# A switching algorithm for global exponential stabilization of uncertain chained systems

TL;DR: A novel switching control strategy is proposed involving the use of input/state scaling and integrator backstepping and the ability to achieve Lyapunov stability, exponential convergence, and robustness to a set of uncertain drift terms is proposed.

Abstract: This note deals with chained form systems with strongly nonlinear unmodeled dynamics and external disturbances. The objective is to design a robust nonlinear state feedback law such that the closed-loop system is globally Kexponentially stable. We propose a novel switching control strategy involving the use of input/state scaling and integrator backstepping. The new features of our controllers include the ability to achieve Lyapunov stability, exponential convergence, and robustness to a set of uncertain drift terms.

## Summary (2 min read)

### I. INTRODUCTION

- As explained and illustrated in [11] , [18] , and the references therein, many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to chained form systems via coordinates and state-feedback transformation.
- Exponential convergence is an important performance characteristic for practical applications.
- To date, several important steps have been made toward the design of a continuous time-varying and/or discontinuous feedback law guaranteeing the exponential regulation of nonholonomic systems in chained form [2] , [10] , [16] , and [20] .
- Finally, some conclusions are given in Section V.

### II. PROBLEM FORMULATION

- Throughout this note, the following assumptions will be required.
- The authors first recall a notion of K-exponential stability from [20] , which reduces to exponential stability in the Lyapunov sense when the function is a linear function.
- Notice that the forward invariance property proved in Lemma 1 will be used in the controller design and stability analysis in the next section.

### A. Input-State Scaling and Backstepping Design

- P2) is only convenient for the control design.
- In the remainder of this section, the authors focus on designing the control input u provided that Assumptions 1-3 are satisfied.
- Thanks to Lemma 2, (8) satisfies the "lower-triangularity" condition and therefore, the systematic controller design for u can be obtained using so-called backstepping methods [6] , [10] , [12] .
- Therefore, the following theorem can be obtained.

### B. Switching Scheme

- The purpose of this section is to answer this question by proposing a globally exponentially stabilizing static state feedback.
- To prevent this phenomenon from happening, the following switching control strategy for both control inputs u 0 and u is proposed.
- Then, the following static discontinuous feedback law globally K-exponentially stabilizes the uncertain chained form system (1).
- Proof: Remark 1: It should be emphasized that a feedback controller may become excessively large even for small states.
- Note that along the trajectories of the closed-loop system, when u 0 tends to zero, the state does the same.

### IV. EXAMPLE

- A tricycle-type mobile robot with nonholonomic constraints on the linear velocity has often been used as a benchmark example in the recent literature on nonholonomic control systems design [8] , [10] .
- The convergence rate is not exponential but asymptotic.
- The closed-loop system is not Lyapunov stable although the convergence rate is exponential.
- The authors will design a robust state-feedback controller to drive the states of (21) to the origin with exponential convergence and Lyapunov stability.
- So, the following controller can be obtained.

### V. CONCLUSION

- The problem of global K-exponential stabilization is considered for a class of nonholonomic chained systems with strongly nonlinear input/state driven disturbances and drifts.
- Using input-state scaling and backstepping techniques, a globally exponentially convergent state-feedback control law is designed.
- Using a switching scheme dependent on the initial condition, Lyapunov stability and exponential convergence are guaranteed for the closed-loop system.
- The simulations results in a wheeled mobile robot have demonstrated the effectiveness of the proposed control design approach.

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165 citations

### Cites background from "A switching algorithm for global ex..."

...Recently, many interesting results were presented to provide a solid foundation for the performance analysis of hybrid control [2], [4], [5], [7], [9]–[12], [24], [27]–[29], [31], [33]....

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...Other interesting directions include the study of the limiting systems of switched systems as done in [22] with applications to the design of switching-based stabilizing and tracking controllers for physical systems; see [33] for preliminary results....

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...Step i (2pipn 2): As in [17,23,26], consider the Lyapunov function candidate Vi 1⁄4 V i 1ðt;P; e; z1; ....

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...Then the design of the control input u will be obtained using the standard backstepping method shown in [17,18,23,26] to the transformed system (9), i....

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...We recently proposed a switching scheme to achieve Lyapunov stability and exponential convergence for uncertain chained form systems using state feedback in [23]....

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...As in [26,17,18,23], it is easy to know that some suitable smooth functions jðn 1Þjðt;P; x0;z1; ....

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...In Xi et al. (2003), it is assumed not only that ’i0 ¼ 0 in the inequality (7), but also that the modelled dynamics fi and the dynamics with unknown parameters Ti ð yÞ do not exist, i.e., fi¼ 0 and Ti ð yÞ ¼ 0, although the gains of u0, x2u0, . . . , xn 1u0 and u on the right hand sides of the…...

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### "A switching algorithm for global ex..." refers background in this paper

...Examples of these approaches are open-loop periodic steering control, either smooth or continuous time-varying control, and discontinuous feedback control; see, for example, [1]–[3], [6], [7], [9], [10], [13], and [15]–[20]....

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### "A switching algorithm for global ex..." refers background in this paper

...INTRODUCTION Over the past decade, the control and stabilization of nonholonomic systems has formed an active area within the nonlinear control community; see, for example, the recent survey papers [5], [11], and the references cited therein for an interesting introduction to this quickly expanding area....

[...]

...As explained and illustrated in [11], [18], and the references therein, many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to chained form systems via coordinates and state-feedback transformation....

[...]