Dynamic analysis of the ethanol fermentation with the impulsive state feedback control

To keep a sustainable and steady output of ethanol, ethanol fermentation in a bio-reactor with impulsive state feedback control is formulated. The sufficient conditions for existences of order-1 periodic solution and order-2 periodic solution are obtained by using the properties of the periodic solution. The results imply that ethanol fermentation tends to an order-1 periodic solution or order-2 periodic solution. At the same time, we also give the complete expression of the period of the positive period-1 solution. Finally, discussions and numerical simulations are given.     

Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Dynamical behavior of systems of two phytoplankton and one zooplankton populations with toxin producing phytoplankton

Three‐compartment mathematical models of non‐toxic phytoplankton (NTP), toxin producing phytoplankton (TPP), and zooplankton are proposed to explore the role of TPP in algal blooms. The mutual interference between predator zooplankton and avoidance of TPP by zooplankton are incorporated into the model. The NTP and TPP engage in exploit competition and the toxin produced by TPP has no effect on NTP. Using the concept of uniform persistence, we establish coexistence of NTP, TPP, and zooplankton in certain parameter regimes. We study the effects of mutual interference and avoidance by zooplankton upon the population interactions. In addition to the toxin producing mechanism, it is concluded that mutual interference of zooplankton is an important factor for diminishing harmful blooms. Copyright © 2017 John Wiley & Sons, Ltd.     

Nonlinear incidence rate of a pest management SI model with impulsive control strategy

In this work, we consider a pest management SI model with concerns about releasing of infective pests and spraying pesticides at different fixed moments. We prove that all solutions the investigated system are uniformly ultimately bounded, and there exists globally asymptotic stable pest‐extinction boundary periodic solution when certain condition is satisfied. Furthermore, the permanent condition of the system is also obtained. It is concluded that the approach, which combines releasing infective pests with spraying pesticides in different fixed moments, provides reliable tactic basis for the practical pest management. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear feedback control and systems of partial differential equations

Finding an equivalence between two feedback control systems is treated as a problem in the theory of partial differential equation systems. The mathematical aim is to embed the Jakubzyk-Respondek, Hunt-Meyer-Su work on feedback linearization in the general theory of differential systems due to Lie, Cartan, Vessiot, Spencer, and Goldschmidt. We do this by using the functor taking control systems into differential systems, and studying the equivalence invariants of such differential systems. After discussing the general case, attention is focussed on the special situation of most immediate practical importance, the theory of feedback linearization. In this case, the general system for feedback equivalence becomes a system of linear partial differential equations. Conditions are found that the general solution of this system may be described in terms of a Frobenius system and certain differential-algebraic operations.This work was supported by grant from the Ames Research Center of NASA and the Applied Mathematics Program of the National Science Foundation.     

Nonlinear impulsive Langevin equation with mixed derivatives

In this paper, we consider a nonlocal boundary value problem of nonlinear impulsive Langevin equation with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of Diaz-Margolis' fixed-point approach over generalized complete metric space. We give an example that supports our main result.     

On the impulsive fractional anti-periodic BVP modelling with constant coefficients

This paper is inspired by the recent papers on the BVP for impulsive fractional differential equations. We present a general framework to find the solutions of anti-periodic BVP for linear impulsive fractional differential equations with constant coefficients. Some sufficient conditions for existence of the solutions for nonlinear problem are established. Finally, numerical examples are given to illustrate the results.     

Geometrical analysis of a pest management model in food-limited environments with nonlinear impulsive state feedback control

In this paper, a nonlinear impulsive state feedback control system is proposed to model an integrated pest management in food-limited environments. In the system, impulsive feedback control measures are implemented to control pests on the basis of the quantitative state of pests. Mathematically, an intuitive geometric analysis is used to indicate the existence of periodic solutions. The stability of periodic solutions is investigated by using Analogue of Poincar\''{e} Criterion. At last, numerical simulations are given to verify the theoretical analysis.     

Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control

In this study the optimal control of fed-batch glycerol fermentation is investigated based on an impulsive dynamical system. Considering the sudden increase of the glycerol and alkali in fed-batch culture of biodissimilation of glycerol to 1,3-propanediol, this paper proposes a nonlinear impulsive system of fed-batch culture. The existence, uniqueness and regularity properties of piecewise solution for the system are proved. In view of the controllability of volumes of glycerol added to the reactor instantaneously, the paper constructs an optimal control model based on the nonlinear impulsive system and the existence of the optimal control is obtained. The control variables here are the moments and the sizes of jumps in the states at the discrete instants and the objective is to maximize the productivity of 1,3-propanediol over one cycle.     

