Linear Feedback Controller Design Research Articles

The dependent coordinates in the linearization of constrained multibody systems: Handling and elimination

In this paper, the handling of the dependent coordinates in the linearization of constrained multibody systems is clarified. To this end, a brief but comprehensive overview of linearization approaches used in the field of Multibody System Dynamics is presented. These procedures are illustrated by using a common notation and classified into four groups, depending on the initial form of the nonlinear equations of motion and the selection of the generalized coordinates (a redundant or a minimal set). The handling of the dependent coordinates in each of the procedures is discussed. The linearized equations of motion of constrained multibody systems are of great importance in several applications, such as linear stability and modal analyses, the design of linear state feedback controllers or the building of state and input estimators, like Kalman filters. When modeling a constrained multibody system, a number of coordinates greater than the number of degrees of freedom of the system is usually used. While some linearization approaches make use of the complete set of coordinates, several procedures are based on a coordinate partition in terms of independent and dependent coordinates to reduce the number of linearized equations of motion. In this latter scenario, the role of the dependent coordinates in the linearization is important and, in a general case, these dependent coordinates cannot be ignored to obtain the correct linearized equations of motion. This aspect, which may seem obvious, is not straightforward and is addressed in this work. The importance of considering the set of dependent coordinates in the linearization is demonstrated with a well-acknowledged bicycle benchmark multibody model and an electric kickscooter multibody model with rear and front suspensions. To this end, the Jacobian matrix and the linear stability results of these case studies are computed by considering different choices of independent and dependent coordinates.

Feedback Linearization technique for Speed and Torque control of Switched Reluctance Machine

This paper proposes a feedback linearization controller design for torque ripple minimization and speed controller in switched reluctance motor (SRM) drive systems. A dynamic model system of equations in SRM is used to develop the proposed controller. The control strategy is formed by two loops. The first loop aims to control the machine's speed and generate the torque reference. In addition, the other loop is used to control the current to minimize torque ripple and the error between the desired speed and true speed. In order to validate our proposed control strategy, we used the MATLAB-Simulink toolbox. The proposed controller is being evaluated and compared to a conventional PI regulator to showcase its superior performance and advantages. Finally, the obtained results validate the proposed controller's robustness and efficiency in variable speed and load.

Feedback Linearization Controller Design for Solid Oxide Fuel Cells

Electric vehicles have attracted the attention of users because they do not burn fossil fuels and emit zero greenhouse gas emissions. Fuel cells have shown their potential to power vehicles as well. The most common fuel cell types as power sources for automobiles are i) Proton exchange membrane fuel cell (PEMFC) and ii) Solid oxide fuel cell (SOFC). Normally, the PEMFC is considered main propulsion system. On the other side, the SOFC is generally not considered for propulsion system but considered for auxiliary power unit (APU). In this paper, feedback linearization controller for solid oxide fuel cells is proposed. And the performance of proposed controller is simulated under current disturbances operation condition through Matlab simulation.

Robust Feedback Linearization Control Design for Five-Link Human Biped Robot with Multi-Performances

The study first proposes the difficult nonlinear convergent radius and convergent rate formulas and the complete derivations of a mathematical model for the nonlinear five-link human biped robot (FLHBR) system which has been a challenge for engineers in recent decades. The proposed theorem simultaneously has very distinctive superior advantages including the stringent almost disturbance decoupling feature that addresses the major deficiencies of the traditional singular perturbation approach without annoying “complete” conditions for the discriminant function and the global exponential stability feature without solving the impractical Hamilton–Jacobi equation for the traditional H-infinity technique. This article applies the feedback linearization technique to globally stabilize the FLHBR system that greatly improved those shortcomings of nonlinear function approximator and make the effective working range be global for whole state space, whereas the traditional Jacobian linearization technique is valid only for areas near the equilibrium point. In order to make some comparisons with traditional approaches, first example of the representative ones, that cannot be addressed well for the pioneer paper, is shown to demonstrate the fact that the effectiveness of the proposed main theorem is better than the traditional singular perturbation technique. Finally, we execute a second simulation example to compare the proposed approach with the traditional PID approach. The simulation results show that the transient behaviors of the proposed approach including the peak time, the rise time, the settling time and the maximum overshoot specifications are better than the traditional PID approach.

