# Neuromimetic Control — A Linear Model Paradigm

John Baillieul & Zexin Sun

###### Abstract

Stylized models of the neurodynamics that underpin sensory motor control in
animals are proposed and studied. The voluntary motions of animals are
typically initiated by high level intentions created in the primary cortex
through a combination of perceptions of the current state of the environment
along with memories of past reactions to similar states. Muscle movements are
produced as a result of neural processes in which the parallel activity of
large multiplicities of neurons generate signals that collectively lead to
desired actions. Essential to coordinated muscle movement are intentionality,
prediction, regions of the cortex dealing with misperceptions of sensory cues,
and a significant level of resilience with respect to disruptions in the
neural pathways through which signals must propagate. While linear models of
feedback control systems have been well studied over many decades, this paper
proposes and analyzes a class of models whose aims are to capture some of the
essential features of neural control of movement. Whereas most linear models
of feedback systems entail a state component whose dimension is higher than
the number of inputs or outputs, the work that follows will treat models in
which the numbers of input and output channels greatly exceed the state
dimension. While we begin by considering continuous-time systems governed by
differential equations, the aim will be to treat systems whose evolution
involves classes of inputs that emulate neural spike trains. Within the
proposed class of models, the paper will study resilience to channel dropouts,
the ways in which noise and uncertainty can be mitigated by an appropriate
notion of consensus among noisy inputs, and finally, by a simple model in
which binary activations of a multiplicity of input channels produce a dynamic
response that closely approximates the dynamics of a prescribed linear system
whose inputs are continuous functions of time.

††footnotetext:
John Baillieul is with the Departments of Mechanical Engineering, Electrical
and Computer Engineering, and the Division of Systems Engineering at Boston
University, Boston, MA 02115. Zexin Sun is with the Division of Systems
Engineering at Boston University. The authors may be reached at {johnb,
<EMAIL_ADDRESS>
Support from various sources including the Office of Naval Research grants
N00014-10-1-0952, N00014-17-1-2075, and N00014-19-1-2571 is gratefully
acknowledged.
A condensed version of this paper has been submitted to the 60thIEEE
Conference on Decision and Control.

Keywords: Neuromimetic control, parallel quantized actuation, channel
intermittency, neural emulation

##  I Introduction

The past two decades have seen dramatic increases in research devoted to
information processing in networked control systems, [1]. At the same time,
rapidly advancing technologies have spurred correspondingly expanded research
activity in cyber-physical systems and autonomous machines ranging from
assistive robots to self-driving automobiles, [2]. Against this backdrop, a
new focus of research that seeks connections between control systems and
neuroscience has emerged, ([3, 6, 4]), The aim is to understand how
neurobiology should inform the design of control systems that can not only
react in real-time to environmental stimuli but also display predictive and
adaptive capabilities, [5]. More fundamentally, however, the work described
below seeks to understand how these capabilities might emerge from parallel
activity of sets of simple inputs that play a role analogous to sets of
neurons in biological systems. What follows are preliminary results treating
linear control systems in which simple inputs steer the system by collective
activity. Principles of resilience with respect to channel dropouts and
intermittency of channel availability are explored, and the stage is set for
further study of prediction and learning. The paper is organized as follows.
Section II introduces problems in designing resilient feedback stabilized
motions of linear systems with (very) large numbers of inputs. (Resilience
here means that the designs still achieve design objective if a number of
input channels become unavailable.) Section III introduces an approach to
resilient designs based on an approach we call parameter lifting. Section IV
briefly discusses how adding input channels which transmit noisy signals
inputs can have the net effect of reducing uncertainty in achieving a control
objective. Section V takes up the problem of control with quantized inputs,
and Section VI concludes with a discussion of ongoing work that is aimed at
making further connections with problems in neurobiology.

Figure 1: The defining characteristics of neuromimetic linear systems are
large numbers of input and output channels, all of which carry simple signals
comprising possibly only a single bit of information and which collectively
steer the system to achieve system goals.

