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This paper presents an algorithm that finds a centroidal motion and footstep plan for a Spring-Loaded Inverted Pendulum (SLIP)-like bipedal robot model substantially faster than real-time. This is achieved with a novel representation of the dynamic footstep planning problem, where each point in the environment is considered a potential foothold that can apply a force to the center of mass to keep it on a desired trajectory. For a biped, up to two such footholds per time step must be selected, and we approximate this cardinality constraint with an iteratively reweighted $l_1$-norm minimization. Along with a linearizing approximation of an angular momentum constraint, this results in a quadratic program can be solved for a contact schedule and center of mass trajectory with automatic gait discovery. A 2 s planning horizon with 13 time steps and 20 surfaces available at each time is solved in 142 ms, roughly ten times faster than comparable existing methods in the literature. We demonstrate the versatility of this program in a variety of simulated environments., Comment: Accepted to IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2024

We present a sample-based motion planning algorithm specialised to a class of underactuated systems using path parameterisation. The structure this class presents under a path parameterisation enables the trivial computation of dynamic feasibility along a path. Using this, a specialised state-based steering mechanism within an RRT motion planning algorithm is developed, enabling the generation of both geometric paths and their time parameterisations without introducing excessive computational overhead. We find with two systems that our algorithm computes feasible trajectories with higher rates of success and lower mean computation times compared to existing approaches., Comment: 8 pages, accepted for publication at the 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2024)

This paper presents a study of robust policy networks in deep reinforcement learning. We investigate the benefits of policy parameterizations that naturally satisfy constraints on their Lipschitz bound, analyzing their empirical performance and robustness on two representative problems: pendulum swing-up and Atari Pong. We illustrate that policy networks with smaller Lipschitz bounds are more robust to disturbances, random noise, and targeted adversarial attacks than unconstrained policies composed of vanilla multi-layer perceptrons or convolutional neural networks. However, the structure of the Lipschitz layer is important. We find that the widely-used method of spectral normalization is too conservative and severely impacts clean performance, whereas more expressive Lipschitz layers such as the recently-proposed Sandwich layer can achieve improved robustness without sacrificing clean performance.

In this paper, we introduce a novel class of neural differential equation, which are intrinsically Lyapunov stable, exponentially stable or passive. We take a recently proposed Polyak Lojasiewicz network (PLNet) as an Lyapunov function and then parameterize the vector field as the descent directions of the Lyapunov function. The resulting models have a same structure as the general Hamiltonian dynamics, where the Hamiltonian is lower- and upper-bounded by quadratic functions. Moreover, it is also positive definite w.r.t. either a known or learnable equilibrium. We illustrate the effectiveness of the proposed model on a damped double pendulum system.

In this paper, we extend the control contraction metrics (CCM) approach, which was originally proposed for the universal tracking control of nonlinear systems, to those that evolves on Lie groups. Our idea is to view the manifold as a constrained set that is embedded in Euclidean space, and then propose the sufficient conditions for the existence of a CCM and the associated controller design. Notably, we demonstrate that the search for CCM on Lie groups can be reformulated as convex conditions. The results extend the applicability of the CCM approach and provide a framework for analyzing the behavior of control systems with Lie group structures.

This paper presents a new bi-Lipschitz invertible neural network, the BiLipNet, which has the ability to smoothly control both its Lipschitzness (output sensitivity to input perturbations) and inverse Lipschitzness (input distinguishability from different outputs). The second main contribution is a new scalar-output network, the PLNet, which is a composition of a BiLipNet and a quadratic potential. We show that PLNet satisfies the Polyak-Lojasiewicz condition and can be applied to learn non-convex surrogate losses with a unique and efficiently-computable global minimum. The central technical element in these networks is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build the BiLipNet. The certification of these properties is based on incremental quadratic constraints, resulting in much tighter bounds than can be achieved with spectral normalization. Moreover, we formulate the calculation of the inverse of a BiLipNet -- and hence the minimum of a PLNet -- as a series of three-operator splitting problems, for which fast algorithms can be applied., Comment: International Conference on Machine Learning, Vienna, Austria, July 21 -- 17, 2024

