A Hybrid Observer for Estimating the State of a Distributed Linear System

In [2] a distributed observer is described for estimating the state of an $m>0$ channel, $n$-dimensional, continuous-time, jointly observable linear system of the form $\dot{x}=Ax,\;y_{i}=C_{i}x,\;i\in\{1,2,\ldots,m\}$. The state $x\in{\rm I\!R}^{n}$ is simultaneously estimated by $m$ agents assuming that each agent $i$ senses $y_{i}$ and receives the state of each of its neighbors’ estimates. An attractive feature of the observer described in [2] is that it is able to generate an asymptotically correct estimate of $x$ exponentially fast at a pre-assigned rate, if each agent’s set of neighbors do not change with time and the neighbor graph characterizing neighbor relations is strongly connected. However, a shortcoming of the observer in [2] is that it is unable to function correctly if the network changes with time. Changing neighbor graphs will typically occur if the agents are mobile. A second shortcoming of the observer described in [2] is that it is “fragile” by which we mean that the observer is not able to cope with the situation when there is an arbitrary abrupt change in the topology of the neighbor graph such as the loss or addition of a vertex or an arc. For example, if because of a component failure, a loss of battery power, or some other reasons, an agent drops out of the network, what remains of the overall observer will typically not be able to perform correctly and may become unstable, even if joint observability is not lost and what remains of the neighbor graph is still strongly connected.

This paper breaks new ground by introducing a hybrid distributed observer which overcomes the aforementioned difficulties without making restrictive assumptions. To the best of our knowledge, this observer is the first provably correct distributed algorithm capable of generating an asymptotically correct estimate of a jointly observable linear system’s state in the presence of a neighbor graph which changes with time under reasonably general assumptions. Although the observer is developed for continuous-time systems, it can very easily be modified in the obvious way to deal with discrete-time systems.

Notation: Given a collection of $n\times n$ matrices, $A_{1},\;A_{2},\;\ldots,\;A_{m}$, let $\text{diagonal}\{A_{1},A_{2},\ldots,A_{m}\}$ be the block diagonal matrix with $A_{k}$ as its $k$th diagonal block. Given a collection of $n\times 1$ vectors, $v_{1},\;v_{2};\;\ldots,\;v_{m}$, let $\text{column}\{v_{1},v_{2},\ldots,v_{\}}$ be the stacked vector with $v_{k}$ as its $v$th sub-vector. For an $n\times n$ matrix $A$, we let $\text{image}A$ denote the linear subspace spanned by matrix $A$. For two $n\times n$ matrix $A_{1}$, and $A_{2}$, we let $\text{image}A_{2}\cap\text{image}A_{2}$ denote the intersection of the two images.

The overall hybrid observer to be considered consists of $m$ private estimators, one for each agent. Agent $i$’s private estimator, whose function is to generates an estimate $x_{i}$ of $x$, is a hybrid dynamical system consisting of a “local observer” and a “local parameter estimator.” The purpose of local observer $i$ is to generate an asymptotically correct estimate of $L_{i}x$ where $L_{i}$ is any pre-specified, full-rank matrix whose kernel equals the kernel of the observability matrix of the pair $(C_{i},A)$; roughly speaking, $L_{i}x$ can be thought of as that “part of $x$” which is observable to agent $i$. Agent $i$’s local observer is then an $n_{i}$-dimensional continuous-time, linear system of the form

where $n_{i}={\rm rank\;}L_{i}$, $K_{i}$ is a gain matrix to be specified, and $\bar{C}_{i}$ and $\bar{A}_{i}$ are unique solutions to the equations $C_{i}=\bar{C}_{i}L_{i}$ and $L_{i}A=\bar{A}_{i}L_{i}$, respectively. As is well known, the pair $(\bar{C}_{i},\bar{A}_{i})$ is observable and the local observer estimation error $\bar{e}_{i}\stackrel{{\scriptstyle\Delta}}{{=}}w_{i}-L_{i}x$ satisfies

Since $(\bar{C}_{i},\bar{A}_{i})$ is an observable pair, $K_{i}$ can be selected so that $\bar{e}_{i}(t)$ converges to $0$ exponentially fast at any pre-assigned rate. We assume that each $K_{i}$ is so chosen. Since

$w_{i}$ can be viewed as a signal which approximates $L_{i}x$ in the face of exponentially decaying additive noise, namely $\bar{e}_{i}$.

