Quantization rate is a crucial measure of complexity in determining stabilizability of control systems subject to quantized state measurements. This paper investigates quantization complexity for a class of nonlinear systems which are subjected to disturbances of unknown statistics and unknown bounds. This class of systems includes linear stablizable systems as special cases. Two lower bounds on the quantization rates are derived which guarantee input-to-state stabilizability for continuous-time and sampled-data feedback strategies, respectively. Simulation examples are provided to validate the results.
Appendix
Quantizer for Theorem
In general, we need three parameters $M,~\Delta$ and $\Delta_0$ for the quantizer: $M$ is the quantization range, $\Delta$ the quantization error, and $\Delta_0$ the minimal quantization resolution. More precisely, we choose $~M>\Delta>0~$ and $~\Delta_{0}>0~$ such that
\begin{equation}\|z\|\leq M \Rightarrow \|q(z)-z\|\leq \Delta, \tag{35}\end{equation}
\begin{equation}\|z\|> M \Rightarrow \|q(z)\|> M-\Delta, \tag{36}\end{equation}
\begin{equation}\|z\|\leq \Delta_{0} \Rightarrow q(z)=0, \tag{37}\end{equation}
\begin{equation}M>5\Delta+\frac{2\|BK\|}{N}\Delta. \tag{38}\end{equation}
Let $~T_\text{in},~T_{c},~T_\text{out},~\Omega_\text{in},~\Omega_\text{out}~$ be some positive numbers satisfying $~T_\text{in}\leq~T_\text{out}~$, $~T_{c}~<~\frac{1}{2}T_\text{out}$, $~\Omega_\text{in}<1~$ with
\begin{equation*}\Omega_\text{in}(M-2\Delta)-3\Delta>\frac{2\|BK\|}{N}\Delta.\end{equation*}
Moreover, $~T_\text{out}<\log\Omega_\text{out}/L~$ with\begin{equation*}\Omega_\text{out}> \frac{M}{M-2\Delta}.\end{equation*}
Note that $T_\text{in}$ is the unit of time after the last zoom-in or zoom-out before executing another zoom-in, $T_\text{out}$ is the unit of time after a zoom-out before executing another zoom-out, $\Omega_\text{in}$ is the zoom-in factor and $\Omega_\text{out}$ is the zoom-out factor, respectively.
Define
\begin{equation}\ell_\text{in}:=\Omega_\text{in}(M-2\Delta)-2\Delta, \\ \ell_\text{out}:=M-\Delta. \tag{39}\end{equation}
In the control strategy to be developed below, all system variables will be continuous from the right by construction. Variables which are not mentioned remain constants in the following algorithm.
We use the one-parameter family of quantizers
\begin{equation} q_{\mu}(x): =\mu q\left(\frac{x}{\mu}\right), \mu>0.\end{equation}
Here $~\mu~$, called “zoom" variable, is an adjustable scaling parameter with initial value $~\mu_{0}~$. It is known to both the sender and receiver and updated at discrete instants of time by Algorithm A1.Based on the quantized signal, the feedback control law is given by
