Linear Feedback Control Systems Research Articles

Dynamic Feedback Linearization of Control Systems with Symmetry

Control systems of interest are often invariant under Lie groups of transformations. For such control systems, a geometric framework based on Lie symmetry is formulated, and from this a sufficient condition for dynamic feedback linearizability obtained. Additionally, a systematic procedure for obtaining all the smooth, generic system trajectories is shown to follow from the theory. Besides smoothness and the existence of symmetry, no further assumption is made on the local form of a control system, which is therefore permitted to be fully nonlinear and time varying. Likewise, no constraints are imposed on the local form of the dynamic compensator. Particular attention is given to the consideration of geometric (coordinate independent) structures associated to control systems with symmetry. To show how the theory is applied in practice we work through illustrative examples of control systems, including the vertical take-off and landing system, demonstrating the significant role that Lie symmetry plays in dynamic feedback linearization. Besides these, a number of more elementary pedagogical examples are discussed as an aid to reading the paper. The constructions have been automated in the Maple package DifferentialGeometry.

Integrating dynamic economic optimization and encrypted control for cyber‐resilient operation of nonlinear processes

AbstractThis article proposes a two‐layer framework to maximize economic performance through dynamic process economics optimization while addressing fluctuating real‐world economics and enhancing cyberattack resilience via encryption in the feedback control layer for nonlinear processes. The upper layer employs a Lyapunov‐based economic model predictive control scheme, receiving updated economic information for each operating period, while the lower layer utilizes an encrypted linear feedback control system. Encrypted state information is decrypted in the upper layer to determine the economically optimal dynamic operating trajectory through nonlinear optimization. Conversely, the lower layer securely tracks this trajectory in an encrypted space without decryption. To mitigate the cyber vulnerability of the upper layer, we integrate a cyberattack detector that utilizes sensor‐derived data for attack detection. We quantify the errors stemming from quantization, disturbances, and sample‐and‐hold controller implementation. Simulation results of a nonlinear chemical process highlight the robustness and economic benefits of this new control architecture.

Performance-Driven Fault Detection for Uncertain Takagi–Sugeno Fuzzy Feedback Control Systems

This paper mainly studies the performance-driven fault detection for general uncertain Takagi-Sugeno fuzzy feedback control systems, where the infinite-horizon quadratic index is introduced to describe the concerned system performance. Different from the existing output-driven fault detection methods, the proposed approach aims to detect the performance degradations induced by faults. For this purpose, the performance index embedded with the information of system dynamics is reformulated for linear feedback control systems with reference inputs. The performance residual, which is adopted as the evaluation function, is derived based on the Bellman equation and further represented into an explainable form. Then, for uncertain linear feedback control systems, the boundaries of the performance residual are analyzed via the linear matrix inequality technique and an advantaged threshold setting scheme is proposed with the aid of randomized algorithm. Concerning the main objective, a novel approximation method which combines the fuzzy blend of local performance indices and the radial basis function neural network is developed to approximate the global performance index for Takagi-Sugeno fuzzy systems. Based on the result in linear case, the performance residual is constructed along the closed-loop system trajectory and the threshold is determined for the fuzzy control systems with reference inputs and model uncertainties. Finally, simulation studies are provided to show the effectiveness of the proposed theoretical results.

A robust control approach to asymptotic optimality of the heavy ball method for optimization of quadratic functions

Among first order optimization methods, Polyak’s heavy ball method has long been known to guarantee the asymptotic rate of convergence matching Nesterov’s lower bound for functions defined in an infinite-dimensional space. In this paper, we use results on the robust gain margin of linear uncertain feedback control systems to show that the heavy ball method is provably worst-case asymptotically optimal when applied to quadratic functions in a finite dimensional space.

Output Feedback Control for Linear Systems Under Measurement Disturbances

In this brief, the output feedback control issue is investigated for linear systems under unmeasured states with the influences of measurement disturbances. Due to these unmeasured states, the measurement outputs are used for designing the controller, while, they are mixed with measurement disturbances. In order to construct the feedback controller, the states are observed using a state observer, which is driven via the measurement outputs. A disturbance observer is employed to estimate the measurement disturbances. Then, the linear matrix inequations technique is adopted for disposing the coupling between these observers. Furthermore, the Lyapunov stability theory is applied to prove the stability analysis of closed-loop systems. Finally, a numerical example is given to show the availability of proposed control method.

