Essential Linearization: A Synthesis Technique for P.W.M. Control System

Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate relevant matrices and some relevant quantities such as the topological entropy. An application to stabilization via algebraic Riccati equations is included by viewing a control system as an autonomous system in an extended space of states and control inputs. Kernel methods are the main techniques used in this paper and the approach is illustrated via a series of numerical examples. The advantage of using kernel methods is that they allow to perform function approximation from data and, as illustrated in this paper, allow to approximate linear discrete-time autonomous and control systems from data.

The controlled atmospheric re-entry associated with the precision soft-landing of Reusable Launch Vehicles (RLVs) on Earth is very challenging as it depends on multiple parameters [1]. Over the last decade, the cost-effectiveness of such a technology has been finally demonstrated with the successful recoveries of SpaceX’s Falcon 9 first-stage rocket first [2], then followed by other companies such as the Rocket Lab’s Electron micro-launcher [3]. This breakthrough has been made possible by the development of advanced and robust computational methods able to generate in real time the flight conditions and to command the optimal vehicle's deflections accordingly to achieve a safe pinpoint landing. Indeed, during an Earth atmospheric re-entry, the vehicle is subjected to fast system dynamics changes partly induced by external loads associated with the terrestrial environment (e.g., lift, drag, wind and gusts), but also by the actuation commands to answer the landing constraints satisfaction and the vehicle integrity preservation. All those involve uncertainties and nonlinearities, which lead to vehicle’s instability and therefore give reason why for a highly performant Guidance, Navigation and Control system implementation [4]. More particularly, one of the critical aspects is the design of a robust control strategy capable of counteracting the previously defined disturbances and uncertainties while satisfying the strict accuracy requirements associated with the pinpoint landing [5]. As demonstrated by the current state-of-the-art on control design for launchers [6-7], the classical linear control theory represents a rich heritage with a lot of applications. This choice was motivated by its relative easiness of implementation and the possibility to use gain-scheduling techniques to adapt to nonlinear systems. Nevertheless, these techniques are well-adapted to the control system design of single-input single-output systems, such as for example a reusable rocket using a Thrust Vector Control (TVC) system as the unique actuator. The implementation of multiple-input multiple-output control systems becomes then complex since every channel is addressed in a single-loop fashion. This capability is however required for the future generation of reusable rocket, using also aerodynamic steering based on fins in addition to the TVC system for enhancing control authorities. Moreover, model uncertainties are not accurately considered in the design process, developed only with nominal conditions and stability margin requirements. For all these reasons, it results in an extensive (both in terms of time and cost) Verification and Validation campaign with many iterations and Monte-Carlo analyses to assess the performance and robustness of the control system. To overcome these drawbacks, the H-infinity family of methods, introduced a few years ago [8], provides with a powerful solution for robust control design. It relies on defining the control requirements in the frequency domain in terms of weighting functions and minimising the maximum gain of the resulting weighted system from the exogenous inputs to the outputs to be controlled. Moreover, the control-plant interaction is modelled through a Linear Fractional Transformation (LFT) representing the feedback action. Finally, the structured H-infinity method [9] allows to directly impose a specific control structure – like a Proportional-Integral-Derivative (PID), enabling the re-use of gain-scheduling techniques – and to consider parametric uncertainties for enhanced robustness. This paper studies the synthesis of a robust control system via structured H-infinity for the RLV atmospheric re-entry problem. First, the nonlinear 6-Degree-of-Freedom (6-DoF) RLV re-entry dynamics are simplified into a linear model and then linearised along a reference trajectory to get the nominal LFT of the system, then augmented with parametric uncertainties. The model covers the atmospheric re-entry and vertical landing of a first-stage rocket equipped with a TVC system and steerable planar fins. The controllers are built at different points of the re-entry trajectory, using the structured H-infinity framework through PID-like structures. Weighting functions considering the control objectives and requirements of a realistic RLV re-entry scenario are implemented. A robust stability analysis of the obtained system is performed via classical stability margins and structured singular value. Finally, the controllers are gain-scheduled and validated via Monte-Carlo analyses, using a nonlinear 6-DoF RLV re-entry dynamics simulator equipped with successive convex optimisation guidance. The performance of the resulting guidance and control architecture is compared with the baseline developed in previous works [10], using a TVC system only and classical linear feedback control through gain-scheduled PID controllers. This study lies within the ASCenSIon (Advancing Space Access Capabilities - Reusability and Multiple Satellite Injection) project, an innovative training network funded within H2020.

