Linear State Feedback Research Articles

Design of optimal controller for photovoltaic maximum power point tracking applications

ABSTRACT Photovoltaic (PV) maximum power point tracking (MPPT) technology is one of the key technologies that affect the energy utilization of PV power generation systems. How to achieve PV maximum power output quickly, accurately, and simply during the process of changing insolation and temperature conditions is a key problem in current research. At present, the traditional model-free maximum power tracking algorithm suffers from slow response, steady-state oscillations, and large influence of control parameters, whereas the model-based maximum power point tracking algorithm has the advantages of a simple control rate calculation and excellent control performance; however, the nonlinear characteristics of PV cells and power electronic converters lead to difficult controller design and often require a reference value of the maximum power point to be given. In this study, a mathematical model of a PV system was used, and a new PV maximum power tracking algorithm that does not require a reference value to be specified was proposed. The algorithm maps the nonlinear state-space description of the PV power generation system into a linear space through the state feedback linearization design. This solves the problem of difficult controller design for a nonlinear PV power generation system. Starting with the PV maximum power point as the extreme value point of the U-P curve, optimal control theory is applied to derive the PV maximum power tracking control rate, which overcomes the difficulty caused by the model-based control method requiring a reference to be specified. Compared with the traditional method, the proposed maximum power point tracking method improves the response speed by approximately 3–20 times and the tracking accuracy by approximately 3% or less. In addition, the method has a certain amount of immunity to DC bus voltage fluctuations.

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.

Suboptimal control synthesis for state and input delayed quadratic systems

In this article a suboptimal linear-state feedback controller for multi-delay quadratic system is investigated. Optimal state and input coefficients resulting from the expansion over a hybrid basis of block pulse and Legendre polynomials are first obtained by formulating a nonlinear programming problem. Afterwards, suboptimal control gains are found by solving a least square problem constructed with optimal coefficients of the open loop study. A sufficient condition for the exponential stability of the closed loop is obtained from generalized Grönwall–Bellman lemma. The Van de Vusse chemical reactor case is handled allowing to validate the proposed technique.

Control of LPV Modeled AC-Microgrid Based on Mixed H2/H∞ Time-Varying Linear State Feedback and Robust Predictive Algorithm

This paper presents a robust model predictive control (RMPC) method with a new mixed H2/<inline-formula> <tex-math notation="LaTeX">$\text{H}\infty $ </tex-math></inline-formula> linear time-varying state feedback design. In addition, we propose a linear parameter-varying model for inverters in a microgrid (MG), in which disturbances and uncertainty are considered, where the inverters connect in parallel to renewable energy sources (RES). The proposed RMPC can use the gain-scheduled control law and satisfy both the H2 and <inline-formula> <tex-math notation="LaTeX">$\text{H}\infty $ </tex-math></inline-formula> proficiency requirements under various conditions, such as disturbance and load variation. A multistep control method is proposed to reduce the conservativeness caused by the unique feedback control law, enhance the control proficiency, and strengthen the RMPC feasible area. Furthermore, a practical and efficient RMPC is designed to reduce the online computational burden. The presented controller can implement load sharing among distributed generators (DGs) to stabilize the frequency and voltage of an entire smart island. The proposed strategy is implemented and studied in a MG with two DG types and various load types. Specifically, through converters, one type of DGs is used to control frequency and voltage, and the other type is used to control current. These two types of DGs operate in a parallel mode. Simulation results show that the proposed RMPCs are input-to-state practically stable (ISpS). Compared with other controllers in the literature, the proposed strategy can lead to minor total harmonic distortion (THD), lower steady-state error, and faster response to system disturbance and load variation.

Local practical stabilization for a class of discrete-time switched affine systems

This paper deals with the study of discrete-time switched affine systems. While generally the literature is focused on the global (practical) stabilization problem, there are classes of switched affine systems that can be stabilized only locally. In this paper we deal with such a class of switched affine systems. More precisely, we consider switched affine systems with decoupled switching in the state matrix and the affine input term. For this particular class of systems, a switching control law is designed and conditions ensuring the system local practical stability are provided. A qualitative approach is given based on the existence of a stabilizing linear state feedback. These results are illustrated using a numerical example.

