Funnel control of higher relative degree systems

Chowdhury, Dhrubajit; Khalil, Hassan K.

doi:10.23919/acc.2017.7963018

Cited by 9 publications

(9 citation statements)

References 9 publications

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“…45 To circumvent the peaking, saturation is commonly used. However, we stress that a saturation of the observer states as suggested in the work of Chowdhury and Khalil 42 is not possible here, since this only leads to infeasibility of the controller (the closed-loop system does not have a global solution). Instead, the control input must be directly saturated using…”

Section: Funnel Control Via Funnel Pre-compensatormentioning

confidence: 96%

“…To circumvent the peaking, saturation is commonly used. However, we stress that a saturation of the observer states as suggested in the work of Chowdhury and Khalil is not possible here, since this only leads to infeasibility of the controller (the closed‐loop system does not have a global solution). Instead, the control input must be directly saturated using

u false(t false) = sat false(- k_{2} false(t false) e_{2} false(t false), ū false),

where

ū > 0

is the saturation level and

sat : R \times R \to R, (v, \bar{v}) \mapsto \{\begin{matrix} v, & false| v false| \leq \overset{v}{true}, \\ sign false(v false) \overset{v}{true}, & false| v false| > \overset{v}{true} . \end{matrix}

While the peaking phenomenon can be avoided by the saturation of the input, care must be taken with the choice of the saturation level, because values that are too small may lead to infeasibility of the controller in the sense that the solution leaves the domain of the closed‐loop differential equation in finite time.…”

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 98%

“…Remark We compare the controllers (31) and (32) to an alternative approach presented in the recent conference paper of Chowdhury and Khalil; a similar approach, based on prescribed performance control, can be found in the work of Bechlioulis et al In the work of Chowdhury and Khalil, the funnel controller from the work of Ilchmann et al is combined with a high‐gain observer. For nonlinear SISO systems with higher relative degree, a virtual (weighted) output is defined as

s (t) = e (t) + k_{2} μ ė (t) + … + k_{r} μ^{r - 1} e^{(r - 1)} (t)

for some design parameters k i >0 and a scaling parameter μ >0.…”

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 99%

“…In particular, this approach is not model‐free like standard funnel control approaches and the controller is not robust in the sense that, when all its parameters are fixed, it works for a class of systems satisfying some structural assumption. Since the aforementioned approach still involves output derivatives, in a second step presented in the work of Chowdhury and Khalil, the output derivatives are estimated using a high‐gain observer as follows:

\begin{matrix} right {\dot{z}}_{1} (t) & left = z_{2} (t) + \frac{α_{1}}{ε} (e (t) - z_{1} (t)), & right \\ right {\dot{z}}_{2} (t) & left = z_{3} (t) + \frac{α_{2}}{ε^{2}} (e (t) - z_{1} (t)), \\ right & left ⋮ \\ right {\dot{z}}_{r - 1} (t) & left = z_{r} (t) + \frac{α_{r - 1}}{ε^{r - 1}} (e (t) - z_{1} (t)), \\ right {\dot{z}}_{r} (t) & left = F (t, z_{1} (t), …, z_{r} (t)) + Γ u (t) + \frac{α_{r}}{ε^{r}} (e (t) - z_{1} (t)), \end{matrix}

where α 1 ,…, α r are such that the matrix

([], \begin{matrix} - α_{1} & 1 \end{matrix})

…”

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 99%

See 3 more Smart Citations

The funnel pre‐compensator

Berger

Reis

2018

Intl J Robust & Nonlinear

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Summary We introduce the funnel pre‐compensator as a novel and simple adaptive pre‐compensator of “high‐gain type”. We show that this pre‐compensator is feasible for a large class of signal pairs, which satisfy a certain relationship. We show that the funnel pre‐compensator guarantees prescribed transient behavior of the compensator error, it is of low complexity and inherently robust since its design is model‐free. As an application in adaptive control of nonlinear systems, a cascade of funnel pre‐compensators is exploited to obtain an artificial output with explicitly known derivatives, which tracks the system output with prescribed transient behavior. In some important cases the interconnection of the system with the pre‐compensator cascade is shown to have input‐to‐state stable internal dynamics. This guarantees feasibility of a novel funnel controller that consists of a funnel pre‐compensator cascade in conjunction with a recently developed funnel controller for systems with arbitrary relative degree. We illustrate the application of this interconnection for some mechanical systems.

