Adaptive Internal Models in Neuroscience

Disturbance cancellation for linear systems by adaptive internal models

In this paper we solve the tracking and disturbance rejection problem for an unstable wave equation with reference and disturbance signals that are generated by a linear exosystem whose eigenvalues are located on the imaginary axis. We consider both unmatched disturbance at the boundary and also in the domain, and also matched disturbance at the control boundary. To make the control design more easy, the in-domain disturbance is first shifted to the control boundary by introducing a new variable transformation, and the anti-stable boundary term is then transformed into a dissipative boundary term by constructing a transport equation. We find an ordinary differential equation (ODE) with delay that describes the relationship between the output tracking error and the coupled signals consisting of the disturbance and reference signal, by using Riemann variables. Based on this ODE with delay, we apply the internal model principle to find the dynamic controller that achieves the output regulation while rejecting the disturbances, and uses only the tracking error as input.

Repetitive controllers have been shown to be effective for tracking periodic reference commands or for rejecting periodic disturbances. Typical repetitive controllers are synthesized in temporal domain where the periods of the reference or disturbance signals are assumed to be known and stationary. For periodic references and disturbances with varying periods, researchers usually resort to adaptive and robust control approaches. For rotational motion systems where the disturbances or reference signals are spatially periodic (i.e., periodic with respect to angular displacement), the temporal period of the disturbance and reference signals will be inversely proportional to the rotational speed and vary accordingly. Motivating by reducing halftone banding for laser printers, we propose a design framework for synthesizing spatially sampled repetitive controller by reformulating a linear time-invariant system subject to spatially periodic disturbances using angular displacement as the independent variable. The resulting nonlinear system can be represented as a quasi-linear parameter-varying (quasi-LPV) system with the angular velocity as one of the varying state-dependent parameters. An LPV self-gain–scheduling controller that includes a spatially sampled repetitive control can be designed to take into consideration bounded model uncertainty and input nonlinearity, such as actuator saturation. Using the signal from an optical encoder pulse as a triggering interrupt, experimental results verified the effectiveness of the proposed approach in rejecting spatially periodic disturbances that cannot be compensated with fixed period temporal repetitive controllers.

One of the classical problems in control theory is the robust output regulation problem. The output of a system is required to asymptotically follow a given reference signal, despite disturbance signals and perturbations in the system's parameters. Wonham & Francis, and Davison on the other hand, developed independently the theory and construction of finite-dimensional robust controllers. Wonham's approach was based on geometric theory while Davison introduced a combination of servo-compensator and stabilizing compensator. They discovered the famous Internal Model Principle, which states that a robust feedback controller must contain a copy of the dynamics of the exosystem generating the reference and disturbance signals.

In analogy to the Kucera–Youla parametrization, we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption, the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case, the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogs for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita’s theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.

The authors previously (2000) showed that a low-gain controller of the form C/sub /spl epsiv//(s)=/spl Sigma//sub k=-n//sup n/ /spl epsiv/K/sub k//(s-i/spl omega//sub k/) is able to track and reject constant and sinusoidal reference and disturbance signal for a stable plant in the Callier-Desoer (CD) algebra. In this note, we investigate the optimal tuning of the matrix gains K/sub k/ of the controller C/sub /spl epsiv//(s) as the scalar gain /spl epsiv//spl darr/0. The cost function is the maximum error between the reference signal and the measured output signal over all frequencies and bounded reference and disturbance signal amplitudes. Closed forms for asymptotically globally optimal solutions are given. The optimal matrix gains K/sub k/ are expressed in terms of the values of the plant transfer matrix at the reference and disturbance signal frequencies. Thus the matrices K/sub k/ can be tuned with input-output measurements made from the open loop plant without knowledge of the plant model. Although the analysis is in the CD-algebra, to the authors' knowledge the main results are new even for finite-dimensional systems.

The analysis method of optimal tracking performance is proposed for multiple‐input multiple‐output (MIMO) linear time‐invariant (LTI) systems under disturbance rejection. AnH2criterion of the error signal between the output of the plant and the reference signal is used as a measure for the tracking performance. Spectral factorization is applied to obtain the optimal solution of the system tracking error. The explicit expressions are derived for this minimal tracking error with respect to random reference signals under disturbance rejection. It is shown that the nonminimum phase zeros, the zero direction, the unstable poles, the pole direction of a given plant, statistical characteristics of the reference input signal, and disturbance signal have a negative effect on a feedback system's ability to reduce the system error with disturbance rejection. The results show that the optimal tracking performance will further be damaged because of disturbance rejection. Some typical examples are given to illustrate the theoretical results.

