Disturbance rejection in robust PdE-based MRAC laws for uncertain heterogeneous multiagent networks under boundary reference

Disturbance is a pervasive source of uncertainty in most applications. This paper presents model reference adaptive control (MRAC) laws for uncertain multiagent networks with a disturbance rejection capability. The algorithms proposed can also be viewed as the extension of the robust model reference adaptive control (MRAC) laws with disturbance rejection recently derived for systems described by parabolic and hyperbolic partial differential equations (PDEs) with spatially-varying parameters under distributed sensing and actuation to heterogeneous multiagent networks characterized by parameter uncertainty. The latter extension is carried out using partial difference equations (PdEs) on graphs that preserve parabolic and hyperbolic like cumulative network behavior. Unlike in the PDE case, only boundary input is specified for the reference model. The algorithms proposed directly incorporate this boundary reference input into the reference PdE to generate the distributed admissible reference evolution profile followed by the agents. The agent evolution thus depends only on the interaction with the adjacent agents, making the system fully decentralized. Numerical examples are presented as well. The resulting PdE MRAC laws inherit the robust linear structure of their PDE counterparts.     

Partial difference equation based model reference control of a multiagent network of underactuated aquatic vehicles with strongly nonlinear dynamics

In a recent work, the authors presented an extension of robust model reference adaptive control (MRAC) laws for spatially varying partial differential equations (PDEs) proposed by them earlier for the decentralized adaptive control of heterogeneous multiagent networks with agent parameter uncertainty using the partial difference equations (PdEs) on graphs framework. The examples provided demonstrated the capabilities of this approach under the assumption that individual vehicles executing coordinated maneuvers were fully actuated and characterized by linear dynamics. However, detailed models for autonomous vehicles–whether terrestrial, aerial, or aquatic–are often underactuated and strongly nonlinear. Using this approach, but assuming the plant parameters to be known, this work presents the model reference (MR) control laws without adaptation for the coordination of underactuated aquatic vehicles modeled individually in terms of strongly nonlinear dynamic equations arising from ideal planar hydrodynamics. The case of unknown plant parameters for this class of underactuated agents with complex dynamics is an open problem. The paper is based on an invited talk on adaptive control presented at the 2008 World Congress of Nonlinear Analysts.     

Numerical simulation and linear well-posedness analysis for a class of three-phase boundary motion problems

We investigate the linear well-posedness for a class of three-phase boundary motion problems and perform some numerical simulations. In a typical model, three-phase boundaries evolve under certain evolution laws with specified normal velocities. The boundaries meet at a triple junction with appropriate conditions applied. A system of partial differential equations and algebraic equations (PDAE) is proposed to describe the problems. With reasonable assumptions, all problems are shown to be well-posed if all three boundaries evolve under the same evolution law. For problems involving two or more evolution laws, we show the well-posedness case by case for some examples. Numerical simulations are performed for some examples.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

输入通道有干扰多变量MRAC系统全局稳定化控制

对具有未建模动态且输入通道存在干扰的动态不确定多输入多输出(MIMO)模型参考自适应控制(MRAC) 系统，仅应用系统的输入输出量测数据给出了一种变结构模型跟踪控制器设计机制.通过辅 助信号和带有记忆功能的正规化信号,并适当选择控制器参数, 所提出的变结构控制 (VSC)能保证闭环系统的全局稳定性,且跟踪误差可调整到任意小.     

Coupled McKean–Vlasov diffusions: wellposedness,propagation of chaos and invariant measures

In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the model. We prove the wellposedness of the SDEs, the propagation of chaos of the particle system, and the existence and (non)-uniqueness of invariant measures of the PDE system.     

Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions , and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.     

Existence of local‐in‐time classical solutions of a model of flow in a bounded elastic tube

This paper studies the local‐in‐time existence of classical solutions to a hyperbolic system with differential boundary conditions modelling a flow in an elastic tube. The well‐known Lax transformations used for obtaining a priori estimates for conservation laws are difficult to apply here because of the inhomogeneity of the partial differential equations (PDE). Rather, our method relies on a suitable splitting of the original system into the PDE part and the ODE part, the characteristics for the PDE part, appropriate modulus of continuity estimates and a compactness argument. Copyright © 2016 John Wiley & Sons, Ltd.     

Preconditioning for boundary control problems in incompressible fluid dynamics

PDE‐constrained optimization problems arise in many physical applications, prominently in incompressible fluid dynamics. In recent research, efficient solvers for optimization problems governed by the Stokes and Navier–Stokes equations have been developed, which are mostly designed for distributed control. Our work closes a gap by showing the effectiveness of an appropriately modified preconditioner to the case of Stokes boundary control. We also discuss the applicability of an analogous preconditioner for Navier–Stokes boundary control and provide some numerical results.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

Designing SISO observers with unknown input

In this paper we investigate the observer design for Single-InputSingle-Output (SISO) systems with unknown input. We find thatthe design conditions proposed by several authors in the pastlead to unity relative degree of the plant and zeros of theplant in the open left half of the complex plane, for SISO systems.In other words an observer for a SISO plant with unknown inputcan be designed if and only if the plant is of unity relativedegree and the numerator of the transfer function is Hurwitz.These conditions have an interesting consequence in the contextof model reference adaptive control (MRAC) for the case of relativedegree one, in the sense that strictly real positive (SPR) conditionfor the model reference is not needed.     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

A dissipative model for hydrogen storage: existence and regularity results

We prove global existence of a solution to an initial and boundary‐value problem for a highly nonlinear PDE system. The problem arises from a thermo‐mechanical dissipative model describing hydrogen storage by use of metal hydrides. In order to treat the model from an analytical point of view, we formulate it as a phase transition phenomenon thanks to the introduction of a suitable phase variable. Continuum mechanics laws lead to an evolutionary problem involving three state variables: the temperature, the phase parameter and the pressure. The problem thus consists of three coupled partial differential equations combined with initial and boundary conditions. The existence and regularity of the solutions are here investigated by means of a time discretization—textita priori estimates—passage to the limit procedure joined with compactness and monotonicity arguments. Copyright © 2010 John Wiley & Sons, Ltd.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

关于发展方程的守恒率的一个标记

本文使用微分代数的技巧，研究了发展方程的守恒率与对应的行波所满足方程的首次积分之间的关系．通过文本给出的结果，我们研究了Burgers方程和Burgers—KdV方程的可积性，证明了这两类方程都只有一个守恒率．利用本文给出的方法，可以通过常微分方程的研究方法来研究某些非线性发展方程．     

Soil water redistribution and extraction flow models: Conservation laws

We determine all the nontrivial conservation laws for soil water redistribution and extraction flow equations which are modelled by a class of (2+1) nonlinear evolution partial differential equations with three arbitrary elements. It is shown that for arbitrary elements in the model equation there exist trivial conservation laws. We point out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries.     

An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems

We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is described by the dynamics of a pendulum that is fixed to a trolley moving along the crane beam. The resulting problem serves as a case study for optimal control of fully coupled partial and ordinary differential equations (ODEs). This particular type of coupled systems arises from many applications involving mechanical multi-body systems. We motivate the coupled ODE-PDE model, show its analytical well-posedness locally in time and examine the corresponding optimal control problem numerically by means of a projected gradient method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) update.