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 considers the analysis of continuous time gradient-based optimization algorithms through the lens of nonlinear contraction theory. It demonstrates that in the case of a time-invariant objective, most elementary results on gradient descent based on convexity can be replaced by much more general results based on contraction. In particular, gradient descent converges to a unique equilibrium if its dynamics are contracting in any metric, with convexity of the cost corresponding to the special case of contraction in the identity metric. More broadly, contraction analysis provides new ins…
In the past decade, we had developed a series of splitting contraction algorithms for separable convex optimization problems, at the root of the alternating direction method of multipliers. Convergence of these algorithms was studied under specific model-tailored conditions, while these conditions can be conceptually abstracted as two generic conditions when these algorithms are all unified as a prediction-correction framework. In this paper, in turn, we showcase a constructive way for specifying the generic convergence-guaranteeing conditions, via which new splitting contraction algorithms ca…
This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between…
We derive an exact deterministic nonlinear observer to compute the continuous state of an inertial navigation system based on partial discrete measurements, the so-called strapdown problem. Nonlinear contraction is used as the main analysis tool, and the hierarchical structure of the system physics is sytematically exploited. The paper also discusses the use of nonlinear measurements, such as distances to time-varying reference points.
We investigate the incremental stability properties of Itô stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of stochastic nonlinear observers design and stochastic synchronization.
This paper is motivated by the problem of quantitatively bounding the convergence of adaptive control methods for stochastic systems to a stationary distribution. Such bounds are useful for analyzing statistics of trajectories and determining appropriate step sizes for simulations. To this end, we extend a methodology from (unconstrained) stochastic differential equations (SDEs) which provides contractions in a specially chosen Wasserstein distance. This theory focuses on unconstrained SDEs with fairly restrictive assumptions on the drift terms. Typical adaptive control schemes place constrain…
A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning
Center for Control, Dynamical Systems, and Computation, UC Santa Barbara（控制、动态系统与计算中心，加州大学圣巴巴拉分校）；Department of Electronics and Telecommunications, Politecnico di Torino（电子与电信系，托里诺理工学院）；Graduate School of Advanced Science and Engineering, Hiroshima University（先进科学与工程研究生院，广岛大学）
This paper presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Ito stochastic nonlinear systems and Lagrangian systems. Its innovation lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is non-convex, its equivalent convex formulation is proposed utilizing state-dependent coefficient parameterizations of the nonlinear …
This paper considers the safety-critical control design problem with output measurements. An observer-based safety control framework that integrates the estimation error quantified observer and the control barrier function (CBF) approach is proposed. The function approximation technique is employed to approximate the uncertainties introduced by the state estimation error, and an adaptive CBF approach is proposed to design the safe controller which is obtained by solving a convex quadratic program (QP). Theoretical results for CBFs with a relative degree 1 and a higher relative degree are given…
The problem of unknown input observer design is considered for coupled PDE/ODE systems subject to incremental sector bounded nonlinearities and unknown boundary inputs. Assuming available measurements at the boundary of the distributed domain, the synthesis of the unknown input observer is based on Lyapunov methods and convex optimization. Numerical simulations support and confirm the theoretical findings, illustrating the robust estimation performances of the proposed nonlinear unknown input observer.
We present a framework which incorporates three aspects of the estimation problem, namely, sparse sensor configuration, optimal precision, and robustness in the presence of model uncertainty. The problem is formulated in the $\mathcal{H}_{\infty}$ optimal observer design framework. We consider two types of uncertainties in the system, i.e. structured affine and unstructured uncertainties. The objective is to design an observer with a given $\mathcal{H}_{\infty}$ performance index with minimal number of sensors and minimal precision values, while guaranteeing the performance for all admissible …
Decentralized Nonlinear Feedback Design with Separable Control Contraction Metrics
The Australian Centre for Field Robotics（澳大利亚野外机器人中心）；School of Aerospace Mechanical and Mechatronical Engineering, The University of Sydney（悉尼大学航空航天机械与机电工程学院）
Observer design is considered for a class of non-linear systems whose non-linear part is energy preserving. A strategy to construct convergent observers for this class of non-linear system is presented. The approach has the advantage that it is possible, via convex programming, to prove whether the constructed observer converges, in contrast to several existing approaches to observer design for non-linear systems. Finally, the developed methods are applied to the Lorenz attractor and to a low order model for shear fluid flow.
