Low-dimensional Control Research Articles

SECOND ORDER SLIDING MODE ADAPTIVE NEUROCONTROL FOR ROBOT ARMS WITH FINITE TIME CONVERGENCE

In this paper we present a low dimensional adaptive neural network controller for robot manipulators with fast convergence of tracking error. Its novelty lies in the low dimensional network, smooth control input and very fast convergence that reduce the computational cost that face the problem of over parameterization. The control strategy is based on a second order sliding surface which drives the controller and the online computation of weights with a chattering-free control output. Furthermore, a time base generator induces well-posed finite time convergence of tracking errors for any initial condition. We validate our approach including experimental results obtained in a planar 2 dgf manipulator.

Solution of low-dimensional constrained model predictive control problems

Large benefits are possible by utilizing the solution of the constrained optimization problem involved in model predictive control. For a special case of these problems, the solution can be obtained relatively easily from its relationship with the unconstrained optimum. In this paper, a visualization of the relationship between the constrained and unconstrained optimum is presented. Based upon this relationship, a method for finding the constrained optimum is proposed that is suitable for low-dimensional control systems. A comparison with a linear programming formulation on 2 × 2 and 3 × 3 problems shows that the computational effort can be 10–35 times lower. For such processes, the proposed approach may allow one to avail the benefits of optimization by using the small process control systems already present in many plants.

Inertial Manifolds and Gevrey Regularity for the Moore-Greitzer Model of an Axial-Flow Compressor

In this paper, we study the regularity and long-time behavior of the solutions to the Moore-Greitzer model of an axial-flow compressor. In particular, we prove that this dissipative system of evolution equations possesses a global invariant inertial manifold, and therefore its underlying long-time dynamics reduces to that of an ordinary differential system. Furthermore, we show that the solutions of this model belong to a Gevrey class of regularity (real analytic in the spatial variables). As a result, one can show the exponentially fast convergence of the Galerkin approximation method to the exact solution, an evidence of the reliability of the Galerkin method as a computational scheme in this case. The rigorous results presented here justify the readily available low-dimensional numerical experiments and control designs for stabilizing certain states and traveling wave solutions for this model.

Low-dimensional control: tonus (1963).

In 1963, an article on "Tonus" (tone), written by Nikolai A. Bernstein and Yakov M. Kots, appeared in the second edition of the Bols'aja Medicinskaja Enciclopedija [Grand Medical Encyclopedia]. The paper is now published for the first time in the English language, with Mark L. Latash as translator. In accordance with then contemporary neurophysiology and neuropsychology, the paper presented "tone" as a graded phenomenon (as opposed to all-or-none), serving to prepare the segmental level for phasic contractions. Influenced by Granit and Matthews, the authors proposed that the suprasegmental level controls the threshold and the slope of the stretch reflex. In their introduction to the present edition, the editors understand this proposal in the context of low-dimensional control, that is, control in terms of one or a few variables (as opposed to central commands specifying all the details). Selected episodes from the history of low dimensional control and its logical counterpart, spinal intelligence, are used to illustrate how difficult these ideas were to accept. As so often in new scientific developments, confusion was the rule, and in this respect the paper on "Tonus" is no exception. In the epilogue, Kots gives his personal memories of the context in which the paper was written. At the time, he was working on "equitonometry" (equi-tono-metry), measuring tonic balance with gravity eliminated. Results of equitonometric research quite naturally led to the idea that suprasegmental centers control the threshold and the slope of the tonic stretch reflex. As Kots remembers, that was "no big deal".

Finite Difference Output Feedback Control of a Class of Distributed Parameter Processes

Low-dimensional nonlinear feedback control schemes for distributed parameter processes are addressed. The main design procedure involves the use of a feasible numerical method to establish a finite difference-based differential equation system and to utilize the pure nonlinear feedback design for achieving uniform output regulation. When the almost uniform input distribution at steady-state conditions can be planed, the controller reduction can be involved in the proposed control design guidelines. Based on the nonlinear inverse design procedure, the low-dimensional and approximate feedback control law is developed such that the stable and bounded output tracking can be guaranteed. With the aid of internal model control strategy, the low-dimensional output feedback control scheme can ensure the asymptotic output regulation. Finally, those control methodologies are shown to be robust and effective in controlling a nonisothermal tubular reactor at possibly realistic situations.

Wave suppression by nonlinear finite-dimensional control

Korteweg–de Vries–Burgers (KdVB) and Kuramoto–Sivashinsky (KS) equations are two nonlinear partial differential equations (PDEs) which can adequately describe motion of waves in a variety of fluid flow processes. We synthesize nonlinear low-dimensional output feedback controllers for the KdVB and KS equations that enhance convergence rate and achieve stabilization to spatially uniform steady states, respectively. The approach used for controller synthesis employs nonlinear Galerkin's method to derive low-dimensional approximations of the PDEs, which are subsequently used for controller synthesis via geometric control methods. The controllers use measurements obtained by point sensors and are implemented through point control actuators. The performance of the proposed controllers is successfully tested through simulations.