ABSTRACTHeart rate can be used to define exercise intensity; feedback control systems for treadmills which automatically adjust speed to track arbitrary heart rate target profiles are therefore of interest. The aim of this study was to compare linear (L) and nonlinear (NL) controllers using quantitative performance measures. Sixteen healthy male subjects participated in the experimental L vs. NL comparison. The linear controller was calculated using a direct analytical design that employed an existing approximate plant model. The nonlinear controller had the same linear component, but it was augmented using static plant-nonlinearity compensation. At moderate-to-vigorous intensities, no significant differences were found between the linear and nonlinear controllers in mean RMS tracking error (2.34 vs. 2.25 bpm [L vs. NL], p=0.26) and average control signal power (51.7 vs. m2/s2, p=0.16), but dispersion of the latter was substantially higher for NL (range 45.2 to 56.8 vs. 30.7 to m2/s2, L vs. NL). At low speed, RMS tracking errors were similar, but average control signal power was substantially and significantly higher for NL (28.1 vs. m2/s2 [L vs. NL], p<0.001). The performance outcomes for linear and nonlinear control were not significantly different for moderate-to-vigorous intensities, but NL control was overly sensitive at low running speed. Accurate, stable and robust overall performance was achieved for all 16 subjects with the linear controller. This points to disturbance rejection of very-low-frequency heart rate variability as the overriding challenge for design of heart rate controllers.

The question addressed is placed in the broad context of fuzzy control, based on fuzzy PI controller, of industrial processes, the equivalence of fuzzy PI controllers with linear PI controllers, tuned with given methods, for the improvement of performance of control systems. The purpose of the study is to analyze the possibility of these implementation and to find the behavior of the resulted fuzzy control systems to compare with the linear control systems. The tuning criteria chosen are the modulus criterion in Kessler’s variant for second-order processes with two time constants, the symmetric criterion in Kessler’s variant, for the second order processes with purely integrator character and Ziegler-Nichols criterion for slow processes. The equivalence of the fuzzy controllers with the linear controllers was made using the input-output transfer characteristics of the fuzzy block and its linearization in the origin to obtain the gain in the origin. The results consist of an equivalence formula for the controller, simulation characteristics for the control systems, comparisons and analyses. The main conclusions are that fuzzy regulators can be designed by equating linear regulators with the presented method and better regulation performances can be obtained.

Parametric identification for robust control -- Flow process models for pipeline diagnosis -- Output observers for linear infinite-dimensional control systems -- Non-Gaussian noise reduction in measurement signal processing -- Fractional order models of dynamic systems -- Switched models of non-integer order -- Nonlinear predictive control -- Positive linear control systems -- Controllability and stability of semilinear fractional order systems -- Computer simulation in analysis and design of control systems.

Fluid Power Control Systems are mostly used in power plants. System design may change according to the required application and specified output quality. Mathematical model is important to get accurate expectation for system performance. Static and dynamic response of a fluid power control system under different operating conditions, considering supply pressure and temperature variation has been investigated. Different system designs implementing different system stiffness and different types of control valves have been constructed in a linear displacement control system, then experimented and evaluated. Sequence and time of operations has been controlled using PLC controller. Conclusions reveal that the power consumed could be decreased by simplifying system design using one high-response valve instead of two low-response valves. In addition, system dynamics could be improved from 50 % up to 79 % depending on working conditions. Mathematical model for the control system has been predicted, descending from the linear theoretical relationship between flow rate and valve opening according to Bernoulli equation. Supply pressure variation and flow force effects have been considered. The predicted relationship achieved remarkable agreement with the experimental performance of the system.Copyright © 2012 by ASME

Workload control (WLC) systems can be seen as a natural extension of input-output control systems which allow both the control of jobs into the shop and the control of capacity which affects the flow of finished goods out of the shop. In previous research, it has been shown that WLC can help to reduce manufacturing lead times, even if the total workload processed by the shop in a given time period is maintained at a constant rate. However, the relative contributions of the input control mechanisms versus the output control mechanisms have not been clearly determined. This paper seeks to explore this which affects the flow of finished goods out of the shop. In previous research, it has been shown that WLC can help to reduce manufacturing lead times, even if the total workload processed by the shop in a given time period is maintained at a constant rate. However, the relative contributions of the input control mechanisms versus the output control mechanisms have not been clearly determined. This paper seeks to explore this issue using a simulation of a real job shop, in which the WLC system that has been developed at Lancaster University is used. It explores the use of input control alone and the use of input control plus output control. In addition, the relation between workload and various performance measures, such as lead time, queueing time and capacity utilisation, is explored. The results of the experiments show that one should consider input control and output control as two levers of control that are both to be pulled rather than relying on the single lever of stopping the input when the WIP reaches some preset level.

