Dynamic State Feedback Research Articles

Finite-time H∞ control of switched systems with mode-dependent average dwell time

This paper is concerned with finite-time H∞ control problem for a class of switched linear systems by using a mode-dependent average dwell time (MDADT) method. The switching signal used in this paper is more general than the average dwell time (ADT), in which each mode has its own ADT. By combining the MDADT and Multiple Lyapunov Functions (MLFs) technologies, some sufficient conditions, which can guarantee that the corresponding closed-loop system is finite-time bounded with a prescribed H∞ performance, are derived for the switched systems. Moreover, a set of mode-dependent dynamic state feedback controllers are designed. Finally, two examples are given to verify the validity of the proposed approaches.

Spectral reduction, complete damping, and stabilization of a delay system by a single controller

For a spectrally controllable linear autonomous differential-difference system of retarded type, we construct a dynamic state feedback that provides the complete damping of the original system and a finite spectrum of the closed-loop system. The closed-loop system can be made asymptotically stable by an appropriate choice of the spectrum. The results are illustrated by examples.

Output regulation of state-coupled linear multi-agent systems with globally reachable topologies

This paper investigates output regulation problem of state-coupled linear certain and uncertain multi-agent systems with globally reachable topologies. Distributed dynamic state feedback control law is introduced to realize the regulator problem and a general global method for error regulation is established. The Jordan canonical form is used to stabilize the closed-loop control system. Sylvester equation and internal model theory are adopted to achieve the objectives of output regulation for every initial condition in the state space. Finally, numerical simulations are utilized to show the effectiveness of the obtained results.

Command filtered back-stepping control of a flexible air-breathing hypersonic flight vehicle

<AQ*/>Abstract A theoretical framework of nonlinear flight control is exploited and applied to nonlinear longitudinal dynamics of a generic air-breathing hypersonic flight vehicle. A combination of novelty command filtered back-stepping technology and dynamic inversion methodology is adopted for designing a dynamic state-feedback controller that provides stable tracking of the altitude and velocity reference commands. The novel command filtered back-stepping altitude control obviates the need to compute analytic derivatives in the traditional back-stepping design, providing a simple and effective way for controlling non-linear hypersonic flight vehicle. An input-to-state stability-modular approach is presented by combining command filtered back-stepping method with sliding-mode-based integral filters, input-to-state stability analysis, and small-gain theorem. The stability analysis of the closed-loop system including the flexible dynamic, and the convergence of the system outputs are derived. The proposed control scheme is verified in simulations in a climbing maneuver case of separate velocity and altitude reference commands.

A DYNAMICAL STATE FEEDBACK CHAOTIFICATION METHOD WITH APPLICATION ON LIQUID MIXING

This paper introduces chaotic reference model-based dynamical state feedback chaotification method which can be applied to any input-state linearizable (nonlinear) system including linear controllable ones as special cases. In the developed method, any chaotic system of arbitrary dimension can be used as the reference model with no need to transform it into a special form, so providing the advantage of exploiting the vast amount of information on chaotic systems and their implementations available in the literature. To demonstrate the potential effective applications of the method, a permanent magnet dc motor is chaotified by the proposed dynamical state feedback as matching the closed loop dynamics to the well-known Chua's chaotic circuit. Then, an impeller mounted on the chaotified dc motor is used for mixing a corn syrup added acid–base mixture. It is observed in a nonintrusive way that mixing actuated by the chaotified dc motor is more efficient than constant and also periodical motor speed cases for the consideration of neutralization time and power consumption together.

Input-to-state-stability modular command filtered back-stepping attitude control of a generic reentry vehicle

A command filtered back-stepping attitude controller is exploited and analyzed to design a dynamic state-feedback controller for a generic Reentry Vehicle. A novel back-stepping control that obviates the need to compute analytic derivatives in the traditional back-stepping design is presented by combining command filtered back-stepping method, sliding-mode-based integral filters and inputto-state stability (ISS) analysis. The ISS-modular approach provides a simple and effective way for controlling non-linear Reentry Vehicle satisfying the strict-feedback form, simultaneously solving the problem of “explosion of complexity” in traditional back-stepping approach. The stability analysis of the closed-loop system and the convergence of the aerodynamic angles are verified based on the smallgain theorem. Numerical simulations are included to demonstrate the effectiveness of the proposed control scheme.

A leader-following rendezvous problem of double integrator multi-agent systems

This paper studies a leader-following rendezvous problem for a double integrator multi-agent system where the leader system can generate a class of signals such as ramp signal and sinusoidal signals with arbitrary amplitudes and initial phases. Motivated by some techniques in output regulation theory, we propose a dynamic distributed state feedback control protocol that is able to maintain the connectivity of an initially connected communication network, and, at the same time, achieve the objective of the asymptotic tracking of all followers to the leader regardless of external disturbances.