Linear Feedback Control Design for Fractional Order Cryptovirology Stabilization in a Blockchain System

This paper proposes a linear feedback control of design in order to stabilize the fractional order chaotic cryptovirology in a blockchain. An optimized feedback controller is obtained by the use of Particle Swarm Optimization algorithm (PSO). A numerical simulation shows the effectiveness of the proposed controller in stabilizing the system to an unstable fixed point.

Robust Feedback Control for Discrete-Time Systems Based on Iterative LMIs with Polytopic Uncertainty Representations Subject to Stochastic Noise

This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.

Observer-based gust load alleviation via reduced-order models*

In this paper, we investigate the design of linear active output feedback control for aircraft gust load alleviation, involving standard sensors and control surfaces. The control architecture is composed of (i) a robust observer, allowing for estimating some wing bending moments despite the wind perturbation, and (ii) a structured H∞-norm oriented control law exploiting this additional information. To cope with the complexity of the aeroelastic model and make the control and observer synthesis computationally tractable, a model reduction step, based on Petrov-Galerkin projections, is first applied. One specificity of this work is that the obtained projector is exploited in the observer design, linking the reduction and the estimation. The effectiveness of the methodology is illustrated through numerical simulation involving an open access high complexity aeroelastic model.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

Tensor product‐based model transformation approach to cart position modeling and control in pendulum‐cart systems

AbstractThe paper presents the application of the tensor product (TP)‐based model transformation technique to model and control the cart position of single‐input multi‐output pendulum‐cart systems (PCSs). The modeling is first carried out. The derived TP model, the nonlinear model of PCS, and the laboratory equipment are tested in the same open‐loop scenario, and their corresponding outputs are collected, measured, and compared. The parallel distributed compensation (PDC) technique is next applied to TP‐based cart position controller design. A linear state feedback control design is also designed in order to conduct a comparative analysis. The two proposed control system structures are experimentally validated and tested in the same scenarios on the laboratory equipment, and their performance indices are analyzed and compared.

Output feedback control for quadratic systems: A Lyapunov function approach

AbstractThis article deals with the design of quadratic and linear dynamic output feedback controllers for quadratic systems to ensure, on the one hand, local exponential stability of the origin in the closed‐loop form and, on the other hand, to increase an estimate of the basin of attraction of the origin as large as possible. The introduction of a quadratic term in the controller allows us to consider a controller in the same class of dynamics as the studied system. Here, our approach is to consider a Lyapunov function with respect to the extended state gathering the states of the system and the controller. The induced nonlinear inequalities are treated thanks to an auxiliary vector repeating the extended state adequately and by combining distinct linearization techniques to finally obtain linear matrix inequalities. For comparison purposes, we also provide another approach characterizing the extended state as belonging to a polytope in the state space. Numerical examples illustrate the results.

Structural Sparsity in Networked Control Systems

A great deal of research is being conducted to define control strategies that involve a level of sparsity either in the communication network, number of sensors, or the number of active actuators during the control process. These sparse control strategies are typically formulated as an optimization problem that has a criterion with a component associated with the dynamics of the controlled system, and a component based on the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm that induces sparsity on the optimization variables. Although this approach addresses the problem of obtaining sparse approximations, solving this optimization problem can be seen as a design process that is conducted simultaneously along with a variable selection process that does not take into account the inherent structural relationships between the components of the networked system to be controlled, such as spatial relations associated with the physics of the system or communication constraints. In this paper, we incorporate the concept of structural sparsity into the mathematical framework of control design in networked systems. Under this extended framework, it is possible to define control strategies that lead to sparse designs considering a prior knowledge on the structural relations between the variables of the system. This can potentially improve the system's performance, the consistency of the solution with physical constraints, and the interpretability of the design. First, we show how structural sparsity inducing norms can be constructed as a combination of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> - and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norms, encoding structures, such as overlapping groups, nonoverlapping groups, and hierarchies. Also, we show how these norms can induce both individual and group variable selection. Then, we present two different control design techniques, where structural sparsity is induced during the design process: linear feedback control design and dynamic resource allocation. We show through practical applications and simulation results the usefulness of structural sparsity in the control system design process.

