Related Experiment Videos
Hamilton-Jacobi-Bellman equations and approximate dynamic programming on time scales
John Seiffertt1, Suman Sanyal, Donald C Wunsch
1Applied Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA. jes0b4@mst.edu
Summary
This study extends the time scales calculus to approximate dynamic programming, enabling new mathematical approaches for complex problems. The research bridges calculus of time scales and stochastic control for broader applications.
Area of Science:
- Mathematics
- Applied Mathematics
- Dynamic Systems
Background:
- The calculus of time scales is an emerging mathematical field with broad multidisciplinary potential.
- Approximate dynamic programming (ADP) is a powerful technique for solving complex control problems.
Purpose of the Study:
- To extend the calculus of time scales to approximate dynamic programming.
- To develop new mathematical frameworks for dynamic programming on isolated time scales.
Main Methods:
- Extending the backward induction algorithm of dynamic programming to all isolated time scales.
- Motivating and proving Hamilton-Jacobi-Bellman equations on time scales.
Main Results:
- The traditional discrete dynamic programming is generalized to the time scales calculus.
- Hamilton-Jacobi-Bellman equations are established and proven on time scales.
Conclusions:
- This work connects the calculus of time scales with stochastic control via ADP.
- The findings open new avenues for research in both theoretical mathematics and applied control theory.
Related Concept Videos
01:21Linear Approximation in Time Domain
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
01:24Fast Decoupled and DC Powerflow
The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
01:29Application of Linearization and Approximation
A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
01:23Linear time-invariant Systems
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
01:30Lagrange Multipliers: Problem Solving
A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.The total volume of the silo is obtained by...