Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles

This paper investigates the problem of feedback control design with vibration suppression for a flexible air-breathing hypersonic vehicle （FAHV）. FAHV includes intricate coupling between the engine and flight dynamics, as well as complex interplay between flexible and rigid modes, which results in an intractable system for the control design. In this paper, a longitudinal model, which is described by a coupled system of ordinary differential equations （ODEs） and partial differential equations （PDEs）, is adopted. Firstly, a linearized ODE model for the rigid part is established around the trim condition, while vibration of the fuselage is described by PDEs. Secondly, based on the Lyapunov direct method, a control law via ODE state feedback and PDE boundary output feedback is designed for the system such that the closed-loop exponential stability is ensured. Finally, simulation results are given to illustrate the effectiveness of the proposed design method.     

Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems.

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Boundary control for a constrained two-link rigid–flexible manipulator with prescribed performance

In this paper, we consider a boundary control problem for a constrained two-link rigid–flexible manipulator. The nonlinear system is described by hybrid ordinary differential equation–partial differential equation (ODE–PDE) dynamic model. Based on the coupled ODE–PDE model, boundary control is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. With the help of prescribed performance functions, the tracking error can converge to an arbitrarily small residual set and the convergence rate is no less than a certain pre-specified value. Asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle extended to infinite-dimensional system. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.     

Stabilisation for cascade of nonlinear ODEs and counter-convecting transport dynamics

We consider stabilisation for a nonlinear ordinary differential equation (ODE) and counter-convecting transport partial differential equations (PDEs) cascaded system in which the transport coefficients depend on the ODE state. Stability analysis of the closed-loop system is based on the infinite-dimensional backstepping transformations and a Lyapunov functional. A predictor control is proposed such that the closed-loop system is globally asymptotically stable. The proposed design method is illustrated by a single-link manipulator.     

Vibration control of a flexible aerial refuelling hose with input saturation

In this study, we consider a boundary control problem of a flexible aerial refuelling hose in the presence of input saturation. To provide an accurate and concise representation of the hose's behaviour, the flexible hose is modelled as a distributed parameter system described by partial differential equations (PDEs). By using the backstepping method, a boundary control scheme is proposed based on the original PDEs to regulate the hose's vibration. An auxiliary system based on a smooth hyperbolic function and a Nussbaum function is designed to handle the effect of the input saturation. Then based on Lyapunov's direct method, the state of the system is proven to converge to a small neighbourhood of zero by appropriately choosing design parameters. Finally, the results are illustrated using numerical simulations for control performance verification.     

具有空间连续行为的微型压力传感器系统级建模

大多数MEMS器件(如梁、膜等)的动态特性方程为偏微分方程,因此建立对应组件的可重用参数化行为模型是一个难题.本文通过有限差分法把偏微分方程转化为常微分方程组,然后采用混合信号硬件描述语言进行描述,解决了该问题.针对电容式微型压力传感器,专门考虑膜片的空间连续行为以及结构、静电力的耦合作用,建立了包含接口电路在内的系统模型,据此进行了动态行为仿真.通过结果对比,验证了方法的实用性.相对于通用的参数化组件模型,当前MEMS商业化软件多采用逐个器件进行宏模型抽取的方式实现系统级建模和仿真.     

Modeling and boundary control of a flexible marine riser coupled with internal fluid dynamics

In this paper, vibration reduction of a flexible marine riser with time-varying internal fluid is studied by using boundary control method and Lyapunov’s direct method. To achieve more accurate and practical riser’s dynamic behavior, the model of marine riser with time-varying internal fluid is modeled by a distributed parameter system (DPS) with partial differential equations (PDEs) and ordinary differential equations (ODEs) involving functions of space and time. The dynamic responses of riser are completely different if the time-varying internal fluid is considered. Boundary control is designed at the top boundary of the riser based on original infinite dimensionality PDEs model and Lyapunov’s direct method to reduce the riser’s vibrations. The uniform boundedness and closed-loop stability are proved based on the proposed boundary control. Simulation results verify the effectiveness of the proposed boundary control.     

