arXiv subjects

Evangelos A. Theodorou

Publications and source records attributed to Evangelos A. Theodorou.

At least 73 records · Page 4Linked to original sources

arXiv2022

Decentralized Safe Multi-agent Stochastic Optimal Control using Deep FBSDEs and ADMM

In this work, we propose a novel safe and scalable decentralized solution for multi-agent control in the presence of stochastic disturbances. Safety is mathematically encoded using stochastic control barrier functions and safe controls are computed by solving quadratic programs. Decentralization is achieved by augmenting to each agent's optimization variables, copy variables, for its neighbors. This allows us to decouple the centralized multi-agent optimization problem. However, to ensure safety, neighboring agents must agree on "what is safe for both of us" and this creates a need for consensus. To enable safe consensus solutions, we incorporate an ADMM-based approach. Specifically, we propose a Merged CADMM-OSQP implicit neural network layer, that solves a mini-batch of both, local quadratic programs as well as the overall consensus problem, as a single optimization problem. This layer is embedded within a Deep FBSDEs network architecture at every time step, to facilitate end-to-end differentiable, safe and decentralized stochastic optimal control. The efficacy of the proposed approach is demonstrated on several challenging multi-robot tasks in simulation. By imposing requirements on safety specified by collision avoidance constraints, the safe operation of all agents is ensured during the entire training process. We also demonstrate superior scalability in terms of computational and memory savings as compared to a centralized approach.

cs.MA↗

arXiv2022

Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control

In this paper, we present a scalable deep learning approach to solve opinion dynamics stochastic optimal control problems with mean field term coupling in the dynamics and cost function. Our approach relies on the probabilistic representation of the solution of the Hamilton-Jacobi-Bellman partial differential equation. Grounded on the nonlinear version of the Feynman-Kac lemma, the solutions of the Hamilton-Jacobi-Bellman partial differential equation are linked to the solution of Forward-Backward Stochastic Differential Equations. These equations can be solved numerically using a novel deep neural network with architecture tailored to the problem in consideration. The resulting algorithm is tested on a polarized opinion consensus experiment. The large-scale (10K) agents experiment validates the scalability and generalizability of our algorithm. The proposed framework opens up the possibility for future applications on extremely large-scale problems.

cs.MA↗

arXiv2022

Parameterized Differential Dynamic Programming

Differential Dynamic Programming (DDP) is an efficient trajectory optimization algorithm relying on second-order approximations of a system's dynamics and cost function, and has recently been applied to optimize systems with time-invariant parameters. Prior works include system parameter estimation and identifying the optimal switching time between modes of hybrid dynamical systems. This paper generalizes previous work by proposing a general parameterized optimal control objective and deriving a parametric version of DDP, titled Parameterized Differential Dynamic Programming (PDDP). A rigorous convergence analysis of the algorithm is provided, and PDDP is shown to converge to a minimum of the cost regardless of initialization. The effects of varying the optimization to more effectively escape local minima are analyzed. Experiments are presented applying PDDP on multiple robotics systems to solve model predictive control (MPC) and moving horizon estimation (MHE) tasks simultaneously. Finally, PDDP is used to determine the optimal transition point between flight regimes of a complex urban air mobility (UAM) class vehicle exhibiting multiple phases of flight.

math.OC↗

arXiv2022

Opinion Dynamics Steering using Stochastic Search

In this paper, we apply the stochastic search dynamic optimization framework for steering opinion dynamics in a partially active population. The framework is grounded on stochastic optimization theory and relies on sampling of candidate solutions from distributions of the exponential family. We derive the distribution parameter update for an open loop, a feedback, and a novel adaptive feedback policy. All three policies are tested on two different opinion dynamics steering scenarios in simulation and compared against a hand designed feedback policy. The results showcase the effectiveness the framework for opinion dynamics control and the robustness of the adaptive feedback policy with respect to the active agent set size.

math.OC↗

arXiv2022

Barrier States Embedded Iterative Dynamic Game for Robust and Safe Trajectory Optimization