Decentralized Disturbance Accommodation with Limited Plant Model Information

The design of optimal disturbance accommodation and servomechanism controllers with limited plant model information is studied in this paper. We consider discrete-time linear time-invariant systems that are fully actuated and composed of scalar subsystems, each of which is controlled separately and influenced by a scalar disturbance. Each disturbance is assumed to be generated by a system with known dynamics and unknown initial conditions. We restrict ourselves to control design methods that produce structured dynamic state feedback controllers where each subcontroller, at least, has access to the state measurements of those subsystems that can affect its corresponding subsystem. The performance of such control design methods is compared using a metric called the competitive ratio, which is the worst-case ratio of the cost of a given control design strategy to the cost of the optimal control design with full model information. We find an explicit minimizer of the competitive ratio and show that it is undominated, that is, there is no other control design strategy that performs better for all possible plants while having the same worst-case ratio. This optimal controller can be separated into a static feedback law and a dynamic disturbance observer. For step disturbances, it is shown that this structure corresponds to proportional-integral control.

Approximate Finite-Horizon Optimal Control for Input-Affine Nonlinear Systems with Input Constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optimal input is in general not known. The control input can be found solving an Hamilton-Jacobi-Bellman (HJB) partial differential equation (pde): it typically consists of a combination of bang-bang arcs and singular arcs. In the paper a methodology to approximate the solution of the HJB pde arising in the finite-horizon optimal control problem with input constraints is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of an example: the minimum time optimal control problem for an industrial wastewater treatment plant.

Cooperative Control for Uncertain Multiagent Systems via Distributed Output Regulation

The distributed robust output regulation problem for multiagent systems is considered. For heterogeneous uncertain linear systems and a linear exosystem, the controlling aim is to stabilize the closed-loop system and meanwhile let the regulated outputs converge to the origin asymptotically, by the help of local interaction. The communication topology considered is directed acyclic graphs, which means directed graphs without loops. With distributed dynamic state feedback controller and output feedback controller, respectively, the solvability of the problem and the algorithm of controller design are both investigated. The solvability conditions are given in terms of linear matrix inequalities (LMIs). It is shown that, for polytopic uncertainties, the distributed controllers constructed by solving LMIs can satisfy the requirements of output regulation property.

Asynchronous finite-time [formula omitted] control for switched linear systems via mode-dependent dynamic state-feedback

This paper concerns the asynchronous finite-time H∞ control problem for a class of switched linear systems with time-varying disturbances. The asynchronous switching means that the switchings between the candidate controllers and system modes are asynchronous. By using the Average Dwell Time (ADT) and Multiple Lyapunov Functions (MLFs) technologies, some sufficient conditions which can guarantee that the corresponding closed-loop system is finite-time bounded with a prescribed H∞ performance index via asynchronously switched control, are derived for the switched linear systems. Unlike the traditional Lyapunov asymptotic stability, there is no requirement of negative definiteness (or semidefiniteness) on the derivative of Lyapunov-like function. Moreover, a set of mode-dependent dynamic state feedback controllers are designed. Finally, two examples are provided to verify the efficiency of the proposed method.

Approximate finite-horizon optimal control without PDEs

The problem of controlling the state of a system, from a given initial condition, during a fixed time interval minimizing at the same time a criterion of optimality is commonly referred to as finite-horizon optimal control problem. One of the standard approaches to the finite-horizon optimal control problem relies upon the solution of the Hamilton–Jacobi–Bellman (HJB) partial differential equation, which may be difficult or impossible to obtain in closed-form. Herein we propose a methodology to avoid the explicit solution of the HJB pde exploiting a dynamic extension. This results in a dynamic time-varying state feedback yielding an approximate solution to the finite-horizon optimal control problem.

Lyapunov-based hybrid loops for stability and performance of continuous-time control systems

We construct hybrid loops that augment continuous-time control systems. We consider a continuous-time nonlinear plant in feedback with a (possibly non stabilizing) given nonlinear dynamic continuous-time state feedback controller. The arising hybrid closed loops are guaranteed to follow the underlying continuous-time closed-loop dynamics when flowing and to jump in suitable regions of the closed-loop state space to guarantee that a positive definite function V of the closed-loop state and/or a positive definite function Vp of the plant-only state is non-increasing along the hybrid trajectories. Sufficient conditions for the construction of these hybrid loops are given for the nonlinear case and then specialized for the linear case with the use of quadratic functions. For the linear case we illustrate specific choices of the functions V and Vp which allow for the reduction of the overshoot of a scalar output. The proposed approaches are illustrated on linear and nonlinear examples.

