Analysis of an iterative scheme for approximate regulation for nonlinear systems

This paper concerns the analysis of an iterative scheme delivering approximate control laws for the tracking regulation problems for nonlinear systems. The procedure can be applied to finite‐ and infinite‐dimensional systems, and the underlying methodology derives from the geometric methods, which have been developed for both linear and nonlinear systems. In the nonlinear case, the main tool is the center manifold theorem. Indeed, in the geometric methodology, under the assumption that the signals to be tracked are generated by a finite‐dimensional exo‐system, the desired control is obtained by solving a pair of operator equations called the regulator equations. In this paper, we extend the concept of regulator equations to what we refer to as the dynamic regulator equations. Just as it is generally quite difficult to solve the regulator equations, it can be equally difficult to solve the dynamic regulator equations. As the authors have already shown in the linear case, a straightforward attempt to solve the dynamic regulator equations leads to a singular system, which can be regularized to obtain an iterative scheme that provides approximate control laws that provide accurate tracking with very a small tracking error after only a couple of iterations. In this paper, we generalize the iterative scheme to nonlinear systems and provide error estimates for the first 3 iterations. Both finite‐ and infinite‐dimensional examples are given to validate the estimates. We comment that the method has also been applied to a wide range of nonlinear distributed parameter examples described in the references.     

双曲线型分布参数系统边界控制下的干扰解耦

针对干扰输入对系统输出的影响, 研究双曲线型分布参数系统在边界控制下的干扰解耦问题. 首先应用有界控制算子等价转换方法, 获取边界控制下双曲线型分布参数系统的有界拓展形式; 然后通过伽辽金近似法将有界增广系统转换成有限维系统; 接着对等价有限维系统进行干扰解耦分析, 推导出系统可干扰解耦的充分条件; 最后通过数值分析和仿真图例表明了所给出条件的有效性.

     

Some new applications of Russell’s principle to infinite dimensional vibrating systems

The aim of this work is to highlight the interest of a by now classical methodology, commonly called Russell’s principle, in proposing new control strategies and estimates for infinite dimensional vibrating systems. After describing (with complete proofs) a particular version, of interest for our work, of Russell’s principle, we consider two main applications. The first one, which mainly contains results which are new, studies the approximation of a class of boundary control systems by systems with controls distributed in an open set (internal controls), with the support shrinking to the boundary. These approximations are interesting since for the approximating systems we have bounded input operators, which makes easier the use of many control theoretic tools. The second application concerns the approximation of exact controls for infinite dimensional systems using their projections on finite dimensional spaces. We propose here an alternative, based on Russell’s principle, of the existing approximation methods, often based on inverting the Gramian.     

Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

State-space models for infinite-dimensional systems

Distributed effects are present in almost all physical systems. In some cases these can be safely ignored but there are many interesting problems where these effects must be taken into account. Most infinite dimensional systems which are important in control theory are specifiable in terms of a finite number of parameters and hence are, in principle, amenable to identification. The state-space theory of infinite dimensional systems has advanced greatly in the last few years and is now at a point where real applications can be contemplated. The realizability criteria provided by this work can be employed effectively in the first step of the identification procedure, i.e., in the selection of an appropriate infinite dimensional model. We show that there exists a natural classification of nonrational transfer functions, which is based on the character of their singularities. This classification has important implications for the problem of finite dimensional approximations of infinite dimensional systems. In addition, it reveals the class of transfer functions for which there exist models with spectral properties closely reflecting the properties of the singularities of the transfer functions. The study of models with infinitesimal generators having a connected resolvent sheds light on some open problems in classical frequency response methods. Finally, the methods used here allow one to see the finite dimensional theory itself more clearly as the result of placing it in the context of a larger theory.     

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

The paper focuses on the numerical approximation of discrete fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed model of discrete fractional order system is introduced. Then, resorting the nabla discrete Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.     

Ordered sequential filtering for finite dimensional systems with distributed observations

This short paper treats the topic of suboptimal linear estimation for systems which have a finite dimensional state vector but a distributed set of observations. Such systems are of some practical interest, and are indeed considerably easier to work with than most distributed parameter systems. The method proposed here is that a filter structure of linear form, driven by a statistic which is a weighted integral of the distributed observation, be used to generate estimates of the state of the system. The filter parameters, and the weighting function for the integral are selected so that the estimate is unbiased and results in minimum mean-square error, subject to the structural constraints.     

Optimal control of switched distributed parameter systems with spatially scheduled actuators

In many areas of control there are gaps between the existing theory and applications. This is more so in hybrid infinite dimensional systems and in particular hybrid systems in which both the actuator and the controller are switched. The main objective of this paper is to start filling in one of these gaps. We present a theoretical formulation and provide methodologies for implementing optimal and switching policies of spatially scheduled actuators for a class of distributed parameter systems (DPS). The optimization method employed is based on finite horizon LQR optimal control. Well posedness and optimality, pertaining to the switching policies of spatially scheduled actuators, are presented and proven. Tutorial examples motivated by thermal manufacturing applications along with extensive simulation results of the proposed actuator-plus-controller switching scheme are presented.     