\begin{eqnarray}u(t)= \left\{ \begin{array}{ll} 0,& capture=``no", \\ Kq_{\mu}(x),& capture=``yes". \end{array} \right. \tag{40} \end{eqnarray}
\begin{eqnarray}\dot{\tau}_\text{out} = \left\{ \begin{aligned} 1,& \tau_\text{out} < T_\text{out}, \\ 0,& \tau_\text{out} = T_\text{out}, \end{aligned} \right. \tag{41} \end{eqnarray}
and\begin{eqnarray}\dot{\tau}_\text{in} = \left\{ \begin{aligned} 1,& \tau_\text{in} < T_\text{in}, \\ 0,& \tau_\text{in} = T_\text{in}. \end{aligned} \right. \tag{42} \end{eqnarray}
Some lemmas and technical proofs
\begin{equation*}R \geq\left(\frac{M}{\Delta}\sqrt{n}\right)^{n}.\end{equation*}
proof Firstly, we divide the the minimum circumscribed hypercube of the ball $~\{z:\|z\|\leq~M\}~$ into $~R~$ equal hypercubic boxes, numbered from $~1~$ to $~R~$ in some specific way. Secondly, for each hypbercubic box, there is an unique ball in $\mathbb{R}^n$ which is minimally circumscribed to the small box. Let $~q(z)~$ be the center of this ball that contains $~z~$. In case $~z~$ lies on the boundary of several balls, the value of $~q(z)~$ can be chosen arbitrarily among the candidates. Then we obtain
\begin{equation*}\|q(z)-z\|\leq \frac{\sqrt{n}M}{R^{\frac{1}{n}}},\end{equation*}
which implies that\begin{equation*}\frac{\sqrt{n}M}{R^{\frac{1}{n}}} \leq\Delta.\end{equation*}
ıtshapeProof of Lemma
\begin{eqnarray*}\left\{ \begin{array}{ll} \dot{y}(t)=Ay+Bu_{k}+Dd,& t\in I_{k}, \\ y_{k}=x_{k}. \end{array} \right. \end{eqnarray*}
It is straightforward that
\begin{equation*}y(t)=\text{e}^{A(t-kT)}x_{k}+\int_{0}^{t-kT}\text{e}^{Ar}B\text{d} ru_{k}+\int_{kT}^{t}\text{e}^{A(t-r)}Dd(r)\text{d} r\end{equation*}
for $~t\in~I_{k}~$. This implies that\begin{gather*}\begin{aligned} \|y_{k+1}\| & \leq \bigg\|\text{e}^{AT}x_{k}+\int_{0}^\text{T}\text{e}^{Ar}B\text{d} rK(x_{k}+e_{k})\bigg\|+\bigg\|\int_{kT}^{(k+1)T}\text{e}^{A((k+1)T-r)}Dd(r)\text{d} r\bigg\| \\ & \leq \bigg\|\text{e}^{AT}+\int_{0}^\text{T}\text{e}^{Ar}B\text{d} rK \bigg\|\|x_{k}\| +\bigg\|\int_{0}^\text{T}\text{e}^{Ar}B\text{d} rK\bigg\|\|e_{k}\| +\bigg\|\int^{T}_{0}\text{e}^{Ar}D\text{d} r\bigg\|\|d_{k}\|. \end{aligned} \end{gather*}
Next, we consider the following nonlinear system
\begin{eqnarray*}\left\{ \begin{array}{ll} \dot{\varphi}(t)=f(\varphi+y)-Ay ,& t\in I_{k}, \\ \varphi_{k}=0. \end{array} \right. \end{eqnarray*}