##  II Linear Systems with Large Numbers of Inputs

The models to be studied have the simple form

$\begin{array}[]{l}\dot{x}(t)=Ax(t)+Bu(t),\ \ \ x\in\mathbb{R}^{n},\ \
u\in\mathbb{R}^{m},\ {\rm and}\\\\[5.05942pt] y(t)=Cx(t),\ \ \ \ \ \ \ \ \ \ \
\ \ \ y\in\mathbb{R}^{q}.\end{array}$ (1)

As in [9, 10]. we shall be interested in the evolution and output of (1) in
which a portion of the input or output channels may or may not be available
over any given subinterval of time. Among cases of interest, channels may
intermittently switch in or out of operation. In all cases, we are explicitly
assuming that $m,q>n$. In [9] we studied the way the structure of a system of
this form might be affected by random unavailability of input channels. The
work in [10] showed the advantages of having large numbers of input channels
(as measured by input energy costs as a function of the number of active input
channels and resilience to channel drop-outs). In what follows we shall show
that having large numbers of parallel input channels provides corresponding
advantages in dealing with noise and uncertainty.

To further explore the advantages of large numbers of inputs in terms of
robustness to model and operating uncertainty and resilience with respect to
input channel dropouts, we shall examine control laws for systems of the form
(1) in which groups of control primitives are chosen from dictionaries and
aggregated on the fly to achieve desired ends. The ultimate goal is to
understand how carefully aggregated finitely quantized inputs can be used to
steer (1) as desired. To lay the foundation for this inquiry, we begin by
focusing on dictionaries of continuous control primitives that we shall call
set point stabilizing.

###  II-A Resilient eigenvalue assignment for systems with large numbers of
inputs

Briefly introduced in [10] our dictionaries will be comprised of set-point
stabilizing control primitives of the form

$u_{j}(t)=v_{j}+k_{j1}x_{1}+\dots+k_{jn}x_{n},\ \ j=1,\dots,m,$

where the gains $k_{ji}$ are chosen to make the matrix $A+BK$ Hurwitz, and the
$v_{j}$’s are then chosen to make the desired goal point
$x_{g}\in\mathbb{R}^{n}$ an equilibrium of (1). Thus, given $x_{g}$ and a
desired gain matrix $K$ the vector $v$ can chosen so as to satisfy the
equation

$(A+BK)x_{g}+Bv=0.$ (2)

######  Proposition 1.

Let $B$ be full rank $=n$, and let the $m\times n$ matrix $K$ be such that the
eigenvalues of $A+BK$ are in the open left half plane. If $v$ is any vector
satisfying (2), then the $m$ control inputs

$u_{j}(t)=v_{j}+k_{j1}x_{1}(t)+\cdots+k_{jn}x_{n}(t)$ (3)

steer (1) toward the goal $x_{g}$.

Since $m>n$, the problem of finding values $k_{ij}$ making $A+BK$ Hurwitz and
$v$ satisfying (2) is underdetermined. We can thus carry out a parametric
exploration of ranges of values $K$ and $v$ that make the system (1) resilient
in the sense that the loss of one (or more) input channels will not prevent
the system from reaching the desired goal point. To examine the effect of
channel unavailability and channel intermittency, let $P$ be an $m\times m$
diagonal matrix whose diagonal entries are $k$ 1’s and $m$-$k$ 0’s. For each
of the $2^{m}$ possible projection matrices of this form, we shall be
interested in cases where $(A,BP)$ is a controllable pair. We have the
following:

######  Definition 1.

Let $P$ be such a projection matrix with $k$ $1^{\prime}s$ on the main
diagonal. The system (1) is said to be $k$-channel controllable with respect
to $P$ if for all $T>0$, the matrix

$W_{P}(0,T)=\int_{0}^{T}\,e^{A(T-s)}BPB^{T}{e^{A(T-s)}}^{T}\,ds.$

is nonsingular.

######  Remark 1.

For the system $\dot{x}=Ax+Bu$, being $k$-channel controllable with respect to
a canonical projection $P$ having $k$ ones on the diagonal is equivalent to
$(A,BP)$ being a controllable pair.

