A. S. Morse

L. Wang, A. S. Morse

A time-invariant, linear, distributed observer is described for estimating the state of an $m>0$ channel, $n$-dimensional continuous-time linear system of the form $ \dot{x} = Ax,\ y_i = C_i x,\ i \in \{1,2,\cdots, m\}$. The state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives the state $z_j$ of each of its neighbors' estimators. Neighbor relations are characterized by a constant directed graph $\mathbb{N}$ whose vertices correspond to agents and whose arcs depict neighbor relations. The overall distributed observer consists of $m$ linear estimators, one for each agent; $m-1$ of the estimators are of dimension $n$ and one estimator is of dimension $n+m-1$. Using results from classical decentralized control theory, it is shown that subject to the assumptions that (i) none of the $C_i$ are zero, (ii) the neighbor graph $\mathbb{N}$ is strongly connected, (iii) the system whose state is to be estimated is jointly observable, and nothing more, it is possible to freely assign the spectrum of the overall distributed observer.

L. Wang, J. Liu, A. S. Morse et al.

A discrete-time modulus consensus model is considered in which the interaction among a family of networked agents is described by a time-dependent gain graph whose vertices correspond to agents and whose arcs are assigned complex numbers from a cyclic group. Limiting behavior of the model is studied using a graphical approach. It is shown that, under appropriate connectedness, a certain type of clustering will be reached exponentially fast for almost all initial conditions if and only if the sequence of gain graphs is "repeatedly jointly structurally balanced" corresponding to that type of clustering, where the number of clusters is at most the order of a cyclic group. It is also shown that the model will reach a consensus asymptotically at zero if the sequence of gain graphs is repeatedly jointly strongly connected and structurally unbalanced. In the special case when the cyclic group is of order two, the model simplifies to the so-called Altafini model whose gain graph is simply a signed graph.

Daniel Fullmer, A. S. Morse

This paper presents a stochastic model for block arrival times based on the difficulty retargeting rule used in Bitcoin, as well as other proof-of-work blockchains. Unlike some previous work, this paper explicitly models the difficulty target as a random variable which is a function of the previous block arrival times and affecting the block times in the next retargeting period. An explicit marginal distribution is derived for the time between successive blocks (the blocktime), while allowing for randomly changing difficulty. This paper also aims to serve as an introduction to Bitcoin and proof-of-work blockchains for the controls community, focusing on the difficulty retargeting procedure used in Bitcoin.

L. Wang, A. S. Morse, D. Fullmer et al.

A hybrid observer is described for estimating the state of an $m>0$ channel, $n$-dimensional, continuous-time, distributed linear system of the form $\dot{x} = Ax,\;y_i = C_ix,\;i\in\{1,2,\ldots, m\}$. The system's state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. Agent $i$ updates its estimate $x_i$ of $x$ at "event times" $t_1,t_2,\ldots $ using a local observer and a local parameter estimator. The local observer is a continuous time linear system whose input is $y_i$ and whose output $w_i$ is an asymptotically correct estimate of $L_ix$ where $L_i$ a matrix with kernel equaling the unobservable space of $(C_i,A)$. The local parameter estimator is a recursive algorithm designed to estimate, prior to each event time $t_j$, a constant parameter $p_j$ which satisfies the linear equations $w_k(t_j-τ) = L_kp_j+μ_k(t_j-τ),\;k\in\{1,2,\ldots,m\}$, where $τ$ is a small positive constant and $μ_k$ is the state estimation error of local observer $k$. Agent $i$ accomplishes this by iterating its parameter estimator state $z_i$, $q$ times within the interval $[t_j-τ, t_j)$, and by making use of the state of each of its neighbors' parameter estimators at each iteration. The updated value of $x_i$ at event time $t_j$ is then $x_i(t_j) = e^{Aτ}z_i(q)$. Subject to the assumptions that (i) the neighbor graph $\mathbb{N}(t)$ is strongly connected for all time, (ii) the system whose state is to be estimated is jointly observable, (iii) $q$ is sufficiently large, it is shown that each estimate $x_i$ converges to $x$ exponentially fast as $t\rightarrow \infty$ at a rate which can be controlled.