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SCIENCE CHINA Information Sciences, Volume 61, Issue 9: 092205(2018) https://doi.org/10.1007/s11432-016-9172-4

How much information is needed in quantized nonlinear control?

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  • ReceivedNov 18, 2016
  • AcceptedJun 20, 2017
  • PublishedJan 4, 2018

Abstract

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.


Supplement

Appendix

Quantizer for Theorem 2.4

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} The first condition (35) ensures that the quantization error is bounded by $~\Delta~$ when the quantizer is not saturated. Saturation of the quantizer is indicated by (36). (37) defines the minimum resolution of the quantizer so that the measurement is zero when the signal magnitude is below this resolution.

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}

Remark 6. The parameter $~``capture&quot;~$ is an auxiliary logical variable which is used to distinguish the open-loop stage and the control stage. It takes values in the set $~\{``yes&quot;,``no&quot;\}~$ and is initialized at $~``no&quot;~$. The parameters $~``\tau_\text{out}&quot;~$ and $~``\tau_\text{in}&quot;~$ are functions of the continuous time $~t~$, called “auxiliary reset clock variables". The clock variables are initialized at $~0~$ and take values in the intervals $~[0,T_\text{out}]~$ and $~[0,T_\text{in}]~$, respectively. Moreover, they satisfy \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

Lemma 5. Assume that the number $~\sqrt[n]{R}~$ is an odd integer (see ([15])). Then the quantization rate $R$, i.e., the number of the elements in $\mathcal{Q}$, satisfies \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*} This completes the proof of Lemma 5.

ıtshapeProof of Lemma 4.2upshape. We consider the following linear system \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*} for all $~t\in~I_{k}~$. This implies that \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*} It is obvious that $~\|x_{k+1}\|\leq~\|y_{k+1}\|+\|\varphi_{k+1}\|~$ and we complete the proof of Lemma 4.2.

The following Lemma 6 indicates that the zoom variable $~\mu~$ is bounded at the end of each zoom-out interval.

Lemma 6. There exists a continuous bounded function $~\rho^\text{out}~_{\mu}$ such that for any $~\mu>0~$ we have $~\rho^\text{out}_{\mu}(\mu,0,0)>0~$ and the following is true for all $~i\in~\{0,1,\dots,P~\}~$ and all $~\mu_{k_{2i}}>0,~x_{k_{2i}}~\in~\mathbb{R}^{n},~d\in~\mathbb{R}^{s}$: \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 [24].

The following Lemma 7 establishes an appropriate bound on the state $~x~$ during the zoom-in intervals.

Lemma 7. There exist $~\lambda,~\gamma~\in~(0,+\infty)~$ such that \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*} and the $~\mu~$-subsystem evolves according to \begin{equation*}\mu_{k+1}=\Omega_\text{in}\mu_{k}\end{equation*} for all $~k~\in~[k_{2i+1},k_{2i+2}-1]~$. This is a cascade of an ISS system and a GAS system, hence the conclusion holds.

The following Lemma 8 establishes a different bound on the state $~x~$ during the zoom-in intervals.

Lemma 8. There exists a continuous function $~\rho^\text{in}_{x}:\mathbb{R}_{>0}\times\mathbb{R}_{\geq~0}\times~\mathbb{R}_{\geq~0}\rightarrow~\mathbb{R}_{\geq~0}~$, with $~\rho^\text{in}_{x}(\mu,0,0)=0~$ for all $~\mu>0~$, and such that for any $~s\geq~0,~\rho^\text{in}_{x}(\cdot,\cdot,s)~$ is nondecreasing in its first two arguments and for any $~i\in~\{0,1,\dots,P~\}~$ the following holds for all $~\mu_{k_{2i+1}},~x_{k_{2i+1}},~d~$: \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 [24] with the difference that $~H:=c_{2}+c_{3}+c_{3}L_{q}~$ in our case.

The following Lemma 9 indicates that if the zoom-in interval is bounded then the state $~x~$ and the zoom variable $~\mu~$ are bounded by the function of the disturbance $~d~$ at the end of the zoom-in interval.

Lemma 9. Consider an arbitrary $~i\in~\{0,1,\dots,$ $~P~\}~$. If $~k_{2i+2}<\infty~$, then $~i<~P-1~$ and there exists a $~\widetilde{\gamma}\in~(0,+\infty)~$ such that \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 [24].


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  • Figure 1

    (Color online) Simulation results for Example 1.

  • Figure 2

    (Color online) Simulation results for Example 2.

  •   

    Algorithm 1 Updating $~\mu~$

    if $~captur\text{e}^{-}=``no&quot;~$ then

    if $~~\tau_\text{out}^-=T_\text{out}~$ then

    $\mu~\Leftarrow~\Omega_\text{out}\mu^-~$;

    $\tau_\text{out}\Leftarrow~0$;

    end if

    if $~\|q_{\mu^-}(x)\|\leq~\ell_\text{out}\mu^-~$ and $~\tau_\text{out}^-~\in~[T_{c},T_\text{out}-T_{c}]~~$ then

    $~\mu\Leftarrow~\Omega_\text{out}\mu^-~$;

    $~capture\Leftarrow``yes&quot;~$;

    end if

    else

    if $~\|q_{\mu^-}(x)\|\geq~\ell_\text{out}\mu^-~$ then

    $\mu~\Leftarrow~\Omega_\text{out}\mu^-~$;

    $\tau_\text{out}~\Leftarrow~0~$;

    end if

    if $~\|q_{\mu^-}(x)\|\leq~\ell_\text{in}\mu^-~$ and $~\min\{\tau_\text{in}^-,\tau_\text{out}^-\}\geq~T_\text{in}~$ then

    $\mu~\Leftarrow~\Omega_\text{in}\mu^-$;

    $\tau_\text{in}~\Leftarrow~0$;

    end if

    end if

  •   

    Algorithm 1 Updating $~\mu~$

    if $~captur\text{e}^{-}=``no&quot;~$ then

    if $~~\tau_\text{out}^-=T_\text{out}~$ then

    $\mu~\Leftarrow~\Omega_\text{out}\mu^-~$;

    $\tau_\text{out}\Leftarrow~0$;

    end if

    if $~\|q_{\mu^-}(x)\|\leq~\ell_\text{out}\mu^-~$ and $~\tau_\text{out}^-~\in~[T_{c},T_\text{out}-T_{c}]~~$ then

    $~\mu\Leftarrow~\Omega_\text{out}\mu^-~$;

    $~capture\Leftarrow``yes&quot;~$;

    end if

    else

    if $~\|q_{\mu^-}(x)\|\geq~\ell_\text{out}\mu^-~$ then

    $\mu~\Leftarrow~\Omega_\text{out}\mu^-~$;

    $\tau_\text{out}~\Leftarrow~0~$;

    end if

    if $~\|q_{\mu^-}(x)\|\leq~\ell_\text{in}\mu^-~$ and $~\min\{\tau_\text{in}^-,\tau_\text{out}^-\}\geq~T_\text{in}~$ then

    $\mu~\Leftarrow~\Omega_\text{in}\mu^-$;

    $\tau_\text{in}~\Leftarrow~0$;

    end if

    end if

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