Fengjiao Liu

Fengjiao Liu, A. Stephen Morse

It is well known that the "fixed spectrum" {i.e., the set of fixed modes} of a multi-channel linear system plays a central role in the stabilization of such a system with decentralized control. A parameterized multi-channel linear system is said to be "structurally complete" if it has no fixed spectrum for almost all parameter values. Necessary and sufficient algebraic conditions are presented for a multi-channel linear system with dependent parameters to be structurally complete. An equivalent graphical condition is also given for a certain type of parameterization.

George Rapakoulias, Ali Reza Pedram, Fengjiao Liu et al.

Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SBs between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SBs, and various other examples. The implementation is publicly available at https://github.com/georgeRapa/GMMflow.

Fengjiao Liu, Yixiao Zhang, Panagiotis Tsiotras

In this paper, we study the reachability of two closely related matrices appearing in the analysis of linear time-varying (LTV) systems over a finite time interval, namely, its closed-loop state transition matrix via a state feedback control and its state covariance matrix starting from some given initial state covariance matrix. Under a mild assumption, we first characterize the set of closed-loop terminal state transition matrices reachable from the identity matrix using controls of the state feedback form. Then, we provide the set of terminal state covariance matrices reachable from any given positive definite initial state covariance matrix when the LTV system is not necessarily controllable. Both results are based on the solutions of corresponding matrix Riccati differential equations (RDE).

George Rapakoulias, Fengjiao Liu, Panagiotis Tsiotras

In this work, we revisit the discrete-time Schrödinger Bridge (SB) and Density Steering (DS) problems for Gaussian mixture model (GMM) boundary distributions. Building on the existing literature, we construct a set of feasible Markovian policies that transport the initial distribution to the final distribution, and are expressed as mixtures of elementary component-to-component optimal policies. We then study the policy optimization within this feasible set in the context of discrete-time SBs and density-steering problems, respectively. We show that for minimum-effort density-steering problems, the proposed policy achieves the same control cost as existing approaches in the literature. For discrete-time SB problems, the proposed policy yields a cost smaller than or equal to that in the literature, resulting in a less conservative approximation. Finally, we study the continuous-time limit of our proposed discrete-time approach and show that it agrees with recently proposed approximations to the continuous-time SB for GMM boundary distributions. We illustrate this new result through two numerical examples.

Fengjiao Liu, A. Stephen Morse

One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair $(A, B)$ whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters which cause the resulting matrix pair to be controllable. In this paper the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair $(A, B)$. Subject to a certain condition on the parameterization called the "binary assumption", an explicit graph-theoretic characterization of such matrix pairs is derived.

Fengjiao Liu, A. Stephen Morse

Conditions for the existence of a fixed spectrum \{i.e., the set of fixed modes\} for a multi-channel linear system have been known for a long time. The aim of this paper is to reestablish one of these conditions using a new and transparent approach based on matroid theory.