Complete damping of a linear autonomous differential-difference system by a controller of the same type

For a spectrally controllable linear autonomous system of retarded type with commensurable delays, we construct a dynamic state feedback ensuring a complete damping of the original system. The closed-loop system is of retarded type as well. The results are illustrated by an example.

Dynamic state feedback and its application to linear optimal control

We investigate the linear optimal control problems in the context of dynamic state feedback configuration. The dynamic state feedback is a dual structure of the dynamic observer. In conjunction with the well known arguments on linear matrix differential and Lyapunov equations, we elicit the fact that the quadratic performance index is always computable with this configuration. Based on this property, we suggest a nonlinear optimization programming method to get suboptimal or near optimal time-invariant dynamic state feedback controls. One can also evaluate the efficacy of pre-designed dynamic state feedback controllers utilizing this property. Two illustrative examples are given to show the effectiveness of the proposed approach.

Finite-Time $H_{\infty}$ Fuzzy Control of Nonlinear Jump Systems With Time Delays Via Dynamic Observer-Based State Feedback

This paper studies the finite-time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control problem for time-delay nonlinear jump systems via dynamic observer-based state feedback by the fuzzy Lyapunov-Krasovskii functional approach. The Takagi-Sugeno (T-S) fuzzy model is first employed to represent the presented nonlinear Markov jump systems (MJSs) with time delays. Based on the selected Lyapunov-Krasovskii functional, the observer-based state feedback controller is constructed to derive a sufficient condition such that the closed-loop fuzzy MJSs is finite-time bounded and satisfies a prescribed level of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> disturbance attenuation in a finite time interval. Then, in terms of linear matrix inequality (LMIs) techniques, the sufficient condition on the existence of the finite-time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based controller is presented and proved. The controller and observer can be obtained directly by using the existing LMIs optimization techniques. Finally, a numerical example is given to illustrate the effectiveness of the proposed design approach.

Cooperative Output Regulation With Application to Multi-Agent Consensus Under Switching Network

In this paper, we consider the cooperative output regulation of linear multi-agent systems under switching network. The problem can be viewed as a generalization of the leader-following consensus problem of multi-agent systems. Due to the limited information exchanges of different subsystems, the problem cannot be solved by the decentralized approach and is not allowed to be solved by the centralized control. By devising a distributed observer network, we can solve the problem by both dynamic state feedback control and dynamic measurement output feedback control. As an application of our main result, we show that a special case of our results leads to the solution of the leader-following consensus problem of linear multi-agent systems.

Analysis of a Class of Fermentation Models by Optimal Control Methods

Fed batch fermentation is analyzed by means of numerical optimal control. The structures of optimal controls are studied. The ceiling of performance of fed batch mode is compared with that of batch fermentation. State feedback and dynamic (extended) state feedback representations of singular optimal controls are discussed. The dependence of optimal solutions on process parameters, in particular on terminal time, is described which allows the analysis of minimum time problems. Attention is given to computational aspects.

A new stability analysis and controller design method for discrete-time linear systems with saturation nonlinearities

The problems of stability analysis and controllers design for discrete-time linear systems subject to state saturation nonlinearities are investigated in this paper. Both full state saturation and partial state saturation are considered. It is well known to all that the controller design problem under state saturation is very difficult and complex to deal with. In order to overcome the difficulty, a new and tractable system is constructed, and it can be proved that the constructed system is with the same domain of attraction as the original system. With the aid of this property, to estimate the domain of attraction of the original system, an LMI-based method is presented for estimating the domain of attraction of the origin for the new constructed system under state saturation. Further, two optimization algorithms are developed for constructing dynamic output-feedback controllers and state feedback controllers, respectively, which guarantee that the domain of attraction of the origin for the closed-loop system is as ‘large’ as possible. An example is provided to demonstrate the effectiveness of the new method.

Finite-time stabilizing dynamic control of uncertain multi-input linear systems

In this paper, a global robust stabilizer is proposed for a class of uncertain linear systems. The stabilizer is a dynamic state feedback controller, which guarantees finite-time convergence to the origin, as well as Lyapunov stability. We first augment the given plant by one additional dynamics and design the stabilizer by combining backstepping and the idea of a dynamic exponent scaling method. Validating the design only on a half-space of the extended state space, the closed-loop system becomes smooth on the space, so that the solution is unique there. In addition, we prove that any solution will reach the origin (by escaping the half-space) in finite time and that the control signals converge to zero by adopting the new concept of degree indicator.