State Feedback Law Research Articles

Disturbance decoupling for biological fermentation systems with single input single output

The biological fermentation process has the characteristics of nonlinearity and multivariable coupling. To improve the performance of decoupling control in the fermentation process, a disturbance decoupling control based on the Lie symmetry method is proposed to obtain the analytical feedback control for a class of biological fermentation systems with single input single output (SISO). Firstly, the state-space equations and the disturbance decoupling model for a class of SISO biological fermentation systems are defined; Secondly, the key technologies and algorithm approaches of Lie symmetry theory for differential equations are introduced, and the conditions and the properties of Lie symmetry for nonlinear control systems under group action are given in detail; Finally, the derived distribution of Lie symmetric infinitesimal generators is used to prove the sufficient conditions for local disturbance decoupling in the system, and the closed-loop state feedback analytical law of the system is constructed. The proposed control method is applied to the disturbance decoupling control of mycelium concentration and substrate concentration in the biological fermentation process. Numerical simulation results show that the proposed control method can effectively improve the system decoupling control performance. Meanwhile, using Lie symmetry, the cascade decoupling standard form and the static state feedback law of biological fermentation systems can be constructed.

Robust bilinear tracking control of a parabolic trough solar collector via saturation

This paper investigates the problem of robust tracking control of heat transport in a Parabolic Trough Solar Collector (PTSC), where the output has to track a desired reference trajectory. In this work, the PTSC is modeled by state-space bilinear dynamics. The manipulated variable is the pump volumetric flow rate, and the source term, i.e., solar irradiance, is assumed to be unmeasured. In addition, the actuator’s physical constraints induce saturation bounds on the manipulated variable and need to be considered explicitly in the controller design. To deal with these challenges, we first propose a saturated state-feedback law that meets the control objectives. Then, we reconstruct the unknown time-varying source term using an adaptive estimator. Later, through Lyapunov stability analysis, we prove that the closed-loop system and the output tracking error are uniformly ultimately stable. Numerical simulations attest to the performance of the proposed control strategy.

Hybrid feedback control of PMSG-based WECS with multilevel flying-capacitor inverter enhanced by fractional order extremum seeking

This paper proposes an original hybrid control strategy for a three-phase multilevel flying capacitor inverter (FCI) used in wind energy conversion system (WECS). A state feedback law, computed using a Lyapunov function and an invariance principle, is designed to manage grid current control and capacitor voltage balancing, guaranteeing the asymptotic stability of the closed-loop system. Additionally, a fractional-order extremum seeking control (FOESC) algorithm for Maximum Power Point Tracking (MPPT) is introduced, which optimizes power capture by adjusting turbine speed from the DC side without requiring a detailed system model (model-free approach). The use of fractional-order calculus leads to improved convergence and more seamless performance while preserving the robust characteristics of the system. The proposed control strategies are validated through simulation, demonstrating improved transient response and stability under varying wind conditions and parameter uncertainties. These results highlight the potential of advanced control algorithms to enhance the efficiency and reliability of wind energy systems, contributing to global efforts to achieve sustainable energy goals.

L2-gain analysis and fault-tolerant control of nonlinear discrete time-delay saturated switched systems with memory state feedback

For nonlinear discrete time-delay saturated switched systems with memory feedback, the problem of actuator failure with external disturbances is studied. First, we propose a sufficient condition for the existence of a fault-tolerant state feedback law with memory by using the switched Lyapunov function (SLF) strategy, which ensures that the closed-loop system has a maximum tolerant disturbance capacity by solving the optimization problem. Then, the minimum upper bound on the [Formula: see text]-gain is estimated by solving the optimization problem. Finally, the reliability of the proposed method is verified by examples.

Design of reduced-order controllers for fluid flows using full-order controllers and Gaussian process regression

We propose a method to design reduced-order output-feedback controllers for fluid flows with the use of data produced by full-order controllers. First, the full-order controller is obtained by combining an ensemble Kalman filter (EnKF) and a model predictive controller (MPC) that are designed based on the Navier–Stokes equations. The full-order controller has high computational complexity and, therefore, is not suitable for real-time implementation. Hence, we use the full-order controller in offline numerical simulations to generate data for data-driven design of the reduced-order controller with low computational complexity. We find a reduced-order subspace of a closed-loop system under the full-order control from the data. This subspace underlies the reduced-order output-feedback controller. The reduced-order state-feedback law is obtained by approximating the full-order MPC with the use of its input/output data. The reduced-order observer is designed for a reduced-order model that is derived by using the Gaussian process regression (GPR). The GPR enables us to design the reduced-order observer which can evaluate uncertainty due to state-dependent residuals of the reduced-order model. We demonstrate the proposed method for a control problem of a flow around a cylinder at the Reynolds number 100. Numerical simulations reveal that the reduced-order controller performs as almost well as the full-order controller for a set of initial states. In addition, robustness of the reduced-order controller to a temporal disturbance that is not considered in the control design is confirmed in the simulations.

