Linear Feedback Systems Research Articles

Some new rearrangements in sensitivity integrals and concerning inequalities with their application in control

In this paper, we extend the sensitivity integrals for linear feedback systems via a non-analytic sensitivity function in Poisson–Jensen’s terminology. First, we sketch a generalized sensitivity integral accounting for the NMP zeros and the RHP poles of the sensitivity function. Interestingly, the sensitivity integral accounting for the NMP zeros becomes a special case of the generalized integral. Then the concerning sensitivity inequality and the associated lower bound are achieved. We also sketch a sensitivity integral that exploits a modified sensitivity function-based approach and stabilizes the linear feedback system. Their concerning sensitivity inequality and the associated lower bound are achieved as well. An illustrative case study example is employed to demonstrate the idea of the ‘sensitivity lower bound’ to choose the ability of controllers for the given linear feedback systems. The rearranged sensitivity lower bound offers simplified analysis involving fewer computation efforts.

Stochasticity in Feedback Loops: Great Expectations and Guaranteed Ruin

Stochastic components in a feedback loop introduce state behaviors that are fundamentally different from those observed in a deterministic system. The effect of injecting a stochastic signal additively in linear feedback systems can be viewed as the addition of filtered stochastic noise. If the stochastic signal enters the feedback loop in a multiplicative manner, a much richer set of state behaviors emerges. These phenomena are investigated for the simplest possible system: a multiplicative noise in a scalar, integrating feedback loop. The same dynamics arise when considering a first-order system in feedback with a stochastic gain. The dynamics of this form arise naturally in a number of domains, including compound investments in finance, chemical reaction dynamics, population dynamics, epidemiology, control over lossy communication channels, and adaptive control. Understanding the nature of such dynamics in a simple system is a precursor to recognizing them in more complex stochastic dynamical systems.

Model-free adaptive control for the PEMFC air supply system based on interval type-2 fuzzy logic systems

Control aims to avoid oxygen starvation and maximize the net power output by maintaining the optimal oxygen excess ratio (OER), which varies between 1.8 and 2.5. Because of the nonlinearity of the proton exchange membrane fuel cell (PEMFC) air supply system and the different conditions, ensuring an optimal OER is still a challenge. In this study, a model-free adaptive controller is presented for the PEMFC air supply system based on feedback linearization and interval type-2 fuzzy logic systems (IT2 FLSs). Theoretical analysis and experimental results verify the effectiveness of the proposed control scheme. For the theoretical analysis, first, the PEMFC air supply system is transformed into a canonical form with the feedback linearization technique. Then, zero-dynamics stability is discussed in detail to determine the stability of the internal dynamics. Finally, an adaptive interval type-2 fuzzy logic system controller (AIT2FLSC) is designed on the basis of the Lyapunov stability theory, which does not require complete a priori knowledge of the system dynamics. For the experimental results, the root mean square error (RMSE), variance, and standard deviation (SD) of the tracking error are used as tracking performance metrics to evaluate the control accuracy of the proposed AIT2FLSC, which are 0.0968, 0.0093, and 0.0962, respectively. Compared with the traditional proportion integration differentiation controller (RMSE 0.1119, variance 0.0122, and SD 0.1105), this proposed algorithm obtains better adaptability and the RMSE of the tracking error improves 13.48%. Compared with the adaptive type-1 fuzzy logic system controller (AT1FLSC) (RMSE 0.1076, variance 0.0113, and SD 0.1063), this AT2FLSC has a stronger ability to deal with uncertainty and the RMSE of the tracking error improves 10% when the stack temperature is fixed (353.15 K). Furthermore, when the stack temperature is time-varying, the RMSE, variance, and SD of the tracking error under the AIT2FLSC are 0.0966, 0.0092, and 0.0960, respectively, which is less than AT1FLSC (0.1085, 0.0115, and 0.1073) and the RMSE of the tracking error improves 10.99%.

