Tryphon Georgiou

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We consider the problem to steer a linear dynamical system with full state observation from an initial gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise" forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitely by solving a pair of differential Lyapunov equations that are coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schrödinger bridges) and therefore, the present case of steering a linear stochastic system constitutes a Schrödinger bridge for possibly degenerate diffusions. Our results, however, provide the first implementable form of the optimal control for a general Gauss-Markov process. Illustrative examples of the optimal evolution and control for inertial particles and a stochastic oscillator are provided. A final result establishes directly the property of Schrödinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ($\dot x(t) = u(t)$) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrödinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.

Yongxin Chen, Tryphon Georgiou, Allen Tannenbaum

We consider damped stochastic systems in a controlled (time-varying) quadratic potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermodynamics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via and the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.

Yongxin Chen, Tryphon Georgiou

We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed to have known states at the boundary. Thus, the movement of the particles can be modeled as an Ornstein-Uhlenbeck process conditioned on position and velocity measurements at the two end-points. It is shown that optimal stochastic control provides a stochastic differential equation (SDE) that generates such a bridge as a degenerate diffusion process. Generalizations to higher order linear diffusions are considered.

Lipeng Ning, Xianhua Jiang, Tryphon Georgiou

We consider problems of estimation of structured covariance matrices, and in particular of matrices with a Toeplitz structure. We follow a geometric viewpoint that is based on some suitable notion of distance. To this end, we overview and compare several alternatives metrics and divergence measures. We advocate a specific one which represents the Wasserstein distance between the corresponding Gaussians distributions and show that it coincides with the so-called Bures/Hellinger distance between covariance matrices as well. Most importantly, besides the physically appealing interpretation, computation of the metric requires solving a linear matrix inequality (LMI). As a consequence, computations scale nicely for problems involving large covariance matrices, and linear prior constraints on the covariance structure are easy to handle. We compare this transportation/Bures/Hellinger metric with the maximum likelihood and the Burg methods as to their performance with regard to estimation of power spectra with spectral lines on a representative case study from the literature.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We consider a variant of the classical linear quadratic Gaussian regulator (LQG) in which penalties on the endpoint state are replaced by the specification of the terminal state distribution. The resulting theory considerably differs from LQG as well as from formulations that bound the probability of violating state constraints. We develop results for optimal state-feedback control in the two cases where i) steering of the state distribution is to take place over a finite window of time with minimum energy, and ii) the goal is to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the distribution of noise and state are Gaussian. In the first case, we show that provided the system is controllable, the state can be steered to any terminal Gaussian distribution over any specified finite time-interval. In the second case, we characterize explicitly the covariance of admissible stationary state distributions that can be maintained with constant state-feedback control. The conditions for optimality are expressed in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. In the case where the noise and control share identical input channels, the Riccati equations for finite-horizon steering become homogeneous and can be solved in closed form. The present paper is largely based on our recent work in arxiv.org/abs/1408.2222, arxiv.org/abs/1410.3447 and presents an overview of certain key results.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We present an overview of our recent work on implementable solutions to the Schroedinger bridge problem and their potential application to optimal transport and various generalizations.

Shreyas Fadnavis, Hamza Farooq, Maryam Afzali et al.

Fitting multi-exponential models to Diffusion MRI (dMRI) data has always been challenging due to various underlying complexities. In this work, we introduce a novel and robust fitting framework for the standard two-compartment IVIM microstructural model. This framework provides a significant improvement over the existing methods and helps estimate the associated diffusion and perfusion parameters of IVIM in an automatic manner. As a part of this work we provide capabilities to switch between more advanced global optimization methods such as simplicial homology (SH) and differential evolution (DE). Our experiments show that the results obtained from this simultaneous fitting procedure disentangle the model parameters in a reduced subspace. The proposed framework extends the seminal work originated in the MIX framework, with improved procedures for multi-stage fitting. This framework has been made available as an open-source Python implementation and disseminated to the community through the DIPY project.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We study feedback control of coupled nonlinear stochastic oscillators in a force field. We first consider the problem of asymptotically driving the system to a desired {\em steady state} corresponding to reduced thermal noise. Among the feedback controls achieving the desired asymptotic transfer, we find that the most efficient one {from an energy point of view} is characterized by {\em time-reversibility}. We also extend the theory of Schrödinger bridges to this model, thereby steering the system in {\em finite time} and with minimum effort to a target steady-state distribution. The system can then be maintained in this state through the optimal steady-state feedback control. The solution, in the finite-horizon case, involves a space-time harmonic function $φ$, and $-\logφ$ plays the role of an artificial, time-varying potential in which the desired evolution occurs. This framework appears extremely general and flexible and can be viewed as a considerable generalization of existing active control strategies such as macromolecular cooling. In the case of a quadratic potential, the results assume a form particularly attractive from the algorithmic viewpoint as the optimal control can be computed via deterministic matricial differential equations. An example involving inertial particles illustrates both transient and steady state optimal feedback control.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schrödinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schrödinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.

Yongxin Chen, Tryphon Georgiou, Michele Pavon

We consider the problem of minimum energy steering of a linear stochastic system to a final prescribed distribution over a finite horizon and to maintain a stationary distribution over an infinite horizon. We present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and algebraic in the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation. We finally present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.