Gain-Scheduled Structured Control in DC Microgrids

Power buffers are electronic converters that shield direct current (dc) microgrids from the effects of abrupt load changes. By promoting collective behaviors, distributed control solutions can improve the overall grid performance, as power buffers can collectively react to abrupt load changes. In this article, a novel gain-scheduled structured control scheme for power buffers is proposed. The control objective is to regulate both the stored energy and the input impedance of the power buffers in a distributed fashion, while guaranteeing, at the same time, the closed loop stability regardless of load changes occurring within predefined ranges. The proposed controller is expressed as a gain-scheduled state feedback law, whose coefficients depend affinely on the scheduling parameters. The structure of the controller gain is chosen to reflect the topology of the communication network implementing the distributed controller. In this article, the existing theory is extended to deal with such control problem, providing a novel strategy, which can be used to design gain-scheduled controllers for the considered class of structured feedback systems. Simulation results using a high-fidelity dc microgrid simulator verify the effectiveness of the proposed approach.

Robust output regulation of a class of smart actuators described by a minimum phase LPV dynamics

This paper presents a new control strategy for a class of minimum phase linear parameter-varying (LPV) systems representative of smart material actuators. In position control applications of such devices, one is typically interested in achieving output regulation with an arbitrary exponential decay rate. In principle, this problem can be tackled by assigning an exponential decay rate to the full system state, by means of well-established linear matrix inequalities (LMI) methods. For the class of systems under investigation, however, this approach leads to excessively large controller gains in case the desired decay rate is faster than the open-loop zero dynamics. In addition, the existence of unmeasurable state variables related to the material microstructure makes it not possible to directly implement full state feedback laws which are commonly adopted in LPV theory. To overcome these issues, a new design strategy is proposed. The key idea relies on arbitrarily shaping the output exponential decay rate without requiring fast convergence of the full state vector. This goal is achieved by means of a partial state feedback control law which solely depends on the measurable system states. A LMI algorithm is also developed to systematically address the design of the partial state feedback controller. The method is validated by means of simulations on generic examples, as well as via experiments on a mechatronic positioning system based on a dielectric elastomer membrane. In case the desired output convergence speed is faster than the open-loop zero dynamics, it is shown that the new approach leads to better transient behaviors and significantly smaller controller gains than standard LPV design techniques.

Synchronization of stochastic fractional-order model of muscular blood vessels

This study deals with the synchronization problem of the stochastic fractional-order biomathematical model of muscular blood vessels (SFOMBVs) with nonlinear inputs using a state-feedback adaptive control law, in which the effects of uncertainties and external disturbances are considered in the modeling. The linear state feedback and adaptive laws are combined via an adaptive update law, leading to a simple and robust adaptive feedback scheme. The synchronization and tracking performances of the SFOMBV master and slave systems are investigated, and the stability of the proposed control system is proved based on stochastic analysis techniques and Lyapunov theory. The simulation results show that the proposed control method can lead an abnormal muscular blood vessel to a normal trajectory with good robustness by reducing the tracking error.

Design and Verification of Offline Robust Model Predictive Controller for Wheel Slip Control in ABS Brakes

Wheel slip control is a critical aspect of vehicle safety systems, notably the antilock braking system (ABS). Designing a robust controller for the ABS faces the challenge of accommodating its strong nonlinear behavior across varying road conditions and parameters. To ensure optimal performance during braking and prevent skidding or lock-up, the ideal wheel slip value can be determined from the peak of the tire–road friction curve and maintained throughout the braking process. Among various control approaches, model predictive control (MPC) demonstrates superior performance and robustness. However, online MPC implementation encounters significant computational burdens and real-time limitations, particularly when dealing with larger problem sizes. To address these issues, this study introduces an offline robust model predictive control (RMPC) methodology. The proposed approach is based on the robust asymptotically stable invariant ellipsoid methodology, which employs linear matrix inequalities (LMIs) to calculate a collection of invariant state feedback laws associated with a sequence of nested invariant stable ellipsoids. Simulation results indicate a significant reduction in computational burden with the offline RMPC approach compared to online implementation, while effectively tracking the desired wheel slip reference values and system constraints. Moreover, the offline RMPC design aligns well with the online MPC design and verifies its effectiveness in practice.

