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SCIENCE CHINA Information Sciences, Volume 61, Issue 7: 070213(2018) https://doi.org/10.1007/s11432-017-9297-1

Almost sure stabilization of hybrid systems by feedback controlbased on discrete-time observations of mode and state

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  • ReceivedAug 22, 2017
  • AcceptedOct 27, 2017
  • PublishedJun 13, 2018

Abstract

Although the mean square stabilization of hybrid systems byfeedback control based on discrete-time observations of state and mode has been studied by severalauthors since 2013,the corresponding almost sure stabilization problemhas received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system$\dot~x(t)=f(x(t))$ by a linear discrete-time stochastic feedback control $Ax([t/\T]\T){\rm~d}B(t)$ (namely the stochastically controlled systemhas the form ${\rm~d}x(t)=f(x(t)){\rm~d}t~+~Ax([t/\T]\T){\rm~d}B(t)$), where $B(t)$ is a scalar Brownian, $\T~>0$, and $[t/\T]$ is the integer part of $t/\T$. In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system$\dot~x(t)=f(x(t),r(t))$ by nonlinear discrete-time stochastic feedback control $u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$ (so the stochastically controlled system is a hybrid stochastic system of the form ${\rm~d}x(t)=f(x(t),r(t)){\rm~d}t~+~u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$), where $B(t)$ is a multi-dimensional Brownian motion and $r(t)$ is a Markov chain.


Acknowledgment

The authors would like to thank Leverhulme Trust (Grant No. RF-2015-385), Royal Society (Grant No. WM160014, Royal Society Wolfson Research Merit Award), Royal Society and Newton Fund (Grant No. NA160317, Royal Society–Newton Advanced Fellowship), Engineering and Physics Sciences Research Council (Grant No. EP/K503174/1), National Natural Science Foundation of China (Grant Nos. 61503190, 61473334, 61403207), Natural Science Foundation of Jiangsu Province (Grant Nos. BK20150927, BK20131000), and Ministry of Education (MOE) of China (Grant No. MS2014DHDX020) for their financial support. The first author would like to thank Chinese Scholarship Council for awarding him the scholarship to visit the University of Strathclyde for one year.


Supplement

Appendix

Once Assumption A3.1 holds, the verification of Assumption A3.2 depends very much on the choice of $p\in~(0,1)$. In this appendix, we give some easier qualifications that guarantee the existence of such a $p$ and, hence, for Assumption A3.2 to hold. For this purpose, we need the theory of $M$-matrices (see [2,27]). For a vector or matrix $A$, $A>~0$ means that all elements of $A$ are positive. Moreover, a square matrix $A=[a_{ij}]_{N\K~N}$ is called a $Z$-matrix if it has nonpositive off-diagonal entries, namely $$ a_{ij}\le 0 \hbox{ for all } i\not =j. $$

Here, we cite a useful lemma on $M$-matrices.

Lemma A1 ( [20]) . If $A=[a_{ij}]_{N\K~N}$ is a $Z$-matrix, the following statements are equivalent:

(1) $A$ is a nonsingular $M$-matrix.

(2) $A$ is semi-positive; that is, there exists $x~>~0$ in $\RR^N$ such that $Ax~>~0$.

(3) $A^{-1}$ exists and its elements are all nonnegative.

(4) All the leading principal minors of $A$ are positive; that is $$ \left| \begin{matrix} a_{11} & \cdots & a_{1k} \\ \vdots & &\vdots \\ a_{k1} & \cdots & a_{kk} \end{matrix} \right| > 0, \hbox{\rm for every } k=1, 2,\ldots, N. $$

We also need another classical result.

Lemma A2 (Minkowski

Minkowski H. Diophantische Approximationen. Teubner, 1907.)

. If a $Z$-matrix $A=[a_{ij}]_{N\K~N}$ has all positive row sums, that is $$ \sum_{j=1}^N a_{ij} >0, \forall i=1,2,\ldots, N, $$ $\hbox{\rm~det}~A~>~0$.

Now, let us propose a condition that \begin{equation} \left| \begin{array}{cccc} -(\a_1 + 0.5\r_1^2 -\s_1^2) & -\g_{12} & \cdots & -\g_{1N} \\ -(\a_2 + 0.5\r_2^2 -\s_2^2) & -\g_{22} & \cdots & -\g_{2N} \\ \vdots & \vdots & \cdots &\vdots \\ -(\a_N + 0.5\r_N^2 -\s_N^2) & -\g_{N2} & \cdots & -\g_{NN} \end{array} \right| >0, \tag{50}\end{equation} where $\a_i,~\r_i,~~\s_i$ are the constants specified in Assumption A3.1. This condition can be verified straightaway once Assumption A3.1 holds.