The combined operation of optokinetic reflex (OKR) and vestibular-ocular reflex (VOR) is essential for image stability during self-motion. Retinal slip signals, which provide neural substrate for OKR and VOR plasticity, are delivered to the inferior olive. Although it has been assumed that the neural circuitry and mechanisms underlying OKR and VOR plasticity are shared, differential role of the inferior olive in the plasticity of OKR and VOR has not been clearly established. To investigate the differential effect of inferior olive lesion on OKR and VOR plasticity, we examined the change of OKR and VOR gains after gain-up and gain-down VOR training. The results demonstrated that inferior olive-lesion differentially affected cerebellum-dependent motor learning. In control mice, OKR gain increased after both gain-up and gain-down VOR training, and VOR gain increased after gain-up VOR training and decreased after gain-down VOR training. In inferior olive-lesioned mice, OKR gain decreased after both gain-up and gain-down VOR training, and while VOR gain did not significantly change after gain-up VOR training, VOR gain decreased after gain-down VOR training. We suggest that multiple mechanisms of plasticity are differentially involved in VOR and OKR adaptation, and gain-up and gain-down VOR learning rely on different plasticity mechanisms.

Selective Engagement of Plasticity Mechanisms for Motor Memory Storage

The internal model principle of control theory

The influence of bilateral labyrinthectomy on horizontal and vertical optokinetic reflexes in the rabbit

Compensatory eye movements (CEM) maintain a stable image on the retina by minimizing retinal slip. The optokinetic reflex (OKR) and vestibulo-ocular reflex (VOR) compensate for low and high velocity stimuli, respectively. The OKR system is known to be highly nonlinear. The VOR is generally modeled as a linear system and assumed to satisfy the superposition and homogeneity principles. To probe CEM violation of the superposition principle, we recorded eye movement responses in C57BL/6 mice to sum of sine (SoS) stimulation, a combination of multiple nonharmonic inputs. We tested the VOR, OKR, VVOR (visually enhanced VOR), and SVOR (suppressed VOR). We used stimuli containing 0.6 Hz, 0.8 Hz, 1.0 Hz, and 1.9 Hz. Power spectra of SoS stimuli did not yield distortion products. Gains and delays of SoS and single sine (SS) responses were compared to yield relative gains and delays. We find the superposition principle is violated primarily in the OKR, VOR, and SVOR conditions. In OKR, we observed relative gain suppression of the lower SoS stimulus frequency component irrespective of the absolute frequency. Conversely, SVOR and VOR results showed gain enhancement of the lower frequency component and overall decrease in lead. Visually enhanced VOR results showed trends for overall gain suppression and delay decrease. Compensatory eye movements arguably depend on predictive signals. These results may reflect better prediction for SS stimuli. Natural CEM system stimulation generally involves complex frequency spectra. Use of SoS stimuli is a step toward unravelling the signals that really drive CEM and the predictive algorithms they depend on.

This paper concerns the problem of adaptive output regulation for multivariable nonlinear systems in normal form. We present a regulator employing an adaptive internal model of the exogenous signals based on the theory of nonlinear Luenberger observers. Adaptation is performed by means of discrete-time system identification schemes, in which every algorithm fulfilling some optimality and stability conditions can be used. Practical and approximate regulation results are given relating the prediction capabilities of the identified model to the asymptotic bound on the regulated variables, which become asymptotic whenever a "right" internal model exists in the identifier's model set. The proposed approach, moreover, does not require "high-gain" stabilization actions.

In this article, we present a computational model of the oculomotor system and the cerebellum. In contrast with prevailing theories of cerebellar function, we propose the cerebellum embodies adaptive internal models of all persistent, exogenous, deterministic signals acting on the body and observable through the error signals it receives. Our model is validated by simulations, recovering results from a number of oculomotor experiments.

Adaptation of vestibulo-ocular and optokinetic reflexes after massed and spaced vestibulo-ocular motor learning