China
Africa
Explore chapters and articles related to this topic
Introduction
Published in Nayef Ghasem, Modeling and Simulation of Chemical Process Systems, 2018
Models where the dependent state variables have no spatial location obligation and are uniform over the entire system are called lumped parameter systems where time is the only independent variable. For lumped parameter systems, the process state variables are uniform over the entire system; that is, each state variable does not depend on the spatial variables (i.e., x,y, and z in Cartesian coordinates) but only on time t. In this case, the balance equation is written over the whole system using macroscopic modeling. By contrast, when the process dependent state variables vary with time and position, the process is measured as distributed parameter system. Modeling of distributed parameter systems must consider the variation of these variables throughout the entire system.
Intermediate Numerical Integration
Published in Harold Klee, Randal Allen, Simulation of Dynamic Systems with MATLAB® and Simulink®, 2018
Dynamic systems involving variables that exhibit both spatial and temporal variations are modeled by partial differential equations and referred to as distributed parameter systems. The introductory section in Chapter 1 cited the example of a room temperature T(x, y, z, t) that varies as a function of the point coordinates (x, y, z) as well as time t. Analytical solutions of partial differential equation models subject to various boundary conditions are rare in all but the simplest of examples. Numerical solutions are based on a partitioning of the entire volume and surface areas within the system into meshes comprising finite-sized triangular elements with interior and exterior nodes at the vertices. Difference equations, sometimes numbering in the hundreds of thousands depending on the size and shape of the finite elements, are written for the dependent variable(s) at a subset of the nodes. Accurate approximations to the continuous solutions of the partial differential equation models are possible using this “finite element analysis” approach. Examples include the temperature distribution and heat flows from irregular-shaped cooling surfaces, structural analysis, fluid dynamics, and so forth.
Stochastic Optimal Control for Estuarine Management
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
Recently, some optimization methods have been applied to optimal control problems such as groundwater management problems (Wanakule, 1984; Wanakule et al., 1985, 1986; Chang et al., 1992; Culver and Shoemaker, 1992, 1993; Whiffen and Shoemaker, 1993), reservoir operation (Murray and Yakowitz, 1979; Unver, 1987; Unver et al., 1987; Carriaga and Mays, 1994), water distribution systems design and operation (Lansey, 1987; Lansey and Mays, 1989; Brion, 1990; Brion and Mays, 1989), and estuarine system management (Bao, 1992; Bao and Mays, 1994a, 1994b; Li and Mays, 1995). This approach is more general compared with the classic approach, in that it considers the distributed-parameter system as a simulator or a function. The simulator is usually a numerical model for solving the partial differential equations. Because the distributed-parameter system is considered as a function, optimization methods can be used directly. Generally speaking, this approach consists of two different methods in that the optimization methods are different. One method uses the GRG2 optimization code (Lasdon and Waren, 1989) to “communicate” with a simulator (or a numerical model for solving the partial differential equation). Another method is based on the differential dynamic programming (Jacobson and Mayne, 1970; Yakowitz and Brian Rutherford, 1984). The first method is more general than the second one because the differential dynamic programming requires that the performance index be a summation of functions over the time.
Adaptive bounded bilinear control of coupled first-order 1-D hyperbolic PDEs and infinite ODEs with unknown time-varying source term
Published in International Journal of Control, 2022
In the modern industry, many complex processes, such as flexible structures, chemical reactors, and diesel engines belong to the realm of distributed parameter systems (DPSs). They are systems which states are space and time-dependent and thus their dynamics can be described by integral and partial differential equations (PDEs). These PDEs are sometimes coupled with ODEs to portray, for example, transport delay or heat transport in materials with memory (see e.g. L. G. Zhang et al., 2019 and examples therein). The coupling of PDE-ODE systems occurs either at the boundary (like in Cai & Krstic, 2016; Krstic, 2009) or in the PDE domain (Karafyllis et al., 2015; Mohammadi et al., 2015). There are two classes of in-domain coupled PDE-ODE systems. Either the PDE parameters are described by ODEs or the PDE is combined with an infinite-dimensional ODE. This latter occurs when the infinite ODE originates from a PDE with a vanishing propagation speed (see Belkhatir et al., 2019; Mechhoud & Laleg-Kirati, 2018).
Optimal control and duality-based observer design for a hyperbolic PDEs system with application to fixed-bed reactor
Published in International Journal of Systems Science, 2021
Partial differential equations (PDEs) represent the best way to model many dynamical distributed parameter systems in the area of science and engineering. Control problems associated with those systems can be investigated using two types of approaches: discretisation-based approaches that are used to convert the PDEs model into a finite-dimensional system (see e.g. French & King, 1991; Lasiecka, 1980; McKnight & Bosarge, 1973). The main advantage of these approaches is the fact that control theory of finite-dimensional systems is very well developed, however a main disadvantage of these approaches is the fact that they neglect the infinite-dimensional character and also they neglect the distributed feature of the original system in case the number of discretisation points is not high enough. This may lead to the loss of some of the controller performances. On the other hand, infinite-dimensional representation is the second type of approaches to control distributed parameter systems. These approaches reformulate a system of PDEs by using appropriate operators as an abstract differential equation in a state-space of infinite dimension (Aksikas et al., 2017, Bensoussan et al., 2007; Curtain & Zwart, 1995; Xu & Dubljevic, 2016). The fact that these approaches is only a reformulation of the system and not an approximation, then the distributed nature of the system is preserved. In this paper, infinite-dimensional state space representation is used to design both a compensator and a controller for a class of hyperbolic PDEs.
A unified Lyapunov-based design for a dynamic compensator of linear parabolic MIMO PDEs
Published in International Journal of Control, 2021
Most systems arising in engineering applications are spatiotemporal in nature so that their behaviour must depend on time as well as spatial position, for example, some mechanical systems related with heat flows, fluid flows, elastic wave, flexible structure (Balas, 1978; He, Ge, How, & Choo, 2014), chemical engineering (Christofides, 2001; Li & Qi, 2010; Ray, 1981), biodynamics (Banks, 1975), ecosystems, and social systems, etc. Such systems are named as distributed parameter systems and are in general modelled by partial differential equations (PDEs). Commonly, PDEs are equipped with several actuators and sensors in case where some form of automatic control is required for these engineering systems. PDEs with more than one actuation control input and more than one observation output may be considered as multi-input-multi-output (MIMO) PDEs. Hence, it is appealing to design appropriate compensators exponentially stabilising the MIMO PDEs.