, while sliding modes and adaptive schemes address uncertainties and external disturbances. Hybrid approaches marry linear and nonlinear elements, modelling systems that switch between modes or exhibit discontinuities. Across both domains, robustness to modelling errors and rejection of perturbations remain central challenges, driving the development of modern methods for networked systems, distributed control and data-driven design.

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Research from all publishers

Recent advances in multivariable linear control have introduced a geometric procedure for efficient computation of transmission zeros in high-dimensional MIMO systems. By exploiting a novel factorisation of system matrices, the approach dramatically reduces computational effort compared with classical Smith-McMillan techniques, enabling real-time analysis in large-scale sensor networks and communication channels. In optimal control, explicit symplectic-precise iteration algorithms have been proposed for the time-varying linear quadratic regulator and its associated matrix differential Riccati equation. These algorithms avoid long-term numerical drift by preserving underlying Hamiltonian structure and sidestep stiffness issues through a precise iteration scheme. The result is high-fidelity solutions that scale cubicly with system order yet remain robust over extended time horizons. On the nonlinear side, a stochastic tracking framework for quadrotor vehicles has been developed to reject random perturbations while following random reference signals. By integrating an exosystem to model disturbances, a Kalman filter for state and disturbance estimation and a nonlinear observer, the controller achieves reliable performance in discrete time under measurement constraints. This work demonstrates how modern estimation and control tools combine to address complex, uncertain environments in aerial robotics.

Control Theory for Linear and Nonlinear Systems publication trend

The graph below shows the total number of articles in control theory for linear and nonlinear systems across all publications each year (not limited to Nature Index journals).

Technical terms

Controllability: The property that determines whether an external input can steer the internal state of a system to a desired configuration within finite time.

Transmission zero: A frequency at which the transfer function matrix loses rank, causing input excitations to be completely attenuated at the output.

Lyapunov function: A scalar energy-like function whose decrease along system trajectories certifies stability of an equilibrium.

Matrix Riccati equation: A nonlinear algebraic or differential equation whose solution underpins optimal state-feedback gains in quadratic optimal control.

References

1. Examination of Transmission Zeros in the MIMO Sensor-Based Propagation Environment Using a New Geometric Procedure. Sensors (2024).
2. Explicit Symplectic-Precise Iteration Algorithms for Linear Quadratic Regulator and Matrix Differential Riccati Equation. IEEE Access (2021).
3. On the Rejection of Random Perturbations and the Tracking of Random References in a Quadrotor. Complexity (2022).