Robust Regulation of a Power Flow Controller via Nonlinear Integral Action

This article considers a robust output set-point tracking problem for an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$</tex-math> </inline-formula> -terminal power flow controller (PFC) for meshed dc micro-grids. The PFC is a power electronics device used to control the power flow at a node in the meshed grid and may act as a dc circuit breaker. The system is modeled by state-space bilinear dynamics coupled with a polynomial output. In the proposed design, the plant is first extended with an integral action processing the regulation error. The cascaded model composed of the plant and the integrator is then stabilized using a saturated state-feedback law, designed with a forwarding approach. An anti-windup function is added to cope with transient saturations. A tuning method is proposed to set the controller gains along the available degrees of freedom with respect to a cost function. The stability of the closed loop is guaranteed for any compact set of initial conditions and for small parametric variations around the nominal set point. Experiments have been carried out and these properties are successfully assessed on a tenth-scale experimental setup.

Discovering efficient periodic behaviors in mechanical systems via neural approximators

AbstractIt is well known that conservative mechanical systems exhibit local oscillatory behaviors due to their elastic and gravitational potentials, which completely characterize these periodic motions together with the inertial properties of the system. The classification of these periodic behaviors and their geometric characterization are in an ongoing secular debate, which recently led to the so‐called eigenmanifold theory. The eigenmanifold characterizes nonlinear oscillations as a generalization of linear eigenspaces. With the motivation of performing periodic tasks efficiently, we use tools coming from this theory to construct an optimization problem aimed at inducing desired closed‐loop oscillations through a state feedback law. We solve the constructed optimization problem via gradient‐descent methods involving neural networks. Extensive simulations show the validity of the approach.

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.

Finite-time state feedback stabilization of strict-feedback switched systems with asymmetric output-constraints

Finite-time stabilization of strict-feedback switched systems with asymmetric output constraints (AOCs) via state feedback is investigated. First, an elaborately constructed fraction-type barrier Lyapunov function (BLF) is presented. Then, with a mild assumption on strict-feedback switched systems, state feedback laws are constructed by revamping the adding a power integrator approach (AAPIA) and meanwhile a common Lyapunov function (CLF) is also obtained. The resultant closed-loop systems are finite-time stable (FTS) and the asymmetric output constraint is satisfied too. The approach, proposed in this note, can make strict-feedback switched systems with/without AOCs finite-time stable in a unified frame.

Explaining the “Mystery” of Periodicity in Inter-Transmission Times in Two-Dimensional Event-Triggered Controlled Systems

Motivated by scenarioswhere the communication or the computation resources are limited, event-triggered control consists of transmitting data between the plant and the controller according to the actual system needs and not the elapsed time since the last transmission instant as in traditional sampled-data control, so that the desired control objective is achieved. A range of techniques are nowadays available to design event-triggered controllers. However, we generally have only very little information about the actual behavior of the transmission instants and thus about the amount of transmissions being actually generated, though this is a key feature of the design. In this article, we analyze the inter-event times, i.e., the times between two successive transmission instants, when the plant is modeled as a two-dimensional linear time-invariant system. The controller is a state-feedback law and the triggering rule is the relative threshold policy, which is allowed to be time-regularized. One of the main results in this article is the explanation of the oscillatory behavior of the inter-event times when the constant used to define the threshold is small relative to 1, a phenomenon commonly observed in simulations but never explained so far. More generally, the presented results help to understand the behavior of the inter-event times, instead of solely relying on numerical simulations, and thereby can be exploited to rigorously evaluate the performance of the considered triggering condition in terms of (average) inter-transmission times.

Tumour growth control: analysis of alternative approaches

In this work we address the problem of tumour growth control by properly exploiting a low-dimensional model that grounds on the Chemical Reaction Network (CRN) formalism. Originally conceived to work both in deterministic and stochastic frameworks, it is shown that, except for the case of very low number of tumour cells, the deterministic approach is appropriate to characterize the system behaviour, especially for control planning purposes. Two alternative control approaches are here investigated. One trivially assumes a constant infusion of external drug administration, the other is designed according to a state-feedback control scheme, with complete or partial knowledge of the state. Pros and cons of both control laws are investigated, showing that the tumour size at the beginning of the therapy plays a role of paramount importance for fixed infusion therapies, whilst only state-feedback laws can eradicate arbitrarily large tumours.

Output-feedback control of viscous liquid–tank system and its numerical approximation

We solve the output-feedback stabilization problem for a tank with a liquid modeled by the viscous Saint-Venant PDE system. The control input is the acceleration of the tank and a Control Lyapunov Functional methodology is used. The measurements are the tank position and the liquid level at the tank walls. The control scheme is a combination of a state feedback law with functional observers for the tank velocity and the liquid momentum. Four different types of output feedback stabilizers are proposed. A full-order observer and a reduced-order observer are used in order to estimate the tank velocity while the unmeasured liquid momentum is either estimated by using an appropriate scalar filter or is ignored. The reduced order observer differs from the full order observer because it omits the estimation of the measured tank position. Exponential convergence of the closed-loop system to the desired equilibrium point is achieved in each case. An algorithm is provided that guarantees that a robotic arm can move a glass of water to a pre-specified position no matter how full the glass is, without spilling water out of the glass, without residual end point sloshing and without measuring the water momentum and the glass velocity. Finally, the efficiency of the proposed output feedback laws is validated by numerical examples, obtained by using a simple finite-difference numerical scheme. The properties of the proposed, explicit, finite-difference scheme are determined.

Invariant sets for a class of nonlinear control systems tractable by symbolic computation

A set S is said to be controlled invariant with respect to a control system if a state feedback law exists such that the closed loop system has S as an invariant set. In the present paper we generalise results on input-affine polynomial control systems and algebraic varieties (i.e. sets described by the zeros of polynomial equations) considered in Zerz and Walcher (2012) to an extended class of vector fields. More precisely, we consider vector fields of the form f = F ° h, where F is a polynomial vector and h is a continuously differentiable function with certain (algebraic) properties, as well as sets Vh as the preimages of varieties under h. We will see that for example polynomial expressions in sine and cosine satisfy the mentioned properties. The main advantage of the considered function class is that it is accessible to symbolic computation. We give computational methods (based on the theory of Gröbner bases) to decide the controlled invariance of Vh.

Robust Data-Driven Safe Control Using Density Functions

This paper presents a tractable framework for data-driven synthesis of robustly safe control laws. Given noisy experimental data and some priors about the structure of the system, the goal is to synthesize a state feedback law such that the trajectories of the closed loop system are guaranteed to avoid an unsafe set even in the presence of unknown but bounded disturbances (process noise). The main result of the paper shows that for polynomial dynamics, this problem can be reduced to a tractable convex optimization by combining elements from polynomial optimization and the theorem of alternatives. This optimization provides both a rational control law and a density function safety certificate. These results are illustrated with numerical examples.

Gaussian inference for data-driven state-feedback design of nonlinear systems

Data-driven control of nonlinear systems with rigorous guarantees is a challenging problem as it usually calls for nonconvex optimization and often requires the knowledge of the true basis functions of the unknown system dynamics. To tackle these drawbacks, this work is based on a data-driven polynomial representation of general nonlinear systems exploiting Taylor polynomials. Thereby, we design state-feedback laws that render a known equilibrium globally asymptotically stable while operating with respect to a desired quadratic performance criterion. The calculation of the polynomial state feedback boils down to a sum-of-squares optimization problem, and hence to computationally tractable linear matrix inequalities. Moreover, we examine state-input data in presence of Gaussian noise by Bayesian inference to overcome the conservatism of deterministic noise characterizations from recent data-driven control approaches for Gaussian noise.

Motion planning problem for a class of hyperbolic PDEs

The purpose of this paper is to investigate the motion planning problem (MPP) for a class of hyperbolic PDEs with a boundary control. Precisely, we aim to construct an input control leading the state output to track a given (sufficiently smooth) target trajectory. First, we establish a solution of the MPP in the form of a boundary state-feedback law. Second, we provide an explicit expression of the whole state of the resulting closed-loop. Third, we furnish an explicit formula of the MP control via the target trajectory and the solution of an appropriate PDE. Besides, the provided control leads to a complete matching of the output with the predefined trajectory at any instant. The control problem is reformulated in the context of operator semigroup theory and solved using the Laplace transform of a suitable input-output function.