[Paper Review] On the optimal control problem for a class of monotone bilinear systems

Neil K. Dhingra, Marcello Colombino+3 more|arXiv (Cornell University)|Nov 30, 2016

Diabetes and associated disorders1 references3 citations

TL;DR

This paper formulates an infinite-horizon optimal control problem for monotone bilinear systems, proving that the optimal control input is constant over time and can be computed via finite-dimensional non-smooth convex optimization. The key contribution is a subgradient algorithm for computing the optimal controller and an extension to robust control under model uncertainty, with applications to HIV combination drug therapy design.

ABSTRACT

We consider a class of monotone systems in which the control signal multiplies the state. Among other applications, such bilinear systems can be used to model the evolutionary dynamics of HIV in the presence of combination drug therapy. For this class of systems, we formulate an infinite horizon optimal control problem, prove that the optimal control signal is constant over time, and show that it can be computed by solving a finite-dimensional non-smooth convex optimization problem. We provide an explicit expression for the subdifferential set of the objective function and use a subgradient algorithm to design the optimal controller. We further extend our results to characterize the optimal robust controller for systems with uncertain dynamics and show that computing the robust controller is no harder than computing the nominal controller. We illustrate our results with an example motivated by combination drug therapy.

Motivation & Objective

• To address the optimal control of monotone bilinear systems arising in biological and physical processes with nonnegative states, such as HIV viral dynamics.
• To formulate an infinite-horizon optimal control problem using the induced power norm as a performance metric, generalizing H∞ performance.
• To prove that the optimal control input is constant in time, simplifying controller design for such systems.
• To develop a subgradient algorithm for solving the resulting non-smooth convex optimization problem.
• To extend the framework to robust control under model uncertainty, showing that robust controller design is no more complex than nominal design.

Proposed method

• Formulates the optimal control problem for a class of monotone bilinear systems where control multiplies the state, with dynamics governed by a Metzler matrix and control input affecting the system via a diagonal matrix.
• Defines the objective function as the induced power norm of the system transfer function, leading to a non-smooth convex optimization problem in the scalar control gain.
• Derives an explicit expression for the subdifferential of the objective function to enable subgradient-based optimization.
• Applies a subgradient algorithm to compute the optimal constant control input that minimizes the objective function.
• Extends the framework to handle model uncertainty by incorporating structured perturbations in the system matrix, using a robust optimization approach.
• Demonstrates that robust controller design reduces to solving a similar convex problem as the nominal case, preserving computational tractability.

Experimental results

Research questions

• RQ1 Can the optimal control input for a monotone bilinear system be shown to be constant over time under an infinite-horizon performance criterion?
• RQ2 Is the optimal control problem for such systems amenable to finite-dimensional convex optimization despite non-smoothness?
• RQ3 Can robust control be designed for systems with uncertain dynamics without increasing computational complexity beyond the nominal case?
• RQ4 How does the optimal control input derived via convex optimization compare to heuristic or time-varying approaches in biological applications like HIV therapy?
• RQ5 What is the impact of structural uncertainty—such as an uncertain mutation pathway in a viral network—on controller performance and stability?

Key findings

• The optimal control input for the studied class of monotone bilinear systems is constant over time, simplifying implementation and analysis.
• The optimal controller can be computed by solving a finite-dimensional non-smooth convex optimization problem, with an explicit subdifferential expression enabling subgradient methods.
• For the HIV-inspired example with n=10 mutants, r=1, and ρ=3, the nominal optimal control input is u_nom = 1.5936, while the robust controller is u_rob = 1.9413.
• The nominal controller becomes unstable when the uncertain mutation pathway c ≥ 0.0054, whereas the robust controller remains stable up to c ≥ 0.5461.
• The robust controller is designed to handle uncertainty in the mutation pathway (e.g., a cyclic link from x_n to x_1), with the uncertainty bounded by α_1n = 0.1.
• The impulse response analysis confirms that the robust controller is significantly more resilient to model perturbations than the nominal controller, especially under uncertainty.