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Synthesis of continuous-time dynamic quantizers for LFT type quantized feedback systems

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Abstract

This paper focuses on analysis and synthesis methods of continuous-time dynamic quantizers for LFT type quantized control systems. Our aim is to propose a numerical optimization design method of multiple (decentralized) quantizers such that a given linear system is optimally approximated by the given linear system with the multiple quantizers. Our method is based on the invariant set analysis and the LMI technique. In addition, we clarify that our proposed method naturally extends to multiobjective control problems similar to linear control. For implementation, this paper presents an analysis condition of the applicable interval of switching process of quantizer. Finally, it is pointed out that the proposed method is helpful through a numerical example.

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Correspondence to Kenji Sawada.

Additional information

This work was presented in part at the 18th International Symposium on Artificial Life and Robotics, Daejeon, Korea, January 30–February 1, 2013.

Appendices

A: Derivation of \({\mathsf{G}}(s):\)

For discrete-valued input v,

$$v=e+v_q+u$$

holds. For continuous-valued output u of G(s), we have

$$u=C_2x+D_{21}r+D_{22}(e+v_q+u).$$

From the non-singularity of I − D 22 (Assumption 1), we have

$$\begin{aligned} u=&\;(I-D_{22})^{-1}C_2x+(I-D_{22})^{-1}D_{21}r\\ &+(I-D_{22})^{-1}D_{22}(e+v_q)\\ =&\;{{\mathsf{C}}}_2x+{{\mathsf{D}}}_{21}r+(I-D_{22})^{-1}D_{22}(e+v_q),\\ u_q=&\; u+v_q={\mathsf{C}}_2x+{{\mathsf{D}}}_{21}r+{{\mathsf{D}}}_{22}e+((I-D_{22})^{-1}D_{22}+I)v_q\\ =&\; {{\mathsf{C}}}_2x+{{\mathsf{D}}}_{21}r+{{\mathsf{D}}}_{22}e+{{\mathsf{D}}}v_q \end{aligned}$$

Along with this, we thus have the realization of \({\mathsf{G}}(s).\)

B: Derivation of (20)

From \({\Upxi_{\mathsf{P}}=\widehat{{\mathcal{Q}}}}\) and \([\widehat{F}\,\widehat{F}]=[0\,C_q]\Upxi_{\mathsf{P}},\) the following relation holds

$$(20)\,\Leftrightarrow\, \left[ \begin{array}{ll} \widehat{{{\mathcal{Q}}}}&\widehat{{{\mathcal{Q}}}}\left[0\,C_q\right]^{\rm T}\\ \left[0\,C_q\right]\widehat{{{\mathcal{Q}}}} &\delta^2 I_m\\ \end{array} \right]\!\ge 0$$

and we have

$$\begin{array}{l} \left[ \begin{array}{ll} \xi^{\rm T}{{\mathcal{Q}}}^{-1}\widehat{T} & 0\\ 0 & I\\ \end{array} \right] \left[ \begin{array}{ll} \widehat{{{\mathcal{Q}}}} & \widehat{{{\mathcal{Q}}}}\left[0\,C_q\right]^{\rm T} \\ \left[0\,C_q\right]\widehat{{{\mathcal{Q}}}} &\delta^2 I_m\\ \end{array} \right] \left[ \begin{array}{ll} \widehat{T}^{\rm T}{{\mathcal{Q}}}^{-1}\xi & 0\\ 0 & I\\ \end{array} \right]\\ = \left[ \begin{array}{ll} \xi^{\rm T}{{\mathcal{P}}}\xi & \xi^{\rm T}\left[0\,C_q\right]^{\rm T}\\ \left[0 C_q\right]\xi &\delta^2 I_m\\ \end{array} \right]\ge0 \end{array}$$

for any vector \(\xi\in{\rm I\!R}^n\) where \({{\mathcal{Q}}=\widehat{T}\widehat{{\mathcal{Q}}}\widehat{T}^{\rm T}}\) and \({{\mathcal{P}}={\mathcal{Q}}^{-1}.}\) Applying schur complement for the above inequality yields

$$\xi^{\rm T}{{\mathcal{P}}}\xi \ge \frac{1}{\delta^2}\xi^{\rm T}\left[0\,C_q\right]^{\rm T} \left[0\,C_q\right]\xi.$$

For vector \({\xi\in\mathbb{E}({\mathcal{P}})}\) and ξ = [* x T q ]T, we therefore have

$$\max_i\sup_{\xi\in{\mathbb{E}}({{\mathcal{P}}})}|c_{qi}^{\rm T} x_q|=\delta.$$

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Sawada, K., Shin, S. Synthesis of continuous-time dynamic quantizers for LFT type quantized feedback systems. Artif Life Robotics 18, 117–126 (2013). https://doi.org/10.1007/s10015-013-0108-y

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