Constrained LQR Using Online Decomposition Techniques

This paper presents an algorithm to solve the infinite horizon constrained linear quadratic regulator (CLQR) problem using operator splitting methods. First, the CLQR problem is reformulated as a (finite-time) model predictive control (MPC) problem without terminal constraints. Second, the MPC problem is decomposed into smaller subproblems of fixed dimension independent of the horizon length. Third, using the fast alternating minimization algorithm to solve the subproblems, the horizon length is estimated online, by adding or removing subproblems based on a periodic check on the state of the last subproblem to determine whether it belongs to a given control invariant set. We show that the estimated horizon length is bounded and that the control sequence computed using the proposed algorithm is an optimal solution of the CLQR problem. Compared to state-of-the-art algorithms proposed to solve the CLQR problem, our design solves at each iteration only unconstrained least-squares problems and simple gradient calculations. Furthermore, our technique allows the horizon length to decrease online (a useful feature if the initial guess on the horizon is too conservative). Numerical results on a planar system show the potential of our algorithm.Comment: This technical report is an extended version of the paper titled "Constrained LQR Using Online Decomposition Techniques" submitted to the 2016 Conference on Decision and Contro

Momentum Control with Hierarchical Inverse Dynamics on a Torque-Controlled Humanoid

Hierarchical inverse dynamics based on cascades of quadratic programs have been proposed for the control of legged robots. They have important benefits but to the best of our knowledge have never been implemented on a torque controlled humanoid where model inaccuracies, sensor noise and real-time computation requirements can be problematic. Using a reformulation of existing algorithms, we propose a simplification of the problem that allows to achieve real-time control. Momentum-based control is integrated in the task hierarchy and a LQR design approach is used to compute the desired associated closed-loop behavior and improve performance. Extensive experiments on various balancing and tracking tasks show very robust performance in the face of unknown disturbances, even when the humanoid is standing on one foot. Our results demonstrate that hierarchical inverse dynamics together with momentum control can be efficiently used for feedback control under real robot conditions.Comment: 21 pages, 11 figures, 4 tables in Autonomous Robots (2015

Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing nonlinear observable functions to form an invariant subspace where it is possible to obtain linear models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis using a new algorithm to determine terms in a dynamical system by sparse regression of the data in a nonlinear function space [Brunton et al. 2015, arxiv]; we show how this algorithm is related to DMD. Finally, we demonstrate how to design optimal control laws for nonlinear systems using techniques from linear optimal control on Koopman invariant subspaces.Comment: 20 pages, 5 figures, 2 code

System Level Synthesis

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty -- such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.Comment: To appear in Annual Reviews in Contro