A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic neural contraction metric, as illustrated in simulation results.

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“…by Lemma 2 and use it in Theorem 2, although (14) has to be solved implicitly as B depends on u in this case [12], [13].…”

Section: A Stability Of Generalized Sdc Control and Estimationmentioning

confidence: 99%

Neural Stochastic Contraction Metrics for Learning-Based Control and Estimation

Tsukamoto

Chung

Slotine

2021

IEEE Control Syst. Lett.

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“…This approach is not pursued here since most systems require a polynomial of high order to approximate the nonlinear dynamics and the computational complexity grows exponentially in n d , thus prohibiting the practical application. The connection between CCM and LPV gain-scheduling design is discussed in [24].…”

Section: B Quasi-lpv Based Proceduresmentioning

confidence: 99%

A Nonlinear Model Predictive Control Framework Using Reference Generic Terminal Ingredients

Köhler

Müller

Allgöwer

2020

IEEE Trans. Automat. Contr.

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In this paper, we present a quasi infinite horizon nonlinear model predictive control (MPC) scheme for tracking of generic reference trajectories. This scheme is applicable to nonlinear systems, which are locally incrementally stabilizable. For such systems, we provide a reference generic offline procedure to compute an incrementally stabilizing feedback with a continuously parameterized quadratic quasi infinite horizon terminal cost. As a result we get a nonlinear reference tracking MPC scheme with a valid terminal cost for general reachable reference trajectories without increasing the online computational complexity. As a corollary, the terminal cost can also be used to design nonlinear MPC schemes that reliably operate under online changing conditions, including unreachable reference signals. The practicality of this approach is demonstrated with a benchmark example.

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“…Specifically, in [13][14][15]17], a convex optimization-based framework for robust feedback control and state estimation, named ConVex optimization-based Steady-state Tracking Error Minimization (CV-STEM), is derived to find a contraction metric that minimizes an upper bound of the steady-state distance between perturbed and unperturbed system trajectories. In this context, we could utilize Control Contraction Metrics (CCMs) [33,[77][78][79][80][81] for extending contraction theory to the systematic design of differential feedback control δu = k(x, δx, u, t) via convex optimization, achieving greater generality at the expense of computational efficiency in obtaining u. Applications of the CCM to estimation, adaptive control, and motion planning are discussed in [82], [83][84][85], and [78,[86][87][88][89], respectively, using geodesic distances between trajectories [49].…”

Section: Construction Of Contraction Metrics (Sec 3-4)mentioning

confidence: 99%

“…3, 4, and 6-8. This is primarily because the controller design techniques for control-affine nonlinear systems are less complicated than those for control non-affine systems (which often result in u given implicitly by u = k(x, u, t) [80,81]), but still utilizable even for the latter, e.g., by treating u as another control input (see Example 3.1), or by solving the implicit equation u = k(x, u, t) iteratively with a discrete-time controller (see Example 3.2 and Remark 3.3).…”

Section: Robust Nonlinear Control and Estimation Via Contraction Theorymentioning

confidence: 99%

“…We could also consider using the partial derivative of f of the dynamical system directly for control synthesis through differential state feedback δu = k(x, δx, u, t). This idea, formulated as the concept of a CCM [33,[77][78][79][80][81], constructs contraction metrics with global stability guarantees independently of target trajectories, achieving greater generality while requiring added computation in evaluating integrals involving minimizing geodesics. Similar to the CCM, we could design a state estimator using a general formulation based on geodesics distances between trajectories [49,82].…”

Section: Control Contraction Metric (Ccm) Formulationmentioning

confidence: 99%

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Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Tsukamoto,

Chung,

Slotine

2021

Preprint

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Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is therefore to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

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A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control

Cited by 12 publications

References 21 publications

Neural Stochastic Contraction Metrics for Learning-Based Control and Estimation

Neural Stochastic Contraction Metrics for Learning-Based Control and Estimation

A Nonlinear Model Predictive Control Framework Using Reference Generic Terminal Ingredients

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

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