Geometrical Method on Quantized Feedback Control Design for a Class of Nonlinear Systems
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摘要: 本文考虑了一类非线性系统的对数量化反馈控制器的设计问题.假定该非线性系统满足一定扇形条件, 在此条件下, 通过引入闭环系统函数, 利用函数平移的方法, 给出了文中非线性系统具体的对数量化反馈控制器, 可做到稳定和渐近稳定两种控制目标.所得结论可以推广到线性系统, 与已有的结论相吻合, 说明了我们结论的正确性.同时, 所得结论也可应用于一般非线性系统量化控制器的设计.最后, 数值例子验证了所得结论.Abstract: In this paper, logarithmic quantized feedback for a class of nonlinear systems is considered and designed explicitly. Assume that nonlinear systems of our interest satisfies a certain kind of sector condition. Under this assumption, based on the geometrical method, a closed-loop map is introduced and a logarithmic quantizer is designed explicitly based on translation of functions, to stabilize the nonlinear systems. Moreover, it is worthy to point out that the designed quantized feedback can achieve stability and asymptotically stability, respectively. Meanwhile, our result can be applied to the linear case, which is consistent with the existed result. It can also be applied to the general nonlinear systems. Finally, two examples are given to illustrate our result.
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Key words:
- Nonlinear systems /
- quantized feedback /
- logarithmic quantizer /
- piecewise functions
1) 本文责任编委 高会军 -
图 2 分段闭环系统函数$\Gamma(x)$和开环函数$f(x)$的关系
Fig. 2 The graphs of the closed-loop function $\Gamma(x)$ and the open-loop function $f(x)$
图 3 数$y = \sin{\frac{1}{x}}$与直线$y = \pm x$的图($x\in[0.1, 0.5]$)
Fig. 3 The graphs of $y = \sin{\frac{1}{x}}$ and $y = \pm x$ over $x\in[0.1, 0.5]$
图 4 非线性系统状态空间的稳定对数划分($s_1$的确定)
Fig. 4 The state partition of a stable logarithmic quantizer for nonlinear systems (Determine $s_1$)
图 5 非线性系统状态空间的稳定对数划分($r_1$的确定)
Fig. 5 The state partition of a stable logarithmic quantizer for nonlinear systems (Determine $r_1$)
图 6 非线性系统(27)量化控制器下的闭环响应
Fig. 6 Response of the closed-loop nonlinear system (27) with quantized controller
图 8 函数$y = x^3-3x^2+2x$, $x\in[0, 3.5]$
Fig. 8 The curve of $y = x^3-3x^2+2x$, $x\in[0, 3.5]$
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