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Direct data-driven state-feedback control of general nonlinear systems

Chris Verhoek, Patrick J. W. Koelewijn, Sofie Haesaert, Roland Tóth

TL;DR

This work extends the data-driven control paradigm to a broad class of discrete-time nonlinear systems by leveraging a velocity-form representation that yields a linear parameter-varying (LPV) surrogate. By applying the LPV Fundamental Lemma to the velocity-form and using a basis-induced scheduling map $p_k=\\psi(x_k,u_k,x_{k-1},u_{k-1})$, the authors synthesize a data-driven velocity-state-feedback controller $K^\mathrm{v}(p_k)$ from a single data sequence, and realize it as a NL state-feedback law $K^\mathrm{NL}$ for the original system with universal shifted stability and dissipativity guarantees. A fully data-driven closed-loop velocity representation is obtained via a persistently exciting data-dictionary $\\mathcal{D}_{N}^{\Delta}$ and a quadratic performance objective, solved through SDP/LMIs to yield stability and near-optimal infinite-horizon cost. The approach is validated on a simulated unbalanced disc, showing that the universal shifted controller achieves tracking and stability across equilibria, whereas a direct LPV controller may fail to track and an LTI controller can diverge outside a local region. Overall, the work provides a principled, data-driven pathway to equilibrium-independent control of general NL systems with quantifiable guarantees, using velocity-dissipativity as the central theoretical bridge.

Abstract

Through the use of the Fundamental Lemma for linear systems, a direct data-driven state-feedback control synthesis method is presented for a rather general class of nonlinear (NL) systems. The core idea is to develop a data-driven representation of the so-called velocity-form, i.e., the time-difference dynamics, of the NL system, which is shown to admit a direct linear parameter-varying (LPV) representation. By applying the LPV extension of the Fundamental Lemma in this velocity domain, a state-feedback controller is directly synthesized to provide asymptotic stability and dissipativity of the velocity-form. By using realization theory, the synthesized controller is realized as a NL state-feedback law for the original unknown NL system with guarantees of universal shifted stability and dissipativity, i.e., stability and dissipativity w.r.t. any (forced) equilibrium point, of the closed-loop behavior. This is achieved by the use of a single sequence of data from the system and a predefined basis function set to span the scheduling map. The applicability of the results is demonstrated on a simulation example of an unbalanced disc.

Direct data-driven state-feedback control of general nonlinear systems

TL;DR

This work extends the data-driven control paradigm to a broad class of discrete-time nonlinear systems by leveraging a velocity-form representation that yields a linear parameter-varying (LPV) surrogate. By applying the LPV Fundamental Lemma to the velocity-form and using a basis-induced scheduling map , the authors synthesize a data-driven velocity-state-feedback controller from a single data sequence, and realize it as a NL state-feedback law for the original system with universal shifted stability and dissipativity guarantees. A fully data-driven closed-loop velocity representation is obtained via a persistently exciting data-dictionary and a quadratic performance objective, solved through SDP/LMIs to yield stability and near-optimal infinite-horizon cost. The approach is validated on a simulated unbalanced disc, showing that the universal shifted controller achieves tracking and stability across equilibria, whereas a direct LPV controller may fail to track and an LTI controller can diverge outside a local region. Overall, the work provides a principled, data-driven pathway to equilibrium-independent control of general NL systems with quantifiable guarantees, using velocity-dissipativity as the central theoretical bridge.

Abstract

Through the use of the Fundamental Lemma for linear systems, a direct data-driven state-feedback control synthesis method is presented for a rather general class of nonlinear (NL) systems. The core idea is to develop a data-driven representation of the so-called velocity-form, i.e., the time-difference dynamics, of the NL system, which is shown to admit a direct linear parameter-varying (LPV) representation. By applying the LPV extension of the Fundamental Lemma in this velocity domain, a state-feedback controller is directly synthesized to provide asymptotic stability and dissipativity of the velocity-form. By using realization theory, the synthesized controller is realized as a NL state-feedback law for the original unknown NL system with guarantees of universal shifted stability and dissipativity, i.e., stability and dissipativity w.r.t. any (forced) equilibrium point, of the closed-loop behavior. This is achieved by the use of a single sequence of data from the system and a predefined basis function set to span the scheduling map. The applicability of the results is demonstrated on a simulation example of an unbalanced disc.
Paper Structure (11 sections, 3 theorems, 27 equations, 3 figures)

This paper contains 11 sections, 3 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Data-based velocity representations
  4. LPV embedding of the velocity-form
  5. Data-driven closed-loop velocity representations
  6. Data-driven state-feedback control of NL systems with guarantees
  7. Data-driven velocity state-feedback synthesis
  8. Realization of the NL controller
  9. Stability and performance guarantees
  10. Simulation study
  11. Conclusions

Key Result

Corollary 1

Given a PE $\mathcal{D}_N^\Delta$ generated by eq:NL_sys based on which $\overrightarrow{X}_{\!\Delta}$ and $\mathcal{G}_\Delta$ are constructed. Then, the interconnection of eq:NL_sys_LPV, i.e., eq:NL_sys_velocity, and a given $K^\mathrm{v}(p_k)$ under the feedback law eq:fblaw is represented equiv where $\mathcal{V}\in\mathbb{R}^{N-1 \times n_\mathrm{x}(1+n_\mathrm{p}+n_\mathrm{p}^2) }$ is any m

Figures (3)

  • Figure 1: Realization of the controller.
  • Figure 2: Data-dictionary $\mathcal{D}_{N\!+\!1}^{\textsc{nl}}$ used for the NL and LPV control synthesis with $N=8$. The extra gray ( ) data-points are required for the direct LPV representation because of the added state for the integrator behavior.
  • Figure 3: Response of the unbalanced disc with the universal shifted controller ( ) and the LPV controller ( ) in closed-loop for a step reference ( ). An LTI controller ( ) designed with the same specifications diverges directly.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Remark 2
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • ...and 3 more