Numerical simulations of the Boussinesq equation by He's variational iteration method

In this paper, we present a reliable algorithm to study the well-known model of nonlinear dispersive waves proposed by Boussinesq. We solve the Cauchy problem of Boussinesq equation using variational iteration method (VIM). The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that VIM is a powerful mathematical tool for solving nonlinear problems.     

Pollution's ambient problems and regularity of optimal cost function

We study pollution's ambient problems by using the optimal control theory applied to partial differential equations. We consider the problem to find the optimal way to eliminate pollution in the time, such that the concentration is close to a standard level which does not affect the ecological equilibrium when the source is pointwise. We consider a quadratic cost functional and we prove the existence and uniqueness of optimal control. We find a characterisation which makes possible the computing of optimal control. Additionally, we consider the problem moving the pointwise source. So, we define a function j(b) that associates to any point b ∈ Ω the optimal cost functional applied to the optimal control. We show that j is differentiable, provided that the controls are taken in a convenient subset of admissible functions satisfying the cone properties. We also find the point in Ω, for which the cost functional is minimum.     

Dynamical properties of an adaptive control system with a nonlinear reference model. I

We show the results of solving the problem of constructing an adaptive control system with nonlinear reference model for a nonstationary dynamical object with a nonlinear actuator. To analyze the dynamical properties of a nonlinear nonstationary control system we use the method of pointwise transformations. We show the results of analysis and mathematical modeling of the considered control system with the adaptation circuit switched off.     

NONLINEAR OPTIMAL CONTROL: A CONTROL LYAPUNOV FUNCTION AND RECEDING HORIZON PERSPECTIVE

Two well known approaches to nonlinear control involve the use of control Lyapunov functions (CLFs) and receding horizon control (RHC), also known as model predictive control (MPC). The on-line Euler-Lagrange computation of receding horizon control is naturally viewed in terms of optimal control, whereas researchers in CLF methods have emphasized such notions as inverse optimality. We focus on a CLF variation of Sontag's formula, which also results from a special choice of parameters in the so-called pointwise minnorm formulation. Viewed this way, CLF methods have direct connections with the Hamilton-Jacobi-Bellman formulation of optimal control. A single example is used to illustrate the various limitations of each approach. Finally, we contrast the CLF and receding horizon points of view, arguing that their strengths are complementary and suggestive of new ideas and opportunities for control design. The presentation is tutorial, emphasizing concepts and connections over details and technicalities.     

Admissible simulation relations,set-valued feedback,and controlled invariance

This work is a continuation of the author’s study of simulation relations between nonlinear input–output systems with disturbances. Previously we derived a criterion under which a “pointwise” simulation condition implies simulation by so-called “admissible” inputs and disturbances (that is, inputs and disturbances that yield time-dependent vector fields satisfying C ¹ Carathéodory conditions). This criterion included a certain constant-rank assumption. In this paper we use the theory of set-valued mappings and differential inclusions to derive analogous results in which the constant-rank assumption is replaced by a compactness provision that augments the pointwise simulation condition. We illustrate our simulation results by deriving a sufficient condition for achieving global controlled invariance of a (possibly singular) nonlinear system through the use of a set-valued feedback law.     

基于匝道调节的快速路交通密度的无模型周期自适应控制方法

针对快速路交通这一类复杂的MIMO非线性系统, 提出了一种新的无模型周期自适应匝道调节方法. 该方法本身是无模型的, 控制输入信息和伪Jacobi参数可在整个周期上逐点利用以前周期获得的I/O数据周期地进行更新. 通过严格的数学分析证明了算法的几何收敛性. 仿真结果也进一步说明了所提出方法的有效性.     

Spatial periodic adaptive control approach for rotary systems in sampled time

A periodic adaptive control approach is proposed for a class of nonlinear sampled‐time systems with varying parametric uncertainties that are periodic with respect to angular displacement, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a pointwise manner between two consecutive spatial cycles and in the sequel achieves the asymptotic tracking convergence. Rigorous analysis are presented along with a numerical example to show the effectiveness of the proposed controller. Copyright © 2012 John Wiley & Sons, Ltd.     