######  Example 1.

Consider the three input system

$\left(\begin{array}[]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right)=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)+\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&1\\\
\end{array}\right)u.$ (4)

Adopting the notation

$P[i,j,k]=\left(\begin{array}[]{ccc}i&0&0\\\ 0&j&0\\\
0&0&k\end{array}\right),$

the system (4) is 3-channel controllable with respect to $P[1,1,1]$; it is
2-channel controllable with respect to $P[1,1,0],P[1,0,1],$ and $P[0,1,1]$. It
is 1-channel controllable with respect to $P[1,0,0]$ and $P[0,0,1]$, but it
fails to be 1-channel controllable with respect to $P[0,1,0]$.

Within classes of system (1) that are $k-$channel controllable, for $1\leq
k\leq m$, we wish to characterize control designs that achieve set point goals
despite different sets of $j$ control channels ($1\leq j\leq m-k$) being
either intermittently or perhaps even entirely unavailable. When $m>n$, the
problem of finding $K$ and $v$ such that $A+BK$ is Hurwitz and (2) is
satisfied leaves considerable room to explore solutions $(K,v)$ such that
$A+BK$ is Hurwitz and $(A+BPK)x_{g}+BPv=0$ for various coordinate projections
of the type considered in Definition 1.

To take a deeper dive into the theory of LTI systems with large numbers of
input channels, we introduce some notation. Fix integers $0\leq k\leq m$, and
let $[m]:=\\{1,\dots,m\\}$. Let $[m]\choose k$ be the set of $k$-element
subsets of [m]. In Example 3, for instance, ${[3]\choose
2}=\left\\{\\{1,2\\},\\{1,3\\},\\{2,3\\}\right\\}$. Extending the $2\times 3$
example, we consider matrix pairs $(A,B)$ where $A$ is $n\times n$ and $B$ is
$n\times m$. We consider the following:

Problem A. Find an $m\times n$ gain matrix $K$ such that $A+BP_{I}K$ is
Hurwitz for projections $P_{I}$ onto the coordinates corresponding to the
index set $I$ for all $I\in{[m]\choose j}$ with $n\leq j\leq m$.

Problem B. Find an $m\times n$ gain matrix $K$ that assigns eigenvalues of
$A+BP_{I}K$ to specific values for each $I\in{[m]\choose n}$.

Both of these problems can be solved if $m=n+1$, but when $m>n+1$, there are
more equations than unknowns, making the problems over constrained and thus
generally not solvable. We consider the following:

######  Definition 2.

Given a system (1) where $A$ and $B$ are respectively $n\times n$ and $n\times
m$ matrices with $n\leq m$, for each $I\in{[m]\choose n}$, the $I$-th
principal subsystem is given by the state evolution

$\dot{x}(t)=Ax(t)+BP_{I}u(t).$

$\square$

Problem B requires solving $n$ equations for each $I\in{[m]\choose n}$, and if
we seek an $m\times n$ gain matrix $K$ which places eigenvalues of $A+BP_{I}K$
for every$I\in{[m]\choose n}$, then a total of $n{m\choose n}$the simultaneous
equations must be solved to determine the $nm$ entries in $K$. Noting that
$nm\leq n{m\choose n}$, with the inequality being strict if $m>n+1$, we see
that solving Problem B cannot be solved in general. Problem A is less exacting
in that it only requires eigenvalues to lie in the open left half plane, but
it carries the further requirement that a single gain $K$ places all the
closed-loop eigenvalues of all $I-th$ subsystems in the open left half plane
for $I\in{[m]\choose n}$ and all $k$ in the range $n\leq k\leq m$. The
following example shows that solutions to Problem B are not necessarily
solutions to Problem A as well and illustrates the complexity of the resilient
eigenvalue placement problem.

