Stabilizing State Feedback Control Law Research Articles

Robust state-feedback stabilization of an underactuated network of interconnected [formula omitted] hyperbolic PDE systems

We detail in this article the development of a delay-robust stabilizing state feedback control law for an underactuated network of two subsystems of heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the two subsystems is actuated. The proposed approach is based on the backstepping methodology. A backstepping transform allows us to construct a first feedback to tackle in-domain couplings present in the actuated PDE subsystem. Then, we introduce a predictive tracking controller to stabilize the second PDE subsystem. The stabilization of this subsystem implies the stabilization of the whole network. Finally, the proposed control law is combined with a low-pass filter to become robust with respect to small delays in the control signal and uncertainties on the system parameters.

A novel control Lyapunov-Krasovskii functional methodology for simultaneous stabilization of a set of multiple time-delays nonlinear systems

This paper considers the simultaneous stabilization of a set of nonlinear systems, that involve uncertain nonlinearities besides multiple time-varying delays in the states. Under the assumption that the upper bounds of delays are known, a memoryless simultaneously stabilizing state feedback controller is presented by proposing a control Lyapunov-Krasovskii functional (CL-KF) method. As required to establish the CL-KF approach, a systematic procedure is given to construct CL-KFs for the systems under consideration. By the obtained CL-KFs, a common stabilizing state feedback control law is established to drive all the systems to the origin. Examples are finally given to verify the benefit of the proposed design method.

Robust Positively Invariant Polyhedral Sets and Constrained Control using Fuzzy T-S Models: a Bilinear Optimization Design Strategy

Abstract We propose a numerical method to compute stabilizing state feedback control laws and associated polyhedral invariant sets for nonlinear systems represented by Fuzzy Takagi-Sugeno (T-S) models, subject to state and control constraints, and persistent disturbances. Sufficient conditions are derived under which a given polyhedral set is positively invariant under a Parallel Distributed Compensation (PDC), in the form of bilinear algebraic inequalities. Then, a bilinear programming (BP) problem is proposed to compute the state feedback gains and an associated positively invariant polyhedron, with predefined complexity, which solve a constrained regulation problem for the Fuzzy T-S system. A numerical example illustrates the effectiveness of the method.

An Algebraic Approach to the Observer-Based Feedback Stabilization of Linear 2D Discrete Models

In a recent work, we have investigated the state feedback structural stabilization of linear 2D discrete models. Our main tools are a procedure which enables the state feedback stabilization of Roesser models and equivalence transformations in the sense of the algebraic analysis approach to linear systems theory. Starting from a Fornasini-Marchesini model, we can compute (if it exists) a stabilizing state feedback control law for an equivalent Roesser model and we can then recast it in order to obtain a stabilizing control law that can be applied to the original Fornasini-Marchesini model. Extending the same idea, we here focus on the observer-based dynamic output structural stabilization of linear 2D discrete models within the algebraic analysis framework. Given a Fornasini-Marchesini model, we briefly show how an equivalent Roesser model can be stabilized (if possible) by an observer-based controller computed through existing techniques. Then, we show how to interpret the obtained control law so that it stabilizes the original Fornasini-Marchesini model. Finally, we discuss the cases where our procedure succeeds and we point out some cases where it fails, giving insights to tackle the difficulties encountered in the latter cases.

Predictor based control of linear systems with state, input and output delays

In this paper we develop a predictor-based controller for linear systems with state, input and output delays. First, a state predictor is developed for state feedback control. This predictor is formulated recursively over the prediction time by partitioning the input delay into sections smaller than the state delays. The partitioning of the input delay ensures that the resulting predictor equation only depends on the past values of the state and input. It is shown that the proposed predictor gives an exact prediction of the future states. This recursive predictor is then reformulated into a cascade form in order to reduce the number of redundant calculations and to simplify the predictor equation for practical implementation. We construct a predictor based state feedback control law and show that the spectrum of the closed-loop time-delay system under the constructed control law is the same as that of an equivalent time-delay system without input delay and under the nominal state feedback control. Therefore, the proposed predictor based solution can stabilize the delay system if a stabilizing state feedback control law exists for the input delay free system. These state feedback results are then extended to the case of delayed output feedback. The theoretical derivation is verified through numerical examples.

Contribution to the stabilization of stochastic nonlinear systems with time delays

AbstractIn this paper, the problem of feedback stabilization of stochastic differential delay systems is considered. The systems under study are also nonlinear and nonaffine. By using a LaSalle-type theorem for stochastic systems, general conditions for stabilizing the closed-loop system with delays are obtained. In addition, stabilizing state feedback control laws are proposed. An example illustrates the proposed approach.

A remark on stabilization of nonlinear systems with discrete and distributed delays

In this paper, the problem of feedback stabilization of systems with delays is addressed. The systems under consideration are nonlinear and nonaffine in control. We first consider the case of system with discrete delay and then proceed to develop our results in the case of system with both discrete and distributed delays. By using the Lyapunov approach, we deduce general conditions for stabilizing the closed-loop system and derive stabilizing state feedback control laws for the latter.