Stanford University（斯坦福大学）；Google Brain Robotics（谷歌大脑机器人团队）；Massachusetts Institute of Technology（麻省理工学院）
We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in Reproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning as a functional optimization with convex objective and bi-conv…
This paper analyzes the contraction of the primal-dual gradient optimization via contraction theory in the context of discrete-time updating dynamics. The contraction theory based on Riemannian manifolds is first established for convergence analysis of a convex optimization algorithm. The equality and inequality constrained optimization cases are studied, respectively. Under some reasonable assumptions, we construct the Riemannian metric to characterize a contraction region. It is shown that if the step-sizes of the updating dynamics are properly designed, the convergence rates for both cases …
We study distributed composite optimization over networks: agents minimize the sum of a smooth (strongly) convex function, the agents' sum-utility, plus a non-smooth (extended-valued) convex one. We propose a general algorithmic framework for such a class of problems and provide a unified convergence analysis leveraging the theory of operator splitting. Our results unify several approaches proposed in the literature of distributed optimization for special instances of our formulation. Distinguishing features of our scheme are: (i) when the agents' functions are strongly convex, the algorithm c…
Graduate Aerospace Laboratories, Caltech（加州理工学院研究生航空航天实验室）；Nonlinear Systems Laboratory, MIT（麻省理工学院非线性系统实验室）；Reliable Autonomous System Laboratory, MIT（麻省理工学院可靠自主系统实验室）
This paper presents a theoretical overview of a Neural Contraction Metric (NCM): a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, the existence of which is a necessary and sufficient condition for incremental exponential stability of non-autonomous nonlinear system trajectories. Its innovation lies in providing formal robustness guarantees for learning-based control frameworks, utilizing contraction theory as an analytical tool to study the nonlinear stability of learned systems via convex optimization. In particular, we rigorously show …
The Primal-Dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, strict contraction is nontrivial to establish in most cases. This work focuses on continuous, possibly non-autonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple time-scales. We show that the PD algorithm is indeed strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive e…
Contraction analysis is a stability theory for nonlinear systems where stability is defined incrementally between two arbitrary trajectories. It provides an alternative framework in which to study uncertain interconnections or systems with external inputs, where it offers several significant advantages when compared with traditional Lyapunov analysis. Contraction-based methods are particularly useful for analyzing systems with uncertain parameters and for proving synchronization properties of nonlinear oscillators. Existence of a contraction metric for a given system is a necessary and suffici…
This article considers a model formulation and an adaptive observer design for the simultaneous estimation of the state of charge and crossover flux in disproportionation redox flow batteries. This novel nonaqueous battery chemistry allows a simple isothermal lumped parameter model to be formulated. The transport of vanadium through the porous separator is a key unknown function of battery variables and it is approximated in the space of continuous functions. The state and parameter observer adaptation laws are derived using Lyapunov analysis applied to the estimation error, the stability and …
In this paper we present a first-order method that admits near-optimal convergence rates for convex/concave min-max problems while requiring a simple and intuitive analysis. Similarly to the seminal work of Nemirovski and the recent approach of Piliouras et al. in normal form games, our work is based on the fact that the update rule of the Proximal Point method (PP) can be approximated up to accuracy $ε$ with only $O(\log 1/ε)$ additional gradient-calls through the iterations of a contraction map. Then combining the analysis of (PP) method with an error-propagation analysis we establish that t…
We propose a formulation for nonlinear recurrent models that includes simple parametric models of recurrent neural networks as a special case. The proposed formulation leads to a natural estimator in the form of a convex program. We provide a sample complexity for this estimator in the case of stable dynamics, where the nonlinear recursion has a certain contraction property, and under certain regularity conditions on the input distribution. We evaluate the performance of the estimator by simulation on synthetic data. These numerical experiments also suggest the extent at which the imposed theo…