Dynamical system design from a control perspective: finite frequency positive-realness approach

Nonminimum phase zeros are well known to limit the best achievable control performance when the control gain is allowed to be arbitrarily high. On the other hand, the phase crossover appears to be a limiting factor for performance when high-gain controllers are not allowed. In particular, the positive-realness in a finite frequency range seems crucial for achieving good performance in the presence of control constraints. This paper will first give multiple reasons to support this conjecture, and then develop a systematic method for designing mechanical systems to achieve the finite frequency positive-real (FFPR) property. Specifically, we present a state-space characterization of the FFPR property by generalizing the well known Kalman-Yakubovich-Popov lemma to deal with a class of frequency domain inequalities that are required to hold within a finite frequency interval. The result is further extended for uncertain systems to give a sufficient condition for satisfaction of a robust FFPR property. The (nominal) FFPR result is interpreted in the time-domain in terms of input/output signals. Finally, we show that certain sensor/actuator placement problems to achieve the FFPR property can be reduced to finite dimensional convex problems involving linear matrix inequalities. The method is applied to the shape design of a swing-arm for magnetic storage devices with the objective of maximizing the control bandwidth achievable with a limited actuator power.     

Parameter Reconstruction for Biochemical Networks Using Interval Analysis

In recent years, the modeling and simulation of biochemical networks has attracted increasing attention. Such networks are commonly modeled by systems of ordinary differential equations, a special class of which are known as S-systems. These systems are specifically designed to mimic kinetic reactions, and are sufficiently general to model genetic networks, metabolic networks, and signal transduction cascades. The parameters of an S-system correspond to various kinetic rates of the underlying reactions, and one of the main challenges is to determine approximate values of these parameters, given measured (or simulated) time traces of the involved reactants.Due to the high dimensionality of the problem, a straight-forward optimization strategy will rarely produce correct parameter values. Instead, almost all methods available utilize genetic/evolutionary algorithms to perform the non-linear parameter fitting. We propose a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. The proposed method can in principle be applied to any system of finitely parameterized differential equations, and, as we demonstrate, yields encouraging results for low dimensional S-systems.     

Analyzing Asynchronous Pipeline Schedules

Asynchronous pipelining is a form of parallelism that is useful in both distributed and shared memory systems. We show that asynchronous pipeline schedules are a generalization of both noniterative DAG (directed acyclic graph) schedules as well as simpler pipeline schedules, unifying these two types of scheduling. We generalize previous work on determining if a pipeline schedule will deadlock, and generalize Reiter's well-known formula for determining the iteration interval of a deadlock-free schedule, which is the primary measure of the execution time of a schedule. Our generalizations account for nonzero communication times (easy) and the assignment of multiple tasks to processors (nontrivial). A key component of our generalized approach to pipeline schedule analysis is the use of pipeline scheduling edges with potentially negative data dependence distances. We also discuss implementation of an asynchronous pipeline schedule at runtime; show how to efficiently simulate pipeline execution on a sequential processor; define and derive bounds on the startup time of a schedule, which is a secondary schedule performance measure; and describe a new algorithm for evaluating the iteration interval formula.     

一类分布参数系统的反馈能稳性

研究了一类分布参数系统在有限维输出反馈下的指数能稳性,用构造有限维观测输出反馈控制器的方法,得到这一类系统反馈指数能稳的充分条件.     

Invertibility of switched nonlinear systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.     

Computable convergence bounds of series expansions for infinite dimensional linear-analytic systems and application

This paper deals with the convergence of series expansions of trajectories for semi-linear infinite dimensional systems, which are analytic in state and affine in input. A special case of such expansions corresponds to Volterra series which are extensively used for the analysis, the simulation and the control of weakly nonlinear finite dimensional systems. The main results of this paper give computable bounds for both the convergence radius and the truncation error of the series. These results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality. They are available for distributed and boundary control systems. As an illustration, these results are applied to an epidemic population dynamic model. In this example, it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues. The relevance of the method is illustrated by simulations.     

On the computation of the real Hurwitz-stability radius

Recently Qiu et al. (1995) obtained a computationally attractive formula for the evaluation of the real stability radius. This formula involves a global maximization over frequency. Here, for the Hurwitz stability case, we show that the frequency range can be limited to a certain finite interval. Numerical experimentation suggests that this interval is often reasonably small     

Identification of finite dimensional models of infinite dimensional dynamical systems

The identification of finite dimensional discrete-time models of deterministic linear and nonlinear infinite dimensional systems from pointwise observations is investigated. The input and output observations are used to construct finite dimensional approximations of the solution and the forcing function which are expanded in terms of a finite element basis. An algorithm to determine a minimal basis to approximate the data is introduced. Subsequently, the resulting coordinate vectors are used to identify a finite dimensional discrete-time model. Theoretical results concerning the existence, stability and convergence of the finite dimensional representation are established. Numerical results involving identification of finite dimensional models for both linear and nonlinear infinite dimensional systems are presented.     

Convergence of discrete-time Kalman filter estimate to continuous time estimate

This article is concerned with the convergence of the state estimate obtained from the discrete-time Kalman filter to the continuous time estimate as the temporal discretisation is refined. The convergence follows from Martingale convergence theorem as demonstrated below; however, surprisingly, no results exist on the rate of convergence. We derive convergence rate estimates for the discrete-time Kalman filter estimate for finite and infinite dimensional systems. The proofs are based on applying the discrete-time Kalman filter on a dense numerable subset of a certain time interval [0, T].     

A PI-controller for distributed delay systems

A finite dimensional control theory is developed for the design of a proportional plus integral control law for tracking step command inputs in general autonomous time-lag systems with distributed heredity in state, control and output variables. With respect to an ordinary differential equation describing the delay system unstable modes, an auxiliary tracking problem is posed. This auxiliary problem derives its importance from the fact that its solution directly yields a solution to the original tracking problem. This point of contact with a finite dimensional system permits the application of well-tested ordinary systems tracking theory to the tracking problem in time-lag systems.