Robust output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities

Li, Guangqi; Lin, Yan

doi:10.1002/rnc.3389

Cited by 14 publications

(13 citation statements)

References 19 publications

(71 reference statements)

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“…We now prove for k = n . By the definition of

u (\overset{x}{true})

and , and similar to the proof of (80) in , it can be shown that

\begin{matrix} | v (\overset{x}{true}) | & ⩽ c_{2} {falsefalse}_{\sum}^{j = 1} 1.61em (| ξ_{j} |^{r_{n + 1}} + | ξ_{j} |^{\frac{g_{n + 1}}{g_{j} / r_{j}}} 1.61em) + {falsefalse}_{\sum}^{j = 2} h_{n - 1, j} 1.19em (M_{[j, n - 1]} 1.19em) 1.61em (| e_{j} |^{\frac{r_{n + 1}}{r_{j - 1}}} + | e_{j} |^{\frac{g_{n + 1}}{g_{j - 1}}} 1.61em) \\ + c_{2} 1.61em (| e_{n} |^{\frac{r_{n + 1}}{r_{n - 1}}} + | e_{n} |^{\frac{g_{n + 1}}{g_{n - 1}}} 1.61em), \end{matrix}

where c 2 is a known positive constant, and h n −...…”

Section: Detailed Proofs Of Technical Claimsmentioning

confidence: 73%

“…Introduce unmeasured signals z i 's as follows, motivated by :

x_{i} = 1.61em (1.19em (z_{i} + M_{i - 1} x_{i - 1} {1.19em)}^{\frac{g_{i} / r_{i}}{g_{i - 1} / r_{i - 1}}} + 1.19em (z_{i} + M_{i - 1} x_{i - 1} 1.19em) {1.61em)}^{\frac{r_{i}}{r_{i - 1}}}, i = 2, \dots, n,

where M i 's are to‐be‐determined positive constants. Then, by , and noting that

\frac{g_{i} / r_{i}}{g_{i - 1} / r_{i - 1}}

's and

\frac{r_{i - 1}}{r_{i}}

's belong to

{boldR}_{⩾ 0}^{odd}

and are larger than 1, it can be verified that

\begin{matrix} right ż_{i} & left = - M_{i - 1} L (t) x_{i} + M_{i - 1} (() (i - 2) \frac{\dot{L} (t)}{L (t) ...} \end{matrix}

…”

Section: Finite‐time Stabilization Controlmentioning

confidence: 99%

“…Define

{\overset{x}{true}}_{k} = 1.19em ({({\overset{z}{true}}_{k} + M_{k - 1} x_{k - 1})}^{\frac{g_{k} / r_{k}}{g_{k - 1} / r_{k - 1}}} + ({\overset{z}{true}}_{k} + M_{k - 1} x_{k - 1}) {1.19em)}^{\frac{r_{k}}{r_{k - 1}}}

. Then, quite similar to the proof of (35) in , we can derive

\begin{matrix} right - M_{k - 1} L (t) (x_{k} - {\bar{x}}_{k}) H_{k} (\cdot) & left ⩽ L (t) (() - c_{k, 1} M_{k - 1}^{\frac{2 r_{k}}{r_{k} + r_{k - 1}}} + c_{k, 2} ()) (() e_{k}^{\frac{2 + τ}{r_{k - 1}}} + e_{k}^{\frac{μ}{g_{k - 1}}} ()) & right \\ right & left +<...> \end{matrix}

…”

Section: Detailed Proofs Of Technical Claimsmentioning

confidence: 99%

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Global finite‐time stabilization via time‐varying output‐feedback for uncertain nonlinear systems with unknown growth rate

Liu

2017

Intl J Robust & Nonlinear

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Summary This paper considers the global finite‐time output‐feedback stabilization for a class of uncertain nonlinear systems. Comparing with the existing related literature, two essential obstacles exist: On the one hand, the systems in question allow serious parametric unknowns and serious time variations coupling to the unmeasurable states, which is reflected in that the systems have the unmeasurable states dependent growth with the rate being an unknown constant multiplying a known continuous function of time. On the other hand, the systems possess remarkably inherent nonlinearities, whose growth allows to be not only low‐order but especially high‐order with respect to the unmeasurable states. To effectively cope with these obstacles, we established a time‐varying output‐feedback strategy to achieve the finite‐time stabilization for the systems under investigation. First, a time‐varying state‐feedback controller is constructed by adding an integrator method, and by homogeneous domination approach, a time‐varying reduced‐order observer is designed to precisely rebuild the unmeasurable states. Then, by certainty equivalence principle, a desired time‐varying output‐feedback controller is constructed for the systems. It is shown that, as long as the involved time‐varying gain is chosen fast enough to overtake the serious parametric unknowns and the serious time variations, the output‐feedback controller renders that the closed‐loop system states converge to zero in finite time. Copyright © 2017 John Wiley & Sons, Ltd.

show abstract

“…We now prove for k = n . By the definition of

u (\overset{x}{true})

and , and similar to the proof of (80) in , it can be shown that

\begin{matrix} | v (\overset{x}{true}) | & ⩽ c_{2} {falsefalse}_{\sum}^{j = 1} 1.61em (| ξ_{j} |^{r_{n + 1}} + | ξ_{j} |^{\frac{g_{n + 1}}{g_{j} / r_{j}}} 1.61em) + {falsefalse}_{\sum}^{j = 2} h_{n - 1, j} 1.19em (M_{[j, n - 1]} 1.19em) 1.61em (| e_{j} |^{\frac{r_{n + 1}}{r_{j - 1}}} + | e_{j} |^{\frac{g_{n + 1}}{g_{j - 1}}} 1.61em) \\ + c_{2} 1.61em (| e_{n} |^{\frac{r_{n + 1}}{r_{n - 1}}} + | e_{n} |^{\frac{g_{n + 1}}{g_{n - 1}}} 1.61em), \end{matrix}

where c 2 is a known positive constant, and h n −...…”

