Stochastic dissipativity : Many physical and engineering systems are open systems, that is, the system behavior is described by an evolution law that involves the system state and the system input with, possibly, an output equation wherein past trajectories together with the knowledge of any inputs define future trajectories (uniquely or nonuniquely) and the system output depends on the instantaneous (present) values of the system state. Dissipativity theory is a system-theoretic concept that provides a powerful framework for the analysis and control design of open dynamical systems based on generalized system energy considerations. In particular, dissipativity theory exploits the notion that numerous physical dynamical systems have certain input-output and state properties related to conservation, dissipation, and transport of mass and energy. In light of the fact that energy notions involving conservation, dissipation, and transport also arise naturally for dissipative diffusion processes, it seems natural that dissipativity theory can play a key role in the analysis and control design of stochastic dynamical systems. In addition, robust stability for stochastic dynamical systems with stochastic uncertainty can be analyzed by viewing the uncertain stochastic dynamical system as an interconnection of stochastic dissipative dynamical subsystems. Alternatively, stochastic dissipativity theory can be used to design feedback controllers that add dissipation and guarantee stability robustness in probability allowing stochastic stabilization to be understood in physical terms.

As part of this research we extended classical dissipativity theory to stochastic dissipativity for nonlinear stochastic dynamical systems to address the design of feedbackcontrollers that add dissipation and guarantee stability robustness in probability allowing stochastic stabilization to be understood in physical terms. In addition, we used stochastic dissipativity to quantify stochastic robustness as well as address risk-sensitive disturbance rejection, stability in probability of feedback interconnections, and optimality with averaged performance measures for stochastic dynamical systems. In future research, the plan is to develop connections between stochastic dissipativity and stochastic optimal control to address robust stability and robust stabilization problems involving both stochastic and deterministic uncertainty as well as both averaged and worst-case performance criteria.

Nonlinear-nonquadratic stochastic optimal control:Under certain conditions nonlinear controllers offer significant advantages over linear controllers. This has motivated us to develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems, wherein we considered a feedback stochastic optimal control problem over an infinite horizon involving nonlinear-nonquadratic performance functionals. The performance functionals can be evaluated in closed-form as long as the nonlinear-nonquadratic cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability in probability of the nonlinear closed-loop system. This Lyapunov function is shown to be the solution of the steady-state stochastic Hamilton-Jacobi-Bellman equation. These results were then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. To show the efficacy of the framework, the results were applied to a pitch axis longitudinal dynamics model of an F-16 Fighter aircraft system with stochastic disturbances.

Stochastic finite-time and partial-state stabilization: The problem of finite-time optimal control has received very little attention in the literature. When the performance functional involves subquadratic terms, the cost criterion pays close attention to the behavior of the state near the origin. My interest in subquadratic cost criteria stems from the fact that optimal controllers for such criteria are sublinear, and thus, exhibit finite settling time behavior. This phenomenon was studied in my dissertation as well as and applied to a spacecraft control problem with one axis of symmetry and with solar pressure captured as a stochastic disturbance.

Our stochastic finite-time stabilization results yield finite interval controllers even though the original cost criterion was defined on the infinite horizon. Hence, one advantage of this approach for certain applications is to obtain finite-interval controllers without the computational complexities of two-point boundary value problems. It is also important to note that if the order of the subquadratic state terms appearing in the cost functional is sufficiently small, then the controllers actually optimize a minimum-time cost criterion. Currently, such results are only obtainable using the maximum principle, which generally does not yield feedback controllers.

Another important extension is the consideration of optimal partial-state stabilization and partial finite-time stabilization, that is, closed-loop stability with respect to part of the closed-loop systems state, which arises in many engineering applications. Specifically, in spacecraft stabilization via gimballed gyroscopes asymptotic stability of an equilibrium position of the spacecraft is sought while requiring Lyapunov stability of the axis of the gyroscope relative to the spacecraft. Alternatively, in the control of rotating machinery with mass imbalance, spin stabilization about a nonprincipal axis of inertia requires motion stabilization with respect to a subspace instead of the origin. The most common application where partial stabilization is necessary is adaptive control, wherein asymptotic stabilization of the closed-loop plant states is guaranteed without necessarily achieving parameter error convergence. As part of our work on stochastic stabilization, we have developed a unified framework to address the problem of optimal nonlinear feedback control for partial stability and partial-state stabilization of stochastic dynamical systems.

Adaptive control with low-frequency learning and fast adaptation: While adaptive control has been used in numerous applications to achieve system performance without excessive reliance on system models, the necessity of high-gain learning rates for achieving fast adaptation can be a serious limitation of adaptive controllers. Specifically, in safety-critical systems involving large system uncertainties and abrupt changes in system dynamics, fast adaptation is required to achieve stringent tracking performance specifications. However, fast adaptation using high-gain learning rates can cause high frequency oscillations in the control response resulting in system instability. To address the problem of achieving fast adaptation using high-gain learning rates for systems with partial state information, we developed an output feedback adaptive control framework for continuous-time, minimum phase multivariable dynamical systems for output stabilization and command following. The approach is based on a nonminimal state-space realization that generates an expanded set of states using the filtered inputs and filtered outputs (along with their derivatives) of the original system, and requires knowledge of only the open-loop system’s relative degree and a bound on the system’s order.