Considering uncertainties and disturbances is an important, yet challenging, step in successful decision making. The problem becomes more challenging in safety-constrained environments. In this paper, we propose a robust and safe trajectory optimization algorithm through solving a constrained min-max optimal control problem. The proposed method leverages a game theoretic differential dynamic programming approach with barrier states to handle parametric and non-parametric uncertainties in safety-critical control systems. Barrier states are embedded into the differential game's dynamics and cost to portray the constrained environment in a higher dimensional state space and certify the safety of the optimized trajectory. Moreover, to find a convergent optimal solution, we propose to perform line-search in a Stackleberg (leader-follower) game fashion instead of picking a constant learning rate. The proposed algorithm is evaluated on a velocity-constrained inverted pendulum model in a moderate and high parametric uncertainties to show its efficacy in such a comprehensible system. The algorithm is subsequently implemented on a quadrotor in a windy environment in which sinusoidal wind turbulences applied in all directions.

eess.SY↗

arXiv2022

Maximum Entropy Differential Dynamic Programming

In this paper, we present a novel maximum entropy formulation of the Differential Dynamic Programming algorithm and derive two variants using unimodal and multimodal value functions parameterizations. By combining the maximum entropy Bellman equations with a particular approximation of the cost function, we are able to obtain a new formulation of Differential Dynamic Programming which is able to escape from local minima via exploration with a multimodal policy. To demonstrate the efficacy of the proposed algorithm, we provide experimental results using four systems on tasks that are represented by cost functions with multiple local minima and compare them against vanilla Differential Dynamic Programming. Furthermore, we discuss connections with previous work on the linearly solvable stochastic control framework and its extensions in relation to compositionality.

math.OC↗

arXiv2022

HJB Based Optimal Safe Control Using Control Barrier Functions

This work proposes an optimal safe controller minimizing an infinite horizon cost functional subject to control barrier functions (CBFs) safety conditions. The constrained optimal control problem is reformulated as a minimization problem of the Hamilton-Jacobi-Bellman (HJB) equation subjected to the safety constraints. By solving the optimization problem, we are able to construct a closed form solution that satisfies optimality and safety conditions. The proposed solution is shown to be continuous and thus it renders the safe set forward invariant while minimizing the given cost. Hence, optimal stabilizability and safety objectives are achieved simultaneously. To synthesize the optimal safe controller, we present a modified Galerkin successive approximation approach which guarantees an optimal safe solution given a stabilizing safe initialization. The proposed algorithm is implemented on a constrained nonlinear system to show its efficacy.

eess.SY↗

arXiv2022

Multimodal Maximum Entropy Dynamic Games

Environments with multi-agent interactions often result a rich set of modalities of behavior between agents due to the inherent suboptimality of decision making processes when agents settle for satisfactory decisions. However, existing algorithms for solving these dynamic games are strictly unimodal and fail to capture the intricate multimodal behaviors of the agents. In this paper, we propose MMELQGames (Multimodal Maximum-Entropy Linear Quadratic Games), a novel constrained multimodal maximum entropy formulation of the Differential Dynamic Programming algorithm for solving generalized Nash equilibria. By formulating the problem as a certain dynamic game with incomplete and asymmetric information where agents are uncertain about the cost and dynamics of the game itself, the proposed method is able to reason about multiple local generalized Nash equilibria, enforce constraints with the Augmented Lagrangian framework and also perform Bayesian inference on the latent mode from past observations. We assess the efficacy of the proposed algorithm on two illustrative examples: multi-agent collision avoidance and autonomous racing. In particular, we show that only MMELQGames is able to effectively block a rear vehicle when given a speed disadvantage and the rear vehicle can overtake from multiple positions.

math.OC↗

arXiv2022

Safety Embedded Differential Dynamic Programming Using Discrete Barrier States

Certified safe control is a growing challenge in robotics, especially when performance and safety objectives must be concurrently achieved. In this work, we extend the barrier state (BaS) concept, recently proposed for safe stabilization of continuous time systems, to safety embedded trajectory optimization for discrete time systems using discrete barrier states (DBaS). The constructed DBaS is embedded into the discrete model of the safety-critical system integrating safety objectives into the system's dynamics and performance objectives. Thereby, the control policy is directly supplied by safety-critical information through the barrier state. This allows us to employ the DBaS with differential dynamic programming (DDP) to plan and execute safe optimal trajectories. The proposed algorithm is leveraged on various safety-critical control and planning problems including a differential wheeled robot safe navigation in randomized and complex environments and on a quadrotor to safely perform reaching and tracking tasks. The DBaS-based DDP (DBaS-DDP) is shown to consistently outperform penalty methods commonly used to approximate constrained DDP problems as well as CBF-based safety filters.

