Input Hard Constrained Optimal Covariance Steering

Okamoto, K.; Tsiotras, Panagiotis

doi:10.1109/cdc40024.2019.9029353

Cited by 18 publications

(24 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We mention here that chance state constraints were considered for the covariance control problem in [181]. Moreover, input constraints for discrete and continuous time models have been considered in [14,183]. In [207,251], a nonlinear covariance control problem was studied by iteratively solving an approximate linearized problem and by differential dynamic programming, respectively.…”

Section: Stochastic Control and General Bridge Problemsmentioning

confidence: 99%

“…In a twin paper[61], we review the by now vast literature[125,126,222,115,254,50,52,55,56,57,60,118,13,14,15,16,17,114,205,206,181,182,208,183,3,67] on optimal steering of probability distributions for Gauss--Markov models in continuous and discrete time, over a finite or infinite time horizon, with or without state and/or control constraints and applications.Downloaded 11/09/21 to 147.162.213.111 Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms…”

mentioning

confidence: 99%

See 1 more Smart Citation

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Chen¹,

Georgiou²,

Pavon³

2021

SIAM Rev.

View full text Add to dashboard Cite

Schr\" odinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\" odinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\' ar. The problem, known as the Schr\" odinger bridge problem (SBP) with ``uniform"" prior, was more recently recognized as a regularization of the Monge--Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\" odinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938--1940 specifically for Schr\" odinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\" odinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric.In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\" odinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou--Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview \ast

show abstract

Section: Stochastic Control and General Bridge Problemsmentioning

confidence: 99%

mentioning

confidence: 99%

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Chen¹,

Georgiou²,

Pavon³

2021

SIAM Rev.

View full text Add to dashboard Cite

show abstract

“…Lorenzen et al 18 suggested an approach that combines the works of Korda et al 15 and Kouvaritakis et al 16 where a tuning parameter is introduced that allows for shifting priority between performance and an increased feasible region. Covariance steering‐based SMPC 19,20 is a further SMPC approach that ensures recursive feasibility for linear systems with unbounded noise. Recursive feasibility in SMPC for probabilistically constrained Markovian jump linear systems is addressed by Lu et al 21 Recursively feasible SMPC with closed‐loop chance constraint satisfaction for potentially unbounded disturbance distributions is presented in Hewing et al 22 …”

Section: Introductionmentioning

confidence: 99%

Minimization of constraint violation probability in model predictive control

Brüdigam

Gaßmann

Wollherr

et al. 2021

Intl J Robust & Nonlinear

View full text Add to dashboard Cite

For Model Predictive Control in safety‐critical systems it is not only important to bound the probability of constraint violation but to reduce this constraint violation probability as much as possible. Therefore, an approach is necessary that minimizes the constraint violation probability while ensuring that the Model Predictive Control optimization problem remains feasible even under changing uncertainty. We propose a novel two‐step Model Predictive Control scheme that yields a solution with minimal constraint violation probability for a norm constraint in an environment with uncertainty. After minimal constraint violation is guaranteed, the solution is then also optimized with respect to other control objectives. Recursive feasibility and convergence of the method are proved. A simulation demonstrates the effectiveness of the proposed method for a collision avoidance example.

show abstract

“…In general, the theory of steering marginal distributions has a long history stemming from the problem of Schrödinger bridges and optimal mass transport [8], [12]- [14]. Recent work has focused on incorporating physical constraints on the system, such as state chance constraints [15], obstacles in path-planning environments [11], input hard constraints [16], incomplete state information [17], and extensions in the context of stochastic model predictive control [18] and nonlinear systems [19]- [21].…”

Section: Introductionmentioning

confidence: 99%

“…For example, if there are M chance constraints and N time steps, there would be N M total allocations for the whole problem. Previous works [9], [11], [15], [16], [18] have assumed a constant risk allocation, so that the resulting problem can be turned into a semi-definite program (SDP).…”

Section: Introductionmentioning

confidence: 99%

Chance-Constrained Optimal Covariance Steering with Iterative Risk Allocation

Pilipovsky

Tsiotras

2021

2021 American Control Conference (ACC)

Self Cite

View full text Add to dashboard Cite

This paper extends the optimal covariance steering problem for linear stochastic systems subject to chance constraints to account for optimal risk allocation. Previous works have assumed a uniform risk allocation to cast the optimal control problem as a semi-definite program (SDP), which can be solved efficiently using standard SDP solvers. We adopt the Iterative Risk Allocation (IRA) formalism from [1], which uses a twostage approach to solve the optimal risk allocation problem for covariance steering. The upper-stage of IRA optimizes the risk, which is proved to be a convex problem, while the lowerstage optimizes the controller with the new constraints. This is done iteratively so as to find the optimal risk allocation that achieves the lowest total cost. The proposed framework results in solutions that tend to maximize the terminal covariance, while still satisfying the chance constraints, thus leading to less conservative solutions than previous methodologies. We also introduce two novel convex relaxation methods to approximate quadratic chance constraints as second-order cone constraints. We finally demonstrate the approach to a spacecraft rendezvous problem and compare the results.

show abstract

Input Hard Constrained Optimal Covariance Steering

Cited by 18 publications

References 14 publications

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Minimization of constraint violation probability in model predictive control

Chance-Constrained Optimal Covariance Steering with Iterative Risk Allocation

Contact Info

Product

Resources

About