Previous Theoretical Work

• Details on Closed Loop Reference Models
• Time Delay Systems

Control problems due to unknown delay are ubiquitous in engineering. In particular, several problems that occur in power-train systems in automobiles, combustion systems in propulsion and power generation, and ground vehicles that rely on communications have significantly large and uncertain delays. A unique methodology termed Adaptive Posicast Control, has been developed in our research group that has shown enormous improvement in gas turbine systems which include large time-delays and have stringent performance specifications of safety and emissions. Application of this methodology to different problem areas where time-delay plays a major role is currently being investigated. Adaptive control of systems with unknown time-delay using novel strategies is being explored as well.

• Control of Systems with Unknown Delay
• Adaptive Switched Systems

Switched control systems and related areas of hybrid systems and supervisory control have received increased attention in the last decade and are used in several applications. Adaptive switched and tuned systems have been studied as well. The combined presence of uncertainties and switching delays makes a direct application of these existing results inadequate in distributed embedded systems (DES) which inherently have a switching structure induced both due to the DES and controllers. The overall objective is to realize high control performance in these kinds of applications. In order to show global boundedness of the states of the adaptive system, we utilize fundamental properties of the adaptive system with persistent excitation, and derive additional properties in the presence of reference signals with an invariant persistent excitation property. These properties in turn are suitably exploited and linked with the switching instants. For the other states we use a Multiple Lyapunov function approach to show boundedness. The fact that different techniques are used to show boundedness of the additional states added by the adaptive controller distinguishes our approach from existing switching controllers and their proofs in the literature. References can be found here.

Accelerated Learning in the Presence of Time Varying Features with Applications to Machine Learning and Adaptive Control

Features in machine learning problems are often time varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current accelerated gradient descent methods unstable or weakens their convergence guarantees. This work proposes algorithms for the case when time varying features are present, and demonstrates provable performance guarantees. We develop a variational perspective within a continuous time algorithm. This variational perspective includes, among other things, higher-order learning concepts and normalization, both of which stem from adaptive control, and allows stability to be established for dynamical machine learning problems. These higher-order algorithms are also examined for achieving accelerated learning in adaptive control.

Actuation Nonlinearities

Two common nonlinearities that occur in a dynamic system are due to magnitude and rate saturation in the actuator. While introducing complexities in the analysis of the underlying system, they also introduce significant challenges in the design of an advanced controller. The methodology introduced in earlier works produced by this laboratory has shown that a systematic adaptive control design that accommodates saturation leading to stability and satisfactory tracking can be carried out and have led to successful implementation in a variety of applications including flight control, and combustion control. Recent efforts are directed towards the integration of resetting and accommodation of rate saturation into the adaptive control design.

Nonlinearly Parametrized Systems

An assumption that has been systematically adopted in the adaptive control design is that the unknown parameters enter linearly in the dynamic equations describing the plant. However, it is well known that there exist many practical examples whose corresponding models are nonlinearly parameterized: distillation columns, bioreactors, robot dynamics, friction compensation. Adaptive control theory for systems where parameters occur nonlinearly is currently being developed.

Parametric uncertainties in adaptive estimation and control have been dealt with, by and large, in the context of linear parametrizations. Algorithms based on the gradient descent method either lead to instability or inaccurate performance when the unknown parameters occur nonlinearly. Complex dynamic models are bound to include nonlinear parametrizations which necessitate the need for new adaptation algorithms that behave in a stable and accurate manner. Friction dynamics, magnetic bearings, chemical reactors, and bio-chemical processes are some examples where more than one physical parameter occurs nonlinearly in the underlying dynamic model. Nonlinear model structures such as neural networks, wavelets, Hammerstein models, and Uryson models routinely include nonlinear parametrization for reasons of parsimony. We have been developing an adaptive systems theory for problems in dynamic systems where parameters occur nonlinearly. Stability, control, convergence, and robustness of adaptive systems that arise in this context are being investigated. Results that have been derived thus far related to adaptive control of nonlinearly parametrized systems can be found in the publications listed below.

• Nonlinearly Parametrized Systems