# Research

#### Tensegrity Systems

A tensegrity system is a network of axially loaded elements. The compressive elements are bars and the tensile elements are cables. None of the elements bend, even though the network or the structure bends. The framework is quite sophisticated in realizing minimum mass structures, given desired structural characteristics. The tensions in the cables can be actively controlled to achieve desired deformation or dynamic stiffness properties. Our research here is focussed on developing convex optimization based formulations for structure, sensing & actuation architecture and control law codesign in the presence of operational uncertainty. We also include the effect of finite precision computation in the design and control of these systems. Tensegrity systems embody the essential characteristics of cyber physical systems. Applications include design of morphing airframes, robotic manipulators and general tensegrity based actuators and sensors.

#### Integrated Design and Engineering of Aerospace Systems (IDEAS)

IDEAS is a multidisciplinary applied research initiative that integrates aerodynamics, structural design and flight control design in a single unified framework. The objective is to develop next generation tools for rapid custom design of high confidence unmanned air vehicles for various industries including defence, oil & gas, and agriculture. The vision is to codesign much of the system engineering aspect by integrating state-of-the-art in computational fluid dynamics, structural mechanics, robust control theory, CAD software and 3D printing. The application focus is currently on aerospace systems, but can be extended to general autonomous systems.

#### Optimal Transport in Dynamics and Control

Our research on optimal transport and the associated Wasserstein distance between probability density functions, provides an interesting framework for data driven modeling, model validation (both time and frequency domain), controller verification, and stability analysis of stochastic jump systems. Our focus is on development of a systems theory within the OT framework with applications including cyber physical systems, space situational awareness and other data driven systems.

#### Asynchronous Numerical Algorithms

With the advent of massively parallel (exascale) machines, there is a renewed interest in asynchronous numerical methods. Conventionally, parallel computations are synchronized after every iteration across processing elements. The communication overhead reduces processor utilization significantly (50% or more). In this new paradigm, the synchronization between processing elements are relaxed and computation proceeds with the most recent inputs. In our work, the arrival of new data from neighboring processing elements is modeled as a random process and stability and convergence of the asynchronous numerical algorithm is analyzed in a stochastic jump dynamical system framework. We have showed considerable speedup in solving large scale linear PDEs, quadratic programming problems, and for general asynchronous linear fixed-point iterations.

#### Uncertainty Propagation Algorithms

Given the system dynamics $$\dot{x} = F(x,\Delta)+Bw(t),$$ with uncertain parameters $$\Delta\in\mathbb{R}^d$$, process noise $$w\in\mathbb{R}^m$$, and uncertain initial condition $$x_0\in\mathbb{R}^n$$, we are interested in the time evolution of the state uncertainty. Here we consider $$\Delta$$ and $$x_0$$ to be random variables with known probability density functions (PDF), and $$w(t)$$ to be a random process with known statistical properties. The research focus is to develop algorithms for determining $$f(t,x)$$, the state PDF, for high dimensional nonlinear dynamical systems. The algorithms rely on polynomial chaos and transfer operator theory. These algorithms are critical for nonlinear estimation problems.

#### Probabilistic Robust Control

In this research effort we are interested in extending classical robust control to include probabilistic uncertainty. Consider a linear system with probabilistic uncertainty

$$\dot{x} = A(\Delta) x + B(\Delta)u,$$

where the system matrices are dependent on parameter $$\Delta\in\mathcal{D}_\Delta :=[\Delta_\text{min},\Delta_\text{max}]^d$$. In classical robust control, uncertainty is treated in the worst-case sense. Here, we treat the uncertainty in a probabilistic sense, with given probability density function for $$\Delta$$, and expect to reduce the conservatism in the worst-case framework. Additionally, these algorithms are based on deterministic methods and scale better than randomized algorithms.

#### Anytime Control Algorithms

Control algorithms that are implemented in embedded processors have well defined, fixed time constraints and are considered to be functioning properly, only if it produces correct output within a defined time constraint. Scheduling of these tasks is a fundamental problem for real-time systems and robustness to uncertain transient overloads is critical. Failure to provide enough CPU for computation of control law for long time can result in system instability. In this work, we seek linear control algorithms that compromise performance to accommodate reduction in computational time. We adopt deterministic and stochastic switching between high complexity (high performance and more CPU time) and low complexity (low performance and less CPU time) control algorithms so that close-loop stability is guaranteed in the presence of uncertain CPU availability. These ideas have been succesfully applied to transport aircraft control and quadrotor control systems.

#### Model Predictive Control

Model predictive control has been popular in the process control industry for several years. It is based on the simple idea of repetitive solution of an optimal control problem and updating states with the first input of the optimal command sequence. The repetitive nature of the algorithm results in a state dependent feedback control law. We are interested in application of MPC algorithms to flight control problems in the presence of probabilistic uncertainty.