Inherent size constraints on prokaryote gene networks due to "accelerating" growth

Networks exhibiting "accelerating" growth have total link numbers growing faster than linearly with network size and can exhibit transitions from stationary to nonstationary statistics and from random to scale-free to regular statistics at particular critical network sizes. However, if for any reason the network cannot tolerate such gross structural changes then accelerating networks are constrained to have sizes below some critical value. This is of interest as the regulatory gene networks of single celled prokaryotes are characterized by an accelerating quadratic growth and are size constrained to be less than about 10,000 genes encoded in DNA sequence of less than about 10 megabases. This paper presents a probabilistic accelerating network model for prokaryotic gene regulation which closely matches observed statistics by employing two classes of network nodes (regulatory and non-regulatory) and directed links whose inbound heads are exponentially distributed over all nodes and whose outbound tails are preferentially attached to regulatory nodes and described by a scale free distribution. This model explains the observed quadratic growth in regulator number with gene number and predicts an upper prokaryote size limit closely approximating the observed value.Comment: Corrected error in biological input parameter: 15 pages, 10 figure

LQR Control with Sparse Adversarial Disturbances

Recent developments in cyber-physical systems and event-triggered control have led to an increased interest in the impact of sparse disturbances on dynamical processes. We study Linear Quadratic Regulator (LQR) control under sparse disturbances by analyzing three distinct policies: the blind online policy, the disturbance-aware policy, and the optimal offline policy. We derive the two-dimensional recurrence structure of the optimal disturbance-aware policy, under the assumption that the controller has information about future disturbance values with only a probabilistic model of their locations in time. Under mild conditions, we show that the disturbance-aware policy converges to the blind online policy if the number of disturbances grows sublinearly in the time horizon. Finally, we provide a finite-horizon regret bound between the blind online policy and optimal offline policy, which is proven to be quadratic in the number of disturbances and in their magnitude. This provides a useful characterization of the suboptimality of a standard LQR controller when confronted with unexpected sparse perturbations.Comment: 61st IEEE Conference on Decision and Contro

Contact-Implicit Trajectory Optimization Based on a Variable Smooth Contact Model and Successive Convexification

In this paper, we propose a contact-implicit trajectory optimization (CITO) method based on a variable smooth contact model (VSCM) and successive convexification (SCvx). The VSCM facilitates the convergence of gradient-based optimization without compromising physical fidelity. On the other hand, the proposed SCvx-based approach combines the advantages of direct and shooting methods for CITO. For evaluations, we consider non-prehensile manipulation tasks. The proposed method is compared to a version based on iterative linear quadratic regulator (iLQR) on a planar example. The results demonstrate that both methods can find physically-consistent motions that complete the tasks without a meaningful initial guess owing to the VSCM. The proposed SCvx-based method outperforms the iLQR-based method in terms of convergence, computation time, and the quality of motions found. Finally, the proposed SCvx-based method is tested on a standard robot platform and shown to perform efficiently for a real-world application.Comment: Accepted for publication in ICRA 201

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

Predictive control using an FPGA with application to aircraft control

Alternative and more efficient computational methods can extend the applicability of MPC to systems with tight real-time requirements. This paper presents a “system-on-a-chip” MPC system, implemented on a field programmable gate array (FPGA), consisting of a sparse structure-exploiting primal dual interior point (PDIP) QP solver for MPC reference tracking and a fast gradient QP solver for steady-state target calculation. A parallel reduced precision iterative solver is used to accelerate the solution of the set of linear equations forming the computational bottleneck of the PDIP algorithm. A numerical study of the effect of reducing the number of iterations highlights the effectiveness of the approach. The system is demonstrated with an FPGA-inthe-loop testbench controlling a nonlinear simulation of a large airliner. This study considers many more manipulated inputs than any previous FPGA-based MPC implementation to date, yet the implementation comfortably fits into a mid-range FPGA, and the controller compares well in terms of solution quality and latency to state-of-the-art QP solvers running on a standard PC