The other sub-system comprising agent $i$’s private estimator, is a “local parameter estimator” whose function is to generate estimates of $x$ at each of agent $i$’s preselected event times $t_{i1},t_{i2},\ldots$. Here $t_{i0},t_{i1},t_{i2},\ldots$ is an ascending sequence of event times with a fixed spacing of $T>0$ time units between any two successive event times. In this section it is assume that $t_{i0}=0,\;i\in\mathbf{m}$, and consequently that all event time sequences are the same²²2It is easy to generalize the results in this section to the case when event times are not evenly spaced provided that the spacings between successive pairs of event times remains positive and bounded.. Thus $t_{is}=sT,\;s\geq 0,\;\;\;i\in\mathbf{m}.$ Between event times, each $x_{i}$ is generated using the equation

Motivation for the development of the local parameter estimator whose purpose is to enable agent $i$ to estimate $x(t_{i(s-1)})$ over the event time interval $[t_{i(s-1)},\;t_{is})$, stems from the fact that the equations

admit a unique solution, namely $p=x(t_{i(s-1)})$. Existence follows from (4) whereas uniqueness is a consequence of the assumption of joint observability.

The existence and uniqueness of $p$ suggest that an approximate value of $x(t_{i(s-1)})$ can be obtained after a finite number of iterations - say $q$ - using the linear equation solver discussed in[32]. Having obtained such an approximate value of $x(t_{i(s-1)})$, denoted below by $z_{is}(q)$, the desired estimate of $x(t_{is})$ can be taken as

This is the architecture which will be considered.
The computations needed to update agent $i$’s estimate of $x(t_{i(s-1)})$ are carried out by agent $i$ during the event time interval $[t_{i(s-1)},t_{is})$. This is done using a local parameter estimator which generates a sequence of $q$ auxiliary states $z_{is}(1),z_{is}(2),\ldots,z_{is}(q)$ where $q>0$ is a positive integer to be specified below. The sequence is initialized by setting

and is recursively updated by agent $i$ at local iteration times $\tau_{is}(k),k\in\mathbf{q}\stackrel{{\scriptstyle\Delta}}{{=}}\{1,2,\ldots,q\},$ known only to agent $i$. It is assumed that the $\tau_{is}(k)$ together with the initialization $\tau_{is}(0)$ are of the form

where $\delta_{is}(0),\delta_{is}(1),\delta_{is}(2),\ldots,\delta_{is}(q)$ is a sequence of small deviations which satisfy

Here $\epsilon_{i}$ is a small nonnegative number whose constraints will be described below and $\Delta$ is a positive number satisfying

The signal $z_{is}(q)$ is agent $i$’s updated estimate of $x(t_{i(s-1)})$ and is used to define $x_{i}(t_{is})$ as in (6).
The transfer of information between agents which is needed to generate the $z_{is}(k)$, is carried out asynchronously as follows. For $k\in\mathbf{q}$ and $j\in\mathbf{m}$, agent $j$ broadcasts $z_{js}(k-1)$ at time $\tau_{js}(k-1)+\beta$ where $\beta$ is any prescribed nonnegative number chosen smaller than $\Delta$. It is assumed that the bounds $\epsilon_{i},\;i\in\mathbf{m}$, appearing in (9) are small enough so that there exist $\beta$ and $\Delta$ satisfying

These inequalities ensure that for $k\in\mathbf{q}$, a broadcast by any agent $j$ at time $\tau_{js}(k-1)+\beta$ will occur within the reception interval $[\tau_{is}(k-1),\;\;\tau_{is}(k))$ of agent $i$. Fig. 1 provides an example of the update and communication times of two different different agents $i$ and $j$.

Accordingly, agent $j$ is a data source or just source for agent $i$ on $[\tau_{is}(k-1),\;\;\tau_{is}(k))$ if agent $j$ is in the reception range of agent $i$ at time $\tau_{js}(k-1)+\beta$. Let $\mathcal{S}_{is}(k)$ denote the set of labels of such agents; that is

Note that $i\in\mathcal{S}_{is}(k)$, for all $i\in\mathbf{m}$ so $\mathcal{S}_{is}(k)$ is never empty. Clearly agent $i$ can use the signals $z_{js}(k-1),\;j\in\mathcal{S}_{is}(k)$, to compute $z_{is}(k)$.

