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Research Papers

# Sampled-Data Stabilization of a Class of Nonlinear Systems With Application in Robotics

[+] Author and Article Information
A. M. Pertew1

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AL, T6G 2V4, Canadapertew@ece.ualberta.ca

H. J. Marquez

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AL, T6G 2V4, Canadamarquez@ece.ualberta.ca

Q. Zhao

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AL, T6G 2V4, Canadaqingzhao@ece.ualberta.ca

In all the block diagrams used in this paper, solid and dotted lines will represent continuous-time and discrete-time signals, respectively.

$P$ is a hybrid system (mix of continuous-time and discrete-time signals) and has an ill-defined transfer function. However, the equivalence between the $L2$ gain of these two setups was proven. See Ref. 38, Chapter 13, for more details.

The continuous-time controller is to be discretized later on. Its matrices are denoted by the subscript $c$. It is included here for the sake of comparison between the different SD design techniques. Details about the controller design are omitted since the focus of this paper is on the direct SD design technique.

The zoh method is also called step invariant transformation. It can be obtained in MATLAB ® by using the command $[AK,BK]=c2d(AKc,BKc,h)$. Bilinear can be obtained by adding the option “tustin” in the command.

1

Corresponding author. Also at the Computer and Systems Engineering Department, University of Alexandria, Alexandria, Egypt.

J. Dyn. Sys., Meas., Control 131(2), 021008 (Feb 05, 2009) (8 pages) doi:10.1115/1.3023141 History: Received May 26, 2007; Revised August 24, 2008; Published February 05, 2009

## Abstract

The output feedback stabilization problem for the class of nonlinear Lipschitz systems is considered. A discrete-time feedback controller is designed for the sampled-data case, where the output of the plant is only available at discrete points of time and where the objective is to stabilize the system continuously using a discrete-time controller. We show that exact stabilization in this case can be achieved using a direct sampled-data design approach, based on $H∞$ optimization theory, in which neither the plant model nor the controller need to be discretized a priori. The proposed design is solvable using commercially available software and is shown to have important advantages over the classical emulation approach that has been used to solve similar problems. The applicability of the proposed techniques in the robotics field is thoroughly discussed from both the modeling and design perspectives.

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## Figures

Figure 2

(a) Discrete-time standard setup; (b) sampled-data setup

Figure 3

Feedback interconnection

Figure 4

Schematic of the single-link flexible joint system

Figure 5

H∞ continuous stabilization

Figure 6

H∞ SD stabilization for sampling time=0.1 s

Figure 7

(H∞ direct design) versus (emulation design using bilinear discretization)

Figure 8

(State feedback) versus (H∞ direct design)

Figure 9

Output disturbance and its effect on θm

Figure 10

Robustness of the H∞ direct design for different sampling times

Figure 1

The SD stabilization problem

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