A Distributed Luenberger Observer for Linear State Feedback Systems With Quantized and Rate-Limited Communications

This article addresses the problem of simultaneous distributed state estimation, and control of linear systems with linear state feedback, subjected to process, and measurement noise, under the constraints of quantized, and rate-limited network data transmission. In the set-up adopted, sensors and actuators communicate through a network with a strongly connected topology. Unlike the case of centralized linear systems, for which the separation principle holds, the above practical assumption prevents the separate design of observers, and controller because each of the nodes does not necessarily have access to the control inputs generated at all the other nodes. We derive a linear distributed Luenberger observer, and a set of sufficient conditions that guarantee ultimate boundedness of the estimation error, and system state vectors, with bounds that depend on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of the noise signals, and the number of bits used in the transmissions. A numerical example illustrates the performance and effectiveness of the proposed algorithm in controlling a network of open-loop unstable systems.

Stability Analysis of Linear Feedback Systems in Control

This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of structured singular values of Toeplitz and symmetric Toeplitz matrices for linear time invariant systems is presented by means of low rank ordinary differential equations (ODE’s) based methodology. The proposed methodology is based upon the inner-outer algorithm. The inner algorithm constructs and solves a gradient system of ODE’s while the outer algorithm adjusts the perturbation level with fast Newton’s iteration. The comparison of bounds of structured singular values approximated by low rank ODE’s based methodology results tighter bounds when compared with well-known MATLAB routine mussv, available in MATLAB control toolbox.

Separable Control Lyapunov Functions With Application to Prostheses

This letter extends bipedal trajectory tracking methods to prostheses to enable construction of a class of model-dependent prosthesis controllers using locally available sensor information. The rapidly exponentially stabilizing control Lyapunov functions (RES-CLFs) developed for bipedal robots guarantee stability of the hybrid zero dynamics in the presence of impacts that occur in walking. These methods cannot be directly applied to prostheses because of the unknown human dynamics. We overcome this challenge with two RES-CLFs, one for the prosthesis subsystem and another for the remaining human system. Further, we outline a method to construct these RES-CLFs for this type of separable system by first constructing separable CLFs for partially feedback linearizable systems. This letter develops a class of separable subsystem controllers that rely only on local information but provide formal guarantees of stability for the full hybrid system with zero dynamics.

Evolving granular control with high-gain observers for feedback linearizable nonlinear systems

Feedback linearization control is a simple and effective strategy whenever a faithful model of the system and its states are available. Feedback linearization may suffer from the mismatches between the model used in the design and the actual system due to e.g. uncertain parameter values, parasitic dynamics, or because of the impossibility to measure some states of the system. To aleviate such an issue, we suggest a novel robust adaptive control approach using the evolving participatory learning algorithm together with a high-gain observer. The robust evolving granular high-gain observers (RegHGO) controller is suitable to control nonlinear systems that can be input-output linearized by feedback. The approach is robust to modeling mismatches and does not require full state availability because, once the system is in a suitable canonical form, a high gain observer can be constructed to supply the state information required for control. The usefulness and efficacy of the approach is shown using a fan and plate system, and an DC motor driven angular arm-position control. The fan and plate evaluates the controller in a regulation process, and the angular arm position control evaluates reference tracking perfromance. In both cases, time-varying parameter uncertainties disturb the closed-loop control system. Both, qualitative and quantitative performance evaluation of the RegHGO controller are done. Additionally, we compare the performance of the RegHGO controller with well-established methods such as exact feedback linearization with high-gain state observer and extensions. The results show that robust evolving granular control with high-gain observers achieves better performance than its counterparts.

On Instability Analysis of Linear Feedback Systems

The numerical approximation of the μ -value is key towards the measurement of instability, stability analysis, robustness, and the performance of linear feedback systems in system theory. The MATLAB function mussv available in MATLAB Control Toolbox efficiently computes both lower and upper bounds of the μ -value. This article deals with the numerical approximations of the lower bounds of μ -values by means of low-rank ordinary differential equation (ODE)-based techniques. The numerical simulation shows that approximated lower bounds of μ -values are much tighter when compared to those obtained by the MATLAB function mussv.

Comparisons of robust methods on feedback linearization through experimental tests

The feedback linearization is a powerfull nonlinear method based on the principle of canceling the nonlinearities of the system model. However, if the model differs from the real system, the feedback linearization is prone to fail. Several studies look to provide robustness to the feedback-linearized system, but we note a lack of evaluation among such approaches under similar conditions in practical systems. This work contributes to filling such a gap by comparing the performance of three recent approaches proposed to robustify feedback linearization loops. Therefore, we design controllers based on the robust multi-inversion (RMI), the robust dynamic inversion (RDI), and the robust granular feedback linearization (RGFL), and evaluate them through real-time experiments. The test process consists of a nonlinear surge tank where the level must be controlled. Two experiments are performed to evaluate the controllers in the tracking and regulation modes when the system is subjected to disturbances. Classical quantitative indexes evaluate the performance of the closed-loop system. The experimental tests indicate that the RGFL controller outperforms the other approaches in both regulation and tracking.

Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness

In a paper by Willems et al., it was shown that persistently exciting data can be used to represent the input-output behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent linear matrix inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.

Sensitivity analysis of linear continuous-time feedback systems subject to control and measurement noise: An information-theoretic approach

Sensitivity of linear continuous-time SISO feedback systems, subject to control and measurement noise, is analyzed by deriving the lower bounds of Bode-like integrals via an information-theoretic approach. Bode integrals of four different sensitivity-like functions are employed to gauge the performance limitations of feedback systems. When the signals of the control system are stationary Gaussian, these four different Bode-like integrals can be represented as the differences between mutual information rates. These mutual information rates and hence the corresponding Bode-like integrals are proven to be bounded below by the unstable poles and zeros of the plant model, if the signals of the control system are wide-sense stationary.

Closed-Loop Error Learning Control for Uncertain Nonlinear Systems With Experimental Validation on a Mobile Robot

This paper develops a Closed-Loop Error Learning Control (CLELC) algorithm for feedback linearizable systems with experimental validation on a mobile robot. Traditional feedback and feedforward controllers are designed based on the nominal model by using Feedback Linearization Control (FLC) method. Then, an intelligent controller is designed based on sliding mode learning algorithm that utilizes closed-loop error dynamics to learn the system behavior. The controllers are working in parallel, and the intelligent controller can gradually replace the feedback controller from the control of the system. In addition to the stability of the sliding mode learning algorithm, the closed-loop stability of an $n$th order feedback linearizable system is proven. The simulation results demonstrate that CLELC algorithm can improve control performance (e.g., smaller rise time, settling time and overshoot) in the absence of uncertainties, and also provides robust control performance in the presence of uncertainties as compared to traditional FLC method. To test the efficiency and efficacy of CLELC algorithm, the trajectory tracking problem of a tracked mobile robot is studied in real-time. The experimental results demonstrate that CLELC algorithm ensures high-accurate trajectory tracking performance than traditional FLC method.

Sliding Mode Control for Optimal Torque Transmission of Dry Dual Clutch Assembly of A Two-Speed Electric Vehicle During Launch

As fork-lever actuator has significant advantages in structure dimensions and manufacturing prices, and is able to respond to drivers’ demands in very quick and precise manner, it becomes more and more popular in conventional and electric vehicles (EVs) equipped with dry dual clutch transmission (DDCT). To investigate optimal dynamic behaviors of EV equipped with DDCT, displacements of little ends of diaphragm springs and change rates of such displacements are defined key state variables, and dynamic model of such kind of actuator is built and integrated into DDCT driveline dynamic model of an EV with two-speed. Then, an affine nonlinear launch dynamic model for the whole DDCT system is proposed. Further, optimal torques of electric motor and dual clutch are derived, based on Pontryagin’s minimum principle, mainly for the purpose of smooth power transmission from electric motor to dual clutch. Finally, a sliding mode control algorithm for the feedback linearization system is derived to improve tracking accuracy at the presence of strong nonlinearities and modelling uncertainties. Through careful observation of numerical simulation results from MATLAB/Simulink platform, it can be concluded that the proposed optimal control algorithm is able to improve launch quality to fine level, and sliding mode algorithm for tracking control acquires strong robustness to modeling uncertainties and nonlinearities.

Computing μ-Values for Real and Mixed μ Problems

In various modern linear control systems, a common practice is to make use of control in the feedback loops which act as an important tool for linear feedback systems. Stability and instability analysis of a linear feedback system give the measure of perturbed system to be singular and non-singular. The main objective of this article is to discuss numerical computation of the μ -values bounds by using low ranked ordinary differential equations based technique. Numerical computations illustrate the behavior of the method and the spectrum of operators are then numerically analyzed.