The inertial measurement unit (IMU) preintegration approach nowadays is widely used in various robotic applications. In this article, we revisit the preintegration theory and propose a novel interpretation to understand it from a nonlinear observer perspective, specifically the parameter estimation-based observer (PEBO). We demonstrate that the preintegration approach can be viewed as recursive implementation of PEBO in moving horizons, and that the two approaches are equivalent in the case of perfect measurements. We then discuss how these findings can be used to tackle practical challenges in estimation problems. As byproducts, our results lead to a novel hybrid sampled-data observer design and an approach to address statistical optimality for PEBO in presence of noise.

This paper introduces a new linear parameterization to the problem of visual inertial simultaneous localization and mapping (VI-SLAM) -- without any approximation -- for the case only using information from a single monocular camera and an inertial measurement unit. In this problem set, the system state evolves on the nonlinear manifold $SE(3)\times \mathbb{R}^{3n}$, on which we design dynamic extensions carefully to generate invariant foliations, such that the problem can be reformulated into online \emph{constant parameter} identification, then interestingly with linear regression models obtained. It demonstrates that VI-SLAM can be translated into a linear least squares problem, in the deterministic sense, \emph{globally} and \emph{exactly}. Based on this observation, we propose a novel SLAM observer, following the recently established parameter estimation-based observer (PEBO) methodology. A notable merit is that the proposed observer enjoys almost global asymptotic stability, requiring neither persistency of excitation nor uniform complete observability, which, however, are widely adopted in most existing works with provable stability but can hardly be assured in many practical scenarios.

Neural networks are typically sensitive to small input perturbations, leading to unexpected or brittle behaviour. We present RobustNeuralNetworks.jl: a Julia package for neural network models that are constructed to naturally satisfy a set of user-defined robustness constraints. The package is based on the recently proposed Recurrent Equilibrium Network (REN) and Lipschitz-Bounded Deep Network (LBDN) model classes, and is designed to interface directly with Julia's most widely-used machine learning package, Flux.jl. We discuss the theory behind our model parameterization, give an overview of the package, and provide a tutorial demonstrating its use in image classification, reinforcement learning, and nonlinear state-observer design.

This paper presents a policy parameterization for learning-based control on nonlinear, partially-observed dynamical systems. The parameterization is based on a nonlinear version of the Youla parameterization and the recently proposed Recurrent Equilibrium Network (REN) class of models. We prove that the resulting Youla-REN parameterization automatically satisfies stability (contraction) and user-tunable robustness (Lipschitz) conditions on the closed-loop system. This means it can be used for safe learning-based control with no additional constraints or projections required to enforce stability or robustness. We test the new policy class in simulation on two reinforcement learning tasks: 1) magnetic suspension, and 2) inverting a rotary-arm pendulum. We find that the Youla-REN performs similarly to existing learning-based and optimal control methods while also ensuring stability and exhibiting improved robustness to adversarial disturbances.

In this paper, we present a structured solver based on the preconditioned conjugate gradient method (PCGM) for solving the linear quadratic (LQ) optimal control problem for $K \times N$ sub-systems connected in a two-dimensional (2D) grid structure. Our main contribution is the development of a structured preconditioner based on a fixed number of inner-outer iterations of the nested block Jacobi method. We establish that the proposed preconditioner is positive-definite. Moreover, the proposed approach retains structure in both spatial dimensions as well as in the temporal dimension of the problem. The arithmetic complexity of each PCGM step scales as $O(KNT)$, where $T$ is the length of the time horizon. The computations involved at each step of the proposed PCGM are decomposable and amenable to distributed implementation on parallel processors connected in a 2D grid structure with localized data exchange. We also provide results of numerical experiments performed on two example systems., Comment: Submitted to the 62nd IEEE Conference on Decision and Control for possible publication