Extremum-seeking control of state-constrained nonlinear systems

An extremum-seeking control problem is posed for a class of nonlinear systems with unknown dynamical parameters, whose states are subject to convex, pointwise inequality constraints. Using a barrier function approach, an adaptive method is proposed for generating setpoints online which converge to the feasible minimizer of a convex objective function containing the unknown dynamic parameters. A tracking controller regulates system states to the generated setpoint via state feedback, while maintaining feasibility of the state constraints. A simulation example demonstrates application of the method.     

A frequency‐constrained geometric Pontryagin maximum principle on matrix Lie groups

We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).     

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.     

A polynomial approach for optimal control of switched nonlinear systems

Optimal control problems for switched nonlinear systems are investigated. We propose an alternative approach for solving the optimal control problem for a nonlinear switched system based on the theory of moments. The essence of this method is the transformation of a nonlinear, nonconvex optimal control problem, that is, the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high‐performance numerical computing. Consequently, we propose to convexify the control variables by means of the method of moments obtaining semidefinite programs. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems

 We study over- and underdetermined systems of nonlinear differential-algebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems in the behavior framework.?We give a general (local) existence and uniqueness theory and apply the results to analyze when nonlinear implicit control problems can be made regular by state or output feedback.?The theoretical analysis also leads immediately to numerical methods for the simulation as well as the construction of regularizing feedbacks. Date received: February 21, 2000. Date revised: November 14, 2000.     

Network flow control under capacity constraints: A case study

In this paper, we demonstrate how tools from nonlinear system theory can play an important role in tackling “hard nonlinearities” and “unknown disturbances” in network flow control problems. Specifically, a nonlinear control law is presented for a communication network buffer management model under physical constraints. Explicit conditions are identified under which the problem of asymptotic regulation of a class of networks against unknown inter-node traffic is solvable, in the presence of control input and state saturation. The conditions include a Lipschitz-type condition and a “PE” condition. Under these conditions, we achieve either asymptotic or practical regulation for a single-node system. We also propose a decentralized, discontinuous control law to achieve (global) asymptotic regulation of large-scale networks. Our main result on controlling large-scale networks is based on an interesting extension of the well-known Young's inequality for the case with saturation nonlinearities. We present computer simulations to illustrate the effectiveness of the proposed flow control schemes.     

Stabilization of an unstable tubular reactor by nonlinear passive output feedback control

The multiple–input multiple–output (MIMO) output feedback (OF) control problem of an exothermic multi-jacket tubular open-loop unstable reactor is addressed. Over its axial length, the reactor has several equally sized cooling jackets. The controller must adjust the jacket temperatures on the basis of per jacket temperature measurements so that the closed-loop system is robustly stable. The problem is solved within a constructive framework, by combining notions and tools from chemical reactor engineering and partial differential equations (PDEs) control systems theory. The result is a MIMO nonlinear OF dynamic control design with (i) a decentralized MIMO passive state feedback (SF) controller implemented with a pointwise observer (PWO), (ii) closed-loop stability conditions in terms of sensor set and control gains, and (iii) efficient late lumping-based on-line implementation. The design is put in perspective with industrial PI and inventory control, and applied to a representative example through numerical simulation with favorable comparison against adaptive controllers.     

Realization of a bounded feedback in a nonlinear control problem

We substantiate a method for design of a limited feedback to control a system whose mathematical model includes a nonlinear state function. Designing such a feedback is based on a positional solution of auxiliary optimal control problems which provides the transition from the vicinity of one equilibrium state to the vicinity of a new equilibrium state and stabilization of the system in the new state. Preassigned additional constrains on the trajectory of the control system are satisfied. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 108–116, January–February 2009.     