In order to complete the definition of the hybrid observer, it is necessary to specify values of the $K_{i}$ and $q$. Towards this end, suppose that as a design goal it is desired to pick the $K_{i}$ and $q$ so that all $m$ state estimation errors

converge to zero as fast as $e^{-\lambda t}$ does where $\lambda>0$ is some desired convergence rate. The $K_{i}$ would then have to be chosen using spectrum assignment or some other technique so that the matrix exponentials $e^{(\bar{A}_{i}+K_{i}\bar{C}_{i})t}$ all converge to zero at least as fast as $e^{-\lambda t}$ does. This of course can be accomplished because each matrix pair $(\bar{C}_{i},\bar{A}_{i})$ is observable. In the sequel it will be assumed that for some preselected positive number $\bar{\lambda}>\lambda$, the $K_{i}$ have been chosen so that for $i\in\mathbf{m}$ the local observer estimation errors satisfy

where the $c_{i},\;i\in\mathbf{m}$ are nonnegative constants and $||\cdot||$ denotes the two-norm.

To describe how to define an appropriate value of $q$ to attain the desired convergence rate for the state estimation errors $e_{i},\;i\in\mathbf{m}$, it is necessary to take some preliminary steps. First, for each $i\in\mathbf{m}$, let $P_{i}$ denote the orthogonal projection on the unobservable space of $(C_{i},A_{i})$. It is easy to see that $P_{i}=I-L_{i}^{\prime}(L_{i}L_{i}^{\prime})^{-1}L_{i},\;\;i\in\mathbf{m}$. Moreover, because of the assumption of joint observability,

Next, let $\mathcal{C}$ denote the set of all products of the form $P_{i_{1}}P_{i_{2}}\cdots P_{i_{(m-1)^{2}+1}}$ where each projection matrix in $\{P_{i}:i\in\mathbf{m}\}$ occurs in each of such product at least once. Note that $\mathcal{C}$ is a closed subset of ${\rm I\!R}^{n\times n}$. Since each projection matrix $P_{i},\;i\in\mathbf{m}$ has a two-norm which is no greater than $1$, each matrix $M\in\mathcal{C}$ has a two-norm less than or equal to $1$. Thus $\mathcal{C}$ is also a bounded and thus compact subset. In fact, each product in $\mathcal{C}$ actually has two-norm strictly less than $1$. This is a consequence of (16) and the requirement that each matrix in $\{P_{i}:i\in\mathbf{m}\}$ must occur in each product in $\mathcal{C}$ at least once {Lemma 2, [33]}. These observations imply that maximum of the two-norms of the matrices in $\mathcal{C}$, namely

exists and is a real non-negative number strictly less than $1$ ³³3It is worth noting that although the matrices $L_{i}$ used in defining the $P_{i}$ are not uniquely determined by the unobservable spaces of the pairs $(C_{i},A)$, the orthogonal projection matrices $P_{i}$ nonetheless are. Thus the set $\mathcal{C}$ used in the definition of $\rho$ in (17) ultimately depends only on the family of unobservable spaces of the pairs $(C_{i},A),\;i\in\mathbf{m}$ and not on the particular manner in which the $L_{i}$ are chosen. Just how to explicitly characterize this dependence is a topic for future research.. This in turn implies that the attenuation constant

is also a real non-negative number strictly less than $1$. As will become evident below {cf. (40) and Lemma 5}, in the idealized case when all $\epsilon_{i}$ and $\bar{e}_{i}$ are zero, for any integer $p>0$ and any given value of $q$ satisfying

the value of the signal

is attenuated by at least a factor $\alpha^{p}$ after $q$ iterations during each event - time interval $[t_{i(s-1)},\;t_{is})$; i.e., for $s\geq 1$,

It will soon be apparent, if it is not already from (6), (7) and (14), that over each event- time interval $[t_{i(s-1)},\;t_{is})$,

Since each event - time interval is of length $T$, to achieve an exponential convergence rate of $\lambda$ in the idealized case, it is necessary to pick $q$ so that (19) holds where $p$ is any integer satisfying $e^{||A||T}\alpha^{p}<e^{-\lambda T}$. In other words, the requirement on $q$ is that (19) hold where

with $\lceil r\rceil$ here denoting, for any nonnegative number $r$, the smallest integer $k\geq r$. The following theorem, which applies to the synchronous case when all of the $\epsilon_{i}$ are zero, {but not necessarily the $\bar{e}_{i}$} summarizes these observations.