In this work, we introduce and study a class of Deep Neural Networks (DNNs) in continuous-time. The proposed architecture stems from the combination of Neural Ordinary Differential Equations (Neural ODEs) with the model structure of recently introduced Recurrent Equilibrium Networks (RENs). We show how to endow our proposed NodeRENs with contractivity and dissipativity -- crucial properties for robust learning and control. Most importantly, as for RENs, we derive parametrizations of contractive and dissipative NodeRENs which are unconstrained, hence enabling their learning for a large number of parameters. We validate the properties of NodeRENs, including the possibility of handling irregularly sampled data, in a case study in nonlinear system identification., Comment: Accepted for CDC 2023

We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. In doing so, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian of the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness., Comment: Published as a conference paper at CDC 2023

Spreading processes, e.g. epidemics, wildfires and rumors, are often modeled on static networks. However, their underlying network structures, e.g. changing contacts in social networks, different weather forecasts for wildfires, are due to ever-changing circumstances inherently time-varying in nature. In this paper, we therefore, propose an optimization framework for sparse resource allocation for control of spreading processes over temporal networks with known connectivity patterns. We use convex optimization, in particular exponential cone programming, and dynamic programming techniques to bound and minimize the risk of an undetected outbreak by allocating budgeted resources each time step. We demonstrate with misinformation, epidemic and wildfire examples how the method can provide targeted allocation of resources., Comment: Conference submission, 8 pages. arXiv admin note: text overlap with arXiv:2110.07755

This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed $\ell^2$ Lipschitz bounds, i.e. limited sensitivity to input perturbations. The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP). We provide a ``direct'' parameterization, i.e., a smooth mapping from $\mathbb R^N$ onto the set of weights satisfying the SDP-based bound. Moreover, our parameterization is complete, i.e. a neural network satisfies the SDP bound if and only if it can be represented via our parameterization. This enables training using standard gradient methods, without any inner approximation or computationally intensive tasks (e.g. projections or barrier terms) for the SDP constraint. The new parameterization can equivalently be thought of as either a new layer type (the \textit{sandwich layer}), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers. A comprehensive set of experiments on image classification shows that sandwich layers outperform previous approaches on both empirical and certified robust accuracy. Code is available at \url{https://github.com/acfr/LBDN}., Comment: accepted to ICML 2023

In this paper, we propose a model predictive control (MPC) method for real-time intervention of spreading processes, such as epidemics and wildfire, over large-scale networks. The goal is to allocate budgeted resources each time step to minimize the risk of an undetected outbreak, i.e. the product of the probability of an outbreak and the impact of that outbreak. By using dynamic programming relaxation, the MPC controller is reformulated as a convex optimization problem, in particular an exponential cone programming. We also provide sufficient conditions for the closed-loop risks to asymptotically decrease and a method to estimate the upper bound of when the risk will monotonically decrease. Numerical results are provided for a wildfire example.

The paper addresses the problem of attitude estimation for rigid bodies using (possibly time-varying) vector measurements, for which we provide a necessary and sufficient condition of distinguishability. Such a condition is shown to be strictly weaker than those previously used for attitude observer design. Thereafter, we show that even for the single vector case the resulting condition is sufficient to design almost globally convergent attitude observers, and two explicit designs are obtained. To overcome the weak excitation issue, the first design employs to make full use of historical information, whereas the second scheme dynamically generates a virtual reference vector, which remains non-collinear to the given vector measurement. Simulation results illustrate the accurate estimation despite noisy measurements.

The design of a globally convergent position observer for feature points from visual information is a challenging problem, especially for the case with only inertial measurements and without assumptions of uniform observability, which remained open for a long time. We give a solution to the problem in this paper assuming that only the bearing of a feature point, and biased linear acceleration and rotational velocity of a robot -- all in the body-fixed frame -- are available. Further, in contrast to existing related results, we do not need the value of the gravitational constant either. The proposed approach builds upon the parameter estimation-based observer recently developed in (Ortega et al., Syst. Control Lett., vol.85, 2015) and its extension to matrix Lie groups in our previous work. Conditions on the robot trajectory under which the observer converges are given, and these are strictly weaker than the standard persistency of excitation and uniform complete observability conditions. Finally, as an illustration, we apply the proposed design to the visual inertial navigation problem.

