Moritz Diehl

Quoc Tran Dinh, Suat Gumussoy, Wim Michiels et al.

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem.Applications to various output feedback controller synthesis problems are presented. In these applications the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from COMPleib library.

Tran Dinh Quoc, Moritz Diehl

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.

Alessandro Saviolo, Jonathan Frey, Abhishek Rathod et al.

Model-based control requires an accurate model of the system dynamics for precisely and safely controlling the robot in complex and dynamic environments. Moreover, in the presence of variations in the operating conditions, the model should be continuously refined to compensate for dynamics changes. In this paper, we present a self-supervised learning approach that actively models the dynamics of nonlinear robotic systems. We combine offline learning from past experience and online learning from current robot interaction with the unknown environment. These two ingredients enable a highly sample-efficient and adaptive learning process, capable of accurately inferring model dynamics in real-time even in operating regimes that greatly differ from the training distribution. Moreover, we design an uncertainty-aware model predictive controller that is heuristically conditioned to the aleatoric (data) uncertainty of the learned dynamics. This controller actively chooses the optimal control actions that (i) optimize the control performance and (ii) improve the efficiency of online learning sample collection. We demonstrate the effectiveness of our method through a series of challenging real-world experiments using a quadrotor system. Our approach showcases high resilience and generalization capabilities by consistently adapting to unseen flight conditions, while it significantly outperforms classical and adaptive control baselines.

Tran Dinh Quoc, Carlo Savorgnan, Moritz Diehl

A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main advantage of this algorithm is that it dynamically updates the smoothness parameters which leads to numerically robust performance. The convergence of the algorithm is proved under weak conditions imposed on the original problem. The rate of convergence is $O(\frac{1}{k})$, where $k$ is the iteration counter. In the second part of the paper, the algorithm is coupled with a dual scheme to construct a switching variant of the dual decomposition. We discuss implementation issues and make a theoretical comparison. Numerical examples confirm the theoretical results.

Tran Dinh Quoc, Carlo Savorgnan, Moritz Diehl

This paper proposes real-time sequential convex programming (RTSCP), a method for solving a sequence of nonlinear optimization problems depending on an online parameter. We provide a contraction estimate for the proposed method and, as a byproduct, a new proof of the local convergence of sequential convex programming. The approach is illustrated by an example where RTSCP is applied to nonlinear model predictive control.

Andrea Ghezzi, Jasper Hoffman, Jonathan Frey et al.

This work presents a novel loss function for learning nonlinear Model Predictive Control policies via Imitation Learning. Standard approaches to Imitation Learning neglect information about the expert and generally adopt a loss function based on the distance between expert and learned controls. In this work, we present a loss based on the Q-function directly embedding the performance objectives and constraint satisfaction of the associated Optimal Control Problem (OCP). However, training a Neural Network with the Q-loss requires solving the associated OCP for each new sample. To alleviate the computational burden, we derive a second Q-loss based on the Gauss-Newton approximation of the OCP resulting in a faster training time. We validate our losses against Behavioral Cloning, the standard approach to Imitation Learning, on the control of a nonlinear system with constraints. The final results show that the Q-function-based losses significantly reduce the amount of constraint violations while achieving comparable or better closed-loop costs.

Amon Lahr, Joshua Näf, Kim P. Wabersich et al.

Incorporating learning-based models, such as artificial neural networks or Gaussian processes, into model predictive control (MPC) strategies can significantly improve control performance and online adaptation capabilities for real-world applications. Still, enabling state-of-the-art implementations of learning-based models for MPC is complicated by the challenge of interfacing machine learning frameworks with real-time optimal control software. This work aims at filling this gap by incorporating external sensitivities in sequential quadratic programming solvers for nonlinear optimal control. To this end, we provide L4acados, a general framework for incorporating Python-based dynamics models in the real-time optimal control software acados. By computing external sensitivities via a user-defined Python module, L4acados enables the implementation of MPC controllers with learning-based residual models in acados, while supporting parallelization of sensitivity computations when preparing the quadratic subproblems. We demonstrate significant speed-ups and superior scaling properties of L4acados compared to available software using a neural-network-based control example. Last, we provide an efficient and modular real-time implementation of Gaussian process-based MPC using L4acados, which is applied to two hardware examples: autonomous miniature racing, as well as motion control of a full-scale autonomous vehicle for an ISO lane change maneuver.

Jonathan Frey, Katrin Baumgärtner, Gianluca Frison et al.

The efficient computation of parametric solution sensitivities is a key challenge in the integration of learning-enhanced methods with nonlinear model predictive control (MPC), as their availability is crucial for many learning algorithms. While approaches presented in the machine learning community are limited to convex or unconstrained formulations, this paper discusses the computation of solution sensitivities of general nonlinear programs (NLPs) using the implicit function theorem (IFT) and smoothed optimality conditions treated in interior-point methods (IPM). We detail sensitivity computation within a sequential quadratic programming (SQP) method which employs an IPM for the quadratic subproblems. The publication is accompanied by an efficient open-source implementation within the framework, providing both forward and adjoint sensitivities for general optimal control problems, achieving speedups exceeding 3x over the state-of-the-art solver mpc.pytorch.