Design of a vaccination law for an age-dependent epidemic model using state feedback

An age-dependent epidemic model is studied with the goal of designing a state feedback stabilizing vaccination law to eradicate a disease. This model consists of a set of three nonlinear partial-integro differential equations (PIDE). A salient feature of the dynamical analysis is the fact that, if the basic reproduction number is greater than one, then the disease-free equilibrium is unstable. In view of this, we provide a linearizing state feedback vaccination law that is deduced from the one obtained for the PIDE model discretisation with respect to the age. Conditions guaranteeing stability of the closed-loop system and positivity of the feedback control are obtained using Isidori's theory and semigroup theory. Numerical simulations complete the analysis.

Data-Driven Synthesis of Robust Invariant Sets and Controllers

This paper presents a method to identify an uncertain linear time-invariant (LTI) prediction model for tube-based Robust Model Predictive Control (RMPC). The uncertain model is determined from a given state-input dataset by formulating and solving a Semidefinite Programming problem (SDP), that also determines a static linear feedback gain and corresponding invariant sets satisfying the inclusions required to guarantee recursive feasibility and stability of the RMPC scheme, while minimizing an identification criterion. As demonstrated through an example, the proposed concurrent approach provides less conservative invariant sets than a sequential approach.

Robust Trajectory Tracking for an Uncertain UAV Based on Active Disturbance Rejection

This note addresses the design of a robust trajectory tracking controller based on active disturbance rejection for a quadrotor Unmanned Aircraft Vehicle (UAV). A standard state-feedback linearizing controller, designed using a nominal model, is applied for tracking reference signals. An active disturbance rejection technique based on high-order sliding modes compensates for the uncertainties and nonlinearities. Furthermore, this rejection technique provides robustness against uncertainties and external disturbances after a finite transient time. Outdoors experimental results in a prototype illustrate the workability of the proposed methodology.

Neuroadaptive Robust Speed Control for PMSM Servo Drives with Rotor Failure

In this paper, a neuroadaptive robust trajectory tracking controller is utilized to reduce speed ripples of permanent magnet synchronous machine (PMSM) servo drive under the presence of a fracture or fissure in the rotor and external disturbances. The dynamics equations of PMSM servo drive with the presence of a fracture and unknown frictions are described in detail. Due to inherent nonlinearities in PMSM dynamic model, in addition to internal and external disturbances; a traditional PI controller with fixed parameters cannot correctly regulate the PMSM performance under these scenarios. Hence, a neuroadaptive robust controller (NRC) based on a category of on-line trained artificial neural network is used for this purpose to enhance the robustness and adaptive abilities of traditional PI controller. In this paper, the moth-flame optimization algorithm provides the optimal weight parameters of NRC and three PI controllers (off-line) for a PMSM servo drive. The performance of the NRC is evaluated in the presence of a fracture, unknown frictions, and load disturbances, likewise the result outcomes are contrasted with a traditional optimized PID controller and an optimal linear state feedback method.

A novel multi-objective optimization algorithm for Pareto design of a fuzzy full state feedback linearization controller applied on a ball and wheel system

The control and stabilization of a ball and wheel system around the equilibrium point are challenging tasks because it is an underactuated, nonlinear, and open-loop unstable plant. In this paper, Pareto design of a Fuzzy Full State Feedback Linearization Controller (FFSFLC) for the ball and wheel system based upon a novel multi-objective optimization algorithm is introduced. To this end, at first, a full state feedback linearization approach is employed to stabilize the dynamics of the system. Next, appropriate fuzzy systems are determined to tune the control gains. Then, a new multi-objective optimization algorithm is utilized to promote the proposed control scheme. This optimization algorithm is a combination of Simulated Annealing (SA) and Artificial Bee Colony (ABC) approaches benefiting advantages of the non-dominated Pareto solutions. To evaluate the capabilities of the suggested algorithm, its optimal solutions of several standard test functions are compared with those of five renowned multi-objective optimization algorithms. The results confirm that the proposed hybrid algorithm yields closer non-dominated Pareto solutions to the true optimal Pareto front with shorter runtimes than other algorithms. After selecting proper objective functions, multi-objective optimization of FFSFLC for the ball and wheel system is performed, and the results are compared with previous works. Simulations illustrate that the proposed strategies can accurately converge the system states to the desired conditions and yield superior robustness against disturbance signals in comparison with former studies.