This paper proposes a nonlinear policy architecture for control of partially-observed linear dynamical systems providing built-in closed-loop stability guarantees. The policy is based on a nonlinear version of the Youla parameterization, and augments a known stabilizing linear controller with a nonlinear operator from a recently developed class of dynamic neural network models called the recurrent equilibrium network (REN). We prove that RENs are universal approximators of contracting and Lipschitz nonlinear systems, and subsequently show that the the proposed Youla-REN architecture is a universal approximator of stabilizing nonlinear controllers. The REN architecture simplifies learning since unconstrained optimization can be applied, and we consider both a model-based case where exact gradients are available and reinforcement learning using random search with zeroth-order oracles. In simulation examples our method converges faster to better controllers and is more scalable than existing methods, while guaranteeing stability during learning transients.

This paper presents a parameterization of nonlinear controllers for uncertain systems building on a recently developed neural network architecture, called the recurrent equilibrium network (REN), and a nonlinear version of the Youla parameterization. The proposed framework has "built-in" guarantees of stability, i.e., all policies in the search space result in a contracting (globally exponentially stable) closed-loop system. Thus, it requires very mild assumptions on the choice of cost function and the stability property can be generalized to unseen data. Another useful feature of this approach is that policies are parameterized directly without any constraints, which simplifies learning by a broad range of policy-learning methods based on unconstrained optimization (e.g. stochastic gradient descent). We illustrate the proposed approach with a variety of simulation examples., Comment: submitted to ACC2022

In this paper we propose a method for sparse dynamic allocation of resources to bound the risk of spreading processes, such as epidemics and wildfires, using convex optimization and dynamic programming techniques. Here, risk is defined as the risk of an outbreak, i.e. the product of the probability of an outbreak occurring over a time interval and the future impact of that outbreak, and we can allocate budgeted resources each time step to bound or minimize the risk. Our method in particular provides sparsity of resources, which is important due to the large network structures involved with spreading processes and has advantages when resources can not be distributed widely., Comment: Conference submission. arXiv admin note: text overlap with arXiv:2107.05878, arXiv:2003.07555

In this paper, we present a new data-driven method for learning stable models of nonlinear systems. Our model lifts the original state space to a higher-dimensional linear manifold using Koopman embeddings. Interestingly, we prove that every discrete-time nonlinear contracting model can be learnt in our framework. Another significant merit of the proposed approach is that it allows for unconstrained optimization over the Koopman embedding and operator jointly while enforcing stability of the model, via a direct parameterization of stable linear systems, greatly simplifying the computations involved. We validate our method on a simulated system and analyze the advantages of our parameterization compared to alternatives.

This tutorial paper provides an introduction to recently developed tools for machine learning, especially learning dynamical systems (system identification), with stability and robustness constraints. The main ideas are drawn from contraction analysis and robust control, but adapted to problems in which large-scale models can be learnt with behavioural guarantees. We illustrate the methods with applications in robust image recognition and system identification., Comment: Paper in the invited tutorial session "Contraction Theory for Machine Learning" at 60th IEEE Conference on Decision and Control, 2021

This paper proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting -- a strong form of nonlinear stability -- and/or monotone, i.e. order relations between states are preserved. The main challenges that we address are: simultaneously searching for model parameters and a certificate of stability, and scalability to networks with hundreds or thousands of nodes. We propose a model set that admits convex constraints for stability and monotonicity, and has a separable structure that allows distributed identification via the alternating directions method of multipliers (ADMM). The performance and scalability of the approach is illustrated on a variety of linear and non-linear case studies, including a nonlinear traffic network with a 200-dimensional state space., Comment: Preprint of full paper accepted for publication in IEEE Trans. Automatic Control

