Learning Controllers From Data via Approximate Nonlinearity Cancellation

Persis, Claudio De; Rotulo, Monica; Tesi, Pietro

doi:10.1109/tac.2023.3234889

Cited by 34 publications

(39 citation statements)

References 44 publications

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“…Stability of learning-based MPC was established in [2,72] and followed, for nonlinear systems, by efforts on joint learning of the controller and(or) Lyapunov functions [13-15, 21, 22]. In addition, [64,83] have explored how learning-based control affects systems with known Lyapunov functions, [12,23,68] studied learning of stability certificates and stable controllers from data, and [6] developed a provably stable data-driven algorithm based on system measurements and prior system knowledge.…”

Section: Control Design Problems For Hyperbolic Pdes Are Hyperbolicmentioning

confidence: 99%

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Krstić¹,

Bhan²,

Shi³

2023

Preprint

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Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles-functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite-dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first-order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.

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Section: Control Design Problems For Hyperbolic Pdes Are Hyperbolicmentioning

confidence: 99%

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Krstić¹,

Bhan²,

Shi³

2023

Preprint

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“…[11], [12] generalize the method to learning Lyapunov functions for piecewise linear and hybrid systems, and [13] for learning regions of attraction of nonlinear systems. In addition, [59], [76] have explored how learning-based control will affect nominal systems with known Lyapunov functions, and [9], [22], [62] studied the problem of learning stability certificates and stable controllers directly from data. In a similar vein, [4] has developed a provable stable data-driven algorithm based on system measurements and prior knowledge for linear time-invariant systems.…”

Section: Introductionmentioning

confidence: 99%

Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping

Bhan¹,

Shi²,

Krstić³

2023

Preprint

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We introduce a framework for eliminating the computation of controller gain functions in PDE control. We learn the nonlinear operator from the plant parameters to the control gains with a (deep) neural network. We provide closed-loop stability guarantees (global exponential) under an NN-approximation of the feedback gains. While, in the existing PDE backstepping, finding the gain kernel requires (one offline) solution to an integral equation, the neural operator (NO) approach we propose learns the mapping from the functional coefficients of the plant PDE to the kernel function by employing a sufficiently high number of offline numerical solutions to the kernel integral equation, for a large enough number of the PDE model's different functional coefficients. We prove the existence of a Deep-ONet approximation, with arbitrarily high accuracy, of the exact nonlinear continuous operator mapping PDE coefficient functions into gain functions. Once proven to exist, learning of the NO is standard, completed "once and for all" (never online) and the kernel integral equation doesn't need to be solved ever again, for any new functional coefficient not exceeding the magnitude of the functional coefficients used for training. We also present an extension from approximating the gain kernel operator to approximating the full feedback law mapping, from plant parameter functions and state measurement functions to the control input, with semiglobal practical stability guarantees. Simulation illustrations are provided and code is available on github. This framework, eliminating real-time recomputation of gains, has the potential to be game changing for adaptive control of PDEs and gain scheduling control of nonlinear PDEs.The paper requires no prior background in machine learning or neural networks.

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“…For certain classes of nonlinear systems, various works have proposed extensions and applications of the fundamental lemma using, e.g., basis functions. For instance, Hammerstein-Wiener systems [3], flat and feedback linearizable systems [4], [5], and control design of inputaffine nonlinear systems [6] have been considered. In these works, suitable PE conditions involving the sequence of basis functions (which depend on inputs and/or states/outputs) need to be satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…In these works, suitable PE conditions involving the sequence of basis functions (which depend on inputs and/or states/outputs) need to be satisfied. However, in [3]- [6] these PE conditions could only be verified a posteriori, i.e., after performing an experiment and collecting state/output data, and no a priori input design was proposed to this end.…”

Section: Introductionmentioning

confidence: 99%

On the design of persistently exciting inputs for data-driven control of linear and nonlinear systems

Alsalti¹,

Lopez²,

Müller³

2023

Preprint

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In the context of data-driven control, persistence of excitation (PE) of an input sequence is defined in terms of a rank condition on the Hankel matrix of the input data. For nonlinear systems, recent results employed rank conditions involving collected input and state/output data, for which no guidelines are available on how to satisfy them a priori. In this paper, we first show that a set of discrete impulses is guaranteed to be persistently exciting for any controllable LTI system. Based on this result, for certain classes of nonlinear systems, we guarantee persistence of excitation of a sequence of basis functions a priori, by design of the physical input only. Finally, for nonlinear systems which are locally reachable at the origin, we show that there exist sparse input sequences that guarantee collective PE of sequences of basis functions.

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Learning Controllers From Data via Approximate Nonlinearity Cancellation

Cited by 34 publications

References 44 publications

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping

On the design of persistently exciting inputs for data-driven control of linear and nonlinear systems

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