It was shown in [28] that under an additional condition that \begin{equation} \hbox{ for some $u\in \SM$, $\g_{iu} > 0$, for all $i\not= u$, } \tag{51}\end{equation} condition (50) is equivalent to the following simpler condition \begin{equation} \sum_{i=1}^N \pi_i (\a_i + 0.5\r_i^2 -\s_i^2) < 0, \tag{52}\end{equation} where $(\pi_1,\ldots,~\pi_N)$ are the stationary distribution of the Markov chain as defined in Section 2. In this paper, we replace condition (51) by a slightly weaker one that \begin{equation} \hbox{for some $u\in \SM$, } \g_{iu} \ve (\s_i^2 - 0.5\r_i^2 -\a_i) > 0, \hbox{for all $i\not= u$}. \tag{53}\end{equation} We do not yet know whether (50) is equivalent to (52) under this weaker condition. However, the following proposition is good enough for use in this paper.

Proposition A1. If conditions (50) and (53) hold, Assumption A3.2 is satisfied.

proof Without loss of generality, we may assume that the state $u=N$ in condition (53), namely \begin{equation} \g_{iN} \ve (\s_i^2 - 0.5\r_i^2 -\a_i) > 0, \hbox{for all $1\le i \le N-1$}. \tag{54}\end{equation} If not, by switching state $u$ with $N$, we need to reorder the states of the Markov chain $r(t)$ that is, rename state $u$ as $N$ and $N$ as $u$. Consequently, the determinant in the left-hand side of (50) will switch the $u$th row with the $N$th row and then switch the $u$th column with the $N$th column, but these do not change the value of the determinant, namely the determinant remains positive. Moreover, given a nonsingular $M$-matrix, it is easy to show that the new matrix remains a nonsingular $M$-matrix by switching the $u$th column with the $N$th column and then switching the $u$th row with the $N$th row.

By [27], the derivative ${\rm~d}{\cal~A}(0)/{\rm~d}p~$ is equal to the determinant on the left-hand side of (50), whence ${\rm~d}{\cal~A}(0)/{\rm~d}p~>~0$. It is also easy to see ${\cal~A}(0)=0$. Consequently, for all $p\in~(0,~1)$ sufficiently small, we have \begin{align} \hbox{\rm det} {\cal A}(p) > 0. \tag{55} \end{align} On the other hand, for each $i=1,2,\ldots,~N-1$, either $\g_{iN}~>~0$ or $\g_{iN}=0$. In the case when $\g_{iN}~>~0$, we clearly have $$ \theta_i(p) > -\g_{iN}, \hbox{for all sufficiently small $p\in (0,1)$}; $$ whereas in the case when $\g_{iN}=0$, condition (54) implies $\s_i^2~-~0.5\r_i^2~-\a_i~>0$ whence $$ \theta_i(p) > 0 = -\g_{iN}, \hbox{for all sufficiently small $p\in (0,1)$.} $$ In other words, we always have \begin{align} \theta_i(p) > -\g_{iN}, i=1,2,\ldots, N-1, \tag{56} \end{align} for all $p\in~(0,~1)$ sufficiently small. Fix any $p\in~(0,~1)$ sufficiently small for both (55) and (56) to hold. Consider the deriving principal sub-matrix, $$ {\cal A}_k(p) := \left[ \begin{array}{cccc} \theta_1(p)-\g_{11} & -\g_{12} & \cdots & -\g_{1k} \\ -\g_{21} & \theta_1(p)-\g_{22} & \cdots & -\g_{2k} \\ \vdots & \vdots & \cdots &\vdots \\ -\g_{k1} & -\g_{k2} & \cdots & \theta_k(p)-\g_{kk} \end{array} \right] $$ of ${\cal~A}(p)$, for $k=1,2,\ldots,~N-1$. Obviously, ${\cal~A}_k(p)$ is a $Z$-matrix. Moreover, for every $i=1,2,~\ldots,~k$, the $i$th row of this sub-matrix has its sum $$ \theta_i(p) - \sum_{j=1}^k \g_{ij} = \theta_i(p) + \sum_{j=k+1}^N \g_{ij} \ge \theta_i(p) +\g_{iN} >0 $$ by (56). By Lemma A2, det${\cal~A}_k(p)~>0$. Thus, we should point out that all the deriving principal minors of ${\cal~A}(p)$ are positive. By applying Lemma A1, we can obtain that ${\cal~A}(p)$ is a nonsingular $M$-matrix. This completes the proof.


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