Optimal Control for Pennes' Bioheat Equation

The problem of Pennes' bioheat boundary control through a skin with its inner medium kept at a constant steady‐state temperature where the outer surface subjected to conductive condition is solved by different methods. Analytical and mild solutions of Pennes' boundary control problem yield a discrete optimization problem for temperature profile at the specific depth point as well as temperature profile at all points of the skin in the final time. The pointwise optimal control problem has been solved by discretized form of the unconstrained optimization problem using analytical solution for Pennes' boundary control problem. The analytical solution has been used to solve the optimal control problems via an unconstrained/constrained optimization in Methods I and II. In Method III first, Pennes' boundary control problem is solved using mild solution by strongly continuous semigroup theory. Then, nonlinear optimization problem with linear constraints is proposed to calculate the corresponding piecewise optimal control function. All methods, have been applied to a homogeneous tissue. Mathematical simulation is done to show the optimal controls for the related states when minimum effort is used.     

含有参数化和非参数化不确定性系统重复学习控制

针对含有参数化和非参数化的高阶非线性系统,设计了一种重复学习控制方案.假设未知时变参数和参考信号的共同周期是已知的,通过参数重组技巧,将所有未知时变项合并为一个周期时变向量.将改进Backstepping方法与分段积分机制相结合,构造了微分-差分参数自适应律和重复学习控制律,使跟踪误差在误差平方范数意义下渐近收敛于零.利用Lyapunov泛函,给出了闭环系统收敛的充分条件.仿真结果验证了该方法的有效性.     

Robust nonlinear variable selective control for networked systems

This paper is concerned with the networked control of a class of uncertain nonlinear systems. In this way, Takagi–Sugeno (T-S) fuzzy modelling is used to extend the previously proposed variable selective control (VSC) methodology to nonlinear systems. This extension is based upon the decomposition of the nonlinear system to a set of fuzzy-blended locally linearised subsystems and further application of the VSC methodology to each subsystem. To increase the applicability of the T-S approach for uncertain nonlinear networked control systems, this study considers the asynchronous premise variables in the plant and the controller, and then introduces a robust stability analysis and control synthesis. The resulting optimal switching-fuzzy controller provides a minimum guaranteed cost on an H₂ performance index. Simulation studies on three nonlinear benchmark problems demonstrate the effectiveness of the proposed method.     

Control of a nonlinear liquid level system using a new artificial neural network based reinforcement learning approach

Most industrial processes exhibit inherent nonlinear characteristics. Hence, classical control strategies which use linearized models are not effective in achieving optimal control. In this paper an Artificial Neural Network (ANN) based reinforcement learning (RL) strategy is proposed for controlling a nonlinear interacting liquid level system. This ANN-RL control strategy takes advantage of the generalization, noise immunity and function approximation capabilities of the ANN and optimal decision making capabilities of the RL approach. Two different ANN-RL approaches for solving a generic nonlinear control problem are proposed and their performances are evaluated by applying them to two benchmark nonlinear liquid level control problems. Comparison of the ANN-RL approach is also made to a discretized state space based pure RL control strategy. Performance comparison on the benchmark nonlinear liquid level control problems indicate that the ANN-RL approach results in better control as evidenced by less oscillations, disturbance rejection and overshoot.     

Nonlinear system identification and control of chemical processes using fast orthogonal search

The use of the fast orthogonal search (FOS) method is presented for model estimation and control of nonlinear chemical processes. FOS provides a nonlinear approximation used in an inner-loop that allows for simpler linear control methods to be used as an outer-loop controller. It is a straightforward, simple-to-use method for linearization of systems based on orthogonal system identification. The control concept is derived from the method of inverse dynamic control (IDC). The novel combination of this with the FOS method of system identification results in a very efficient and effective method of control. The method is demonstrated and tested on two nonlinear chemical process control simulations and the results are shown to compare very favourably to published results on the same problems.