Responsive surging, heading and diving controls of autonomous underwater vehicle based on brute forcing and smoothing of controllers

There are many types of controllers had been used to control Autonomous Underwater Vehicle (AUV) such as Proportional Integral Derivative (PID), Linear Quadratic Regulator (LQR), state feedback linearization, integrator back-stepping, and Sliding-Mode Control (SMC). However, for PID and SMC in particular, it is difficult to determine the optimal control design parameters. The objective of this study is to design and develop a responsive motion control system with optimal parameters for an AUV. The contribution of this paper is in term of introducing a filter to smooth reference signal and proposing a brute forcing technique to find optimal controller parameters. The methodology starts with modeling the AUV, estimating the unknown parameters from a real AUV model, designing a control system based on PI and SMC methods, and finally optimizing the controller parameters. The controller design was onto controlling surge speed using PI, heading using SMC, and diving using SMC. Simulation-wise, the developed control system has an average value of 93.89 % of responsiveness to track desired trajectory while 82.33 % of responsiveness without using the smoothing filter. The tested input signals were unit step, ramp, parabolic, and sinusoidal.

The Impact of the Dynamic Model in Feedback Linearization Trajectory Tracking of a Mobile Robot

This paper proposes the impact of the Dynamic model in Input-Output State Feedback Linearization (IO-SFL) technique for trajectory tracking of differential drive mobile robots, which has been restricted to using just the kinematics in most of the previous approaches. To simplify the control problem, this paper develops a novel control approach based on the velocity and position control strategy. To improve the results, the dynamics are taken into account. The objective of this paper is to illustrate the flaws unseen when adopting the kinematics-only controllers because the nonlinear kinematic model will suffice for control design only when the inner velocity (dynamic) loop is faster than the slower outer control loop. This is a big concern when using kinematic controllers to robots that don’t have a low-level controller, Arduino robots for example. The control approach is verified using the Lyapunov stability analysis. MATLAB/SIMULINK is carried out to determine the impact of the proposed controller for the trajectory tracking problem, from the simulation, it was discovered that the proposed controller has an excellent dynamic characteristic, simple, rapid response, stable capability for trajectory-tracking, and ignorable tracking error. A comparison between the presence and absence of the dynamic model shows the error in tracking due to dynamic system that must be taken into account if our system doesn’t come with a built-in one, thus, confirming the superiority of the proposed approach in terms of precision, with a neglectable difference in computations.

Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.

Control and anticontrol of chaos in fractional-order models of Diabetes, HIV, Dengue, Migraine, Parkinson's and Ebola virus diseases

This work investigates new fractional-order (FO) models of six chaotic diseases whose fractional dynamics have not been studied so far in literature. Here, we design and analyse suitable controllers to control chaos in these bio- mathematical models that show chaos and require stability; and anticontrollers to generate chaos where it is absent and turbulence may be sought. The proposed controllers and anticontrollers address the problem of the health hazards arising from the dysfunctionalities due to the impact of chaos in these biological models. Chaotic behaviour of vital parameters is characterised by extreme sensitivity, and their slight variations such as fall in density of β – cells, abnormal secretion of insulin in blood, small drop in the number of CD4+T cells, etc. may lead to unpredictable complexities in the models. Also, the evolution of the chaotic attractor in the disease model is revealed only when the bifurcation against the FO parameter is performed, indicating that it is a significant parameter to analyse the progression of a disease. Controllers to supress chaos in FO models of Diabetes Mellitus, Human Immunodeficiency Virus (HIV), Ebola Virus and Dengue models, are designed using Back-stepping, Adaptive Feedback and Sliding Mode Control strategies. Anticontrollers to introduce chaos in FO models of Parkinson's illness and Migraine, are designed using Linear State Feedback, Single State Sinusoidal Feedback and Sliding Mode Anticontrol strategies. The simulation results in terms of bifurcation diagrams, time series plots and phase portraits confirm the successful accomplishment of the control objectives.