######  Example 2.

For the system (4) of Example 1, we consider the three 2-channel controllable
pairs

$(A,BP[110]),\ \ \ (A,BP[101]),\ \ \ (A,BP[011]).$

We look for a $3\times 2$ gain matrix $K$ such that $A+BPK$ has specified
eigenvalues for each of these $P$’s. Thus we seek to assign eigenvalues to the
three matrices

$\left(\begin{array}[]{cc}k_{21}&1+k_{22}\\\ k_{11}&k_{12}\end{array}\right),\
\ \ \left(\begin{array}[]{cc}k_{31}&1+k_{32}\\\
k_{11}+k_{31}&k_{12}+k_{32}\end{array}\right),$
$\left(\begin{array}[]{cc}k_{21}+k_{31}&1+k_{22}+k_{32}\\\
k_{31}&k_{32}\end{array}\right)$

respectively. For any choice of LHP eigenvalues, this requires solving six
equations in six unknowns. For all three matrices to have eigenvalues at
(-1,-1) the $k_{ij}$-values
$k_{11}=0,k_{12}=-1,k_{21}=-1,k_{22}=0,k_{31}=-1/2,k_{32}=-1/2$ place these
closed loop eigenvalues as desired. For this choice of $K$, the closed loop
eigenvalues of $(A+BK)$ are $-3/2\pm i/2$. It is not possible to assign all
eigenvalues by state feedback independently for the four controllable pairs
$(A,B)$, $(A,BP[110])$, $(A,B,P[101])$, and $(A,BP[011])$. Moreover, it is
coincidental that the eigenvalues of $A+BK$ are in the left half plane. This
is seen in the following examples. (Consider
$k_{11}=-2,k_{12}=-3,k_{21}=-1,k_{22}=-2.4,k_{31}=0,k_{32}=-1/2$. There are
the following closed-loop eigenvalues: For $A+BP[110]K$: $(-3.95,-0.05)$, for
$A+BP[101]K$: $(-3.19,-0.31)$, and for $A+BP[011]K$: $(-1,-0.5)$; but the
eigenvalues of $A+BK$ are $(-4.57,0.066)$.)

Resilience requires an approach that applies simultaneously to all pairs
($A,BP_{I}$) where $P_{I}$ ranges over the lattice of projections $P_{I}$,
$I\in{[m]\choose j},\ j=n\dots,m$. In view of these examples, it is of
interest to explore conditions under which the stability of feedback designs
will be preserved as control input channels become intermittently or even
permanently unavailable.

##  III Lifted Parameters

To exploit the advantages of a large number of control input channels, we turn
our attention to using the extra degrees of freedom in specifying the control
offsets $v_{1},\dots,v_{m}$ so as to make (1) resilient in the face of
channels being intermittently unavailable. As noted in [10], we seek an offset
vector $v$ that satisfies $B\left[(\hat{A}+K)x_{g}+v\right]=0$ where $\hat{A}$
is any $m\times n$ matrix satisfying $B\hat{A}=A$. If $K$ is chosen such that
$A+BK$ is Hurwitz, the feedback law $u=Kx+v$ satisfies (2) and by Proposition
1 steers the state of (1) asymptotically toward the goal in $x_{g}$. Under the
assumption that $B$ has full rank $n$, such matrix solutions can be
found–although such an $\hat{A}$ will not be unique. Once $\hat{A}$ and the
gain matrix $K$ have been chosen, the offset vector $v$ is determined by the
equation

$(\hat{A}+K)x_{g}+v=0.$ (5)

Conditions under which $\hat{A}$ may be chosen to make (1) resilient to
channel dropouts is the following.

######  Theorem 1.

([10]) Consider the linear system (1) in which the number of control inputs,
$m$, is strictly larger than the dimension of the state, $n$ and in which rank
$B=n$. Let the gain $K$ be chosen such that $A+BK$ is Hurwitz, and assume that
(a) $P$ is a projection of the form considered in Definition 1 and (1) is
${\ell}$-channel controllable with respect to $P$;
(b) $A+BPK$ is Hurwitz;
(c) the solution $\hat{A}$ of $B\hat{A}=A$ is invariant under $P$—i.e.,
$P\hat{A}=\hat{A}$; and
(d) $BP$ has rank $n$. Then the control inputs defined by (3) steer (1) toward
the goal point $x_{g}$ whether or not the $(m-\ell)$ input channels that are
mapped to zero by $P$ are available.

The next two lemmas review simple facts about the input matrices $B$ under
consideration.