To deal with the asynchronous case when at least some of the $\epsilon_{i}$ are nonzero, it is necessary to assume that $\mathbb{N}(t)$ is constant on each of the time intervals

where

For this assumption to make sense, these intervals cannot overlap. The following lemma establishes that this is in fact the case.

From (10), $\epsilon\leq\beta$ and $\epsilon+\beta<\Delta$. These inequalities imply that $-\epsilon+sT+\beta\geq sT$ and $\epsilon+sT+(q-1)\Delta+\beta<(s+1)T$ respectively. From this it follows that $\mathcal{I}_{s}(1)\subset[sT,(s+1)T)$, that $\mathcal{I}_{s}(q)\subset[sT,(s+1)T)$ and thus that $\mathcal{I}_{s}(k)\subset[sT,(s+1)T),\;\;k\in\mathbf{q}$.  

We are led to the asynchronous version of Theorem 22.

The proof of this theorem will be given in the next section. Notice that the asynchronous case here can not be recognized in a discrete-time setting with a discrete-time clock shared by all $m$ agents considering delays[34].

The aim of this section is to analyze the behavior of the hybrid observer defined in the last section. To do this, use will be made of the notion of a “mixed matrix norm” which will now be defined. For any positive integers $k$, $m$, $n$ and $p$, let $\mathcal{M}$ denote the real $kmnp$- dimensional vector space of block partitioned matrices $\mathbf{M}=[M_{ij}]_{k\times m}$ where each block $M_{ij}$ is a $n\times p$ matrix. By the mixed matrix norm of $\mathbf{M}\in\mathcal{M},$ written $|\mathbf{M}|$, is meant the infinity norm of the matrix $[||M_{ij}||]_{k\times m}$ where $||M_{ij}||$ is the two-norm of $M_{ij}$. For example, with $e$ denoting the “stacked” state estimation error $e\stackrel{{\scriptstyle\Delta}}{{=}}\text{column}\{e_{1},e_{2},\ldots,e_{m}\}$ the quantity $\max\{||e_{i}||:i\in\mathbf{m}\}$ mentioned in the last section, is $|e|$, the mixed matrix norm of $e$. It is straight forward to verify that $|\cdot|$ is in fact a norm and that this norm is sub-multiplicative [33].

Recall that the purpose of agent $i$’s local parameter estimator defined by (7), (12), and (13) is to estimate $x(t_{i(s-1)})$ after executing $q$ iterations during the $s$th event time interval of agent $i$. In view of this, we define the parameter error vectors for $i\in\mathbf{m}$,

for all $s\geq 1$. This, (7), and the definition of $e_{i}$ in (14) imply that

In addition, from (5), (6) and (14) it is clear that that

To derive the update equation for $\pi_{is}(k)$ as $k$ ranges from $1$ to $q$, we first note from (13) that

Next note that because of (4) and (12)

From this and (31) it follows that for $k\in\mathbf{q},\;i\in\mathbf{m},\;s\geq 1$,

where as before, $P_{i}=I-Q_{i}L_{i}$. These are the local parameter error equations for the hybrid observer.

The next step in the analysis of the system is studied the evolution of the all-agent parameter error vector

Note first that because of (29) and (30)

and

where $e(s)$ is the all-agent state estimation error vector

and $\tilde{A}=\text{diagonal}\{A,A,\ldots,A\}$.

In order to develop an update equation for $\pi_{s}(k)$ as $k$ ranges from $1$ to $q$, it is necessary to combine the $m$ update equations in (32) into a single equation and to do this requires a succinct description of the graph determined by the sets $\mathcal{S}_{is},\;i\in\mathbf{m}$ defined in (11). There are two cases to consider: the synchronous case which is when all of the $\epsilon_{i}=0$ and the asynchronous case when some or all of the $\epsilon_{i}$ may be non-zero. The following lemmas cover both cases.

The following lemma asserts that (36) still holds in the asynchronous case when some of the $\epsilon_{i}$ are nonzero, provided $\mathbb{N}(t)$ is constant on each interval $\mathcal{I}_{s}(k),\;k\in\mathbf{q},\;s\geq 1$.