An implicit class of continuous dynamical system with data-sample outputs: a robust approach

Abstract This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential-algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-type outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based descriptor techniques from the stability theory of continuous systems. The above combination of the modified invariant ellipsoid approach and descriptor method makes it possible to obtain the robustness of the designed control and to establish some well-known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.

Urban Comprehensive Water Consumption: Nonlinear Control of Production Factor Input Based upon the C-D Function

Utilizing the urban water demand function and the Cobb-Douglas (C-D) production function, an economic control model for the multi-input-multi-output (MIMO) nonlinear system was designed and implemented to describe urban comprehensive water consumption, where the urban water demand function was expressed as the product of the number of water users and per capita comprehensive water consumption, and the urban water supply function was expressed as a C-D production function. The control variables included capital investment and labor input for the urban water supply. In contrast to the Solow model, Shell model and aggregate model with renewable labor resources, the proposed model eliminated value constraints on investment and labor input in the state equations and hence avoided the difficulty in applying these models to urban water supply institutions. Furthermore, the feedback linearization control design (FLCD) method was employed to accomplish stability of the system. In contrast to the optimal control method, the FLCD method possesses an explicit solution of the control law and does not require the solution of a two-point boundary value problem of an ordinary differential equation, making the method more convenient for application. Moreover, two different scenarios of urban water consumption, one for the growth period and the other for the decline period, were simulated to demonstrate the effectiveness of the proposed control scheme.

Robust Nonlinear Controller Design for On-Grid/Off-Grid Wind Energy Battery-Storage System

This paper investigates the design of a unified robust nonlinear multi-input-multi-output controller for a wind energy battery-storage system. The system can be operated either in grid-connected mode or stand-alone mode. Feedback linearization control design method is used to find the controller equations. The main advantage of the proposed control strategy over the existing methods is that only a single controller is used to control the system regardless the mode of operation. With the new approach, the controller is designed so that the grid contribution is significantly reduced and the battery can provide a backup energy or store the surplus wind energy efficiently. In addition, the dc-link voltage and load power are regulated by adjusting the charging and discharging cycle of the battery through a bi-directional buck-boost dc/dc converter in contrast to existing methods where the dc-link is regulated by the grid-side converter. The robustness of the proposed controller is evaluated in simulation with various types of loads, including unbalanced load and dynamic load. The results show that a good performance in closed loop is achieved.

Nano-Scale Positioning Design with Piezoelectric Materials

Piezoelectric materials naturally possess high potential to deliver nano-scale positioning resolution; hence, they are adopted in a variety of engineering applications widely. Unfortunately, unacceptable positioning errors always appear because of the natural hysteresis effect of the piezoelectric materials. This natural property must be mitigated in practical applications. For solving this drawback, a nonlinear positioning design is proposed in this article. This nonlinear positioning design of piezoelectric materials is realized by the following four steps: 1. The famous Bouc–Wen model is utilized to present the input and output behaviors of piezoelectric materials; 2. System parameters of the Bouc–Wen model that describe the characteristics of piezoelectric materials are simultaneously identified with the particle swam optimization method; 3. Stability verification for the identified Bouc–Wen model; 4. A nonlinear feedback linearization control design is derived for the nano-scale positioning design of the piezoelectric material, mathematically. One important contribution of this investigation is that the positioning error between the output displacement of the controlled piezoelectric materials and the desired trajectory in nano-scale level can be proven to converge to zero asymptotically, under the effect of the hysteresis.