######  Lemma 1.

Let $B$ be an $n\times m$ matrix with $m\geq n$. If all principal minors of
$B$ are nonzero, then $B$ has full rank.

###### Proof.

The matrix $B$ has either more or the same number of columns as rows. Its rank
is thus the number of linearly independent rows. If there is a nontrivial
linear combination of rows of $B$, then this linear combination is inherited
by all $n\times n$ submatrices, and thus all principal minors would be zero. ∎

######  Lemma 2.

Let $B$ be an $n\times m$ matrix with $m\geq n$ and having full rank $n$. Then
$BB^{T}$ is positive definite.

###### Proof.

If $B$ has full rank, the $n$ rows are linearly independent. Hence if $x^{T}B$
is the zero vector in $\mathbb{R}^{m}$, then we must have $x=0$. Thus, for all
$x\in\mathbb{R}^{n}$, $x^{T}BB^{T}x=\|B^{T}x\|^{2}\geq 0$ with equality
holding if and only if $x=0$. ∎

######  Lemma 3.

Consider a linear function $B:\mathbb{R}^{m}\to\mathbb{R}^{n}$ given by an
$n\times m$ rank $n$ matrix $B$ with $m\geq n$. This induces a linear function
$\hat{\rm B}$ having rank $n^{2}$ from the space of $m\times n$ matrices,
$\mathbb{R}^{m\times n}$, to the space of $n\times n$ matrices,
$\mathbb{R}^{n\times n}$, which is given explicitly by $\hat{\rm B}$(Y) =
$B\cdot$Y.

###### Proof.

Because $B$ has full rank $n$, it has a rank $n$ right inverse $U$. Given any
$n\times n$ matrix $A$, the image of $UA\in\mathbb{R}^{m\times n}$ under
$\hat{\rm B}$ is $A$, proving the lemma. ∎

This function lifting is depicted graphically by

$\begin{array}[]{clcc}\hat{\rm B}:&\mathbb{R}^{m\times
n}&\rightarrow&\mathbb{R}^{n\times n}\\\ &\ \ \big{|}&&\big{|}\\\
B:&\mathbb{R}^{m}&\rightarrow&\mathbb{R}^{n}.\end{array}$

######  Lemma 4.

Given that the rank of the $n\times m$ matrix $B$ is $n$ with $n<m$, the
dimension of the null space of $B$ is $m-n$. The dimension of the null space
of $\hat{\rm B}$ is $n(m-n)$.

###### Proof.

Let the set column vectors $\left\\{\vec{n}_{1},\dots,\vec{n}_{m-n}\right\\}$
be a basis of the nullspace of $B$. The set of all $m\times n$ matrices of the
form
$N=(\vec{n}_{i_{1}}\vdots\vec{n}_{i_{2}}\vdots\cdots\vdots\vec{n}_{i_{n}})$,
where $i_{j}\in[m-n]$, is a linearly independent set that spans the null space
of $\hat{\rm B}$. There are $m-n$ independent choices of labels for each of
the $n$ columns of $N$, proving the lemma. ∎

######  Lemma 5.

Let $A$ be an $n\times n$ matrix and $B$ be an $n\times m$ matrix with $m>n$
and $B$ having full rank $n$. Then there is an $n(m-n)$-parameter family of
solutions $X=\hat{A}$ to the matrix equation $\hat{\rm B}(X)=A$.

###### Proof.

From Lemma 4, we may find a basis $\\{N_{1},\dots,N_{n(m-n)}\\}$ of the
nullspace of $\hat{\rm B}$, which we denote ${\cal N}(\hat{\rm B})$. A
particular solution of the matrix equation is $\hat{A}=B^{T}(BB^{T})^{-1}A$,
and any other solution may be written as $B^{T}(BB^{T})^{-1}A+N$ where
$N\in{\cal N}(\hat{\rm B})$. This proves the lemma. ∎

######  Lemma 6.

Consider the LTI system $\dot{x}=Ax+Bu$, where $A$ is an $n\times n$ real
matrix and $B$ is an $n\times m$ real matrix with $m>n$. Suppose that all
principal minors of $B$ are nonzero, and further assume that $A=0$. Then, for
any real number $\alpha>0$, the feedback gain $K=-\alpha B^{T}$ is
stabilizing; i.e. $BK$ has all eigenvalues in the open left half plane.
Moreover, for all $n\leq j\leq m$ and $I\in{[m]\choose j}$, all matrices
$BP_{I}K$ have eigenvalues in the open left half plane.

