# 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? • AcceptedJun 20, 2017
• PublishedJan 4, 2018
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### 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 ()). 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 .

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  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 .

### References

 A?str?m K J, Kumar P R. Control: A perspective. Automatica, 2014, 50: 3-43 CrossRef Google Scholar

 Delchamps D F. Stabilizing a linear system with quantized state feedback. IEEE Trans Automat Contr, 1990, 35: 916-924 CrossRef Google Scholar

 De Persis C. Nonlinear stabilizability via encoded feedback: The case of integral ISS systems. Automatica, 2006, 42: 1813-1816 CrossRef Google Scholar

 Persis C D, Jayawardhana B. Coordination of Passive Systems under Quantized Measurements. SIAM J Control Optim, 2012, 50: 3155-3177 CrossRef Google Scholar

 Elia N, Mitter S K. Stabilization of linear systems with limited information. IEEE Trans Automat Contr, 2001, 46: 1384-1400 CrossRef Google Scholar

 Fradkov A L, Andrievsky B, Ananyevskiy M S. Passification based synchronization of nonlinear systems under communication constraints and bounded disturbances. Automatica, 2015, 55: 287-293 CrossRef Google Scholar

 Minyue Fu , Lihua Xie . The sector bound approach to quantized feedback control. IEEE Trans Automat Contr, 2005, 50: 1698-1711 CrossRef Google Scholar

 Kameneva T, Nešić D. Input-to-state stabilization of nonlinear systems with quantized feedback. In: Proceedings of the 17th IFAC World Congress, Seoul, 2008. 12480--12485. Google Scholar

 Matveev A S, Savkin A V. Estimation and Control over Communication Networks. Boston: Birkhäuser, 2009. Google Scholar

 Wang L Y, Li C, Yin G G. State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations. IEEE Trans Automat Contr, 2011, 56: 2639-2654 CrossRef Google Scholar

 Wing Shing Wong , Brockett R W. Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback. IEEE Trans Automat Contr, 1999, 44: 1049-1053 CrossRef Google Scholar

 Nair G N, Evans R J. Stabilization with data-rate-limited feedback: tightest attainable bounds. Syst Control Lett, 2000, 41: 49-56 CrossRef Google Scholar

 Nair G N, Evans R J. Exponential stabilisability of finite-dimensional linear systems with limited data rates. Automatica, 2003, 39: 585-593 CrossRef Google Scholar

 Tatikonda S, Mitter S. Control Under Communication Constraints. IEEE Trans Automat Contr, 2004, 49: 1056-1068 CrossRef Google Scholar

 Liberzon D, Hespanha J P. Stabilization of nonlinear systems with limited information feedback. IEEE Trans Automat Contr, 2005, 50: 910-915 CrossRef Google Scholar

 Hayakawa T, Ishii H, Tsumura K. Adaptive quantized control for nonlinear uncertain systems. Syst Control Lett, 2009, 58: 625-632 CrossRef Google Scholar

 Kang X, Ishii H. Coarsest quantization for networked control of uncertain linear systems. Automatica, 2015, 51: 1-8 CrossRef Google Scholar

 Liu K, Fridman E, Johansson K H. Dynamic quantization of uncertain linear networked control systems. Automatica, 2015, 59: 248-255 CrossRef Google Scholar

 Zhou J, Wen C, Yang G. Adaptive Backstepping Stabilization of Nonlinear Uncertain Systems With Quantized Input Signal. IEEE Trans Automat Contr, 2014, 59: 460-464 CrossRef Google Scholar

 Nair G N, Evans R J. Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates. SIAM J Control Optim, 2004, 43: 413-436 CrossRef Google Scholar

 Sahai A, Mitter S. The necessity and sufficiency of anytime capacity for stabilization of a linear system over a noisy communication link---Part I: scalar systems. IEEE Trans Autom Control, 2006, 52: 3369--3395. Google Scholar

 Tatikonda S, Sahai A, Mitter S. Stochastic Linear Control Over a Communication Channel. IEEE Trans Automat Contr, 2004, 49: 1549-1561 CrossRef Google Scholar

 Brockett R W, Liberzon D. Quantized feedback stabilization of linear systems. IEEE Trans Automat Contr, 2000, 45: 1279-1289 CrossRef Google Scholar

 Liberzon D, Nesic D. Input-to-State Stabilization of Linear Systems With Quantized State Measurements. IEEE Trans Automat Contr, 2007, 52: 767-781 CrossRef Google Scholar

• 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

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

$~capture\Leftarrowyes&quot;~$;

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

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

$~capture\Leftarrowyes&quot;~$;

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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