Data-driven Optimal Output Feedback Control of Linear Systems from Input-Output Data

We design optimal output feedback controllers for linear systems where only outputs are available for measurement and control actuation. The goal is to render the closed-loop system stable while minimizing a quadratic cost function balancing performance and control effort. We first provide a model-based solution to this problem, where an optimal dynamic output feedback controller is computed by solving a semidefinite program. Then, we shift our attention to data-based solutions, bypassing the system identification step. We derive semidefinite programs that are explicitly stated in terms of input-output data. The effectiveness of the method is illustrated via a power system case study.

LMI-Based Stubborn and Dead-Zone Redesign in Linear Dynamic Output Feedback

The redesign of output feedback controllers for linear systems based on adaptive saturation (stubborn) and dead-zone redesign is investigated by showing that input-to-state stability holds in closed loop upon the satisfaction of linear matrix inequalities. Such results are obtained by using sector conditions that are involved in the Lyapunov analysis in order to ensure input-to-state stability. A simulation case study shows the effectiveness of the proposed redesign in denoising and outlier attenuation with increased accuracy and precision.

An Improved Design of Event-Triggered Feedback Controllers for Linear Systems Based on Fast and Slow Dynamics

In this article, an event-triggered technique is proposed to implement state feedback controllers for continuous-time linear systems. Toward this end, the closed-loop dynamic is decomposed into the fast and slow subsystems. Then, two individualized event-triggered mechanisms (ETMs) are designed to compute two sequences of updates. The first ETM determines when the fast subsystem should be sampled and its measured data should be sent to the controller. The second ETM decides the execution times for sampling all the system states, including both the fast and slow states. The proposed event-triggered technique leads to sampling policies in which the data of the slow states are sent to the controller with long interevent intervals whereas the fast dynamics are simultaneously allowed to be sampled with relatively shorter interevent intervals. In comparison with conventional methods with one ETM, the proposed event-triggered technique, equipped with an extra degree of freedom, leads to better results both in achieving smoother closed-loop responses and reducing the number of the transmitted packages from the system to the controller. Theoretical aspects of the proposed method including the boundedness of the system and its Zeno free property are investigated. Two simulation and experimental case studies are provided to show the effectiveness of the proposed event-triggered controller.

Dixon resultant theory for stability analysis of distributed delay systems and enhancement of delay robustness

This study scrutinizes the stability problem of linear time-invariant feedback control systems with a constant-coefficient, partial delay distribution from a new perspective, which is built on an equivalence between the system of interest and the one with two lumped delays. We aim to determine all the potential stability changing curves (PSCC) of the system in the domain of delays in order to make a non-conservative stability assessment. First, we propose the Dixon resultant-based frequency sweeping procedure to calculate the so-called kernel and offspring hypersurfaces (KOH) of the system. The superiority in the computational efficiency of this Dixon-type method is revealed by comparison with the Sylvester-type one. Second, we specifically tackle the standing root case for the singularity at the zero root, leading to what we call the standing root boundary (SRB). Then, we claim that the union of the KOH and SRB constitutes the PSCC of the system. With these, the stability map of the system is then created using the Cluster Treatment of Characteristic Roots paradigm. Furthermore, we declare the delay robustness is enhanced by the proposed control law. Finally, we demonstrate the effectiveness of the presented procedures over two example case studies by the Quasi-Polynomial mapping-based Root-finder routine as well as the Simulink-based simulation.

A Discrete Approach to Feedback Linearization, Yaw Control of An Unmanned Helicopter

Nonlinear control laws often need to be implemented with digital hardware. Use of digital control systems leads to communication/processing delays which are widely neglected in control of mechanical systems. This paper proposes a discrete approach to feedback linearization that considers these commonly overlooked delays in design. The proposed approach is shown to both improve the performance and remove the need for continuous derivative terms. In feedback linearization control systems, designed in the continuous domain, derivative terms are required to speed up the control response of mechanical systems, but disadvantageously cause high sensitivity to noise. The proposed approach was used to design a feedback linearization control system for a common turning maneuver of an unmanned helicopter in yaw. At this maneuver, the helicopter centroid motion and pitch rotational speed are almost zero. Governing differential equations of the helicopter at this maneuver are nonlinear and coupled. A feedback linearization law was proposed to curb nonlinearity and, a discrete control system, considering the inevitable delay due to the use of digital control systems, was adopted to complete the control law. This innovative approach resulted in less sensitivity to noises and performance boost. Practical limits in terms of control input, rotor speed, sampling frequency and noises of the gyroscope, the tachometer and the acceleration sensor were taken into account in this research. The results show that the proposed control system leads to fast and smooth yaw turns even at a high pitch angle (close to vertical) or in the case of being hit by external objects.