###### Proof.

Let $\bar{B}=P_{I}B$. Noting that $\bar{B}$ has full rank $n$ and
$\bar{B}\bar{B}^{T}=BP_{I}B^{T}$, the lemma follows from Lemma 2.

∎

######  Lemma 7.

Consider the LTI system $\dot{x}=Ax+Bu$, where $A$ is an $n\times n$ real
matrix and $B$ is an $n\times m$ real matrix with $m>n$. Suppose that all
principal minors of $B$ are nonzero, and further assume that $\hat{A}$ is a
solution of $\hat{\rm B}(\hat{A})=A$. Then, for any real $\alpha>0$ and
feedback gain

$K=-\alpha B^{T}-\hat{A},$ (6)

the closed loop system $\dot{x}=(A+BK)x$ is exponentially stable—i.e..the
eigenvalues of $A+BK$ lie in the open left half plane.

###### Proof.

Substituting $u=Kx$ into $\dot{x}=Ax+Bu$, we find that the closed-loop system
is given by $\dot{x}=-\alpha BB^{T}x$, and the conclusion follows from Lemma
6. ∎

######  Definition 3.

Let $I\in{[m]\choose j}$. The lattice of index supersets of I in $[m]$ is
given by ${\cal L}_{I}=\\{L\subset[m]:I\subset L\\}$.

######  Theorem 2.

Consider the LTI system $\dot{x}=Ax+Bu$, where $A$ is an $n\times n$ real
matrix and $B$ is an $n\times m$ real matrix with $m>n$. Suppose that all
principal minors of $B$ are nonzero, and further assume that $\hat{A}$ is a
solution of $\hat{\rm B}(\hat{A})=A$, and that for a given $I\in{[m]\choose
j}$, $\hat{A}$ is invariant under $P_{I}$—i.e.$P_{I}\hat{A}=\hat{A}$. Then,
for any real $\alpha>0$ and feedback gain $K=-\alpha B^{T}-\hat{A}$ the closed
loop systems $\dot{x}=(A+BP_{L}K)x$ are exponentially stable for all
$L\in{\cal L}_{I}$.

###### Proof.

The theorem follows from Lemma 6 with $BP_{I}$ and $P_{I}K$ here playing the
roles of $B$ and $K$ in the lemma. ∎

Within the class of resilient (invariant under projections) stabilizing
feedback designs identified in Theorem 2, there is considerable freedom to
vary paramters to meet system objectives. To examine parameter choices, let
$P_{[k]}$ be a diagonal $m\times m$ projection matrix having $k$-0’s and
$m-k$-1’s on the principal diagonal ($1\leq k\leq m-n$). For each such
$P_{[k]}$, the $n(m-n)$ parameter family of solutions $\hat{A}$ to $\hat{\rm
B}(\hat{A})=A$ is further constrained by the invariance
$P_{[k]}\hat{A}=\hat{A}$ which imposes $kn$ further constraints. Hence, the
family of $P_{[k]}$-invariant $\hat{A}$’s is $n(m-n-k)$-dimensional; see Fig.
2. Within this family, design choices may involve adjusting the overall
strength of the feedback (parameter $\alpha$) or differentially adjusting the
influence of each input channel by scaling rows of $B^{T}$ in (6). Such
weighting will be discussed in greater detail in Section V where we note that
to make the models reflective of the parallel actions of many simple inputs,
it is necessary to normalize the weight $\alpha$ of $B^{T}$ in (6) so as to
not let the influence of inputs depend on the total number of those available
(as opposed to the total number acting according to the various projections
$P$). In other words, we want to take care not to overly weight the influence
of large groups versus small groups of channels.