Single-step full-state feedback control design for nonlinear hyperbolic PDEs

The present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation.     

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.     

Modeling and control of flexible space stations

In this article, a preliminary formulation of large space structures and their stabilization is considered. The system consists of a (rigid) massive body and flexible configurations that consist of several beams, forming the space structure. The rigid body is located at the center of the space structure and may play the role of experimental modules. A complete dynamics of the system has been developed using Hamilton's principle. The equations that govern the motion of the complete system consist of six ordinary differential equations and several partial differential equations together with appropriate boundary conditions. The partial differential equations govern the vibration of flexible components. The ordinary differential equations describe the rotational and translational motion of the central body.The dynamics indicate very strong interaction among rigid-body translation, rigid-body rotation, and vibrations of flexible members through nonlinear couplings. Hence, any rotation of the rigid body induces vibration in the beams and vice versa. Also, any disturbance in the orbit induces vibration in the beams and wobbles in the body rotation and vice versa. This makes the system performance unsatisfactory for many practical applications. In this article, stabilization of the above-mentioned system subject to external disturbances is considered. The asymptotic stability of the perturbed system by application of velocity feedback controls is proved using Lyapunov's method.Numerical simulations are carried out in order to illustrate the impact of dynamic coupling or interaction among several members of the system and the effectiveness of the suggested feedback controls for stabilization. This study is expected to provide some insight into the complexity of modeling, analysis, and stabilization of actual space stations.     

Robust adaptive boundary control of a perturbed hybrid Euler-Bernoulli beam with coupled rigid and flexible motion

In this paper, a robust adaptive boundary controller is proposed to stabilize the coupled rigid-flexible motion of an Euler-Bernoulli beam in presence of boundary and distributed perturbations. Applying Hamilton’s principle, the dynamics of the hybrid beam model, including the actuators hub and the payload at its ends, is represented through four nonhomogeneous nonlinear partial differential equations (PDEs) subject to ordinary differential equations (ODEs) of boundary conditions. Using a Lyapunov-based control synthesis procedure, a robust nonlinear boundary controller is established that asymptotically stabilizes the perturbed beam vibration while regulating the rigid motion coordinates. A redesign of the proposed control laws produces a robust adaptive boundary controller that achieves control objectives in the presence of both parametric and modelling uncertainties. Control design is directly based on system PDEs without truncating the model so that instabilities from spillover effects are mitigated. The control inputs to the beam consist of three forces/torque applied to the actuators hub and a transverse force applied to the tip payload. Simulation results are used to investigate the efficiency of the proposed approach.

     

Adaptive actuator fault compensation control for a rigid-flexible manipulator with ODEs-PDEs model

In this paper, the actuator fault problem is studied for a two-link rigid-flexible manipulator system in the presence of boundary disturbance and state constraint. The system consists of a rigid beam, a flexible beam and a payload at the end, which are described by hybrid ordinary differential equations–partial differential equations. The novel controller includes a proportional-derivative feedback structure, a disturbance observer and a fault-tolerant algorithm, which can regulate the joint positions and eliminate vibration of flexible beam, on circumstance of boundary disturbance and actuator fault. With the help of Barrier Lyapunov Function, the states will not be violated. It is proved that the closed-loop system has asymptotic stability by LaSalle Invariance Principle. Simulations are provided to demonstrate the effectiveness of the proposed controller.     

Robust distributed controller design of rotational single-link smart materials beam

A dynamic modelling and controller design were presented for a single-link smart materials beam, a flexible beam bonded with piezoelectric actuators and sensors for better control performance. Taking into account bounded disturbances, a robust distributed controller was constructed based on the system model, which was described by a set of partial differential equations (PDEs) and boundary conditions (BCs) . Subsequently, a finite dimensional controller was further developed, and it was proven that this controller can stabilize the finite dimensional model with arbitrary number of flexible modes.     