This control architecture was applied to a roll autopilot controller of a guided missile and the longitudinal dynamics of a Boeing 747 airplane. In addition, the adaptive control architecture was applied to the stabilization of a flexible spacecraft. Specifically, we show that a standard adaptive architecture cannot simultaneously achieve a fast convergence rate and obtain a smooth system response. Large adaptive gains lead to a highly oscillatory system response, large control inputs, and unrealistic control input rates. On the other hand, small adaptive gains result in slow convergence rates. However, when our adaptive control architecture is used the system is able to achieve a fast convergence rate and obtain a smooth system response, simultaneously. In addition, the maximum control input and control input rate are decreased and the flexible dynamics of the system were less excited.

Stochastic differential games: Differential games have been studied in various contexts including risk-sensitive control, mathematical finance, and communication networks as well as network resource allocation. The pioneering work on the subject involved a deterministic two-player differential game problem whose solution is characterized by the Hamilton-Jacobi-Isaacs equation. In ongoing research, we have developed a framework that focuses on the role of the Lyapunov function guaranteeing stochastic stability of the differential game and its connection to the steady-state solution of the stochastic Hamilton-Jacobi-Isaacs equation characterizing the optimal nonlinear feedback controller and stopper policies. In order to avoid the complexity in solving the stochastic steady-state, Hamilton-Jacobi-Isaacs equation my framework does not attempt to minimize/maximize a given cost functional, but rather, parameterizes a family of stochastically stabilizing controller and stopper policies that minimizes/maximizes a derived cost functional that provides the flexibility in specifying the control and stopper.

Thermodynamics, control, and large-scale, multi-agent systems: My current research has concentrated on coordinated control of large-scale \textit{stochastic} multi-agent networked systems. In many applications involving multiagent systems, a group of agents are required to agree on certain quantities of interest. In such applications, it is important to develop information consensus protocols for a network of dynamic agents. As part of this ongoing research effort, I am addressing semistable (i.e, Lyapunov stable plus convergent) consensus problems for nonlinear stochastic network systems with switching communication topologies and communication dropouts. In particular, I am developing a thermodynamic-based stochastic control framework in order to consider random communication disturbances between agents in the network, wherein the evolution of each link of the random network follows a Markov process. This will necessitate the development of almost sure consensus of multiagent systems with nonlinear stochastic dynamics under distributed nonlinear consensus protocols. Furthermore, I will relate almost sure consensus of multiple agents over random networks to the semistability notion of the nonlinear stochastic systems. Specifically, using my recent results on stochastic semistability for nonlinear stochastic systems with a continuum of equilibria, I will develop consensus control protocols for multiagent systems in the face of probabilistic variations in the interconnections between agents.

System thermodynamics: As part of my long-term research endeavor I plan to concentrate on the amalgamation of thermodynamics, statistical physics, information theory, and control. Professor Haddad and I have been working on unified framework of system thermodynamics. Specifically, building on the results of Professor Haddad’s 2005 monograph, we are working on a system-theoretic foundation for thermodynamics by combining the two universalisms of thermodynamics and dynamical systems theory under a single umbrella so as to harmonize it with classical mechanics. In particular, we are developing a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic and stochastic large-scale dynamical system models as well as infinite-dimensional models that bridge the gap between classical and statistical thermodynamics and includes topics on finite-time thermodynamics, critical phase transitions, fluctuation theorems, chemical thermodynamics, and relativistic effects.

• Nonlinear Analysis and Control
• Stochastic Calculus
• Adaptive Control
• Mathematical Finance
• System Thermodynamics
• Stochatic Differential Games
• Machine Learning
• Robotics
• Biga Data Analytics
• Distributed Network Control
• Information Theory

Adviser: Dr. W. M. Haddad

Dr. Haddad has made numerous contributions to the development of nonlinear control theory and its application to aerospace, electrical, and biomedical engineering. His transdisciplinary research in systems and control is documented in over 600 archival journal and conference publications, and 7 books in the areas of science, mathematics, medicine, and engineering.

Dr. Haddad is an NSF Presidential Faculty Fellow; a member of the Academy of Nonlinear Sciences; an IEEE Fellow; and the recipient of the 2014 AIAA Pendray Aerospace Literature Award. His research is on nonlinear robust and adaptive control, nonlinear systems, large-scale systems, hierarchical control, hybrid systems, stochastic systems, thermodynamics, network systems, systems biology, and mathematical neuroscience.

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