Online Optimal State Feedback Control of Linear Systems Over Wireless MIMO Fading Channels

We consider the optimal control of linear systems over wireless MIMO fading channels, where the MIMO wireless fading and random access of the remote controller may cause intermittent controllability or uncontrollability of the closed-loop control system. We formulate the optimal control design over random access MIMO fading channels as an infinite horizon average cost Markov decision process (MDP), and we propose a novel state reduction technique such that the optimality condition is transformed into a time-invariant reduced-state Bellman optimality equation. We provide the closed-form characterizations on the existence and uniqueness of the optimal control solution via analyzing the reduced-state Bellman optimality equation. Specifically, in the case that the closed-loop system is almost surely controllable, we show that the optimal control solution always exists and is unique. In the case that MIMO fading channels and the random access of the remote controller destroy the closed-loop controllability, we propose a novel controllable and uncontrollable positive semidefinite (PSD) cone decomposition induced by the singular value decomposition (SVD) of the MIMO fading channel contaminated control input matrix. Based on the decomposed fine-grained reduced-state Bellman optimality equation, we further propose a closed-form sufficient condition for both the existence and the uniqueness of the optimal control solution. The closed-form sufficient condition reveals the fact that the optimal control action may still exist even if the closed-loop system suffers from intermittent controllability or almost sure uncontrollability.

Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method

This paper considers the problem of how to control nonlinear systems with a large delay in the input. A linearization method is presented to achieve local asymptotic stabilization (LAS) via memoryless state and/or output feedback, for multiple-input/multiple-output (MIMO) nonlinear systems with a large input delay. The development of the linearization technique is based on the Razumikhin theorem, a forward completeness lemma of the FDE (functional differential equation) on a closed and bounded interval, and the previous work on truncated predictor feedback control of linear systems with input delay.

A non-conservative state feedback control methodology for linear systems with state delay

This paper deals with the state feedback control for linear systems in the presence of state delay using a direct numerical design methodology. The design algorithm iteratively searches the solution space of controller parameters to find a proper controller gain that guarantees the delay-independent stability of the closed-loop system. The algorithm divides the solution space to some simplexes, selects one simplex and evaluates two checking methods on this simplex. The first method checks the stabilisability of the corner points (to establish the feasible point) and the other detects the total infeasibility (total de-stabilisability) of the simplex (to omit the undesired parts). One interesting property of the method is its capability in detecting total de-stabilisability of a simplex through its corner points. If none of these methods is successful, the simplex is divided into two smaller ones which will be checked in the next algorithm’s iterations. According to the direct search nature, the algorithm is non-conservative and assuredly reaches a stabilizable point in the feasible space of the design space. The proposed algorithm and previous methods are evaluated on a set of random generated systems to compare the feasibility of the methods. Simulation results reveal the superiority of the proposed algorithm.

A Control System of Cell Electroporation induced by Double Nanosecond Pulsed Electric Field

A feedback linearization control system, which is used to study and analyze cell electroporation induced by double nanosecond pulsed electric field, is proposed in this paper. Then the effects of the control system which controls the process of electroporation induced by unipolar or bipolar double pulse are investigated to achieve controllability of electroporation. The simulation results show that the output of the control system, namely, the transmembrane potential can reach a stable value of 0.78 (-0.78)V, the radius increases linearly from 0.86nm to 0.88nm and the pore density increases linearly to 2.072×108m−2 (2.077×108m−2) when the input of the control system, the expected transmembrane potential is set to 0.79 (-0.79) V, and these results are in well agreement with the simulation result obtained by computing the related equations of electroporation, demonstrating that the control system developed in study can well regulate cell electroporation under the action of double nanosecond pulsed electric field. Overall, the control system can help the researcher to obtain the corresponding output electroporation parameters by directly inputting the expected transmembrane potential.