Finite-dimensional control of a flexible space structure: A new view based on input-output space

How to design a simple, finite-dimensional controller for complex, infinite-dimensional Flexible Space Structures (FSSs) with uncertain physical parameters is the main topic of this paper. The research shows that a simple controller that makes the controlled FSS system of high stable degree can be obtained by using only the input-output property instead of the poles-zeros property of the FSS system, which is a new result in the control of uncertain FSS systems. The main idea can be summarized as: (1) it is difficult, even impossible, to obtain convergent poles and zeros estimation for an uncertain FSS system, which is different from that in a finite-dimensional system; (2) there is an inherent relation between the input-output property and the poles-zeros property for the FSS system, which means that modelling and controlling, based on the input-output property, may guarantee the high authority control of the FSS system; (3) in the H sense and based on input-output space to make the model error as small as possible and the controller tolerate a model error as large as possible, will result in a very low dimensional controller.

Neurofuzzy state estimators and their applications

Neurofuzzy algorithms have been extensively developed in recent years for the real time/online identification of nonlinear a priori unknown dynamical processes. As with all rule base paradigms they suffer from the curse of dimensonality, restricting their practical use to low dimensional control problems. This paper shows how adaptive construction algorithms based on additive decomposition techniques can overcome this problem, to produce parsimonious neurofuzzy models which retain their transparency or interpretability. Not only does this approach extend the applicability of neurofuzzy algorithms, it also enables low complexity controllers and estimators to be derived. In this context neurofuzzy state estimators are derived, which automatically parameterise a Kalman filter for a process state estimate reconstruction from any input/output data source. This approach avoids pitfalls of the extended Kalman filter, and is optimal for local models. The paper discusses real world applications of this new theory of modelling and estimation to helicopter guidance, intelligent driver warning system, communication antennas, autonomous underwater vehicles, ship collision avoidance guidance, and an IFAC benchmark problem.

Low-dimensional control of the circular cylinder wake

Many wake flows exhibit self-excited flow oscillations which are sustained by the flow itself and are not caused by amplification of external noise. The archetypal example of a self-excited wake flow is the low Reynolds number flow past a circular cylinder. This flow exhibits self-sustained periodic vortex shedding above a critical Reynolds number. In general, control of such flows requires stabilization of many globally unstable modes; the present work describes a multiple-sensor control strategy for the cylinder wake which succeeds in controlling a simplified wake model at a Reynolds number above that at which single-sensor schemes fail.Representation of the flow field by a finite set of coherent structures or modes, which are extracted by proper orthogonal decomposition and correspond to the large-scale wake components, allows the efficient design of a closed-loop control algorithm. A neural network is used to furnish an empirical prediction of the modal response of the wake to external control forcing. This model avoids the need for explicit representation of the control actuator–wake interaction. Additionally, the neural network structure of the model allows the design of a robust nonlinear control algorithm. Furthermore the controller does not necessarily require velocity field information, but can control the wake using other quantities (for example flow visualization pictures) which characterize the structure of the velocity field. Successful control of a simplified cylinder wake model is used to demonstrate the feasibility of the low-dimensional control strategy.

Low-dimensional approximation and control of periodic solutions in spatially extended systems

Nonlinear model reduction is combined with numerical continuation and linear state-space control techniques to design regulators for periodic solutions in a spatially extended system. We address issues of construction and systematic evaluation of low-dimensional dynamic models using Galerkin projections on empirical orthogonal eigenfunctions (also known as proper orthogonal decomposition modes or Karhunen-Lo\`eve modes). The reduced order dynamical systems are used first to compute the open-loop bifurcation diagrams and then to design feedback controllers stabilizing unstable limit cycles. We outline the steps for discrete-time controller design and computational linear stability analysis of the resulting hybrid (continuous-discrete) closed-loop systems.

Neurofuzzy State Estimators and their Applications

Neurofuzzy algorithms have been extensively developed in recent years for the real time/online identification of nonlinear a priori unknown dynamical processes. As with all rule base paradigms they suffer from the curse of dimensionality, restricting their practical use to low dimensional control problems. This paper shows how adaptive construction algorithms based on additive and extended additive decomposition techniques can overcome this problem, to produce parsimonious neurofuzzy models which retain their transparency or interpretability. Not only does this approach extend the applicability of neurofuzzy algorithms, it also enables low complexity controllers, estimators to be derived. In this context neurofuzzy state estimators are derived which automatically parameterise a Kalman filter for a process state estimate reconstruction from any input/output data source. This approach avoids pitfalls of the extended Kalman filter, and is optimal for local models. The paper discusses real world applications of this new theory of modelling and estimation to helicopter guidance, intelligent driver warning system, communication antennas, autonomous underwater vehicles and an IFAC benchmark problem.