Linear control of nonlinear processes: recent developments and future directions

Distributed control and optimization form the backbone of many critical control systems, with typical examples including sensor networks, power grids, and intelligent transportation systems. As these systems grow in scale and complexity, ensuring the privacy of individual agents becomes increasingly essential. Privacy protection is no longer an optional add-on; it is a crucial element in the design of these systems. Sensitive data, such as states, measurements, and decisions, must be shielded from unauthorized access or malicious attacks, while still allowing the system to function effectively. This special issue aims to explore novel techniques and strategies that protect privacy in distributed control and optimization, showcasing how these approaches can be seamlessly integrated into real-world applications. The diverse contributions in this issue demonstrate the growing importance of privacy in the evolving field of distributed control systems. RNC7747 introduces a privacy-preserving method for coverage control in multi-agent systems with limited communication ranges, ensuring efficient operation without compromising agents' privacy. RNC7798 proposes a privacy-preserving consensus mechanism based on pulse-coupled oscillators, helping maintain privacy while agents coordinate in distributed networks. RNC7778 presents privacy preservation techniques for cloud-based cooperative LQG control systems, enabling privacy protection even when leveraging cloud resources. RNC8051 discusses achieving privacy-preserving average consensus in unreliable network environments, ensuring privacy despite communication challenges. RNC7929 introduces a novel approach to open privacy-preserving consensus via state decomposition, allowing agents to reach consensus while maintaining privacy. RNC7966 investigates leader-follower consensus in multi-agent systems with distributed event-triggered observations, ensuring privacy during agent coordination in real-time systems. RNC7789 discusses the tradeoff between privacy protection strength and control performance in control systems. RNC8029 proposes a new privacy mechanism for distributed average consensus on time-varying directed graphs. RNC7776 investigates event-triggered distributed optimization with differential privacy, reducing communication costs while safeguarding sensitive data. RNC7885 develops an event-triggered privacy-preserving optimization method for directed communication networks, emphasizing privacy during optimization tasks. RNC7791 focuses on robust optimization for virtual power plant scheduling under uncertainty, incorporating privacy preservation into the optimization process. RNC7730 explores prescribed-time distributed optimization with set constraints, offering time guarantees while maintaining privacy in dynamic optimization scenarios. RNC7926 provides a privacy-preserving algorithm for constrained resource allocation with communication delays, crucial for real-time decision-making in distributed systems. RNC7848 presents an adaptive event-triggered control strategy for cyber-physical systems under DoS and deception attacks, ensuring privacy and system security. RNC8068 addresses initial state privacy in nonlinear systems on Riemannian manifolds, offering methods to protect privacy in complex, high-dimensional control systems. RNC7906 focuses on dynamic consensus-based formation control for multi-robot systems, preserving privacy through output masking techniques while achieving coordination. RNC7688 introduces a periodic dynamic encoding mechanism for stealthy attack detection in distributed state estimation, focusing on security and privacy under attack scenarios. RNC70033 tackles robust consensus Kalman filtering for distributed state-saturated systems, incorporating dynamic-disturbed saturation levels and censored measurements to ensure privacy under uncertain conditions. The contributions in this special issue span a broad spectrum of topics related to privacy in distributed control and optimization. As distributed systems continue to evolve, these papers contribute to the ongoing conversation about balancing privacy protection with efficient system operation, highlighting the importance of privacy as a key consideration in the design and optimization of future distributed control systems. The authors declare no conflicts of interest.

A control system is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">self-powered</i> if the only energy it requires for operation is that which it absorbs from the plant. For a linear feedback law to be feasible for a self-powered control system, its feedback signal must be colocated with the control inputs, and its input-output mapping must satisfy an associated passivity constraint. The imposition of such a feedback law can be viewed equivalently as the imposition of a linear passive shunt admittance at the actuation ports of the plant. In this paper we consider the use of actively-controlled electronics to impose a self-powered linear feedback law. To be feasible, it is insufficient that the imposed admittance be passive, because parasitic losses must additionally be overcome. We derive sufficient feasibility conditions which explicitly account for these losses. In the finite-dimensional, time-invariant case, the feasibility condition distills to a more conservative version of the Positive Real Lemma, which is parametrized by various loss parameters. Three examples are given, in which this condition is used to determine the least-efficient loss parameters necessary to realize a desired feedback law.

