[Paper Review] Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

Alessandro Saccon, Nathan van de Wouw+1 more|arXiv (Cornell University)|Jun 11, 2014

Advanced Control Systems Optimization12 references4 citations

TL;DR

This paper develops a novel first-order sensitivity analysis framework for hybrid systems with state jumps, introducing a new error notion to approximate perturbed trajectories despite mismatched jump times. It generalizes linear quadratic regulator (LQR) control for trajectory tracking by incorporating jump gains and Riccati equations with state resets, enabling local stabilization of time-varying references in hybrid systems with discontinuous dynamics.

ABSTRACT

This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered jumps in the state evolution, potentially accompanied by mode switching in the control vector field as well. For a given trajectory with state jumps, we show how to construct an approximation of a nearby perturbed trajectory corresponding to a small variation of the initial condition and input. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal to those of the nominal one. The main contribution of this work is the development of a notion of error to clarify in which sense the approximate trajectory is, at each instant of time, a firstorder approximation of the perturbed trajectory. This notion of error naturally finds application in the (local) tracking problem of a time-varying reference trajectory of a hybrid system. To illustrate the possible use of this new error definition in the context of trajectory tracking, we outline how the standard linear trajectory tracking control for nonlinear systems -based on linear quadratic regulator (LQR) theory to compute the optimal feedback gain- could be generalized for hybrid systems.

Motivation & Objective

• To address the challenge of sensitivity analysis in hybrid systems where state jumps cause mismatched jump times between nominal and perturbed trajectories.
• To develop a rigorous first-order approximation of perturbed trajectories in the presence of discontinuous state evolution and mode switching.
• To define a novel error metric that enables local comparison between nominal and perturbed trajectories at each instant, even when jump times differ.
• To generalize linear quadratic regulator (LQR) control for trajectory tracking in hybrid systems by incorporating jump dynamics and time-varying feedback gains.
• To enable local stabilization of time-varying reference trajectories in hybrid systems with state-triggered jumps, particularly when reference and plant jump times do not align.

Proposed method

• Derives the jump gain matrix $ H $, defined in Equation (31), using the implicit function theorem to quantify state jumps under small perturbations.
• Introduces a new error notion that measures the first-order deviation between nominal and perturbed trajectories, accounting for time-shifted jump instants.
• Proposes a two-phase linear quadratic regulator (LQR) formulation for hybrid systems with state jumps, using piecewise-linear dynamics and Riccati equations with jump conditions.
• Models the system dynamics in two intervals: before and after the jump, with state transitions governed by $ z^+ = z^- + Hz^- $, and solves the optimal control problem over a finite horizon.
• Derives the feedback gain $ K(t) = R^{-1}B^T P(t) $, where $ P(t) $ satisfies a Riccati differential equation with a reset condition at the jump time: $ P^-(\tau) = (I+H)^T P^+(\tau)(I+H) $.
• Applies the solution to trajectory tracking by computing a switching feedback law that stabilizes the system around a time-varying reference trajectory, even when jump times differ.

Experimental results

Research questions

• RQ1 How can a first-order approximation of a perturbed trajectory be constructed in hybrid systems with state jumps when the jump times of the nominal and perturbed trajectories do not coincide?
• RQ2 What is a mathematically rigorous and meaningful definition of error that allows local comparison between nominal and perturbed trajectories in hybrid systems with discontinuous dynamics?
• RQ3 How can the standard LQR-based trajectory tracking control be extended to hybrid systems with state-triggered jumps and non-synchronized jump events?
• RQ4 What is the role of the jump gain matrix $ H $ in propagating sensitivity across discontinuities in hybrid systems?
• RQ5 How does the reset condition $ P^-(\tau) = (I+H)^T P^+(\tau)(I+H) $ affect the optimal feedback gain in a hybrid LQR framework?

Key findings

• The paper establishes a new error notion that quantifies the first-order deviation between nominal and perturbed trajectories in hybrid systems with state jumps, even when jump times differ.
• The jump gain $ H $, derived via the implicit function theorem, enables accurate propagation of sensitivity across discontinuous state transitions.
• A two-interval LQR formulation is proposed, with a Riccati equation that includes a reset condition at the jump time, ensuring continuity of the value function across jumps.
• The feedback gain $ K(t) $ is computed via a time-varying Riccati equation with a jump-induced reset, allowing for local stabilization of time-varying reference trajectories.
• The proposed method generalizes standard LQR control to hybrid systems by incorporating jump dynamics, enabling effective trajectory tracking even when plant and reference trajectories have mismatched jump instants.
• The approach is shown to be consistent with trajectory mirroring techniques and opens the door to second-order sensitivity analysis in hybrid systems for numerical optimal control.