Sensor and controller location problems for distributed parameter systems

Optimal location of point sensors and controllers in stochastic regulator problems for evolution equations is considered using a semigroup approach and the separation principle. The existence of an optimal location is established under very general hypotheses. The theory is illustrated by specializing to a diffusion equation and some computational results are presented.     

Multivariable tuning regulators for unknown systems

The problem of determining a multivariable robust PI-controller for an unknown linear multivariable stable plant is discussed. A method for constructing a multivariable P-controller, which uses interactions of the plant, is developed. These interactions are detected by observing the output of the plant subject to step-inputs. The developed P-controller form together with the robust multivariable I-controller (Davison, 1976) a robust multivariable PI-controller, the construction of which is also based on step-responses. The controllers are applied to a laboratory-scaled concentration-flow process.     

Asymptotic methods in the optimal control of distributed systems

A short review of the various problems which arise in connection with the use of asymptotic methods in the optimal control of distributed systems is presented. We consider the cases when the asymptotic analysis comes from the state equation, or from the cost function, or both and also when the state equation is defined in perturbed domains.     

Robust control and tuning problem for distributed parameter systems

In this paper robust multivariable controllers for parabolic distributed parameter systems will be discussed. The purpose of a robust controller is to achieve output regulation, disturbance rejection and insensitivity against some perturbations in the system's and controller's parameters. The robust controller consists of two parts: the unstable servo-compensator and the stabilizing compensator. The servo-compensator will be fixed on the basis of the spectrum of the reference and disturbance signals. The purpose of the stabilizing compensator is to stabilize the extended unstable system that consists of the stable plant and the servo-compensator. In this paper it is proved that the stabilizing compensator can be decomposed into a scalar gain and a matrix gain. A simple sufficient condition for finding stabilizing matrix gains will be given and a straightforward way to compute the gains will be presented. The proposed method is practical in the sense that the dimension of the controller is finite and small, output feedback is used and tuning the controller can be done with the information that can be measured from the stable plant with input-output measurements. To the authors’ knowledge, the main results are new even for finite-dimensional systems.     

Trajectory planning for quasilinear parabolic distributed parameter systems based on finite-difference semi-discretisations

In this contribution, a flatness-based approach is considered for the solution of the trajectory planning problem for quasilinear parabolic distributed parameter systems (DPS) by making use of finite-difference semi-discretisations. It is shown that the method yields solutions which are equivalent to results known from the infinite-dimensional trajectory planning for a certain class of quasi-linear parabolic DPS. Furthermore, the methodology being proposed can also be applied to systems with general analytic nonlinearities. As analytical convergence results are not available in this case, a numerical test criterion for the convergence behaviour is suggested.     

Adaptive open loop control for a class of distributed parameter systems

The real time implementation of closed loop control laws obtained by the minimization of a quadratic criterion with finite time intervals is a difficult problem, if we consider computing time and the eventual determination of the state variables. It would seem useful therefore to define, for complex systems such as distributed parameter systems, a control type which unites the advantages of open loop control, with its relative simplicity of implementation, and those of closed loop control, with its ability to reduce perturbations. It is for this reason that we use the principle of adaptive open loop control. In this paper, we describe the principles of such a method and results obtained from one example studied by numerical computation or hybrid simulation in real time.     

Frequency domain analysis and robust control design for an ideal flexible beam

We study the problem of designing a feedback controller for a highly flexible Euler-Bernoulli beam using a distributed-parameter H^∞-method. Employing skew Toeplitz theory, we derive the H^∞-optimal controller for a weighted mixed sensitivity design for the beam. Based on the structure of the optimal controller we obtain suboptimal, finite-dimensional, linear, time-invariant controllers. With this approach we are able to include the added difficulties of a pure time delay and a noncollocated actuator/sensor pair directly into the design process.     

A robust multi-model predictive controller for distributed parameter systems

In this work a robust nonlinear model predictive controller for nonlinear convection-diffusion-reaction systems is presented. The controller makes use of a collection of reduced order approximations of the plant (models) reconstructed on-line by projection methods on proper orthogonal decomposition (POD) basis functions. The model selection and model update step is based on a sufficient condition that determines the maximum allowable process-model mismatch to guarantee stable control performance despite process uncertainty and disturbances. Proofs on the existence of a sequence of feasible approximations and control stability are given.Since plant approximations are built on-line based on actual measurements, the proposed controller can be interpreted as a multi-model nonlinear predictive control (MMPC). The performance of the MMPC strategy is illustrated by simulation experiments on a problem that involves reactant concentration control of a tubular reactor with recycle.     

退化高阶抛物型分布参数系统的迭代学习控制

研究一类高阶分布参数系统的迭代学习控制问题,该类系统由退化高阶抛物型偏微分方程构成.根据系统所满足的性质,基于P型学习算法构建得到迭代学习控制器.利用压缩映射原理,证明该算法能使得系统的输出跟踪误差于L~2空间内沿迭代轴方向收敛于零.最后,仿真算例验证了算法的有效性.     