Rudolf Reiter, Jasper Hoffmann, Dirk Reinhardt et al.

The fields of MPC and RL consider two successful control techniques for Markov decision processes. Both approaches are derived from similar fundamental principles, and both are widely used in practical applications, including robotics, process control, energy systems, and autonomous driving. Despite their similarities, MPC and RL follow distinct paradigms that emerged from diverse communities and different requirements. Various technical discrepancies, particularly the role of an environment model as part of the algorithm, lead to methodologies with nearly complementary advantages. Due to their orthogonal benefits, research interest in combination methods has recently increased significantly, leading to a large and growing set of complex ideas leveraging MPC and RL. This work illuminates the differences, similarities, and fundamentals that allow for different combination algorithms and categorizes existing work accordingly. Particularly, we focus on the versatile actor-critic RL approach as a basis for our categorization and examine how the online optimization approach of MPC can be used to improve the overall closed-loop performance of a policy.

Rudolf Reiter, Andrea Ghezzi, Katrin Baumgärtner et al.

\Ac{MPC} and \ac{RL} are two powerful control strategies with, arguably, complementary advantages. In this work, we show how actor-critic \ac{RL} techniques can be leveraged to improve the performance of \ac{MPC}. The \ac{RL} critic is used as an approximation of the optimal value function, and an actor roll-out provides an initial guess for primal variables of the \ac{MPC}. A parallel control architecture is proposed where each \ac{MPC} instance is solved twice for different initial guesses. Besides the actor roll-out initialization, a shifted initialization from the previous solution is used. Thereafter, the actor and the critic are again used to approximately evaluate the infinite horizon cost of these trajectories. The control actions from the lowest-cost trajectory are applied to the system at each time step. We establish that the proposed algorithm is guaranteed to outperform the original \ac{RL} policy plus an error term that depends on the accuracy of the critic and decays with the horizon length of the \ac{MPC} formulation. Moreover, we do not require globally optimal solutions for these guarantees to hold. The approach is demonstrated on an illustrative toy example and an \ac{AD} overtaking scenario.

Matilde Gargiani, Andrea Zanelli, Quoc Tran-Dinh et al.

First-order stochastic methods for solving large-scale non-convex optimization problems are widely used in many big-data applications, e.g. training deep neural networks as well as other complex and potentially non-convex machine learning models. Their inexpensive iterations generally come together with slow global convergence rate (mostly sublinear), leading to the necessity of carrying out a very high number of iterations before the iterates reach a neighborhood of a minimizer. In this work, we present a first-order stochastic algorithm based on a combination of homotopy methods and SGD, called Homotopy-Stochastic Gradient Descent (H-SGD), which finds interesting connections with some proposed heuristics in the literature, e.g. optimization by Gaussian continuation, training by diffusion, mollifying networks. Under some mild assumptions on the problem structure, we conduct a theoretical analysis of the proposed algorithm. Our analysis shows that, with a specifically designed scheme for the homotopy parameter, H-SGD enjoys a global linear rate of convergence to a neighborhood of a minimum while maintaining fast and inexpensive iterations. Experimental evaluations confirm the theoretical results and show that H-SGD can outperform standard SGD.

Barbara Barros Carlos, Tommaso Sartor, Andrea Zanelli et al.

The advances in computer processor technology have enabled the application of nonlinear model predictive control (NMPC) to agile systems, such as quadrotors. These systems are characterized by their underactuation, nonlinearities, bounded inputs, and time-delays. Classical control solutions fall short in overcoming these difficulties and fully exploiting the capabilities offered by such platforms. This paper presents the design and implementation of an efficient position controller for quadrotors based on real-time NMPC with time-delay compensation and bounds enforcement on the actuators. To deal with the limited computational resources onboard, an offboard control architecture is proposed. It is implemented using the high-performance software package acados, which solves optimal control problems and implements a real-time iteration (RTI) variant of a sequential quadratic programming (SQP) scheme with Gauss-Newton Hessian approximation. The quadratic subproblems (QP) in the SQP scheme are solved with HPIPM, an interior-point method solver, built on top of the linear algebra library BLASFEO, finely tuned for multiple CPU architectures. Solution times are further reduced by reformulating the QPs using the efficient partial condensing algorithm implemented in HPIPM. We demonstrate the capabilities of our architecture using the Crazyflie 2.1 nano-quadrotor.

Jia-Jie Zhu, Wittawat Jitkrittum, Moritz Diehl et al.

We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization with respect to measures in DRO into the dual program that searches for RKHS functions. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as polynomial losses and knowledge of the Lipschitz constant. We then establish a connection between DRO and stochastic optimization with expectation constraints. Finally, we propose practical algorithms based on both batch convex solvers and stochastic functional gradient, which apply to general optimization and machine learning tasks.

Matilde Gargiani, Andrea Zanelli, Moritz Diehl et al.

