##### Time Delay Systems

A modern challenge in control theory is robust/adaptive/optimal control for nonlinear and time delay systems. Time delay, commonly known as time difference of arrival or dead time, is an extensive phenomenon encountered in many different research areas such as unmanned air vehicles (UAVs), missiles, master-slave robots, haptic systems and biological systems. Leading causes of the time delay phenomena are the system dynamics, communication lines and networks, feedback structures and sensors. Current research from NCR involves designing novel, robust and adaptive controllers for uncertain nonlinear systems subject to known and unknown, constant and time dependent, input and state time delays, with additive disturbances.

##### Intelligent Control of Uncertain Nonlinear Systems

Improved control performance can result from including model knowledge of the system in the control design and stability analysis. Yet, there is some degree of uncertainty in most models of dynamic systems. Different tools may be required, based on the nature of the uncertainty. The focus of intelligent control is to understand how to incorporate different methods to compensate for the uncertainty. Efforts in this topic focus on methods such as

- Extremum Seeking
- Repetitive Learning Control
- Neural Networks
- Adaptive Control
- and Biologically Inspired

to compensate for the uncertainty in the system. These approaches are in contrast to robust control methods that use high gain or high frequency components to reject uncertainty.

##### Robust Integral of the Sign of the Error (RISE)

Robust Integral of the Sign of the Error (RISE) is a recently developed new differentiable high gain feedback control strategy that contains a unique integral signum term which can accommodate for sufficiently smooth bounded disturbances. A significant outcome of this new control structure is that asymptotic stability is obtained despite a fairly general uncertain disturbance. This technique can be used to develop a tracking controller for nonlinear systems in the presence of additive disturbances and parametric uncertainties under the assumption that the disturbances are C2 with bounded time derivatives. The fact that the RISE controller is continuous is especially useful from the implementation point of view. Other robust discontinuous feedback strategies are disadvantageous in that they require infinite bandwidth and cause chattering that may cause undesirable oscillations in the mechanical systems.

##### Corollaries for Nonsmooth Systems

For continuous systems, stability techniques such as the LaSalle-Yoshizawa Theorem provide a convenient analysis tool when the candidate Lyapunov function derivative is upper bounded by a negative semi-deﬁnite function. However, adapting the LaSalle-Yoshizawa Theorem to systems where the time derivative of the system states are not locally Lipschitz remains an open problem. We consider Filippov solutions for nonautonomous nonlinear systems with right-hand side discontinuities utilizing Lipschitz continuous and achieve asymptotic convergence when regular Lyapunov functions whose time derivatives (in the sense of Filippov) can be upper bounded by negative semi-deﬁnite functions.

##### Optimal Control of Nonlinear Systems

Optimal control theory involves the design of controllers that can satisfy some tracking or regulation control objective while simultaneously minimizing some performance metric. A sufficient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. For the special case of linear time-invariant systems, the solution to the HJB equation reduces to solving the algebraic Riccati equation. However, for general systems, the challenge is to find a value function that satisfies the HJB equation. Finding this value function has remained problematic because it requires the solution of a partial differential equation that can not be solved explicitly. To address these issues at NCR we have came up with the following solutions. We have designed an optimal controller in which a system in which all terms are assumed known (temporarily) is feedback linearized and a control law is developed based on the HJB optimization method for a given quadratic performance index. Under the assumption that parametric uncertainty and unknown bounded disturbances are present in the dynamics, the control law is modified to contain the RISE feedback term which is used to identify the uncertainty. Specifically, a Lyapunov stability analysis is included to show that the RISE feedback term asymptotically identifies the unknown dynamics (yielding semi-global asymptotic tracking) provided upper bounds on the disturbances are known and the control gains are selected appropriately. As in previous literature the control law converges to the optimal law, however because our result is asymptotic rather than UUB the control law converges exactly to the optimal law. Likewise, we have developed an adaptive IOC based on the established theoretical foundation. The developed controller minimizes a meaningful performance index as the generalized coordinates of a nonlinear Euler-Lagrange system globally asymptotically track a desired time-varying trajectory despite LP uncertainty in the dynamics. The unique ability to consider the IOC problem for uncertain Euler-Lagrange dynamics is due to a novel optimization analysis. A meaningful cost function (i.e., a positive function of the states and control input) is developed and an analysis is provided to prove the cost is minimized without having to prove the Lyapunov function is a CLF. To develop the optimal controller for the uncertain system, the open loop error system is segregated to include two adaptive terms. One adaptive term is based on the tracking error which contributes to the cost function, and the other adaptive term does not explicitly depend on the tracking error (and therefore does not explicitly contribute to the cost function). A Lyapunov analysis is provided to examine the stability of the developed controller and to determine a respective meaningful cost functional. Preliminary simulation results are included to illustrate the performance of the controller.