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The Miracle of Random Control: Why can you find the perfect control scheme in the noise?
In modern control theory, stochastic control or stochastic optimal control has become an important subfield. This research focuses on uncertainties, including random noise to which observational processes and system evolution are subject. The main purpose of stochastic control is to design a time path of the control variable to achieve the control task as efficiently as possible, despite the presence of various noises and uncertainties.
Stochastic control aims to design the time path of control variables so that even in an uncertain environment, the goal can still be achieved at the lowest cost.
The key to stochastic control technology is the concept of deterministic equivalence. In the linear quadratic Gaussian control model, it is assumed that the model is linear, the objective function is an expectation in quadratic form, and the perturbations are purely additive. For a lumped discrete-time system containing only additive uncertainties, the properties of the optimal control solution are the same as those without considering perturbations.
Deterministic equivalence means that in many cases the optimal control scheme for a stochastic problem can be reduced to an answer to a deterministic problem.
In a discrete-time context, the decision maker observes state variables at each time period and may be affected by observation noise. The goal can be to optimize the expected value for the entire time period, or to optimize the objective function only for the final time period. At each time period, new observations are made and control variables need to be optimally adjusted. It is usually necessary to backtrack and calculate the matrix Riccati equation to find the optimal solution.
Even if the deterministic equivalence does not hold, the optimal control solution for each time period can still be solved back through the Riccati equation.
By applying these concepts, we can actually observe the application of stochastic control in many fields such as finance, engineering and economics. In finance, stochastic control is used to study the allocation of optimal portfolios, taking into account the stochastic returns and risks of different assets.
Stochastic control not only provides solutions in payment structure, but also promotes in-depth thinking about potential risks, thus forming a new way of financial thinking.
As time evolves, continued observation will lead to continuous adjustments of the control variables, with the goal of maximizing the expected value of the function, or its cumulative value, over a period of time. This process involves complex random processes at various points in time, and the search for the optimal solution becomes more difficult. But as technology advances, we now have more powerful computational models to handle these challenges.
In the random control of continuous time, the story becomes more complicated. The control objective here may be the integral of a concave function of the state variable from time zero to the end time T. To achieve these goals, controllers must constantly adjust their strategies amid changes.
The development of stochastic control technology itself has grown rapidly since the 1970s, playing an important role especially in financial applications.
Robert Merton and other scholars use stochastic control to explore the problem of optimal investment portfolios between safe and risky assets. Their research had a profound impact on the development of financial literature and marked the important position of stochastic control in the field of finance. These theories are also applied to the analysis of modern financial crises to help analyze how risks and uncertainties affect economic decision-making.
Furthermore, with the rise of robust model predictive control (MPC) and stochastic model predictive control (SMPC), the practicality and adaptability of modern control theory are being strengthened. This makes the control system more capable of dealing with complex environments and uncertainties than just static or simple models.
Do you also want to know how to use stochastic control technology to help you make more informed decisions in uncertain environments?