Following early work on Hessian-free methods for deep learning, we study a stochastic generalized Gauss-Newton method (SGN) for training DNNs. SGN is a second-order optimization method, with efficient iterations, that we demonstrate to often require substantially fewer iterations than standard SGD to converge. As the name suggests, SGN uses a Gauss-Newton approximation for the Hessian matrix, and, in order to compute an approximate search direction, relies on the conjugate gradient method combined with forward and reverse automatic differentiation. Despite the success of SGD and its first-order variants, and despite Hessian-free methods based on the Gauss-Newton Hessian approximation having been already theoretically proposed as practical methods for training DNNs, we believe that SGN has a lot of undiscovered and yet not fully displayed potential in big mini-batch scenarios. For this setting, we demonstrate that SGN does not only substantially improve over SGD in terms of the number of iterations, but also in terms of runtime. This is made possible by an efficient, easy-to-use and flexible implementation of SGN we propose in the Theano deep learning platform, which, unlike Tensorflow and Pytorch, supports forward automatic differentiation. This enables researchers to further study and improve this promising optimization technique and hopefully reconsider stochastic second-order methods as competitive optimization techniques for training DNNs; we also hope that the promise of SGN may lead to forward automatic differentiation being added to Tensorflow or Pytorch. Our results also show that in big mini-batch scenarios SGN is more robust than SGD with respect to its hyperparameters (we never had to tune its step-size for our benchmarks!), which eases the expensive process of hyperparameter tuning that is instead crucial for the performance of first-order methods.

Jia-Jie Zhu, Wittawat Jitkrittum, Moritz Diehl et al.

In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constraint violation probability in the context of a constrained stochastic control system.

Jia-Jie Zhu, Moritz Diehl, Bernhard Schölkopf

We apply kernel mean embedding methods to sample-based stochastic optimization and control. Specifically, we use the reduced-set expansion method as a way to discard sampled scenarios. The effect of such constraint removal is improved optimality and decreased conservativeness. This is achieved by solving a distributional-distance-regularized optimization problem. We demonstrated this optimization formulation is well-motivated in theory, computationally tractable and effective in numerical algorithms.

Tobias Schoels, Per Rutquist, Luigi Palmieri et al.

Robots have been operating in dynamic environments and shared workspaces for decades. Most optimization based motion planning methods, however, do not consider the movement of other agents, e.g. humans or other robots, and therefore do not guarantee collision avoidance in such scenarios. This paper builds upon the Convex Inner ApprOximation (CIAO) method and proposes a motion planning algorithm that guarantees collision avoidance in predictable dynamic environments. Furthermore, it generalizes CIAO's free region concept to arbitrary norms and proposes a cost function to approximate time optimal motion planning. The proposed method, CIAO$^\star$, finds kinodynamically feasible and collision free trajectories for constrained single body robots using model predictive control (MPC). It optimizes the motion of one agent and accounts for the predicted movement of surrounding agents and obstacles. The experimental evaluation shows that CIAO$^\star$ reaches close to time optimal behavior.

Jia-Jie Zhu, Krikamol Muandet, Moritz Diehl et al.

This work presents the concept of kernel mean embedding and kernel probabilistic programming in the context of stochastic systems. We propose formulations to represent, compare, and propagate uncertainties for fairly general stochastic dynamics in a distribution-free manner. The new tools enjoy sound theory rooted in functional analysis and wide applicability as demonstrated in distinct numerical examples. The implication of this new concept is a new mode of thinking about the statistical nature of uncertainty in dynamical systems.

Tobias Schoels, Luigi Palmieri, Kai O. Arras et al.

Even though mobile robots have been around for decades, trajectory optimization and continuous time collision avoidance remain subject of active research. Existing methods trade off between path quality, computational complexity, and kinodynamic feasibility. This work approaches the problem using a nonlinear model predictive control (NMPC) framework, that is based on a novel convex inner approximation of the collision avoidance constraint. The proposed Convex Inner ApprOximation (CIAO) method finds kinodynamically feasible and continuous time collision free trajectories, in few iterations, typically one. For a feasible initialization, the approach is guaranteed to find a feasible solution, i.e. it preserves feasibility. Our experimental evaluation shows that CIAO outperforms state of the art baselines in terms of planning efficiency and path quality. Experiments on a robot with 12 states show that it also scales to high-dimensional systems. Furthermore real-world experiments demonstrate its capability of unifying trajectory optimization and tracking for safe motion planning in dynamic environments.

Markus Giftthaler, Michael Neunert, Markus Stäuble et al.

This paper introduces a family of iterative algorithms for unconstrained nonlinear optimal control. We generalize the well-known iLQR algorithm to different multiple-shooting variants, combining advantages like straight-forward initialization and a closed-loop forward integration. All algorithms have similar computational complexity, i.e. linear complexity in the time horizon, and can be derived in the same computational framework. We compare the full-step variants of our algorithms and present several simulation examples, including a high-dimensional underactuated robot subject to contact switches. Simulation results show that our multiple-shooting algorithms can achieve faster convergence, better local contraction rates and much shorter runtimes than classical iLQR, which makes them a superior choice for nonlinear model predictive control applications.