Model Predictive Control for Hybrid Levitation Systems of Maglev Trains With State Constraints

Levitation control is a core technique of magnetic levitation (Maglev) trains, whose performance has a significant impact on the driving safety and ride comfort during the operation of Maglev trains. With the rapid development of high temperature superconducting (HTS) materials, a novel hybrid levitation system for Maglev trains, composed of HTS and normal conducting electromagnets, is becoming a more promising alternative to the conventional levitation system, due to a lager levitation air gap and lower energy consumption. The controller of the hybrid levitation system is hard to design due to strong nonlinearity, open-loop instability and fast-response requirements. In this study, we propose a robust model predictive control (MPC) strategy for the novel hybrid levitation system in which state constraints are taken into account, including the air gap, and the velocity and acceleration of the hybrid magnet. In the MPC strategy, the prediction model employs a stabilized model through nonlinear and linear state feedback. With regard to the online computational load, the primal-dual interior-point (PDI) algorithm is adopted to solve the optimization problem with high efficiency to deal with constraints in a real-time manner. Finally, simulation results are demonstrated to verify the effectiveness of the proposed MPC-PDI control strategy for the hybrid levitation system, and the significances considering state constraints are revealed. This study provides an engineering realization solution to the novel hybrid levitation system of a Maglev train.

Physics and differential geometry in modern control theory applied to a synchronous generator

This article will show how to address non-linear systems by means of state feedback linearization, through it we can have the stability of the system, which in most of the industry systems usually occurs. As the main contribution, tools of differential geometry will be presented, which is a recent branch of mathematics and physics which expands the concepts and definitions of classical mathematics, therefore, the model will be presented in terms of the definitions of space Tangent, derivative and Lie bracket all this allows us to understand and propose new solutions more efficiently since traditionally the techniques used revolve around the definitions of fields on finite spaces. The model on which the control techniques will be applied will be implemented on an electrical machine such as the synchronous generator, with the purpose of applying a control law that allows the outputs of the synchronous machine to adjust to the desired references, said the model is of great importance in the electricity sector industry because it can control the variables of the system.

Interval observer-based methodology for passive fault tolerant control of linear parameter-varying systems

The paper deals with passive fault tolerant control for linear parameter varying systems subject to component faults. Under the assumption that the faults magnitudes are considered unknown but bounded, a novel methodology is proposed using interval observer with an [Formula: see text] formalism to attenuate the effects of the uncertainties and to improve the accuracy of the proposed observer. The necessary and sufficient conditions of the control system stability are developed in terms of matrix inequalities constraints using Lyapunov stability theory. Based on a linear state feedback, a fault tolerant control strategy is designed to handle component faults effect as well as external disturbances and preserve the system closed-loop stability for both fault-free and component faulty cases. Two simulation examples are presented to demonstrate the effectiveness of the proposed method.

Chaos control and chaos synchronization of a multi-wing chaotic system and its application in multi-frequency weak signal detection

In this paper, first, a nonlinear feedback controller for achieving chaos control of a novel multi-wing chaotic system is presented. The nonlinear feedback controller has two parts. The first part is used to compensate an equilibrium point for the multi-wing chaotic system. The second part is a linear state feedback controller. The nonlinear feedback controller can globally asymptotically stabilize the multi-wing chaotic system to the equilibrium point. Stability conditions are given by using the Barbashin–Krasovskii theorem. Then, a linear state feedback controller for achieving chaos synchronization of the multi-wing chaotic system is presented. The linear state feedback controller can asymptotically stabilize the chaos synchronization error system to the origin. Stability conditions are given by using the passivity-based theory. Finally, a multi-frequency weak signal detection method is presented based on chaos control of the multi-wing chaotic system. The detection method can detect the frequencies of the weak signal and does not need to determine the critical point.

On the Problem of Robust Assignment of Dynamic System Poles

For a linear time-invariant single-input system with a linear inertia-free state feedback, a problem of such assignment of poles, in which bounded parametric perturbations do not bring these poles outside a certain region is considered. A generalization of the well-known solution to this problem, based on Rouche's theorem and assuming that the structure of system and the input matrices meet the consistency conditions, to linear control systems with matrices of an arbitrary form is proposed. A special parameter is introduced, on the value of which the fulfilment of the conditions of Rouche's theorem, the shape of the pole scatter region, its "volume" and the minimum attainable root measures of the dynamics performance depend. The relationship between the width of parametric uncertainty intervals and the minimum required value of the geometric mean root of the characteristic polynomial of a system with a modal controller is revealed. An example of application of the proposed technique and a computer program developed on its basis in the MATLAB language for the synthesis of a modal controller for an electromechanical system is given.

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.