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Section: Funnel Control Via Funnel Pre-compensatormentioning

confidence: 96%

u false(t false) = sat false(- k_{2} false(t false) e_{2} false(t false), ū false),

where

ū > 0

is the saturation level and

sat : R \times R \to R, (v, \bar{v}) \mapsto \{\begin{matrix} v, & false| v false| \leq \overset{v}{true}, \\ sign false(v false) \overset{v}{true}, & false| v false| > \overset{v}{true} . \end{matrix}

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 98%

s (t) = e (t) + k_{2} μ ė (t) + … + k_{r} μ^{r - 1} e^{(r - 1)} (t)

for some design parameters k i >0 and a scaling parameter μ >0.…”

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 99%

\begin{matrix} right {\dot{z}}_{1} (t) & left = z_{2} (t) + \frac{α_{1}}{ε} (e (t) - z_{1} (t)), & right \\ right {\dot{z}}_{2} (t) & left = z_{3} (t) + \frac{α_{2}}{ε^{2}} (e (t) - z_{1} (t)), \\ right & left ⋮ \\ right {\dot{z}}_{r - 1} (t) & left = z_{r} (t) + \frac{α_{r - 1}}{ε^{r - 1}} (e (t) - z_{1} (t)), \\ right {\dot{z}}_{r} (t) & left = F (t, z_{1} (t), …, z_{r} (t)) + Γ u (t) + \frac{α_{r}}{ε^{r}} (e (t) - z_{1} (t)), \end{matrix}

where α 1 ,…, α r are such that the matrix

([], \begin{matrix} - α_{1} & 1 \end{matrix})

…”

Section: Funnel Control Via Funnel Pre‐compensatormentioning

confidence: 99%

See 2 more Smart Citations

The funnel pre‐compensator

Berger

Reis

2018

Intl J Robust & Nonlinear

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show abstract

“…Although practical tracking has a relatively conservative objective, it requires rather milder restrictions on the systems and the reference signals than asymptotic tracking, and particularly, is adequate for many real applications. Therefore, practical tracking has been an appealing research topic (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references therein). However, in many of the related studies, e.g., [1][2][3][4][5][6][7][8][9], only the basic steady-state tracking performance (i.e., the ultimate convergence of the tracking error) is achieved, which is sometimes far from enough for real applications.…”

Section: Introductionmentioning

confidence: 99%

Global practical tracking with prescribed transient performance for inherently nonlinear systems with extremely severe uncertainties

Liu

2018

Sci. China Inf. Sci.

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This paper considers the global practical tracking for a class of uncertain nonlinear systems. Remarkably, the systems under investigation admit rather inherent nonlinearities, and especially allow arguably the most severe uncertainties: unknown control directions and non-parametric uncertainties. Despite this, a refined tracking objective, rather than a reduced one, is sought. That is, not only pre-specified arbitrary tracking accuracy is guaranteed, but also certain prescribed transient performance (e.g., arrival time and maximum overshoot) is ensured to better meet real applications. To solve the problem, a new tracking scheme is established, crucially introducing delicate time-varying gains to counteract the severe uncertainties and guarantee the prescribed performance. It is shown that the designed controller renders the tracking error to forever evolve within a prescribed performance funnel, through which the desired tracking objective is accomplished for the systems. Particularly, by subtly specifying the funnel, global fixed-time practical tracking (i.e., that with prescribed arrival time) and semiglobal practical tracking with prescribed maximal overshoot can be achieved for the systems. Moreover, the tracking scheme remains valid in the presence of rather less-restrictive unmodeled dynamics.

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Introduction

Gao,

2023

Spacecraft Maneuver With Performance Guaranteed

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Funnel control of higher relative degree systems

Cited by 9 publications

References 9 publications

The funnel pre‐compensator

The funnel pre‐compensator

Global practical tracking with prescribed transient performance for inherently nonlinear systems with extremely severe uncertainties

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