Self-organizing multi-resolution grid for motion planning and control.

A fully self-organizing neural network approach to low-dimensional control problems is described. We consider the problem of learning to control an object and solving the path planning problem at the same time. Control is based on the path planning model that follows the gradient of the stationary solution of a diffusion process working in the state space. Previous works are extended by introducing a self-organizing multigrid-like discretizing structure to represent the external world. Diffusion is simulated within a recurrent neural network built on this multigrid system. The novelty of the approach is that the diffusion on the multigrid is fast. Moreover, the diffusion process on the multigrid fits well the requirements of the path planning: it accelerates the diffusion in large free space regions while still keeps the resolution in small bottleneck-like labyrinths along the path. Control is achieved in the usual way: associative learning identifies the inverse dynamics of the system in a direct fashion. To this end there are introduced interneurons between neighboring discretizing units that detect the strength of the steady-state diffusion and forward control commands to the control neurons via modifiable connections. This architecture forms the Multigrid Position-and-Direction-to-Action (MPDA) map. The architecture integrates reactive path planning and continuous motion control. It is also shown that the scheme leads to population coding for the actual command vector.

Controller Reduction with Consideration of Closed-Loop Performance and Stability.

In this paper we consider controller reduction preserving both performance and stability of the closed-loop system. First, a quadratic cost is introduced which represents the performance of the closed-loop system formed by a linear multivariable plant and high-dimensional dynamic controller. Then the controller reduction problem is formulated as the problem of finding a fixed low-dimensional dynamic controller preserving stability and value of the quadratic cost. Secondly, by using the Lyapunov equation, the problem is converted into the convex optimization problem with constraint described by a linear matrix inequality (LMI), and the sufficient condition for the existence of such an optimal reduced controller is shown. Finally, an example illustrates the methodology.

Necessary conditions for distributed-parameter systems with controls of fewer variables than state variables

Necessary conditions are proved for multi-dimensional control systems, in which the state is a function of several independent variables in a fixed domain, while the controlu is a function of only some of the independent variables (that is,u is a lower-dimensional control). The cost functional can be taken in the usual Lagrange form of a multiple integral or in Mayer form. The state equations are a system of partial differential equations in explicit form with assigned boundary conditions, and constraints may be imposed on the values of the control variables.

Timbre Space as a Musical Control Structure

Research on musical timbre typically seeks representations of the perceptual structure inherent in a set of sounds that have implications for expressive control over the sounds in composition and performance. With digital analysis-based sound synthesis and with experiments on tone quality perception, we can obtain representations of sounds that suggest ways to provide low-dimensional control over their perceptually important properties. In this paper, we will describe a system for taking subjective measures of perceptual contrast between sound objects and using this data as input to some computer programs. The computer programs use multidimensional scaling algorithms to generate geometric representations from the input data. In the timbral spaces that result from the scaling programs, the various tones can be represented as points and a good statistical relationship can be sought between the distances in the space and the contrast judgments between the corresponding tones. The spatial representation is given a psychoacoustical interpretation by relating its dimensions to the acoustical properties of the tones. Controls are then applied directly to these properties in synthesis. The control schemes to be described are for additive synthesis and allow for the manipulation of the evolving spectral energy distribution and various temporal features of the tones. Tests of the control schemes have been carried out in musical contexts.* Particular emphasis will be given here to the construction of melodic lines in which the timbre is manipulated on a note-to-note basis. Implications for the design of human control interfaces and of software for real-time digital sound synthesizers will be discussed.

Perceptual-cognitive explorations of a toroidal set of free-form stimuli

A polar-coordinate analogue of Fourier synthesis generates organic-appearing “free forms” that can be continuously deformed along any desired number of difficult-to-verbalize dimensions. Since the dimensions are also circular, the forms correspond to points on the surface of a torus which, though conveniently finite, is free of bounding edges. Two experiments explore a particular two-dimensional set of 81 such forms. The first shows that perceived pair-wise similarities among the individual forms are well explained purely in terms of the distances among their corresponding points in the toroidal parameter space. The second, however, establishes that forms that tend to be grouped together as having the same cognitive interpretation define regions in parameter space that are variously shaped or even bimodal and, hence, that cannot be explained solely on the basis of the fixed set of pair-wise similarities. The stimuli appear to offer a novel combination of cognitive richness and low-dimensional parametric control.

Existence Theorems for Multidimensional Control Systems with Lower-Dimensional Controls

Existence theorems are proved for multidimensional Lagrange control systems, in which the state is a function of several independent variables in a fixed domain, while in contrast, the control is a function of only some of the independent variables (that is, u is a lower-dimensional control). As usual in Lagrange problems, the cost functional is a multiple integral, the state equation is a system of partial differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables.