Analysis of Hybrid Dynamical Systems With Uncertainty in Initial Conditions

2015 ◽  
Author(s):  
Yuhua He ◽  
Arpan Mukherjee ◽  
Rahul Rai
Keyword(s):  
Dynamical Systems ◽  
Systems Engineering ◽  
Initial Conditions ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Superior Performance ◽  
Hybrid Dynamical Systems ◽  
Unscented Transform ◽  
Output Variability ◽  
Novel Method

Hybrid dynamical systems (HDS) models are backbone of modeling a myriad of systems found in systems engineering, buildings, manufacturing, auto-pilot control, and chemical processes domains among others. Uncertainty quantification (UQ) techniques to ascertain output variability in HDS with parametric uncertainty is relatively understudied topic. In this paper, we present a novel method to enable UQ of HDSs. Specifically, we outline a computational pipeline to solve different types of Stochastic Hybrid Systems (SHS) with uncertainty in initial conditions. The developed method is based on a numerical integration technique Conjugate Unscented Transform (CUT) and discrete UQ technique. The developed method has been applied to different range of problems including theoretical problems and real life mechanical systems modeled in Simulink environment. Performance of the proposed method has been compared against Monte Carlo and Unscented Transform methods. Results indicate superior performance of the proposed technique over the existing methods.

scholarly journals Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Dynamical Systems ◽  
Optimal Design ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Least Square ◽  
Optimization Process ◽  
Entire Family ◽  
New Framework

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear, have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.

scholarly journals Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Dynamical Systems ◽  
Optimal Design ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Least Square ◽  
Optimization Process ◽  
Entire Family ◽  
New Framework

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work, uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.

Motion Planning of Uncertain Fully-Actuated Dynamical Systems: A Forward Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Least Square ◽  
Forward Dynamics ◽  
Optimal Motion ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain fully-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; ignoring uncertainty during design may lead to poor robustness and suboptimal performance. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through a forward dynamics formulation where deterministic actuator inputs are prescribed and uncertain state trajectories are quantified. The benefits of the ability to quantify the resulting state uncertainty are illustrated in an effort optimal motion planning case-study of a serial manipulator pick-and-place application. The resulting design determines a feasible effort optimal motion plan—subject to actuator and obstacle avoidance constraints—for all possible systems within the probability space. Variance of the system’s terminal conditions are also minimized in a Pareto optimal sense. The inverse dynamics formulation (using deterministic state trajectories and uncertain actuator inputs) is presented in a companion paper.

A Novel Sampling Technique for Probabilistic Static Coverage Problems

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Binbin Zhang ◽  
Nagavenkat Adurthi ◽  
Rahul Rai ◽  
Puneet Singla
Keyword(s):  
Resource Allocation ◽  
San Francisco ◽  
Systems Engineering ◽  
Real Life ◽  
Sampling Technique ◽  
Superior Performance ◽  
Coverage Problem ◽  
Area Of Interest ◽  
Wide Range ◽  
Coverage Problems

Resource allocation in the presence of constraints is an important activity in many systems engineering problems such as surveillance, infrastructure planning, environmental monitoring, and cooperative task performance. The resources in many important problems are agents such as a person, machine, unmanned aerial vehicles (UAVs), infrastructures, and software. Effective execution of a given task is highly correlated with effective allocation of resources to execute the task. An important class of resource allocation problem in the presence of limited resources is static coverage problem. In static coverage problems, it is necessary to allocate resources (stationary configuration of agents) to cover an area of interest so that an event or spatial property of the area can be detected or monitored with high probability. In this paper, we outline a novel sampling algorithm for the static coverage problem in presence of probabilistic resource intensity allocation maps (RIAMs). The key intuition behind our sampling approach is to use the finite number of samples to generate an accurate representation of RIAM. The outlined sampling technique is based on an optimization framework that approximates the RIAM with piecewise linear surfaces on the Delaunay triangles and optimizes the sample placement locations to decrease the difference between the probability distribution and Delaunay triangle surface. Numerical results demonstrate that the algorithm is robust to the initial sample point locations and has superior performance in a wide range of theoretical problems and real-life applications. In a real-life application setting, we demonstrate the efficacy of the proposed algorithm to predict the position of wind stations for monitoring wind speeds across the U.S. The algorithm is also used to give recommendations on the placement of police cars in San Francisco and weather buoys in Pacific Ocean.

Motion Planning of Uncertain Under-Actuated Dynamical Systems: A Hybrid Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Design System ◽  
Double Pendulum ◽  
Hybrid Dynamics ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiencies of this approach enable the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through a hybrid dynamics formulation for under-actuated systems where actuated state and unactuated input trajectories are prescribed and uncertain unactuated states and actuated inputs are quantified. The benefits of the ability to quantify the resulting uncertainties are illustrated in a power optimal motion planning case-study for an inverting double pendulum problem. The resulting design determines a motion plan that minimizes the required input power—subject to actuator and terminal condition variance constraints—for all possible systems within the probability space.

scholarly journals Motion Planning of Uncertain Ordinary Differential Equation Systems

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Inverse Dynamics ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Underactuated Systems ◽  
Forward Dynamics ◽  
Hybrid Dynamics

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.

Motion Planning of Uncertain Fully-Actuated Dynamical Systems: An Inverse Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Inverse Dynamics ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Optimal Motion ◽  
Time Optimal ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain fully-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; ignoring uncertainty during design may lead to poor robustness and suboptimal performance. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through an inverse dynamics formulation where deterministic state trajectories are prescribed and uncertain actuator inputs are quantified. The benefits of the ability to quantify the resulting actuator uncertainty are illustrated in a time optimal motion planning case-study of a serial manipulator pick-and-place application. The resulting design determines a feasible time optimal motion plan—subject to actuator and obstacle avoidance constraints—for all possible systems within the probability space. The forward dynamics formulation (using deterministic actuator inputs and uncertain state trajectories) is presented in a companion paper.

Well-posed hybrid systems and their properties

2012 ◽  
Author(s):  
Rafal Goebel ◽  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Dynamical Systems ◽  
Hybrid Systems ◽  
Hybrid System ◽  
Stability Theory ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Hybrid Dynamical Systems ◽  
Uniqueness Of Solutions ◽  
Classical Setting ◽  
Well Posed

This chapter defines nominally well-posed hybrid systems and well-posed hybrid systems to be those hybrid systems, vaguely speaking, for which graphical limits of graphically convergent sequences of solutions, with no perturbations and with vanishing perturbations, respectively, are still solutions. In a classical setting, a well-posed problem is often defined as one in which a solution exists, is unique, and depends continuously on parameters. For hybrid dynamical systems, insisting on uniqueness of solutions and on their continuous dependence on initial conditions is very restrictive and, as it turns out, not necessary to develop a reasonable stability theory. In fact, stability theory results are possible for a quite general class of hybrid systems. The class of well-posed hybrid systems includes the Krasovskii regularization of a general hybrid system and, more generally, it includes every hybrid system meeting some mild regularity assumptions on the data.

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

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