James A. Primbs

Stochastic Receding Horizon Control of Constrained Linear Systems With State and Control Multiplicative Noise

We develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain constraints. Our receding horizon formulation is based upon an on-line optimization that utilizes open-loop plus linear feedback and is solved as a semi-definite programming problem. We also provide a characterization of stability, performance, and constraint satisfaction properties of the receding horizon controlled system under a specific choice of terminal weight and terminal constraint. A simple numerical example is used to illustrate the approach.

A receding horizon generalization of pointwise min-norm controllers

Control Lyapunov functions (CLFs) are used in conjunction with receding horizon control to develop a new class of receding horizon control schemes. In the process, strong connections between the seemingly disparate approaches are revealed, leading to a unified picture that ties together the notions of pointwise min-norm, receding horizon, and optimal control. This framework is used to develop a CLF based receding horizon scheme, of which a special case provides an appropriate extension of Sontags formula. The scheme is first presented as an idealized continuous-time receding horizon control law. The issue of implementation under discrete-time sampling is then discussed as a modification. These schemes are shown to possess a number of desirable theoretical and implementation properties. An example is provided, demonstrating their application to a nonlinear control problem. Finally, stronger connections to both optimal and pointwise min-norm control are proved.

Brief Feasibility and stability of constrained finite receding horizon control

Issues of feasibility and stability are considered for a finite horizon formulation of receding horizon control for linear systems under mixed linear state and control constraints. We prove that given any compact set of initial conditions that is feasible for the infinite horizon problem, there exists a finite horizon length above which a receding horizon policy will provide both feasibility and stability, even when no end or stability constraint is imposed. Finally, computations for determining a sufficient horizon length are carried out on a simple open-loop stable example under control saturation constraints.

Optimal pairs trading: A stochastic control approach

In this paper, we propose a stochastic control approach to the problem of pairs trading. We model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem. We are able to obtain the optimal solution to this control problem in closed form via the corresponding Hamilton-Jacobi-Bellman equation. We also provide closed form maximum-likelihood estimation values for the parameters in the model. The approach is illustrated with a numerical example involving simulated data for a pair of stocks.

Dynamic hedging of basket options under proportional transaction costs using receding horizon control

In this article, we develop a semi-definite programming-based receding horizon control approach to the problem of dynamic hedging of European basket call options under proportional transaction costs. The hedging problem for a European call option is formulated as a finite horizon constrained stochastic control problem. This allows us to develop a receding horizon control approach that repeatedly solves semi-definite programmes on-line in order to dynamically hedge. This approach is competitive with Black–Scholes delta hedging in the one-dimensional case with no transaction costs, but it also applies to multi-dimensional options such as basket options, and can include transaction costs. We illustrate its effectiveness through a numerical example involving an option on a basket of five stocks.

Brief Comparison of nonlinear control design techniques on a model of the Caltech ducted fan

In this paper we compare different nonlinear control design methods by applying them to the planar model of a ducted fan engine. The methods used range from Jacobian linearization of the nonlinear plant and designing an LQR controller, to using model predictive control and linear parameter varying methods. The controller design can be divided into two steps. The first step requires the derivation of a control Lyapunov function (CLF), while the second involves using an existing CLF to generate a controller. The main premise of this paper is that by combining the best of these two phases, it is possible to find controllers that achieve superior performance when compared to those that apply each phase independently. All of the results are compared to the optimal solution which is approximated by solving a trajectory optimization problem with a sufficiently large time horizon.

A soft constraint approach to stochastic receding horizon control

This paper presents a soft constraint approach to constrained stochastic receding horizon control for linear systems with state and control multiplicative noise. We formulate an on-line optimization that penalizes constraint violations and can be solved as a semi-definite program. Additionally, we prove stability results that guarantee asymptotic stability with probability one. A simple numerical example illustrates the approach.

Stochastic Receding Horizon Control of Constrained Linear Systems with State and Control Multiplicative Noise

In this paper we develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain linear and quadratic expectation constraints. Our receding horizon formulation is based upon an online optimization that utilizes open-loop plus linear feedback and is solved as a semi-definite programming problem. We also provide a result characterizing stochastic stability for the receding horizon controlled system under a specific choice of terminal weight and terminal constraint. A numerical example is used to illustrate the approach.

Brief The analysis of optimization based controllers

Many control techniques employ on-line optimization in the determination of a control policy. We develop a framework which provides sufficient convex conditions, in the form of linear matrix inequalities, for the analysis of constrained quadratic based optimization schemes. These results encompass standard robustness analysis problems for a wide variety of receding horizon control schemes, including those with polytopic and structured uncertainty. A simple example illustrates the methodology.

Portfolio Optimization Applications of Stochastic Receding Horizon Control

This paper develops stochastic receding horizon control for constrained dynamic portfolio optimization problems. In particular, we formulate two portfolio optimization problems. The first is that of risk adjusted wealth maximization, while the second is the problem of optimally tracking an index of stocks with fewer stocks. We consider both of these problems subject to probabilistic chance constraints. By modeling the dynamics in the problems as linear systems subject to state and control multiplicative noise, and approximating linear chance constraints with quadratic expectation constraints, we show that both can be approached using stochastic receding horizon control. In particular, we use a closed loop version of stochastic receding horizon control where the on-line optimization is solved as a semi-definite program. Numerical examples demonstrate the computations involved in these problems and indicate that stochastic receding horizon control is a promising new approach to constrained portfolio optimization problems.