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Constructing Dynamic Feedback Linearizable Discretizations

Ashutosh Jindal, Florentina Nicolau, David Martin Diego, Ravi Banavar

TL;DR

This paper addresses preserving feedback linearizability under discretization for nonlinear systems that are linearizable by dynamic feedback. It develops a framework using discretization maps and retraction maps on manifolds, lifting discretizations from the linearized coordinates through a diffeomorphism $\Phi$ to yield a discrete-time system that remains FL, with a discrete feedback $\mu_k=\alpha(\xi_k)+\beta(\xi_k)v_k$. A main result shows that, given a dynamic-FL extended system transformable to a linear model $\dot{z}=Az+Bv$, one can construct first-order accurate discretizations that produce $z_{k+1}=A_hz_k+B_hv_k$, and hence a discrete-time FL controller. The approach is demonstrated on a unicycle-like example and validated via simulations, achieving stabilization with a global error on the order of $10^{-2}$ for a fixed stepsize. Overall, the work extends prior static-FL discretization results to dynamic FL, enabling FL-based control in discrete time for a broader class of nonlinear systems and offering a pathway to higher-order schemes in the future.

Abstract

Dynamic feedback linearization-based methods allow us to design control algorithms for a fairly large class of nonlinear systems in continuous time. However, this feature does not extend to their sampled counterparts, i.e., for a given dynamically feedback linearizable continuous time system, its numerical discretization may fail to be so. In this article, we present a way to construct discretization schemes (accurate up to first order) that result in schemes that are feedback linearizable. This result is an extension of our previous work, where we had considered only static feedback linearizable systems. The result presented here applies to a fairly general class of nonlinear systems, in particular, our analysis applies to both endogenous and exogenous types of feedback. While the results in this article are presented on a control affine form of nonlinear systems, they can be readily modified to general nonlinear systems.

Constructing Dynamic Feedback Linearizable Discretizations

TL;DR

This paper addresses preserving feedback linearizability under discretization for nonlinear systems that are linearizable by dynamic feedback. It develops a framework using discretization maps and retraction maps on manifolds, lifting discretizations from the linearized coordinates through a diffeomorphism to yield a discrete-time system that remains FL, with a discrete feedback . A main result shows that, given a dynamic-FL extended system transformable to a linear model , one can construct first-order accurate discretizations that produce , and hence a discrete-time FL controller. The approach is demonstrated on a unicycle-like example and validated via simulations, achieving stabilization with a global error on the order of for a fixed stepsize. Overall, the work extends prior static-FL discretization results to dynamic FL, enabling FL-based control in discrete time for a broader class of nonlinear systems and offering a pathway to higher-order schemes in the future.

Abstract

Dynamic feedback linearization-based methods allow us to design control algorithms for a fairly large class of nonlinear systems in continuous time. However, this feature does not extend to their sampled counterparts, i.e., for a given dynamically feedback linearizable continuous time system, its numerical discretization may fail to be so. In this article, we present a way to construct discretization schemes (accurate up to first order) that result in schemes that are feedback linearizable. This result is an extension of our previous work, where we had considered only static feedback linearizable systems. The result presented here applies to a fairly general class of nonlinear systems, in particular, our analysis applies to both endogenous and exogenous types of feedback. While the results in this article are presented on a control affine form of nonlinear systems, they can be readily modified to general nonlinear systems.
Paper Structure (8 sections, 3 theorems, 45 equations, 5 figures)

This paper contains 8 sections, 3 theorems, 45 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Static and Dynamic Feedback Linearization
  3. Constructing Feedback Linearizable Discretization
  4. Retraction and Discretization maps
  5. Main result
  6. Illustrating Theorem \ref{['main_result']} on Example \ref{['ex_1']}
  7. Simulation Results
  8. Conclusion

Key Result

Proposition III.1

Consider two $n$-dimensional manifolds $M$ and $N$. Suppose $M\ni x\longmapsto \phi(x)\eqqcolon y\in N$ is a diffeomorphism. Then for a given discretization map $\mathcal{D}$ on $M$, $\mathcal{D}_\phi := (\phi\times\phi)\circ \mathcal{D}\circ T\phi^{-1}$ is a discretization map on $N$ (see Figure fi

Figures (5)

  • Figure 3.1: $\mathcal{D}$ and $\mathcal{D}_\phi$ commute as shown above
  • Figure 3.2: Schematic representation of constructing a feedback linearizable discretization scheme for \ref{['ext_sys']}. Both \ref{['ext_sys']} and its discretization \ref{['retr_desc_lin1']} are linearizable by coordinate change $z\coloneqq \Phi(\xi)$ and feedback $\mu \coloneqq \Psi({\xi},v)= \alpha(\xi)+\beta(\xi)v$ (CTLS- Continuous-time linear system, DTLS - Discrete-time linear system).
  • Figure 5.1: System states $\xi_k\coloneqq (x_k,w_k)$ for \ref{['ex1_disc_nonlin']} for a stepsize $h=10^{-2}$ and $t_k\in[0,10]$.
  • Figure 5.2: Control input $\mu_k\coloneqq (\mu_k^1,\mu_k^2)$ for \ref{['ex1_disc_nonlin']} for a stepsize $h=10^{-2}$ and $t_k\in[0,10]$.
  • Figure 5.3: Global Error $\left\lVert \xi(t_k)-\xi_k \right\rVert$ for \ref{['ex1_disc_nonlin']}, for a stepsize $h=10^{-2}$ and $t_k\in[0,10]$.

Theorems & Definitions (13)

  • Definition II.1: Static Feedback Linearization
  • Remark II.1
  • Example II.1
  • Definition II.2: Dynamic Compensator
  • Remark II.2
  • Definition II.3: Dynamic Feedback Linearization
  • Definition III.1: Retraction Maps AbMaSeBookRetraction
  • Definition III.2: Discretization Maps 21MBLDMdD
  • Remark III.1
  • Proposition III.1: Lift of discretization maps retr_disc_map
  • ...and 3 more