cs.RO↗

arXiv2022

Spatiotemporal Costmap Inference for MPC via Deep Inverse Reinforcement Learning

It can be difficult to autonomously produce driver behavior so that it appears natural to other traffic participants. Through Inverse Reinforcement Learning (IRL), we can automate this process by learning the underlying reward function from human demonstrations. We propose a new IRL algorithm that learns a goal-conditioned spatiotemporal reward function. The resulting costmap is used by Model Predictive Controllers (MPCs) to perform a task without any hand-designing or hand-tuning of the cost function. We evaluate our proposed Goal-conditioned SpatioTemporal Zeroing Maximum Entropy Deep IRL (GSTZ)-MEDIRL framework together with MPC in the CARLA simulator for autonomous driving, lane keeping, and lane changing tasks in a challenging dense traffic highway scenario. Our proposed methods show higher success rates compared to other baseline methods including behavior cloning, state-of-the-art RL policies, and MPC with a learning-based behavior prediction model.

cs.RO↗

arXiv2021

Stochastic optimization for learning quantum state feedback control

High fidelity state preparation represents a fundamental challenge in the application of quantum technology. While the majority of optimal control approaches use feedback to improve the controller, the controller itself often does not incorporate explicit state dependence. Here, we present a general framework for training deep feedback networks for open quantum systems with quantum nondemolition measurement that allows a variety of system and control structures that are prohibitive by many other techniques and can in effect react to unmodeled effects through nonlinear filtering. We demonstrate that this method is efficient due to inherent parallelizability, robust to open system interactions, and outperforms landmark state feedback control results in simulation.

quant-ph↗

arXiv2021

Optimal-Horizon Model-Predictive Control with Differential Dynamic Programming

We present an algorithm, based on the Differential Dynamic Programming framework, to handle trajectory optimization problems in which the horizon is determined online rather than fixed a priori. This algorithm exhibits exact one-step convergence for linear, quadratic, time-invariant problems and is fast enough for real-time nonlinear model-predictive control. We show derivations for the nonlinear algorithm in the discrete-time case, and apply this algorithm to a variety of nonlinear problems. Finally, we show the efficacy of the optimal-horizon model-predictive control scheme compared to a standard MPC controller, on an obstacle-avoidance problem with planar robots.

cs.RO↗

arXiv2021

Second-Order Neural ODE Optimizer

We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient computation by solving a backward ODE, deriving efficient second-order methods becomes highly nontrivial. Nevertheless, inspired by the recent Optimal Control (OC) interpretation of training deep networks, we show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. We further explore a low-rank representation of the second-order derivatives and show that it leads to efficient preconditioned updates with the aid of Kronecker-based factorization. The resulting method -- named SNOpt -- converges much faster than first-order baselines in wall-clock time, and the improvement remains consistent across various applications, e.g. image classification, generative flow, and time-series prediction. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies, strengthening the OC perspective as a principled tool of analyzing optimization in deep learning. Our code is available at https://github.com/ghliu/snopt.

cs.LG↗

arXiv2021

Receding Horizon Differential Dynamic Programming Under Parametric Uncertainty

Generalized Polynomial Chaos (gPC) theory has been widely used for representing parametric uncertainty in a system, thanks to its ability to propagate uncertainty evolution. In an optimal control context, gPC can be combined with several optimization techniques to achieve a control policy that handles effectively this type of uncertainty. Such a suitable method is Differential Dynamic Programming (DDP), leading to an algorithm that inherits the scalability to high-dimensional systems and fast convergence nature of the latter. In this paper, we expand this combination aiming to acquire probabilistic guarantees on the satisfaction of nonlinear constraints. In particular, we exploit the ability of gPC to express higher order moments of the uncertainty distribution - without any Gaussianity assumption - and we incorporate chance constraints that lead to expressions involving the state covariance. Furthermore, we demonstrate that by implementing our algorithm in a receding horizon fashion, we are able to compute control policies that effectively reduce the accumulation of uncertainty on the trajectory. The applicability of our method is verified through simulation results on a differential wheeled robot and a quadrotor that perform obstacle avoidance tasks.