Distributed Proportional-spatial Derivative control of nonlinear parabolic systems via fuzzy PDE modeling approach

In this paper, a distributed fuzzy control design based on Proportional-spatial Derivative (P-sD) is proposed for the exponential stabilization of a class of nonlinear spatially distributed systems described by parabolic partial differential equations (PDEs). Initially, a Takagi-Sugeno (T-S) fuzzy parabolic PDE model is proposed to accurately represent the nonlinear parabolic PDE system. Then, based on the T-S fuzzy PDE model, a novel distributed fuzzy P-sD state feedback controller is developed by combining the PDE theory and the Lyapunov technique, such that the closed-loop PDE system is exponentially stable with a given decay rate. The sufficient condition on the existence of an exponentially stabilizing fuzzy controller is given in terms of a set of spatial differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality (LMI) techniques is also provided to solve these SDLMIs. Finally, the developed design methodology is successfully applied to the feedback control of the Fitz-Hugh-Nagumo equation.     

具有端点质量的一维波动方程半离散格式的一致指数稳定性

无穷维系统主要由偏微分方程描述, 可是大部分用偏微分方程描述的控制系统, 无论是单纯的数值实验还是需要应用到实际的问题中去, 都需要对方程进行有限数值离散. 本文考虑了端点带有质量的波动方程在边界反馈控制下半离散格式的一致指数稳定性. 首先, 原闭环系统通过降阶法变成低阶的等价系统, 通过一种间接Lyapunov函数方法证明了降阶等价的连续系统是一致指数稳定的. 其次, 对等价系统空间变量离散得到半离散的差分格式.平行于连续系统, 间接Lyapunov函数方法证明了半离散系统的一致指数稳定性. 数值实验证明了基于降阶法的一致指数稳定性和经典半离散格式的非一致指数稳定性.     

Adaptive boundary control of a flexible manipulator with input saturation

In this study, we consider the anti-windup design as one of the approaches for the boundary control problem of a flexible manipulator in the presence of system parametric uncertainties, external disturbances and bounded inputs. The dynamics of the system are represented by partial differential equations (PDEs). Using the singular perturbation approach, the PDE model is divided into two simpler subsystems. With the Lyapunov's direct method, an adaptive boundary control scheme is developed to regulate the angular position and suppress the elastic vibration simultaneously and the adaptive laws are designed to compensate for the system parametric uncertainties and the disturbances. The proposed control scheme allows the application of smooth hyperbolic functions, which satisfy physical conditions and input restrictions, be easily realised. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

Direct strain feedback control of flexible robot arms: newtheoretical and experimental results

This paper addresses control problems for flexible robot arms by using direct strain feedback. The purpose is to make clear why direct strain feedback can damp out vibration of flexible arms satisfactorily. We concentrate on one-link flexible robot arms whose dynamic models can be represented by linear partial differential equations with appropriate boundary conditions which have been well examined in a number of papers. A key contribution of this paper is the introduction of the concept of (strict) A-dependent operators, which allows us to prove rigorously the closed loop stability of direct strain feedback and the existence and uniqueness of nonstandard second order abstract differential equations in Hilbert spaces. Several control experiments are performed, verifying the main theoretical points of this paper, demonstrating satisfactory control results of direct strain feedback, and leading to potential application of this simple control method for flexible robot control     

Solidification and control of a liquid metal column

In this paper, two different 1D mechanistic models for the solidification of a pure substance are presented. The first model is based on the two-domain approach, resulting in 2 partial differential equations (PDEs) and one ordinary differential equation (ODE) with 2 boundary conditions, 2 interface conditions, and one initial condition: the Stefan problem.In the second model, the metal column is considered as one-domain, and one PDE is valid for the whole domain. The result is one PDE with two boundary conditions.The models are implemented in MATLAB, and the ODE solver ode23s is used for solving the systems of equations. The models are developed in order to simulate and control the dynamic response of the solidification rate. The control scheme is based on a linear PI controller.     

Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations(ODEs) .Complete rigidity is the approximation of practical models; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Becausse of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.