Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

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

Backstepping controller synthesis and characterizations of incremental stability

Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in either parametric-strict-feedback or strict-feedback form, and they require these control systems to be smooth. In this paper, we propose a design technique that is applicable to larger classes of (not necessarily smooth) control systems. Moreover, we propose a recursive way of constructing contraction metrics (for smooth control systems) and incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems, the approximation of stochastic hybrid systems, source-code model checking for nonlinear dynamical systems and so on. The effectiveness of the proposed results in this paper is illustrated by synthesizing a controller rendering a non-smooth control system incrementally stable as well as constructing its finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure

Finite Horizon Privacy of Stochastic Dynamical Systems:A Synthesis Framework for Dependent Gaussian Mechanisms

We address the problem of synthesizing distorting mechanisms that maximize privacy of stochastic dynamical systems. Information about the system state is obtained through sensor measurements. This data is transmitted to a remote station through an unsecured/public communication network. We aim to keep part of the system state private (a private output); however, because the network is unsecured, adversaries might access sensor data and input signals, which can be used to estimate private outputs. To prevent an accurate estimation, we pass sensor data and input signals through a distorting (privacy-preserving) mechanism before transmission, and send the distorted data to the trusted user. These mechanisms consist of a coordinate transformation and additive dependent Gaussian vectors. We formulate the synthesis of the distorting mechanisms as a convex program, where we minimize the mutual information (our privacy metric) between an arbitrarily large sequence of private outputs and the disclosed distorted data for desired distortion levels -- how different actual and distorted data are allowed to be

Sampled-data extremum-seeking framework for constrained optimization of nonlinear dynamical systems

Most extremum-seeking control (ESC) approaches focus solely on the problem of finding the extremum of some unknown, steady-state input–output map, providing parameter settings that lead to optimal steady-state system performance. However, many industrial applications also have to deal with constraints on operating conditions due to, e.g., actuator limitations, limitations on tunable system parameters, or constraints on measurable variables. In particular, constraints on measurable variables are typically unknown in terms of their relationship with the tunable system parameters. In addition, the constraints on system inputs as a result of the constraints on measurable variables may conflict with the otherwise optimal operational condition, and hence should be taken into account in the data-based optimization approach. In this work, we propose a sampled-data extremum-seeking framework for the constrained optimization of a class of nonlinear dynamical systems with measurable constrained variables. In this framework, barrier function methods are employed, exploiting both the objective function and constraint functions which are available through output measurement only. We show, under the assumption that the parametric initialization yield operating conditions that do not violate the constraints, that (1) the resulting closed-loop dynamics is stable, (2) constraint satisfaction of the inputs is guaranteed for all iterations of the optimization process, and (3) constrained optimization is achieved. We illustrate the working principle of the proposed framework by means of an industrial case study of the constrained optimization of extreme ultraviolet light generation in a laser-produced plasma source within a state-of-the-art lithography system.</p

Gaussian Mechanisms Against Statistical Inference:Synthesis Tools

In this manuscript, we provide a set of tools (in terms of semidefinite programs) to synthesize Gaussian mechanisms to maximize privacy of databases. Information about the database is disclosed through queries requested by (potentially) adversarial users. We aim to keep part of the database private (private sensitive information); however, disclosed data could be used to estimate private information. To avoid an accurate estimation by the adversaries, we pass the requested data through distorting (privacy-preserving) mechanisms before transmission and send the distorted data to the user. These mechanisms consist of a coordinate transformation and an additive dependent Gaussian vector. We formulate the synthesis of distorting mechanisms in terms of semidefinite programs in which we seek to minimize the mutual information (our privacy metric) between private data and the disclosed distorted data given a desired distortion level -- how different actual and distorted data are allowed to be

From Low to High Order Motion Planners:Safe Robot Navigation Using Motion Prediction and Reference Governor

Safe navigation around obstacles is a fundamental challenge for highly dynamic robots. The state-of-the-art approach for adapting simple reference path planners to complex robot dynamics using trajectory optimization and tracking control is brittle and requires significant replanning cycles. In this letter, we introduce a novel feedback motion planning framework that extends the applicability of low-order (e.g. position-/velocity-controlled) reference motion planners to high-order (e.g., acceleration-/jerk-controlled) robot models using motion prediction and reference governors. We use predicted robot motion range for safety assessment and establish a continuous bidirectional interface between high-level planning and low-level control via a reference governor. We describe the generic fundamental building blocks of our feedback motion planning framework and give specific example designs for motion control, prediction, and reference planning. We prove the correctness of our planning framework and demonstrate its performance in numerical simulations. We conclude that accurate motion prediction is crucial for closing the gap between high-level planning and low-level control.</p

Privacy-Preserving Federated Learning via System Immersion and Random Matrix Encryption

Federated learning (FL) has emerged as a privacy solution for collaborative distributed learning where clients train AI models directly on their devices instead of sharing their data with a centralized (potentially adversarial) server. Although FL preserves local data privacy to some extent, it has been shown that information about clients' data can still be inferred from model updates. In recent years, various privacy-preserving schemes have been developed to address this privacy leakage. However, they often provide privacy at the expense of model performance or system efficiency and balancing these tradeoffs is a crucial challenge when implementing FL schemes. In this manuscript, we propose a Privacy-Preserving Federated Learning (PPFL) framework built on the synergy of matrix encryption and system immersion tools from control theory. The idea is to immerse the learning algorithm, a Stochastic Gradient Decent (SGD), into a higher-dimensional system (the so-called target system) and design the dynamics of the target system so that: the trajectories of the original SGD are immersed/embedded in its trajectories, and it learns on encrypted data (here we use random matrix encryption). Matrix encryption is reformulated at the server as a random change of coordinates that maps original parameters to a higher-dimensional parameter space and enforces that the target SGD converges to an encrypted version of the original SGD optimal solution. The server decrypts the aggregated model using the left inverse of the immersion map. We show that our algorithm provides the same level of accuracy and convergence rate as the standard FL with a negligible computation cost while revealing no information about the clients' data

Feedback Motion Prediction for Safe Unicycle Robot Navigation

As a simple and robust mobile robot base, differential drive robots that can be modelled as a kinematic unicycle find significant applications in logistics and service robotics in both industrial and domestic settings. Safe robot navigation around obstacles is an essential skill for such unicycle robots to perform diverse useful tasks in complex cluttered environments, especially around people and other robots. Fast and accurate safety assessment plays a key role in reactive and safe robot motion design. In this paper, as a more accurate and still simple alternative to the standard circular Lyapunov level sets, we introduce novel conic feedback motion prediction methods for bounding the close-loop motion trajectory of the kinematic unicycle robot model under a standard unicycle motion control approach. We present an application of unicycle feedback motion prediction for safe robot navigation around obstacles using reference governors, where the safety of a unicycle robot is continuously monitored based on the predicted future robot motion. We investigate the role of motion prediction on robot behaviour in numerical simulations and conclude that fast and accurate feedback motion prediction is key for fast, reactive, and safe robot navigation around obstacles.Comment: 11 pages, 5 figures, extended version of a paper submitted to a conference publicatio