Peak Estimation

Bound extreme values of state functions 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 may be used to quantify the safety of trajectories. 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.

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 (Miller & Sznaier, 2021).

All plots certify safety of trajectories with respect to the red half-circle unsafe set. Left plot: a barrier certificate of safety, in which the green curve is the 0-level set. Center-plot: a safety margin, along with the optimizing-trajectory in dark blue. Right-plot: the trajectory that achieves a distance of closest approach, along with the contour of all similarly-close points.

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.

Left plot: a distance estimate under an uncertainty process. Center plot: the closest approach with respect to L1 distance. Right plot: closest approach betwen a point on the translating shape (pink square) and the unsafe set.

Relevant Publications

Journal Articles

  1. LCSS
    Peak Estimation Recovery and Safety Analysis
    Miller, JaredHenrion, Didier, and Sznaier, Mario
    IEEE Control Systems Letters 2021

Conference Articles

    Facial Input Decompositions for Robust Peak Estimation under Polyhedral Uncertainty
    Miller, Jared, and Sznaier, Mario
    In 10th IFAC Symposium on Robust Control Design (ROCOND) 2022
  2. CDC
    Bounding the Distance of Closest Approach to Unsafe Sets with Occupation Measures
    Miller, Jared, and Sznaier, Mario
    In 61st IEEE Conference on Decision and Control (CDC) 2022
  3. CDC
    Peak Estimation for Uncertain and Switched Systems
    Miller, JaredHenrion, DidierSznaier, Mario, and Korda, Milan
    In 60th IEEE Conference on Decision and Control (CDC) 2021


  1. Peak Estimation of Hybrid Systems with Convex Optimization
    Miller, Jared, and Sznaier, Mario
    Preprint 2023
  2. Peak Estimation of Time Delay Systems using Occupation Measures
    Miller, JaredKorda, MilanMagron, Victor, and Sznaier, Mario
    Preprint 2023
  3. Quantifying the Safety of Trajectories using Peak-Minimizing Control
    Miller, Jared, and Sznaier, Mario
    Preprint 2023
  4. Analysis and Control of Input-Affine Dynamical Systems using Infinite-Dimensional Robust Counterparts
    Miller, Jared, and Sznaier, Mario
    Preprint 2023
  5. Bounding the Distance to Unsafe Sets with Convex Optimization
    Miller, Jared, and Sznaier, Mario
    Preprint 2021