Singular perturbation analysis of a receding horizon controller

A new receding horizon approach to the linear regulator problem is presented. At each instant of time, the control strategy is based on the minimization of a quadratic performance index extending over an interval of fixed length into the future. Connections with standard results are discussed and, under general assumptions, the closed-loop system is shown to be asymptotically stable. A potential advantage of our method lies in the fact that a singular perturbation analysis is well posed. This indicates that, for plants containing slow and fast modes, the design procedure can be split up into two stages. The resulting composite control exhibits attractive engineering properties. Use and effectiveness of the proposed techniques are demonstrated by investigating a practical example.     

Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems

This paper studies the parameter identification problem of nonlinear abstract parabolic distributed parameter systems via variational method ^[1]. Based on the fundamental optimal control theory and the transposition method studied in ^[2], the existence of optimal parameter is proved, and the necessary condition for the optimal parameter is established.     

Inverse systems for reproducing linear functions of inputs

Most of the previous studies on inverse systems have treated only the case where all components of the input of the original system are recoverable. However, even when not all components are recoverable, there is still a possibility of reproducing part of the input. As a generalization of the inverse of a linear time-invariant dynamical system to include such cases, ‘α-integral F-inverse’ is proposed in this paper which reproduces the αth integral of a linear function Fu of the input vector u. A necessary and sufficient condition for the existence of an α-integral F-inverse is derived. A construction procedure of such an inverse is also given along with a numerical example.     

Modal consensus observers for distributed parameter systems

This article proposes a new type of a consensus protocol for the synchronization of distributed observers in systems governed by parabolic partial differential equations. Addressing the goal of sharing useful information among distributed observers, it delves into the details governing the modal decompositions of distributed parameter systems. Assuming that two different groups of sensors are available to provide process information to the two distributed observers, the proposed modal consensus design ensures that only useful information is transmitted to the requisite modal components of each of the observers. Without any consensus protocol, the observers capture different frequency content of the spatial process in differing degrees, as it relates to the concept of modal observability. Their modal components exhibit different learning behavior toward the process state. In the extreme case, it turns out that certain modal components of the distributed observers occasionally behave as naïve observers. To ensure that, both collectively and modal componentwise, the observers agree both with the process state and with each other, a modal component consensus protocol is proposed. Such a consensus protocol is mono-directional and provides only useful information necessary to the appropriate modal component of the distributed filters that behaves as a naïve modal observer. This protocol, when abstracted and applied to different state decompositions can be viewed as mono-directional projections of information transmitted and received by the participating distributed observers. Detailed numerical studies of advection PDE in one and two spatial dimensions are included to elucidate the details of the proposed modal consensus observers.     

On the 2-induced norm of a sampled-data system

Formulas are given for the norms of SG and GH: G is a linear continuous-time system, S an ideal sampler, and H a zero-order hold; the signal spaces are ₂(−∞, ∞) (continuous-time) and l₂(Z) (discrete-time). These formulas are then applied to problems of uniformly optimal control of sampled-data systems.     

On the optimal control of two classes of non-linear distributed parameter systems†

An optimal control problem, for two classes of non-linear distributed parameter systems, is formulated. Sufficient conditions on the optimal control are derived. These conditions form a two-point boundary value problem of partial differential equations.     

Finite difference implementation of distributed parameter filters

The numerical problems associated with the full three-dimensional (spatial) distributed parameter filters demand a novel technique that maintains accuracy and efficiency for the complex partial differential Riccati equation governing the filter variance. The filters can be successfully implemented by using numerical operators that can, in various ways, be factored. Use of the finite difference method having this property increases the accuracy and efficiency of the implementation.

This paper outlines a time-splitting technique for the numerical implementation of distributed parameter filters. The paper provides rationales to the square root and U-D factorizations of the distributed parameter filter by means of the time-splitting finite difference method.     

Controller design for parabolic distributed parameter systems using finite integral transform techniques

A procedure is presented for designing a control system for distributed parameter systems of parabolic type based on the reduced-order decoupled state-space model obtained by a finite integral-transform technique. A Kalman filter is used as an observer to estimate the state variables, and state feedback control is performed. The method was applied to a one-dimensional heat conduction process and a moving bed adsorber. Both state estimation and control performances were satisfactory in spite of the model and parameter uncertainties. Following this controller design approach, the searching algorithms for the optimal sensors' and the optimal actuators' allocation problems were solved. These algorithms were applied to a one-dimensional heat conduction process in order to confirm the state estimator and controller performance. The fastest state estimation could be achieved by assigning the sensors at the optimal locations and the desired state distribution was realized with a few actuators located at the optimal positions.     

Transfer functions of distributed parameter systems: A tutorial

In this tutorial article the rich variety of transfer functions for systems described by partial-differential equations is illustrated by means of several examples under various boundary conditions. An important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations, properness, relative degree and minimum phase. It is sometimes possible to design a controller using the irrational transfer function, and several such techniques are outlined. More often, the irrational transfer function is approximated by a rational one for the purpose of controller design. Various approximation techniques and their underlying theory are briefly discussed.     

Some recent applications of distributed parameter systems theory—A survey

A survey of some recent applications of distributed parameter systems theory is presented. The practical areas discussed range from process control problems in an industrial plant to the identification, monitoring and control of air and water quality in our environment. Some new, promising areas of application are discussed along with suggestions for future research emphasis.