Figure 2: Solutions to $\hat{\rm B}(\hat{A})=A$ and the lattice of channel
dropouts. At the root, there is an $n(m-n)$-parameter family of solutions, and
if any channel drops out, the solutions under the corresponding projection
$P_{I}$ are an $n(m-n-1)$-parameter family. At the bottom leaf nodes of the
lattice where $m-n$ channels have become unavailable, there will typically be
a single invariant $\hat{A}$. Note that there are $m(m-1)\cdots(m-n+1)$
distinct paths from the root to the ${m\choose n}$ leaves.

######  Example 3.

We end this section with an example of random unavailability of input
channels. Considering still the system of Example 1, define the feedback gain
$K=\alpha B^{T}-\hat{A}$ where

$\hat{A}=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\ 0&0\end{array}\right),$

and $\alpha=2$. Consider channel intermittency in which each of the three
channels is randomly available according to a two-state Markov process defined
by

$\left(\begin{array}[]{c}\dot{p}_{u}(t)\\\
\dot{p}_{a}(t)\end{array}\right)=\left(\begin{array}[]{cc}-\delta&\epsilon\\\
\delta&-\epsilon\end{array}\right)\left(\begin{array}[]{c}p_{u}(t)\\\
p_{a}(t)\end{array}\right),$

where $p_{u}\ =$ probability of channel unavailability; $p_{a}=1-p_{u}\ =$
probability of the channel being available. Assuming the availabilities of the
channels are i.i.d. with this availability law, simulations show that for a
range of parameters $\delta,\epsilon>0$ that

$\dot{x}=(A+BP(t)K)\,x$ (7)

is asymptotically stable with the time-dependent projection $P(t)$ being a
diagonal matrix whose diagonal entries can take any combination of values 0
and 1. The special case $\delta=3,\ \epsilon=3$ is illustrated in Fig. 3. The
hypotheses of Theorem 2 characterize a worst case scenario in terms of
stability since there may be time intervals over which (7) does not satisfy
the hypotheses because $P(t)$ does not leave $\hat{A}$ invariant.

Figure 3: An example shows that when channels are intermittently available,
the designed feedback can still maintain the asymptotic stability of the
system, even if there are some short short periods of instability as exhibited
in the black circle.

##  IV Uncertainty reduction from adding a channel

Suppose that instead of eq. (5), uncertainty in each input channel is taken
into account and the actual system dynamics are governed by

$(\hat{A}+K)x_{g}+v+n_{\epsilon}=0,$ (8)

where $n_{\epsilon}$ is a Gaussian random vector whose entries are $i.i.d\in
N(0,1)$. Assume further that these channel-wise uncertainties are mutually
independent. Then the asymptotic steady state limit $x_{\infty}$ will satisfy

$E[\|x_{\infty}-x_{g}\|^{2}]=tr[(A+BK)^{-1}B\Sigma_{\epsilon}B^{T}((A+BK)^{T})^{-1}],$
(9)

where $\Sigma_{\epsilon}=I$ is the $m\times m$ covariance of the channel
perturbations. The question of how steady state error is affected by the
addition of a noisy channel is partially addressed as follows.

######  Theorem 3.

Suppose the system (1) is controllable, and that $K$ and $\hat{A}$ have been
chosen as in Theorem 2 so that $A+BK$ is Hurwitz and the control input $u(t)=\
Kx(t)+v$ has been chosen to steer the state to $x_{g}$ using $v$ given by (5).
Let $b\in\mathbb{R}^{n}$ and consider the augmented $n\times(m+1)$ matrix
$\bar{B}=(B\vdots b)$. Then with the control input $\bar{u}=\bar{K}x+\bar{v}$,
where $\bar{K}=\left(\begin{array}[]{cc}K\\\ k\end{array}\right)$, $k=-\alpha
b^{T}$, and offset $\bar{v}=\left(\begin{array}[]{cc}v\\\
0\end{array}\right)$, the system $\dot{x}=Ax+\bar{B}\bar{u}$ is steered such
that the steady state error covariance is

$\Sigma_{\bar{B}}=M(B\vdots b)\Sigma_{\epsilon}(B\vdots b)^{T}M^{T},$

where

$M=\left(A+(B\vdots b)(\begin{array}[]{c}K\\\ k\end{array})\right)^{-1}.$

The corresponding steady state error covariance for $\dot{x}=Ax+Bu$ is

$\Sigma_{B}=(A+BK)^{-1}B\Sigma_{\epsilon}B^{T}((A+BK)^{T})^{-1},\ and$

if the mean squared asymptotic errors under the two control laws are denoted
by $err_{\bar{B}}$ and $err_{B}$, then $err_{\bar{B}}<err_{B}$.

