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Some results on the time-domain structure of linear, time-invariant systems over quaternions are presented, both in the continuous and in the discrete-time case. Within the behavioral approach, a system is defined as a set of trajectories (functions or sequences). In this paper, the trajectories are solutions of linear differential or difference equation with constant coefficients which belong to...
In this paper we focus on a particular class of nonlinear dynamical systems given by polynomial vector fields in rectangular domains (boxes). This is a generalization of the work of Belta and Habets dealing with multi-affine dynamical systems on rectangles. The main idea is to use the blossoming principle which allows us to relate our polynomial dynamical system to a multi-affine one. This technique...
Hybrid dynamical systems can exhibit many unique phenomena, such as Zeno behavior. Zeno behavior is the occurrence of infinite discrete transitions in finite time. Zeno behavior has been likened to a form of finite-time asymptotic stability, and corresponding Lyapunov theorems have been developed. In this paper, we propose a method to construct Lyapunov functions to prove Zeno stability of compact...
We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved...
This paper considers the class of deterministic continuous-time optimal control problems (OCPs) with piecewise-affine (PWA) vector field, polynomial Lagrangian and semialgebraic input and state constraints. The OCP is first relaxed as an infinite-dimensional linear program (LP) over a space of occupation measures. This LP is then approached by an asymptotically converging hierarchy of linear matrix...
In this paper a definition for the property of behavioral invariance is proposed with the purpose of generalizing the state space geometric approach to the behavioral setting. Based on this notion together with the well-known notion of behavioral observer, a definition of conditioned invariance is also presented. The results obtained for the characterization of the defined properties put into evidence...
We study Hamiltonian systems, namely, systems comprising of trajectories which are ‘stationary’ with respect to a quadratic performance index: they play a central role in many optimal control problems. A typical assumption in the literature is that of ‘regularity’: the resulting first-order dynamical system is a regular state space system and not a singular descriptor system. In this paper we show...
In this paper, we deal with the real-time dynamic traffic assignment (DTA) problem using a nonlinear traffic flow model. A feedback control based on flatness technique is introduced. The control objective is to minimize the difference of the travel time on alternate routes in a road network which have the same origin and destination. Accordingly, congestion could be reduced or eliminated. Some numerical...
We present a technique for synthesizing switching guards for hybrid systems by using sum of squares (SOS) programming. The guards are defined to be semialgebraic sets calculated from a bilinear SOS program. We present a method for ensuring that synthesized guards satisfy a state-based safety constraint and do not allow Zeno executions.We use an iterative algorithm to solve the bilinear program and...
In this paper we present a novel approach to the analysis of nonnegative dynamical systems whose vector fields are polynomial or rational functions. Our analysis framework is based on results developed and presented in a previous study on general conditions that imply non-vanishing of polynomial functions on the positive orthant. This approach is due to the sparsity of the negative terms in the polynomial,...
Reversibility of dynamical processes arises in many physical dynamical systems. For example, lossless Newtonian and Hamiltonian mechanical systems exhibit trajectories that can be obtained by time going forward and backward, providing an example of time symmetry that arises in natural sciences. Another example of such time symmetry is the phenomenon known as Poincaré recurrence wherein the dynamical...
In this paper two-dimensional (2D) discrete behaviors, defined on the grid ℤ+ × ℤ and having the time as (first) independent variable, are investigated. For these behaviors, by emphasizing the causality notion that is naturally associated with the time variable, we introduce two new concepts of controllability. Algebraic characterizations of time-controllability and of zero-time-controllability are...
Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending...
The problem of time optimal feedback control of a single input, continuous time, linear time invariant (LTI) system is considered. The control input is constrained to obey |u(t)| ≤ 1. It is known that the solution to this problem is bang-bang with the input switching between the extreme values ±1 according to some “switching surfaces” in the state space. It is shown that for a class of LTI systems,...
This paper describes recent progress in the study of switching linear systems i.e. linear systems whose dynamics can switch among a family of possible configurations/modes. We focus the attention on closed-loop mode-observability, namely the problem of identifying the active (unknown) mode of the system from closed-loop data. The analysis focuses on two fundamental questions: i) How the control objectives...
Instances of control systems are presented for which dynamical feedback linearizability can be assessed from differential forms of highest relative degree after some “contact” transformation has been applied. This bypasses the need to find a polynomial differential operator that leads to an integrable co-distribution.
Mission abort scenarios of RLV due to an engine failure which leads to engine cut-off are considered. the abort trajectory optimization problem is treated as a maximum range problem for abort capability analysis. an hp-adaptive pseudospectral method be employed to solve this problem because of it's accuracy and computational efficiency. Fuel jettisoning rate is treated as a optimization parameter...
We propose three novel methods to evaluate a distance function for robotic motion planning based on semiinfinite programming (SIP) framework; these methods include golden section search (GSS), conservative advancement (CA) and a hybrid of GSS and CA. The distance function can have a positive and a negative value, each of which corresponds to the Euclidean distance and penetration depth, respectively...
This work presents a novel approach for collision assessment for automotive environments. As collision prevention and risk analysis are key challenges for today's intelligent transport systems, sophisticated solutions for a collision-free trajectory generation get indispensable. The presented collision checker is integrated into an optimal control based planning framework that generates minimum jerk...
Reinforcement learning techniques have been developed to solve difficult learning control problems having small amount of a priori knowledge about the system dynamics. In this paper, a simple unstable exemplar test problem is proposed to investigate issues in parametric convergence of the value function. A specific closed-form solution for the value function is determined which has a polynomial form...
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