In this paper, we propose an optimization framework that combines surveillance schedules and sparse control to bound the risk of spreading processes such as epidemics and wildfires. Here, risk is considered the risk of an undetected outbreak, i.e. the product of the probability of an outbreak and the impact of that outbreak, and we can bound or minimize the risk by resource allocation and persistent monitoring schedules. The presented framework utilizes the properties of positive systems and convex optimization to provide scalable algorithms for both surveillance and intervention purposes. We demonstrate with different spreading process examples how the method can incorporate different parameters and scenarios such as a vaccination strategy for epidemics and the effect of vegetation, wind and outbreak rate on a wildfire in persistent monitoring scenarios., Comment: Journal submission

This paper introduces recurrent equilibrium networks (RENs), a new class of nonlinear dynamical models} for applications in machine learning, system identification and control. The new model class admits ``built in'' behavioural guarantees of stability and robustness. All models in the proposed class are contracting -- a strong form of nonlinear stability -- and models can satisfy prescribed incremental integral quadratic constraints (IQC), including Lipschitz bounds and incremental passivity. RENs are otherwise very flexible: they can represent all stable linear systems, all previously-known sets of contracting recurrent neural networks and echo state networks, all deep feedforward neural networks, and all stable Wiener/Hammerstein models, and can approximate all fading-memory and contracting nonlinear systems. RENs are parameterized directly by a vector in R^N, i.e. stability and robustness are ensured without parameter constraints, which simplifies learning since \HL{generic methods for unconstrained optimization such as stochastic gradient descent and its variants can be used}. The performance and robustness of the new model set is evaluated on benchmark nonlinear system identification problems, and the paper also presents applications in data-driven nonlinear observer design and control with stability guarantees., Comment: To appear in IEEE Transactions on Automatic Control

In this paper, we present a virtual control contraction metric (VCCM) based nonlinear parameter-varying (NPV) approach to design a state-feedback controller for a control moment gyroscope (CMG) to track a user-defined trajectory set. This VCCM based nonlinear stabilization and performance synthesis approach, which is similar to linear parameter-varying (LPV) control approaches, allows to achieve exact guarantees of exponential stability and $\mathcal{L}_2$-gain performance on nonlinear systems with respect to all trajectories from the predetermined set, which is not the case with the conventional LPV methods. Simulation and experimental studies conducted in both fully- and under-actuated operating modes of the CMG show effectiveness of this approach compared to standard LPV control methods.

In this paper we propose a novel observer to solve the problem of visual simultaneous localization and mapping (SLAM), only using the information from a single monocular camera and an inertial measurement unit (IMU). The system state evolves on the manifold $SE(3)\times \mathbb{R}^{3n}$, on which we design dynamic extensions carefully in order to generate an invariant foliation, such that the problem is reformulated into online \emph{constant parameter} identification. Then, following the recently introduced parameter estimation-based observer (PEBO) and the dynamic regressor extension and mixing (DREM) procedure, we provide a new simple solution. A notable merit is that the proposed observer guarantees almost global asymptotic stability requiring neither persistency of excitation nor uniform complete observability, which, however, are widely adopted in most existing works with guaranteed stability.

In this paper we prove new connections between two frameworks for analysis and control of nonlinear systems: the Koopman operator framework and contraction analysis. Each method, in different ways, provides exact and global analyses of nonlinear systems by way of linear systems theory. The main results of this paper show equivalence between contraction and Koopman approaches for a wide class of stability analysis and control design problems. In particular: stability or stablizability in the Koopman framework implies the existence of a contraction metric (resp. control contraction metric) for the nonlinear system. Further in certain cases the converse holds: contraction implies the existence of a set of observables with which stability can be verified via the Koopman framework. We provide results for the cases of autonomous and time-varying systems, as well as orbital stability of limit cycles. Furthermore, the converse claims are based on a novel relation between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger observer. We also provide a byproduct of the main results, that is, a new method to learn contraction metrics from trajectory data via linear system identification.