math.OC↗

arXiv2021

Generalization of Safe Optimal Control Actions on Networked Multi-Agent Systems

We propose a unified framework to fast generate a safe optimal control action for a new task from existing controllers on Multi-Agent Systems (MASs). The control action composition is achieved by taking a weighted mixture of the existing controllers according to the contribution of each component task. Instead of sophisticatedly tuning the cost parameters and other hyper-parameters for safe and reliable behavior in the optimal control framework, the safety of each single task solution is guaranteed using the control barrier functions (CBFs) for high-degree stochastic systems, which constrains the system state within a known safe operation region where it originates from. Linearity of CBF constraints in control enables the control action composition. The discussed framework can immediately provide reliable solutions to new tasks by taking a weighted mixture of solved component-task actions and filtering on some CBF constraints, instead of performing an extensive sampling to achieve a new controller. Our results are verified and demonstrated on both a single UAV and two cooperative UAV teams in an environment with obstacles.

eess.SY↗

arXiv2021

Deep $\mathcal{L}^1$ Stochastic Optimal Control Policies for Planetary Soft-landing

In this paper, we introduce a novel deep learning based solution to the Powered-Descent Guidance (PDG) problem, grounded in principles of nonlinear Stochastic Optimal Control (SOC) and Feynman-Kac theory. Our algorithm solves the PDG problem by framing it as an $\mathcal{L}^1$ SOC problem for minimum fuel consumption. Additionally, it can handle practically useful control constraints, nonlinear dynamics and enforces state constraints as soft-constraints. This is achieved by building off of recent work on deep Forward-Backward Stochastic Differential Equations (FBSDEs) and differentiable non-convex optimization neural-network layers based on stochastic search. In contrast to previous approaches, our algorithm does not require convexification of the constraints or linearization of the dynamics and is empirically shown to be robust to stochastic disturbances and the initial position of the spacecraft. After training offline, our controller can be activated once the spacecraft is within a pre-specified radius of the landing zone and at a pre-specified altitude i.e., the base of an inverted cone with the tip at the landing zone. We demonstrate empirically that our controller can successfully and safely land all trajectories initialized at the base of this cone while minimizing fuel consumption.

eess.SY↗

arXiv2021

Safety Embedded Control of Nonlinear Systems via Barrier States

In many safety-critical control systems, possibly opposing safety restrictions and control performance objectives arise. To confront such a conflict, this letter proposes a novel methodology that embeds safety into stability of control systems. The development enforces safety by means of barrier functions used in optimization through the construction of barrier states (BaS) which are embedded in the control system's model. As a result, as long as the equilibrium point of interest of the closed loop system is asymptotically stable, the generated trajectories are guaranteed to be safe. Consequently, a conflict between control objectives and safety constraints is substantially avoided. To show the efficacy of the proposed technique, we employ barrier states with the simple pole placement method to design safe linear controls. Nonlinear optimal control is subsequently employed to fulfill safety, stability and performance objectives by solving the associated Hamilton-Jacobi-Bellman (HJB) which minimizes a cost functional that can involve the BaS. Following this further, we exploit optimal control with barrier states on an unstable, constrained second dimensional pendulum on a cart model that is desired to avoid low velocities regions where the system may exhibit some controllability loss and on two mobile robots to safely arrive to opposite targets with an obstacle on the way.

eess.SY↗

arXiv2021

Bayesian Learning-Based Adaptive Control for Safety Critical Systems

Deep learning has enjoyed much recent success, and applying state-of-the-art model learning methods to controls is an exciting prospect. However, there is a strong reluctance to use these methods on safety-critical systems, which have constraints on safety, stability, and real-time performance. We propose a framework which satisfies these constraints while allowing the use of deep neural networks for learning model uncertainties. Central to our method is the use of Bayesian model learning, which provides an avenue for maintaining appropriate degrees of caution in the face of the unknown. In the proposed approach, we develop an adaptive control framework leveraging the theory of stochastic CLFs (Control Lyapunov Functions) and stochastic CBFs (Control Barrier Functions) along with tractable Bayesian model learning via Gaussian Processes or Bayesian neural networks. Under reasonable assumptions, we guarantee stability and safety while adapting to unknown dynamics with probability 1. We demonstrate this architecture for high-speed terrestrial mobility targeting potential applications in safety-critical high-speed Mars rover missions.

eess.SY↗