It is easy to prove that\begin{equation*}\begin{aligned} |\varphi(t)| & \leq \int_{kT}^{t}\|f(\varphi(s)+y(s))-Ay(s)\|\text{d} s \\ & \leq \int_{kT}^{t}L\|\varphi(s)\|+(L+\|A\|)\|y(s)\|\text{d} s \\ & \leq \text{e}^{L(t-kT)}(L+\|A\|)\int_{kT}^{t}\|y(s)\|\text{e}^{L(kT-s)}\text{d} s \\ & \leq \text{e}^{L(t-kT)}(L+\|A\|)\times \int_{kT}^{t} \bigg\|\text{e}^{A(s-kT)}x_{k}+\int_{0}^{s-kT}\text{e}^{Ar}B\text{d} ru_{k}+\int_{kT}^{s}\text{e}^{A(s-r)}Dd(r)\text{d} r\bigg\| \text{e}^{L(kT-s)}\text{d} s \end{aligned}\end{equation*}
\begin{equation*}\begin{aligned} \|\varphi_{k+1}\| & \leq \text{e}^{LT}(L+\|A\|)\int_{kT}^{(k+1)T}\bigg\{\bigg\|\text{e}^{A(s-kT)}x_{k}+\int_{0}^{s-kT}\text{e}^{Ar}B\text{d} rK(x_{k}+e_{k}) \\ & +\int_{kT}^{s}\text{e}^{A(s-r)}Dd(r)\text{d} r\bigg\|\bigg\} \text{e}^{L(kT-s)}\text{d} s \\ &\leq \text{e}^{LT}(L+\|A\|)\int_{kT}^{(k+1)T}\bigg\{ \bigg\|(\text{e}^{A(s-kT)}+\int_{0}^{s-kT}\text{e}^{Ar}B\text{d} rK)x_{k}\bigg\| \\ & +\bigg\|\int_{0}^{s-kT}\text{e}^{Ar}B\text{d} rKe_{k}\bigg\| +\bigg\|\int_{kT}^{s}\text{e}^{A(s-r)}Dd(r)\text{d} r\bigg\|\bigg\}\text{e}^{L(kT-s)}\text{d} s \\ &\leq \text{e}^{LT}(L+\|A\|)\int_{kT}^{(k+1)T}\bigg\{ \bigg\|(\text{e}^{A(s-kT)}+\int_{0}^{s-kT}\text{e}^{Ar}B\text{d} rK)\bigg\|\|x_{k}\| \\ & +\bigg\|\int_{0}^{s-kT}\text{e}^{Ar}BK\text{d} r\bigg\|\|e_{k}\| +\bigg\| \int_{0}^{s-kT}\text{e}^{Ar}D\text{d} r\bigg\|\|d_{k}\|\bigg\}\text{e}^{L(kT-s)}\text{d} s. \end{aligned}\end{equation*}
The following Lemma
\begin{equation*}\mu_{k_{2i+1}} \leq \rho^\text{out}_{\mu}\left({\mu_{k_{2i}}},\|x_{k_{2i}}\|,d_{[k_{2i},k_{2i+1}-1]}\right).\end{equation*}
proof The proof is identical to Lemma IV.6 in
The following Lemma
\begin{equation*}\|x_{k}\| \leq \text{e}^{-\lambda(k-k_{2i+1})}(\|x_{2i+1}\|+\mu_{k_{2i+1}})+\gamma d_{[k_{2i+1},k-1]} , \forall k \in [k_{2i+1},k_{2i+2}].\end{equation*}
proof During the zoom in intervals we have by construction
\begin{gather*}\|x_{k}\|\leq M\mu_{k}, \\ \bigg\|q\left(\frac{x_{k}}{\mu_{k}}\right)-\frac{x_{k}}{\mu_{k}}\bigg\|\leq \Delta. \end{gather*}
Meanwhile, the $~x~$-subsystem satisfies\begin{equation*}\|x_{k+1}\|\leq c_{2}\|x_{k}\|+c_{3}\|e_{k}\| +c_{4}\|d_{k}\|\end{equation*}
\begin{equation*}\mu_{k+1}=\Omega_\text{in}\mu_{k}\end{equation*}
The following Lemma
\begin{equation*}\|x_{k}\|\leq \rho^\text{in}_{x}(\mu_{k_{2i+1}},\|x_{k_{2i+1}}\|,d_{[k_{2i+1},k_{2i+2}-1]}) , \forall k \in [k_{2i+1},k_{2i+2}].\end{equation*}
proof The proof is almost the same as that of Lemma IV.8 in
The following Lemma
\begin{equation*}\max\{\|x_{k_{2i+2}}\|,\mu_{k_{2i+2}}\} \leq \widetilde{\gamma}d_{[k_{2i+1},k_{2i+2}-1]}.\end{equation*}
proof See Lemma IV.9 in
Figure 1
(Color online) Simulation results for Example 1.
Figure 2
(Color online) Simulation results for Example 2.
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