# Peak Estimation

Quantify the safety of trajectories using occupation measure techniques

Peak estimation is the practice of finding the maximum value of a state function over trajectories of a dynamical system. Instances of peak estimation include finding the speed of a car, the height of an aircraft, the voltage in a power line, etc. Peak estimation can be applied towards safety quantification, such as by measuring the safety of a trajectory by its distance of closest approach to an unsafe set. This project extends the occupation measure framework developed for optimal control and peak estimation. The Moment-Sum-of-Squares hierarchy is employed to obtain convergent bounds to the true peak value when all system data is polynomial.

This thesis was defended on April 3, 2023. Information is available at [Thesis], [Slides], [Bonus Slides], [Poster].

Our first step to perform this quantification involved measuring the constraint violation using maximin optimization, yielding a safety margin (with safety verified if this margin is negative) (Miller et al., 2021). Peak estimation may also occur for dynamics with compact-valued time-dependent or time-independent uncertainty (including switching) (Miller et al., 2021). An extension of this includes quantifying the safety of trajectories by finding the distance of closest approach to an unsafe set (missing reference).

Extensions of the peak/distance estimation framework include adding (bounded) uncertainty into dynamics, allowing for piecewise distance functions (e.g. L1 or L-infinity normed distances), and ensuring the safety of all points on a moving shape with respect to the unsafe set.

Peak estimation can be used in a data-driven framework to formulate safety quantification of unknown systems. When the dynamics are input-affine and the corruption is semidefinite-representable (e.g., box-bounded), infinite-dimensional robust counterparts may be applied to drastically reduce the size of PSD matrices (Miller & Sznaier, 2023). This robust counterpart framework can be applied to peak estimation, reachable set estimation, distance estimation, and to the peak-minimizing control problem that finds minimum possible data corruption needed to crash into the unsafe set (Miller & Sznaier, 2023).

Peak and Distance Estimation may be applied to dynamical systems that do not follow Ordinary Differential Equations. Peak estimation may be applied to hybrid systems by modifying hybrid Optimal Control Programs (Miller & Sznaier, 2023). Stochastic Differential Equations (SDEs) can also be analyzed by finding the peak Value-at-Risk (epsilon-quantile) of the state function (Miller et al., 2023).

Peak estimation problems can be posed over Time-Delay systems, in which the trajectory dynamics depend on past and present functions of state (Miller et al., 2023).