**NSF CMMI-1931270: **A Convex Computational Framework for Understanding and
Controlling Nonlinear Systems

**Project Description:**

Complicated engineered systems have become increasingly common and autonomous, with examples including underwater vehicles, UAVs, and self-driving cars, as well as less visible systems such as power inverters, battery storage devices, network routers, and data storage devices. Extended periods of autonomy inevitably lead to such unexpected changes as: altered or degraded system configuration; failures of actuators or sensors; and evolution of the working environment. Without an automated mechanism for identification of and adaptation to such changes, it is likely that the technological foundation of our society will become increasingly unreliable. The goal of the project is to improve autonomy in complicated systems. Specifically, optimization algorithms are used to learn the environment, analyze this information, and design control strategies. Based on fundamental theory of how dynamical systems operate, these algorithms provide a rigorous basis for cognizance and adaptability. The impact of this work will be a more safe and reliable technological infrastructure, both on earth and in space. Increasing the reliability and duration of autonomous systems will lead to, for example, faster and cheaper exploration of space, safer transportation networks, and more reliable communication networks.

Recently, Sum-of-Squares (SOS) algorithms have become a powerful tool for understanding nonlinear dynamical systems. The power of SOS lies in its convex parametrization of non-quadratic Lyapunov functions. Unfortunately, however, this convex formulation assumes the dynamics are known and has not been extended to estimating the Region of Attraction (ROA), Minimum Invariant Set (MIS), and Forward Reachability (FRS). At the core of this project is the novel observation that the ROA/MIS/FRS problems can be expressed using sub- or super-solutions to a value function defined by the solution to a Hamilton-Jacobi-Bellman (HJB) equation. The first part of the project exploits this equivalence to show that the ROA/MIS/FRS problems can be posed as SOS optimization problems wherein the objective is volume minimization or maximization of sublevel sets of a sub/super-value function. The second part of the project uses new convex volume metrics to solve these SOS volume optimization problems. The third part of the project considers the case where the dynamics are unknown and uses trajectory data to directly estimate the ROA and FRS without use of a dynamic model. The algorithms are also applied to control of spacecraft attitude dynamics.

**Project Duration:**

09/01/2019-08/31/2022

**Project Personnel:**

Matthew M. Peet (PI), Arizona State University

Morgan Jones (PhD), Arizona State University

**International Collaborators:**

Pedro P. L. D. Peres, The University of Campinas. Campinas, Brasil

**Software Products:**

Invariant Set Code (Not Ready For Release)

**Associated Publications at ASU:**

Student Theses:

Journal Publications:

B. Colbert and M. Peet

A Convex Parametrization of a New Class of Universal Kernel Functions for use in Kernel Learning

Submitted to the Journal of Machine Learning Research.

[1711.05477]
[.pdf]
[slides]
[CodeOcean]

Summary:We propose a new class of kernel functions for using in learning the kernel. This class is parameterized by positive matrices, dense in the cone of kernels and each element of the class is universal. We donstrate that the resulting algorithms are more accurate than any other result in the literature.

CDC/ACC Conference Publications:

M. Jones and M. Peet

Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets

IEEE Conference on Decision and Control, 2019.

[arXiv:1903.07274]
[.pdf]
[slides]
[CodeOcean]

B. Colbert and M. Peet

Using SDP to Parameterize Universal Kernel Functions

IEEE Conference on Decision and Control, 2019.

[arXiv:1809.10308]
[.pdf]
[slides]
[CodeOcean]

M. Jones and M. Peet

Using SOS and Sublevel Set Volume Minimization for Estimation of Forward Reachable Sets

Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2019).

[arXiv:1901.11174]
[.pdf]
[slides]
[CodeOcean]

B. Colbert and L. Crespo and M. Peet

A Sum of Squares Optimization Approach to Uncertainty Quantification

Proceedings of the American Control Conference, 2019.

[arXiv:1809.10308]
[.pdf]
[slides]

M. Jones and M. Peet

Sublevel Set Volume Minimization for Outer Set Approximations of Attractors

Proceedings of the American Control Conference, 2019.

[arXiv:1809.10308]
[.pdf]
[slides]