Feedback control of linear systems with optimal sensor and actuator selection

This article demonstrates a combined feedback control design for linear time-invariant and linear parameter-varying systems and optimal sensors and actuator selection. The combined design problem is systematically constructed as a mixed Boolean semidefinite programming optimization problem. We impose Big-M reformulations to the non-deterministic polynomial-time-hard coupled problem to be solved as a convex optimization problem using the branch and bound algorithm. The combined design of dynamic output feedback control along with optimal actuator selection for a linear time-invariant seismic rejection controller design serves as an application for validation by simulation. In addition, active vibration control of a smart composite plate along with optimal sensor and actuator selection validates the developed approach for linear parameter-varying controller synthesis. On comparing this approach with exhaustive search, it is observed that mixed Boolean semidefinite programming approaches have faster computation time, and comparing with the iterative reweighted ℓ1 norm algorithm and mixed Boolean semidefinite programming using outer approximations, mixed Boolean semidefinite programming yields a global solution.

Improved co-design of event-triggered dynamic output feedback controllers for linear systems

This paper presents a less conservative co-design method of output feedback law and event-triggering condition for linear systems in a networked control scenario. The previous work added three constraints to obtain linear matrix inequalities (LMI) from bilinear matrix inequalities (BMI). However, the additional constraints introduce the conservatism of the LMI condition. In this study, we relax the conservativeness induced by the additional constraints by replacing it with other constraints with free variables. As a result, the feasible set obtained by the proposed condition encompasses that of the previously published one. A convex optimization problem is provided for the co-design of the output feedback law and event-triggering condition to reduce the number of transmissions. A numerical example shows the effectiveness of the proposed method.

On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming

Abstract Stability and stabilization for linear state feedback control systems in the presence of sensor quantization are studied. As the closed-loop system is described by a discontinuous right-hand side differential equation, Krasovskii solutions (to the closed-loop system) are considered. Sufficient conditions in the form of matrix inequalities are proposed to characterize uniform global asymptotic stability of a compact set containing the origin. Such conditions are shown to be always feasible whenever the quantization-free closed-loop system is asymptotically stable. Building on the obtained conditions, computationally affordable algorithms for the solution to the considered problems are illustrated. The effectiveness of the proposed methodology is shown in three examples.

Infinite Horizon Optimal Output Feedback Control for Linear Systems with State Equality Constraints

This note deals with the infinite horizon linear regulation problem using output feedback with state equality constraints. Similar to the corresponding state feedback problem of a previous work, an existence condition for the output feedback gain and, if it exists, all constrainable output feedback gains are determined. However, different from the fore-mentioned state feedback case, only the necessary conditions for the optimal output feedback gain which minimizes the given standard cost function are determined. The performance of the developed algorithm is demonstrated using numerical simulations for a simple model of divert control system and the lateral dynamics of an F-16 aircraft.

Anti-windup disturbance suppression control of feedback linearizable systems with application to hypersonic flight vehicles

This paper deals with the anti-windup control problem for a class of feedback linearizable systems subject to external disturbances and constrained actuators. The original system is approximately feedback linearized into a tractable form with additive mismatched disturbances. A baseline disturbance suppression control is built on the combination of a disturbance observer and a compensation law. Moreover, this baseline control is strengthened by two anti-windup mechanisms to accommodate the possible input saturation, leading to the anti-windup disturbance suppression control. The resulting control schemes guarantee output tracking errors to be bounded in the presence of input saturation, while the zero steady-state tracking is finally achieved due to the recovery property. The proposed methods are applied to the longitudinal tracking control of hypersonic flight vehicles to highlight their efficiency and superiority.

Nonfragile [formula omitted] output feedback control of linear systems with an event-triggered scheme against unreliable communication links

This paper studies the event-triggered H∞ static output feedback control of linear systems with unreliable communication. The unreliable phenomenon between the event-triggering unit and the controller is described by a stochastic variable with Bernoulli random binary distribution. To cast the considered problem in the robust control framework, the event-triggering scheme is presented by a time-delay form. A vertex structure separation strategy is utilized to handle control gain with interval variations, which could alleviate computation burden heavily. Resorting to a division of control gain from Lyapunov variable, a new method for the non-fragile H∞ controller synthesis is established in the framework of linear matrix inequalities. Simulations are executed to show the validity of the proposed approach.