Observer-based adaptive fuzzy prescribed performance control for feedback linearizable MIMO nonlinear systems with unknown control direction

This paper investigates the prescribed performance tracking control problem for a class of feedback linearizable multi-input and multi-output (MIMO) nonlinear systems with immeasurable states and unknown control direction. The fuzzy systems are employed to approximate the unknown nonlinear functions, an adaptive fuzzy state observer is developed to estimate the unmeasured states. By utilizing the state observer, an adaptive fuzzy output-feedback controller equipped with the Nussbaum-type function is developed to solve the unknown control direction issues. It is proved that all the signals in the closed-loop systems are bounded and that the prescribed performance error constraints can be achieved. Simulation examples are provided to highlight the effectiveness of the proposed approach.

Equivalence Among Flat Filters, Dirty Derivative-Based PID Controllers, ADRC, and Integral Reconstructor-Based Sliding Mode Control

We explore the equivalence among flat filters, dirty derivative-based proportional integral derivative (PID) controllers, active disturbance rejection control, and integral reconstructor-based sliding mode control, in the context of SISO second-order, perturbed, pure integration systems. This is the prevailing paradigmatic class of differentially flat systems or feedback linearizable systems. The equivalence is valid beyond the second-order pure integration systems. However, PID controllers of such plants do not make much sense without imposing assumptions that will move the considerations out of the pure integration systems case. The equivalence among the rest of the controllers is valid for any finite order pure integration system.

A robust performance enhancement of primary H∞ controller based on auto-selection of adjustable fractional weights: Application on a permanent magnet synchronous motor

This paper proposes a new robustification strategy of a primary H∞ controller based on a systematic selection of adequate fractional weights. Its robust performance (RP) margin is well enhanced with respect to unstructured multiplicative uncertainties presented in linear feedback systems. The proposed robustification strategy provides a robustified H∞ controller when following proposed hierarchical control is well respected. First, the primary [Formula: see text] controller is synthesized from solving a weighted-mixed sensitivity H∞ problem using initial integer weights. Thus, an initial RP margin is obtained. Second, an automatic selection of adjustable fractional weights is performed by the particle swarm optimization algorithm, in which some proposed tuning rules are accordingly well satisfied. Third, frequency response data of these weights are computed and then fitted by corresponding approximated integer weights using a frequency identification technique. Finally, these weights reformulate a new weighted-mixed sensitivity problem. The optimal solution to this problem updates the previous initial RP margin. These last three steps are repeated as the updated RP margin is diminished. Otherwise, the proposed hierarchical control is achieved by selecting the best adjustable fractional weights, providing, therefore, the best approximated integer weights and leading, therefore, to the robustified H∞ controller. In order to confirm the effectiveness of our proposed hierarchical control, primary and robustified H∞ controllers are applied on a permanent magnet synchronous motor where its actual behavior is modeled by an unstructured multiplicative uncertain model. The results obtained are compared in frequency domains using the singular value plots of their sensitivity functions. Otherwise, the same results are compared in time domains using the Powersim® software.

Interval State Estimation of Linear Multicellular Systems

Linear multicellular system is a type of differential inclusion system, which can be deemed as an extension of linear control system with set‐valued mapping. As an important issue in existing control systems, interval state estimation has been widely applied in engineering practices. Over the years, the objects of the studies on interval state estimation have been extended from the initial linear time‐invariant systems to linear time‐varying systems, chaotic systems, feedback linearization systems, and nonlinear Lipschitz systems. However, there is no report on the design of interval observer for linear multicellular system. To make up for this gap, this chapter attempts to explore the design of an interval observer for linear multicellular systems.

Fuzzy Logic Deadzone Compensation with Feedback Linearization of Nonlinear Systems

A fuzzy logic compensator is designed for feedback linearizable nonlinear systems with deadzone nonlinearity. The classification property of fuzzy logic systems makes them a natural candidate for the rejection of errors induced by the deadzone, which has regions in which it behaves differently. A tuning algorithm is given for the fuzzy logic parameters, so that the deadzone compensation scheme becomes adaptive, guaranteeing small tracking errors and bounded parameter estimates. Formal nonlinear stability proofs are given to show that the tracking error is small. The fuzzy logic deadzone compensator is simulated on a one-link robot system to show its efficacy.

一类线性负反馈系统的数学描述

一类线性负反馈系统的数学描述