Nonlinear optimal feedback control for lunar module soft landing

In this paper, the task of achieving the soft landing of a lunar module such that the fuel consumption and the flight time are minimized is formulated as an optimal control problem. The motion of the lunar module is described in a three dimensional coordinate system. We obtain the form of the optimal closed loop control law, where a feedback gain matrix is involved. It is then shown that this feedback gain matrix satisfies a Riccati-like matrix differential equation. The optimal control problem is first solved as an open loop optimal control problem by using a time scaling transform and the control parameterization method. Then, by virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, we present a practical method to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics. Simulation results show that the proposed approach is highly effective.     

Interface feedback control stabilization of a nonlinear fluid-structure interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈Rⁿ, n=2, where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of the Navier-Stokes equation coupled on the boundary with the dynamic system of elasticity. We shall consider models where the elastic body exhibits small but rapid oscillations. These are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid. Questions related to the stability of finite energy solutions are of paramount interest.It was shown in Lasiecka and Lu (2011) [14] that all data of finite energy produce solutions whose energy converges strongly to zero. The cited result holds under “partial flatness” geometric condition whose role is to control the effects of the pressure in the NS equation. Related conditions has been used in Avalos and Triggiani (2008) [23] for the analysis of the linear model. The goal of the present work is to study uniform stability of all finite energy solutions corresponding to nonlinear interaction. This particular question, of interest in its own rights, is also a necessary preliminary step for the analysis of optimal control strategies arising in infinite-horizon control problems associated with the structure. It is shown in this paper that a stress type feedback control applied on the interface of the structure produces solutions whose energy is exponentially stable.     

Nonlinear adaptive output feedback robust control of hydraulic actuators with largely unknown modeling uncertainties

In this paper, a nonlinear adaptive output feedback robust controller is proposed for motion control of hydraulic servo systems in the presence of largely unknown matched and mismatched modeling uncertainties. Different from the existing control technologies, the presented hydraulic closed-loop controller which can deal with strong matched and mismatched parametric uncertainties is synthesized via the backstepping technique. Specially, a nonlinear disturbance observer which can estimate the largely mismatched disturbance is integrated into the design of the linear extended state observer to obtain estimation of unmeasurable system states, uncertain parameters and strong disturbances simultaneously. In addition, the projection-type adaptive law is synthesized into the design of the resulting controller. More importantly, the global stability of the whole closed-loop system is strictly guaranteed by the Lyapunov analysis. Furthermore, the effectiveness and practicability of the presented control strategy have been demonstrated by comparative experiments under different working conditions.     

Nonlinear feedback control of chaotic pendulum in presence of saturation effect

In present paper, a feedback linearization control is applied to control a chaotic pendulum system. Tracking the desired periodic orbits such as period-one, period-two, and period-four orbits is efficiently achieved. Due to the presence of saturation in real world control signals, the stability of controller is investigated in presence of saturation and sufficient stability conditions are obtained. At first feedback linearization control law is designed, then to avoid the singularity condition, a saturating constraint is applied to the control signal. The stability conditions are obtained analytically. These conditions must be investigated for each specific case numerically. Simulation results show the effectiveness and robustness of proposed controller. A major advantage of this method is its shorter chaotic transient time in compare to other methods such as OGY and Pyragas controllers.     

Output tracking control of Boolean control networks with impulsive effects

This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state‐space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time‐varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.     

Nonlinear dynamic analysis of 2-DOF nonlinear vibration isolation floating raft systems with feedback control

In this paper, the average method is adopted to analysis dynamic characteristics of nonlinear vibration isolation floating raft system with feedback control. The analytic results show that the purposes of reducing amplitude of oscillation and complicating the motion can be achieved by adjusting properly the system parameters, exciting frequency and control gain. The conclusions can provide some available evidences for the design and improvement of both the passive and active control of the vibration isolation systems. By altering the exciting frequency and control gain, complex motion of the system can be obtained. Numerical simulations show the system exhibits period vibration, double period vibration and quasi-period motion.