Sampled-Data Control Systems Analysis and Optimization

Sampled‐data control systems address the interplay between continuous‐time plant dynamics and discrete‐time control interventions. By employing devices such as a zero‐order hold, digital controllers update inputs at fixed intervals while the process evolves continuously. Analysis and optimisation hinge on closed‐loop stability, disturbance rejection, robustness to model uncertainty and precise trajectory tracking. Central performance measures include the H∞ norm for worst‐case disturbance attenuation, the H2 norm for energy‐based criteria and induced or Hankel norms to quantify hybrid input‐to‐output gains. Modern advances leverage hybrid‐systems theory, variational principles on arbitrary time scales and linear matrix inequalities to reconcile continuous and discrete specifications. Computational techniques such as piecewise polynomial and superellipsoidal approximations provide tight error bounds and tractable solutions. Applications span robotic manipulators, networked control infrastructures, aerospace guidance and industrial automation, where sampling constraints, time‐varying delays and multirate schemes are intrinsic. By uniting theoretical insights with algorithmic innovation, sampled‐data control research continues to enhance the reliability and performance of digitally implemented control systems.

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A recent study offered a thorough characterisation of the L2/L1 induced norm and the quasi‐Hankel norm in sampled‐data frameworks. By taking into account the h‐periodicity of the input‐output mapping, it established that the induced and Hankel norms coincide and that there always exists a critical phase shift at which the Hankel norm is attained. A matrix‐valued function approach was devised to compute these norms efficiently and to relate them directly to the H2 performance metric, yielding a unified analysis tool for hybrid control.

Another contribution developed a hybrid‐systems approach for H∞ and H2 optimal controller synthesis in linear time‐invariant plants under sampled‐data constraints. The method constructs an explicit Riccati‐based generalised plant with mixed discrete and continuous dynamics, formulates linear matrix inequalities and applies classical loop‐shaping within the sampled‐data context. Benchmark examples, including reference tracking of a two‐mass–spring–damper system, demonstrated enhanced stability margins and performance compared with existing designs.

Further work addressed the l∞‐induced norm computation for multivariable discrete‐time systems. By truncating the inherent infinite‐dimensional operator into a finite block and its residual tail, the authors derived upper and lower bounds that converge exponentially fast as the truncation order increases. Numerical simulations confirmed the theoretical convergence rates and underscored the practical viability of the method for high‐dimensional control systems.

Sampled-Data Control Systems Analysis and Optimization publication trend

The graph below shows the total number of articles in sampled-data control systems analysis and optimization across all publications each year (not limited to Nature Index journals).

Technical terms

Sampling period (h): The constant interval between successive measurements or control updates in a digital control scheme.

Zero‐order hold (ZOH): A device that converts discrete control values into a piecewise‐constant continuous signal over each sampling interval.

H∞ norm: A measure of the maximum gain from disturbance to output across all frequencies, indicating worst‐case robustness.

Hankel norm: The supremum of the norms of the Hankel operator evaluated over all phase shifts within the sampling period, reflecting energy transfer from past inputs to future outputs.

References

1. Optimal sampled-data control, and generalizations on time scales. Mathematical Control and Related Fields (2016).
2. $ L_2/L_1 $ induced norm and Hankel norm analysis in sampled-data systems. AIMS Mathematics (2024).
3. H ∞ and H 2 optimal sampled-data controller synthesis: A hybrid systems approach with mixed discrete/continuous specifications. Automatica (2021).
4. The $ l_\infty $-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis. AIMS Mathematics (2023).
5. Computing the L∞-Induced Norm of LTI Systems: Generalization of Piecewise Quadratic and Cubic Approximations. IEEE Access (2020).
6. Construction of the Time-Optimal Bounded Control for Linear Discrete-Time Systems Based on the Method of Superellipsoidal Approximation. Автоматика и телемеханика (2023).