Controlling Chaos—Forced van der Pol Equation

Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error.

Nonlinear Decentralized Feedback Linearizing Controller Design for Islanded DC Microgrids

This paper aims to design decentralized controllers for different components in islanded DC microgrids. The major components in the DC microgrid as considered in this paper include a fuel cell, solar photovoltaic (PV) unit, and battery energy storage system (BESS) along with critical and non-critical loads. The main control objective is to maintain the power balance within the DC microgrid through the regulation of the common DC-bus voltage. The controllers are designed based on the dynamical models of the fuel cell, solar PV unit, and BESS. The feedback linearization technique is employed to obtain the control laws, which simplifies the original dynamical models and decouples different components in the form of several subsystems. In this way, the feedback linearization technique allows different components in DC microgrids to achieve the desired control objectives by using only the local information (i.e., in a decentralized manner). The performance of the proposed decentralized controllers for different components is evaluated on a test DC microgrid under different operating conditions. Simulation results demonstrate that the proposed control scheme performs in a much better way as compared to an existing proportional integral controller.

Three-Stage Feedback Controller Design With Applications to Three Time-Scale Linear Control Systems

This paper presents a new technique for design of full-state feedback controllers for linear dynamic systems in three stages. The new technique is based on appropriate partitioning of the linear dynamic system into linear dynamic subsystems. Every controller design stage is done at the subsystem level using only information about the subsystem (reduced-order) matrices. Due to independent design in each stage, different subsystem controllers can be designed to control different subsystems. Partial subsystem level optimality and partial eigenvalue subsystem assignment can be achieved. Using different feedback controllers to control different subsystems of a system has not been present in any other known linear full-state feedback controller design technique. The new technique requires only solutions of reduced-order subsystem level algebraic equations. No additional assumptions were imposed except what is common in linear feedback control theory (the system is controllable (stabilizable)) and theory of three time-scale linear systems (the fastest subsystem state matrix is invertible)). The local full-state feedback controllers are combined to form a global full-state controller for the system under consideration. The presented results are specialized to the three time-scale linear control systems that have natural decomposition into slow, fast, and very fast subsystems, for which numerical ill conditioning is removed and solutions of the design algebraic equations are easily obtained. The proposed three-stage three time-scale feedback controller technique is demonstrated on the eighth-order model of a fuel cell model.

Feedback linearization and feedforward neural network control with application to twin rotor mechanism

A twin rotor multi-input multi-output system (TRMMS) is a high-order nonlinear system with a significant cross-coupling effect. The control of TRMMSs is considered a markedly challenging topic in the field of robust control. This study proposes a novel feedback linearization and feedforward neural network controller design for a TRMMS with almost disturbance decoupling (ADD) capabilities. The proposed composite controller achieves exponentially global stability and ADD performance without applying any traditional parallel learning algorithms. This study proposes an organization of the feedforward neural network and the weights among the layers to guarantee the stability of the overall system. A number of nonlinear systems, which are too complex to be solved by general ADD studies, are proposed in this study to demonstrate that the proposed methodology can effectively achieve the tracking and ADD performances through Matlab. Moreover, an efficient algorithm is proposed for designing the feedback linearization and feedforward neural network control with ADD and tracking capabilities.

Control de Fuerza de Robots Manipuladores Basado en Observadores Proporcionales Integrales Generalizados

In this work the design of a linear observer–linear controller robust output feedback scheme is introduced for simultaneous trajectory tracking of position and force in fully actuated robot manipulators. The unknown state–dependent additive nonlinearity influencing the input–output description is modeled as an absolutely bounded “time–varying perturbation”. Generalized Proportional Integral (GPI) observers are shown to naturally estimate the unknown perturbation and a certain number of its time derivatives in an arbitrarily close manner. This information is used to advantage on the linear feedback controller design via a simple cancelation effort. To the best of the authors’ knowledge GPI observers have not been used before for robot force control. A comparison experimental analysis is presented to show the good performance of the proposed approach.