Section: Detailed Proofs Of Technical Claimsmentioning

confidence: 73%

“…Introduce unmeasured signals z i 's as follows, motivated by :

x_{i} = 1.61em (1.19em (z_{i} + M_{i - 1} x_{i - 1} {1.19em)}^{\frac{g_{i} / r_{i}}{g_{i - 1} / r_{i - 1}}} + 1.19em (z_{i} + M_{i - 1} x_{i - 1} 1.19em) {1.61em)}^{\frac{r_{i}}{r_{i - 1}}}, i = 2, \dots, n,

where M i 's are to‐be‐determined positive constants. Then, by , and noting that

\frac{g_{i} / r_{i}}{g_{i - 1} / r_{i - 1}}

's and

\frac{r_{i - 1}}{r_{i}}

's belong to

{boldR}_{⩾ 0}^{odd}

and are larger than 1, it can be verified that

\begin{matrix} right ż_{i} & left = - M_{i - 1} L (t) x_{i} + M_{i - 1} (() (i - 2) \frac{\dot{L} (t)}{L (t) ...} \end{matrix}

…”

Section: Finite‐time Stabilization Controlmentioning

confidence: 99%

“…Define

{\overset{x}{true}}_{k} = 1.19em ({({\overset{z}{true}}_{k} + M_{k - 1} x_{k - 1})}^{\frac{g_{k} / r_{k}}{g_{k - 1} / r_{k - 1}}} + ({\overset{z}{true}}_{k} + M_{k - 1} x_{k - 1}) {1.19em)}^{\frac{r_{k}}{r_{k - 1}}}

. Then, quite similar to the proof of (35) in , we can derive

\begin{matrix} right - M_{k - 1} L (t) (x_{k} - {\bar{x}}_{k}) H_{k} (\cdot) & left ⩽ L (t) (() - c_{k, 1} M_{k - 1}^{\frac{2 r_{k}}{r_{k} + r_{k - 1}}} + c_{k, 2} ()) (() e_{k}^{\frac{2 + τ}{r_{k - 1}}} + e_{k}^{\frac{μ}{g_{k - 1}}} ()) & right \\ right & left +<...> \end{matrix}

…”

Section: Detailed Proofs Of Technical Claimsmentioning

confidence: 99%

See 1 more Smart Citation

Global finite‐time stabilization via time‐varying output‐feedback for uncertain nonlinear systems with unknown growth rate

Liu

2017

Intl J Robust & Nonlinear

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show abstract

“…Regrettably, their approaches are only suitable for lower-triangular cascade systems. On the other 2 Mathematical Problems in Engineering hand, some literatures [27] achieved global output feedback stabilization when output function depends only on a state. References [28,29] required that output function be continuous differentiability and initial value equals zero when output is unknown.…”

Section: Introductionmentioning

confidence: 99%

Global Output Feedback Stabilization for a Class of Nonlinear Cascade Systems

Liu

Sun

Meng

et al. 2018

Mathematical Problems in Engineering

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This paper focuses on the problem of global output feedback stabilization for a class of nonlinear cascade systems with timevarying output function. By using double-domination approach, an output feedback controller is developed to guarantee the global asymptotic stability of closed-loop system. The novel control strategy successfully constructs a unified Lyapunov function, which is suitable for both upper-triangular and lower-triangular systems. Finally, two numerical examples are provided to illustrate the effectiveness of a control strategy.

show abstract

“…All of the methods of ESC for stabilization described above assume the unknown systems are affine in control, of the form:

\overset{x}{true} = f false(x, t false) + g false(x, t false) u .

However, in most physical systems, the control effort enters the system's dynamics through a nonlinear function, such as an input with deadzone and saturation. See Hu and Lin for an overview and control approaches, for a dynamic surface approach for control of systems with unknown input dead zone, for a semi‐global tracking result for systems with input saturation, and for robust output feedback stabilization of nonlinear systems with low‐ and high‐order nonlinearities. Thus, it is a major limitation that ESC applies only to systems affine in control.…”

Section: Introductionmentioning

confidence: 99%

Constrained extremum seeking stabilization of systems not affine in control

Scheinker

2017

Intl J Robust & Nonlinear

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Summary Recently, a form of extremum seeking for control (ESC) was developed for the stabilization of uncertain nonlinear systems. In ESC, the extremum seeker itself directly controls the systems through feedback rather than fine tuning a controller. The ESC results, and other related results, apply only to systems affine in control. However, in most physical systems, the control effort enters the system's dynamics through a nonlinear function, such as an input with deadzone and saturation. In this work, we use our previous results on ESC to develop constrained extremum seeking stabilizing controllers for systems of practical interest that are nonaffine in control. Considering that any odd function can be approximated arbitrarily well by an odd polynomial, we present analytic results for designing controllers for systems in which control enters the system dynamics through an odd polynomial. Furthermore, we study the robustness of the scheme to uncertainty in the odd polynomial degree as well as robustness to even‐powered perturbations.

show abstract

Robust output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities

Cited by 14 publications

References 19 publications

Global finite‐time stabilization via time‐varying output‐feedback for uncertain nonlinear systems with unknown growth rate

Global finite‐time stabilization via time‐varying output‐feedback for uncertain nonlinear systems with unknown growth rate

Global Output Feedback Stabilization for a Class of Nonlinear Cascade Systems

Constrained extremum seeking stabilization of systems not affine in control

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