This paper briefly surveys some recent results relevant to the suitability of "neural nets" as models for dynamical systems as well as controllers for nonlinear plants. In particular, it touches upon questions of approximation, identifiability, construction of feedback laws, classification and interpolation, and computational capabilities of nets. No discussion is included of "learning" algorithms, concentrating instead on representational issues. 1. Introduction The basic paradigm for control is that of a "plant" or physical device P interconnected with a controller C. The controller uses measurements from P in order P C Figure 1: Basic Paradigm to compute signals, which are then fed back into the plant so as to attain a given regulation objective. (This description can be extended to incorporate the effect of external disturbances, the specification of desired trajectories, and so forth.) The plant P represents an existing system, and it is essential to have a mathematical model ...

OF THE DISSERTATION Foundations of Recurrent Neural Networks by Hava (Eve) Tova Siegelmann, Ph.D. Dissertation Director: Professor Eduardo D. Sontag "Artificial neural networks" provide an appealing model of computation. Such networks consist of an interconnection of a number of parallel agents, or "neurons." Each of these receives certain signals as inputs, computes some simple function, and produces a signal as output, which is in turn broadcast to the successive neurons involved in a given computation. Some of the signals originate from outside the network, and act as inputs to the whole system, while some of the output signals are communicated back to the environment and are used to encode the end result of computation. In this dissertation we focus on the "recurrent network" model, in which the underlying graph is not subject to any constraints. We investigate the computational power of neural nets, taking a classical computer science point of view. We characterize the language re...

Advances in nonlinear control theory have provided the mathematical foundations necessary to establish conditions for stability of several types of adaptive fuzzy controllers. However, very few, if any, of these techniques have been compared to conventional adaptive or nonadaptive nonlinear controllers or tested beyond simulation; therefore, many of them remain as purely theoretical developments whose practical value is difficult to ascertain. In this paper we will develop three case studies where we perform a comparative analysis between the adaptive fuzzy techniques in [1]–[3] and some conventional adaptive and nonadaptive nonlinear control techniques. In each case, the analysis will be performed both in simulation and in implementation, in order to show practical examples of how the performance of these controllers compares to conventional controllers in real systems.

We study the problem of identification for nonlinear systems in the presence of unknown driving noise, using both feedforward multilayer neural network and radial basis function network models. Our objective is to resolve the difficulty associated with the Persistency of Excitation condition inherent to the standard schemes in the neural identification literature. This difficulty is circumvented here by a novel formulation and by using a new class of identification algorithms recently obtained in [1]. We show how these algorithms can be exploited to successfully identify the nonlinearity in the system using neural network models. By embedding the original problem in one with noise-perturbed state measurements, we present a class of identifiers (under L1 and L2 cost criteria) which secure a good approximant for the system nonlinearity provided that some global optimization technique is used. In this respect, many available learning algorithms in the current neural network literature, e....

This paper explores the stability of a class of feedback fuzzy systems. The class consists of generalized additive fuzzy systems that compute a system output as a convex sum of linear operators. Continuous versions of these systems are globally asymptotically stable if all rule matrices are stable (negative definite). So local rule stability leads to global system stability. This relationship between local and global system stability does not hold for the better known discrete versions of feedback fuzzy systems. A corollary shows that it does hold for the discrete versions in the special but practical case of diagonal rule matrices. The paper first reviews additive fuzzy systems and then extends them to the class of generalized additive fuzzy systems. The Appendix derives the basic ratio structure of additive fuzzy systems and shows how supervised learning can tune their parameters.

In this letter, stable direct and indirect adaptive controllers are presented that use Takagi–Sugeno (T–S) fuzzy systems, conventional fuzzy systems, or a class of neural networks to provide asymptotic tracking of a reference signal vector for a class of continuous time multi-input multi-output (MIMO) square nonlinear plants with poorly understood dynamics. The direct adaptive scheme allows for the inclusion of a priori knowledge about the control input in terms of exact mathematical equations or linguistics, while the indirect adaptive controller permits the explicit use of equations to represent portions of the plant dynamics. We prove that with or without such knowledge the adaptive schemes can “learn” how to control the plant, provide for bounded internal signals, and achieve asymptotically stable tracking of the reference inputs. We do not impose any initialization conditions on the controllers and guarantee convergence of the tracking error to zero.

Abstract—In this note, we present a direct adaptive control method for a class of uncertain nonlinear systems with a time-varying structure. We view the nonlinear systems as composed of a finite number of “pieces, ” which are interpolated by functions that depend on a possibly exogenous scheduling variable. We assume that each piece is in strict-feedback form, and show that the method yields stability of all signals in the closed-loop, as well as convergence of the state vector to a residual set around the equilibrium, whose size can be set by the choice of several design parameters. The class of systems considered here is a generalization of the class of strict-feedback systems traditionally considered in the backstepping literature. We also provide design guidelines based on bounds on the transient. Index Terms—Backstepping, direct adaptive control, interpolation of strict feedback systems, nonlinear systems, time-varying structure. I.

Abstract Stable indirect and direct adaptive controllers are presented for a class of input-output feedbacklinearizable time-varying nonlinear systems. The radial basis function neural networks are used as on-line approximators to learn the time-varying characteristics of system parameters. Stabilityresults are given in the paper and the performance of the indirect and direct adaptive schemes is demonstrated through a fault tolerant engine control problem where the faults are naturallytime varying.

The properties of an adaptive control system based on a general class of nonlinear models are analyzed. The class of models contains NARX models represented by feedforward neural networks with sigmoidal non-linearity, some radial basis-function expansions, local model networks, and some fuzzy systems. We apply a simple adaptive feedback linearizing controller. The analysis takes into account modeling error and slowly time-varying nominal model parameters. We discuss stability of the closed loop, as well as robustness, and derive performance bounds on the tracking error. Due to the general setup, the results are mainly of a qualitative nature. Finally, we discuss the relevance of the results, and outline some practical modifications and possible extensions. 1. Introduction The analytical study of adaptive control loops involving complicated nonlinear model structures and controllers such as neural networks and fuzzy systems has evolved considerably over the past five years. Some stabi...

We show that an adaptive input/output feedback linearization control scheme for minimum phase nonlinear systems is robust with respect to unstructured plant uncertainties that include such as unmodeled dynamics and disturbances, provided the adaptive law is modified in the same fashion as for linear systems, and the plant nonlinearities satisfies some growth constraints. In the analysis we utilize weighted L 2 -norms analogously to the case with a linear model. Keywords: Adaptive Control, Nonlinear Systems, Robustness, Stability, Feedback Linearization. 1 This work was in part supported by the Research Council of Norway under Grant ST.10.12.221718 given to the first author, and in part by the National Science Foundation (NSF) under Grant ECS-9119722. 2 Present address: SINTEF Automatic Control, 7034 Trondheim, Norway. 1 Introduction For adaptive control of linear systems, the robustness issue has turned out to be very important, since it has been demonstrated that even small mode...