State feedback design for input-saturating quadratic systems

This paper proposes a method to design stabilizing state feedback control laws for nonlinear quadratic systems subject to input saturation. Based on a quadratic Lyapunov function, a modified sector condition and a particular representation for the quadratic terms, synthesis conditions in a “quasi”-LMI form are stated in a regional (local) context. An LMI-based optimization problem is then derived for computing the state feedback gains maximizing the estimate of the stability region of the closed-loop system.

ON FEEDBACK STABILIZATION OF NONAFFINE DISCRETE TIME NONLINEAR SYSTEMS WITH TIME DELAYS

In this contribution, the problem of stabilization of nonaffine discrete time nonlinear systems with time delay is addressed. By using the Lyapunov-Razumikhin approach, general conditions for stabilizing the closed loop system are proposed and stabilizing state feedback control laws are derived.

Nonlinear approximate observers for feedback control

A class of approximate nonlinear observers is presented and, when used to implement stabilizing state feedback control laws, shown to yield asymptotically stable output feedback regulation.

Observer design and stabilization of the dominant state of discrete-time linear systems

In this paper, we present necessary and sufficient conditions for both observer design and stabilization of the dominant state of discrete-time linear systems. Our method essentially uses the model reduction of the original linear control system. We also establish a separation principle so that when the reduced-order plant is stabilizable, the state of the reduced order observer may be used in lieu of the state of the reduced order plant for implementing the stabilizing state feedback control law. The model reduction and the observer design detailed in this paper are discrete-time analogs of the results of Aldeen and Trinh (1994) for continuous-time linear systems.

RHYTHMIC STABILIZATION OF PERIODIC ORBITS IN A WEDGE

The paper addresses the rhythmic stabilization of periodic orbits in a wedge billiard with actuated edges. The output feedback strategy, based on the sole measurement of impact times, results from the combination of a stabilizing state feedback control law and a nonlinear deadbeat state estimator. It is shown that the robustness of both the control law and the observer leads to a simple rhythmic controller achieving a large basin of attraction.

STABILIZATION OF NEUTRAL SYSTEMS WITH SATURATING INPUTS

This paper focuses on the stabilization problem of neutral systems in the presence of control saturation. Based on a descriptor approach and the use of a modified sector condition, global and local stabilization conditions are derived using Lyapunov-Krasovskii functionals. These conditions allow to consider systems presenting time-varying delays and are formulated directly as linear matrix inequalities (LMIs). Optimization problems are proposed with the aim of computing stabilizing state feedback control laws.

Robust Global Stabilization of Continuous Bioreactors

This paper studies the problem of designing controllers for enlarging the stability region of continuous stirred microbial bioreactors. A specific application is in anaerobic digestion, where the stability region can be very small if the operating steady state is selected to maximize the methane production rate. A control Lyapunov function approach is followed to construct a globally stabilizing state feedback control law. This turns out to be proportional output feedback in the case of anaerobic digestion, where the measurement is the methane production rate. The robustness properties of the feedback controller are also investigated. For situations of large, measurable changes in the organic load of the bioreactor, a feedforward measurement is incorporated in the control law, leading to improved robustness margin.

Two results for adaptive output feedback stabilization of nonlinear systems

The problem of (adaptive) stabilization by means of output feedback of a class of nonlinear systems is addressed and solved. The proposed method relies on the asymptotic reconstruction of a stabilizing state feedback control law, does not require stable zero dynamics nor the construction of a Lyapunov function for the closed loop system, and treats in a unified way unknown parameters and unmeasured states. The applicability of the proposed method is discussed via theoretical examples. Finally, it is shown that the proposed method yields a solution to the problem of output feedback regulation for a DC-to-DC power converter and the efficacy of the resulting controller is verified via experiments.

Stability and Stabilization of Linear Discrete-Time Systems Subject to Control Saturation

The problems of analysis and synthesis of linear discrete-time control systems subject to actuators amplitude saturation are addressed. The theoretical results are developed by using tools from the absolute stability theory for discrete-time systems. Algorithms based on the solution of linear matrix inequalities (LMIs) are proposed both for computing approximations of the basin of attraction for the closed loop system when the control law is given and for designing stabilizing state feedback control laws. Both the analysis and design procedures take into account the effective occurrence of saturation and the nonlinear behavior of the closed-loop system.

リアプノフ直接法による非線形システムのニューラル安定化制御器の設計

This paper deals with a neural stabilizing controller of general nonlinear systems. The stabilizing state feedback control law is approximated with a multilayer neural network. Connection weights in the neural controller are determined by a min-max algorithm such that the Lyapunov stability theorem holds via a control Lyapunov function.

Integrator backstepping for rounded controls and control rates

We present a backstepping procedure for the design of globally stabilizing state feedback control laws such that the magnitudes of the control signals and their derivatives are bounded by constants which do not depend on the initial conditions. We accomplish this by propagating such boundedness properties through each step of the recursive design.

Algebraic separation in realizing a linear state feedback control law by means of an adaptive observer

Based on a novel adaptive observer, which does not require signal boundedness in its stability proof, an algebraic separation property of linear state feedback control and adaptive state observation is established. This means, whenever a linear, stabilizing state feedback control law is realized with the state replaced by the state estimate of the given stable adaptive observer, then the resulting nonlinear control system is also globally asymptotically Lyapunov stable with respect to the initial state and parameter observation error of the adaptive observer. In particular, no assumptions on the system dynamics nor on the speed of the adaptation are made.