In this paper, we propose a new approach to design globally convergent reduced-order observers for nonlinear control systems via contraction analysis and convex optimization. Despite the fact that contraction is a concept naturally suitable for state estimation, the existing solutions are either local or relatively conservative when applying to physical systems. To address this, we show that this problem can be translated into an off-line search for a coordinate transformation after which the dynamics is (transversely) contracting. The obtained sufficient condition consists of some easily verifiable differential inequalities, which, on one hand, identify a very general class of "detectable" nonlinear systems, and on the other hand, can be expressed as computationally efficient convex optimization, making the design procedure more systematic. Connections with some well-established approaches and concepts are also clarified in the paper. Finally, we illustrate the proposed method with several numerical and physical examples, including polynomial, mechanical, electromechanical and biochemical systems.

This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. This model class includes standard multilayer and residual networks as special cases. The new parameterization admits a Lipschitz bound during training via unconstrained optimization: no projections or barrier functions are required. Lipschitz bounds are a common proxy for robustness and appear in many generalization bounds. Furthermore, compared to previous works we show well-posedness (existence of solutions) under less restrictive conditions on the network weights and more natural assumptions on the activation functions: that they are monotone and slope restricted. These results are proved by establishing novel connections with convex optimization, operator splitting on non-Euclidean spaces, and contracting neural ODEs. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks., Comment: Conference submission, 19 pages

In this paper we address the problem of tracking control of nonlinear systems via contraction analysis. The necessary conditions of the systems which can achieve universal asymptotic tracking are studied under several different cases. We show the links to the well developed control contraction metric, as well as its invariance under dynamic extension. In terms of these conditions, we identify a differentially detectable output, based on which a simple differential controller for trajectory tracking is designed via damping injection. As illustration we apply to electrostatic microactuators.

The nature of information sharing in common distributed consensus algorithms permits network eavesdroppers to expose sensitive system information. An important parameter within distributed systems, often neglected under the scope of privacy preservation, is the influence structure - the weighting each agent places on the sources of their opinion pool. This paper proposes a local (i.e. computed individually by each agent), time varying mask to prevent the discovery of the influence structure by an external observer with access to the entire information flow, network knowledge and mask formulation. This result is produced through the auxiliary demonstration of the preserved stability of a Friedkin-Johnsen system under a set of generalised conditions. The mask is developed under these constraints and involves perturbing the influence structure by decaying pseudonoise. This paper provides the information matrix of the best influence structure estimate by an eavesdropper lacking a priori knowledge and uses stochastic simulations to analyse the performance of the mask against ranging system hyperparameters., Comment: 16 pages, 5 figures, extended version of the paper to be published in the 59th IEEE Conference on Decision and Control (CDC)

This paper studies initial-value privacy problems of linear dynamical systems. We consider a standard linear time-invariant system with random process and measurement noises. For such a system, eavesdroppers having access to system output trajectories may infer the system initial states, leading to initial-value privacy risks. When a finite number of output trajectories are eavesdropped, we consider a requirement that any guess about the initial values can be plausibly denied. When an infinite number of output trajectories are eavesdropped, we consider a requirement that the initial values should not be uniquely recoverable. In view of these two privacy requirements, we define differential initial-value privacy and intrinsic initial-value privacy, respectively, for the system as metrics of privacy risks. First of all, we prove that the intrinsic initial-value privacy is equivalent to unobservability, while the differential initial-value privacy can be achieved for a privacy budget depending on an extended observability matrix of the system and the covariance of the noises. Next, the inherent network nature of the considered linear system is explored, where each individual state corresponds to a node and the state and output matrices induce interaction and sensing graphs, leading to a network system. Under this network system perspective, we allow the initial states at some nodes to be public, and investigate the resulting intrinsic initial-value privacy of each individual node. We establish necessary and sufficient conditions for such individual node initial-value privacy, and also prove that the intrinsic initial-value privacy of individual nodes is generically determined by the network structure. These results may be extended to linear systems with time-varying dynamics under the same analysis framework., Comment: 29 pages, 6 figures

Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems often used to model sequence-to-sequence maps. RNNs have excellent expressive power but lack the stability or robustness guarantees that are necessary for many applications. In this paper, we formulate convex sets of RNNs with stability and robustness guarantees. The guarantees are derived using incremental quadratic constraints and can ensure global exponential stability of all solutions, and bounds on incremental $ \ell_2 $ gain (the Lipschitz constant of the learned sequence-to-sequence mapping). Using an implicit model structure, we construct a parametrization of RNNs that is jointly convex in the model parameters and stability certificate. We prove that this model structure includes all previously-proposed convex sets of stable RNNs as special cases, and also includes all stable linear dynamical systems. We illustrate the utility of the proposed model class in the context of non-linear system identification., Comment: conference submission, 6 pages

This paper presents a systematic approach to nonlinear state-feedback control design that has three main advantages: (i) it ensures exponential stability and $ \mathcal{L}_2 $-gain performance with respect to a user-defined set of reference trajectories, and (ii) it provides constructive conditions based on convex optimization and a path-integral-based control realization, and (iii) it is less restrictive than previous similar approaches. In the proposed approach, first a virtual representation of the nonlinear dynamics is constructed for which a behavioral (parameter-varying) embedding is generated. Then, by introducing a virtual control contraction metric, a convex control synthesis formulation is derived. Finally, a control realization with a virtual reference generator is computed, which is guaranteed to achieve exponential stability and $ \mathcal{L}_2 $-gain performance for all trajectories of the targeted reference behavior. We show that the proposed methodology is a unified generalization of the two distinct categories of linear-parameter-varying (LPV) state-feedback control approaches: global and local methods. Moreover, it provides rigorous stability and performance guarantees as a method for nonlinear tracking control, while such properties are not guaranteed for tracking control using standard LPV approaches.

In this letter we propose a method for sparse allocation of resources to control spreading processes -- such as epidemics and wildfires -- using convex optimization, in particular exponential cone programming. Sparsity of allocation has advantages in situations where resources cannot easily be distributed over a large area. In addition, we introduce a model of risk to optimize the product of the likelihood and the future impact of an outbreak. We demonstrate with a simplified wildfire example that our method can provide more targeted resource allocation compared to previous approaches based on geometric programming., Comment: 6 pages, conference submission

This paper aims to provide a new problem formulation of path following for mechanical systems without time parameterization nor guidance laws, namely, we express the control objective as an orbital stabilization problem. It is shown that, it is possible to adapt the immersion and invariance technique to design static state-feedback controllers that solve the problem. In particular, we select the target dynamics adopting the recently introduced Mexican sombrero energy assignment method. To demonstrate the effectiveness of the proposed method we apply it to control underactuated marine surface vessels.

Stability of recurrent models is closely linked with trainability, generalizability and in some applications, safety. Methods that train stable recurrent neural networks, however, do so at a significant cost to expressibility. We propose an implicit model structure that allows for a convex parametrization of stable models using contraction analysis of non-linear systems. Using these stability conditions we propose a new approach to model initialization and then provide a number of empirical results comparing the performance of our proposed model set to previous stable RNNs and vanilla RNNs. By carefully controlling stability in the model, we observe a significant increase in the speed of training and model performance., Comment: Conference submission

Nonlinear stabilization using control contraction metric (CCM) method usually involves an online optimization problem to compute a minimal geodesic (a shortest path) between pair of states, which is not desirable for real-time applications. This paper introduces a continuous-time dynamic realization which distributes the computational cost of the optimization problem over the time domain. The basic idea is to force the internal state of the dynamic controller to converge to a geodesic using covariant derivative information. A numerical example illustrates the proposed approach., Comment: submitted to ACC2020

Gain-scheduled control based on linear parameter-varying (LPV) models derived from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, a nonlinear control scheme based on Control Contraction Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of these two approaches. We show that the CCM based approach is an extended gain-scheduled control scheme which achieves global reference-independent stability and performance through an exact control realization which integrates a series of local LPV controllers on a particular path between the current and reference states., Comment: IFAC LPVS 2019