The advent of sensor networks, robots, autonomous vehicles and the smart grid have made the dependability of circuits and systems that control them critical to society and national defense. While significant advances in the design of linear and nonlinear control systems have been made to allow modes of operation not possible in the past, the problem of resilience to errors induced by hostile operating environments remains largely unexplored even though the probability of such errors occurring during real-time operation has increased. In this talk we propose mechanisms for detecting transient errors in control systems and circuitry as well as diagnosing and correcting for their effects on overall system operation. It is shown how real-number checksum encodings of circuit function can be used to detect and correct errors in the plant and feedback subsystems of linear control systems. Applications to signal processing and control algorithms are described. It is shown how errors in motor control electronics can be detected and corrected using the proposed methodology. Finally, extensions to nonlinear control systems are presented.

The present study is concerned with the development of a unified approach to the modeling and the design of an effective feedback linearizing and stabilizing nonlinear controller for multibody flexible and articulated systems. Modeling and control system design for such systems is of current interest. The principal objectives are to investigate the attitude regulation of rigid and/or flexible articulated multibody structures and to demonstrate how interacting nonlinear behaviors of flexible and articulated systems affect the stability and robustness properties of the controlled system. A systematic method for the formulation of the equations of motion for the nonlinear dynamics of flexible and articulated systems is employed based on a unified serial chain body algorithm, and this algorithm is integrated into Poincare's form of Lagrange's equations to formulate an innovative new approach to modeling of the system. The closed form solution of partial feedback linearization is derived using simple mathematical techniques and PFL compensation is applied as a general approach for the attitude decoupling and stabilization of the system having perfect knowledge of model uncertainties. The performance of a linear stabilizer, constructed on the same decoupling principle of the PFL stabilizer, is compared to that of PFL stabilizer with respect to attitude decoupling and stabilization. In the presence of model uncertainties, parameter adaptive control method is combined with PFL to enhance stability and robustness properties. Throughout these analyses, it is shown that (1) the developed modeling procedure is an effective tool in constructing feedback linearizing control laws, (2) with the perfect knowledge of parametric uncertainties, PFL controller yields excellent performance with respect to attitude decoupling and stabilization, (3) the domain of attraction of the linear controller applied to nonlinear systems is critically small, and the boundary of stability is explained as a subcritical Hopf bifurcation, and (4) the equilibrium point of the closed-loop adaptive control system is asymptotically stable, but the domain of attraction of adaptive PFL is limited due to nonlinear effects between flexible and articulated bodies.

From the perspective of vehicle dynamics, the four-wheel independent steering vehicle dynamics stability control method is studied, and a four-wheel independent steering varying parameter linear quadratic regulator control system is proposed with the help of expert control method. In the article, a four-wheel independent steering linear quadratic regulator controller for model following purpose is designed first. Then, by analyzing the four-wheel independent steering vehicle dynamic characteristics and the influence of linear quadratic regulator control parameters on control performance, a linear quadratic regulator control parameter adjustment strategy based on vehicle steering state is proposed to achieve the adaptive adjustment of linear quadratic regulator control parameters. In addition, to further improve the control performance, the proposed varying parameter linear quadratic regulator control system is optimized by genetic algorithm. Finally, simulation studies have been conducted by applying the proposed control system to the 8-degree-of-freedom four-wheel independent steering vehicle dynamics model. The simulation results indicate that the proposed control system has better performance and robustness and can effectively improve the stability and steering safety of the four-wheel independent steering vehicle.

It is common practice to use linear plant models and linear controllers in the control systems design. Such approach has simple explanation applying to plants with insignificant nonlinearity or to those, functioning closely to a working point. But linear controllers, indeed with some modification, are used even for plants with significant non-linearities. Because of several reasons the non-linear controllers have not broad application. First, the linear control theory is well developed; while the non-linear control methods are clear for few engineers in practice. Second, there are some technological and economical difficulties to get high quality study of the process to be controlled in order to build detailed (more precise) non-linear plant model. Third, new ideas in the field of the control theory are continuously realized, which expand the span of the linear control systems applications as an alternative to utilizing complicated models at the expense of troubles of theoretical and practical nature.

The designs of two linear control systems approach to stabilize the balance of an unmanned bicycle system are presented. Both approaches are based on the use of a reaction wheel or flywheel to balance the bicycle. The two linear control approaches, based on the linearization of a nonlinear model obtained using Lagrange formalism, are the classic linear controllers, PID and State Feedback control. The performance of both controllers is verified by digital simulation and real-time experimental results.