Unmanned Aerial Vehicle (UAV) path planning algorithms often assume a knowledge reward function or priority map, indicating the most important areas to visit. In this paper we propose a method to create priority maps for monitoring or intervention of dynamic spreading processes such as wildfires. The presented optimization framework utilizes the properties of positive systems, in particular the separable structure of value (cost-to-go) functions, to provide scalable algorithms for surveillance and intervention. We present results obtained for a 16 and 1000 node example and convey how the priority map responds to changes in the dynamics of the system. The larger example of 1000 nodes, representing a fictional landscape, shows how the method can integrate bushfire spreading dynamics, landscape and wind conditions. Finally, we give an example of combining the proposed method with a travelling salesman problem for UAV path planning for wildfire intervention., Comment: Accepted for ICRA 2019

We present a new approach to verifying contraction and $L_2$-gain of uncertain nonlinear systems, extending the well-known method of integral quadratic constraints. The uncertain system consists of a feedback interconnection of a nonlinear nominal system and uncertainties satisfying differential integral quadratic constraints. A pointwise linear matrix inequality condition is formulated to verify the closed-loop differential $ L_2 $ gain, which can lead to global reference-independent $ L_2 $ gain performance of the nonlinear uncertain system. For a polynomial nominal system, the convex verification conditions can be solved via sum-of-squares programming. A simple computational example based on jet-engine surge with input delays illustrates the approach., Comment: submitted to CDC2019

A distributed computing protocol consists of three components: (i) Data Localization: a network-wide dataset is decomposed into local datasets separately preserved at a network of nodes; (ii) Node Communication: the nodes hold individual dynamical states and communicate with the neighbors about these dynamical states; (iii) Local Computation: state recursions are computed at each individual node. Information about the local datasets enters the computation process through the node-to-node communication and the local computations, which may be leaked to dynamics eavesdroppers having access to global or local node states. In this paper, we systematically investigate this potential computational privacy risks in distributed computing protocols in the form of structured system identification, and then propose and thoroughly analyze a Privacy-Preserving-Summation-Consistent (PPSC) mechanism as a generic privacy encryption subroutine for consensus-based distributed computations. The central idea is that the consensus manifold is where we can both hide node privacy and achieve computational accuracy. In this first part of the paper, we demonstrate the computational privacy risks in distributed algorithms against dynamics eavesdroppers and particularly in distributed linear equation solvers, and then propose the PPSC mechanism and illustrate its usefulness.

This paper gives convex conditions for synthesis of a distributed control system for large-scale networked nonlinear dynamic systems. It is shown that the technique of control contraction metrics (CCMs) can be extended to this problem by utilizing separable metric structures, resulting in controllers that only depend on information from local sensors and communications from immediate neighbours. The conditions given are pointwise linear matrix inequalities, and are necessary and sufficient for linear positive systems and certain monotone nonlinear systems. Distributed synthesis methods for systems on chordal graphs are also proposed based on SDP decompositions. The results are illustrated on a problem of vehicle platooning with heterogeneous vehicles, and a network of nonlinear dynamic systems with over 1000 states that is not feedback linearizable and has an uncontrollable linearization, Comment: Conditionally accepted to IEEE Transactions on Control of Networked Systems

Estimation of nonlinear dynamic models from data poses many challenges, including model instability and non-convexity of long-term simulation fidelity. Recently Lagrangian relaxation has been proposed as a method to approximate simulation fidelity and guarantee stability via semidefinite programming (SDP), however the resulting SDPs have large dimension, limiting their utility in practical problems. In this paper we develop a path-following interior point algorithm that takes advantage of special structure in the problem and reduces computational complexity from cubic to linear growth with the length of the data set. The new algorithm enables empirical comparisons to established methods including Nonlinear ARX, and we demonstrate superior generalization to new data. We also explore the "regularizing" effect of stability constraints as an alternative to regressor subset selection., Comment: accepted to IEEE Transactions on Automatic Control