###### Proof.

Let $M_{B}=(A+BK)^{-1}$ and $M_{\bar{B}}=\left(A+(B\vdots
b)(\begin{array}[]{c}K\\\ k\end{array})\right)^{-1}.$ We have $A+(B\vdots
b)\left(\begin{array}[]{c}K\\\ k\end{array}\right)=A+BK+bk=A+B(-\alpha
B^{T}-\hat{A})-\alpha bb^{T}=-\alpha(BB^{T}+bb^{T}$). Hence
$M_{\bar{B}}=-(1/\alpha)(BB^{T}+bb^{T})^{-1}$, whereas
$M_{B}=-(1/\alpha)(BB^{T})^{-1}$.

Whence, in comparing mean square errors,

$\begin{array}[]{ccl}err_{\hat{B}}&=&tr\left\\{M_{\bar{B}}\bar{B}{\bar{B}}^{T}M_{\bar{B}}^{T}\right\\}\\\\[5.05942pt]
&=&(1/\alpha^{2})tr\left\\{(BB^{T}+bb^{T})^{-1}(BB^{T}+bb^{T})\right.\\\ &&\ \
\ \ \ \ \ \ \ \ \left.(BB^{T}+bb^{T})^{-1}\right\\}\\\\[5.05942pt]
&=&(1/\alpha^{2})tr\left\\{(BB^{T}+bb^{T})^{-1}\right\\};\end{array}$
$\begin{array}[]{ccl}err_{B}&=&tr\left\\{M_{B}BB^{T}M_{B}^{T}\right\\}\\\\[5.05942pt]
&=&(1/\alpha^{2})tr\left\\{((BB^{T})^{-1}(BB^{T})(BB^{T})^{-1}\right\\}\\\\[5.05942pt]
&=&(1/\alpha^{2})tr\left\\{(BB^{T})^{-1}\right\\}.\end{array}$

Since $BB^{T}+bb^{T}>BB^{T}$ in the natural ordering of symmetric positive
definite matrices, the conclusion of the theorem follows. ∎

##  V The nuanced control authority of highly parallel quantized actuation

Technological advances have made it both possible and desirable to re-examine
classical linear feedback designs using a new lens as described in the
previous sections. In this section, we revisit the concepts of the previous
sections in terms of quantized control along the lines of [1], [17], [18],
[19], and [25]. The advantages of large numbers of input channels in terms of
reducing energy ([10]) and uncertainty (Theorem 3 above) come at a cost that
is not easily modeled in the context of linear time-invariant (LTI) systems.
Indeed as the number of control input channels increases, so does the amount
of information that must be processed to implement a feedback design of the
form of Theorem 2 in terms of the attention needed (measured in bits per
second) to ensure stable motions. (See [16] for a discussion of attention in
feedback control.) An equally subtle problem with the models of massively
parallel input described in the preceding sections is the need for asymptotic
scaling of inputs as the number of channels grows. If the number of input
channels is large enough, then the feedback control $u=Kx$ for $K=-B^{T}$ will
be stabilizing for any system (1). This is certainly plausible based on
Gershgorin’s Theorem and the observation that if $B_{1}$ has full rank $m$,
and $B_{2}$ is obtained from $B_{1}$ by adding one (or more) columns, then
$B_{2}B_{2}^{T}>B_{1}B_{1}^{T}$ in terms of the standard order relationship on
positive definite matrices. A precise statement of how fast the matrix
$BB^{T}$ grows from adding columns in given by the following.

######  Proposition 2.

Let $B$ be an $2\times m$ matrix with $m>2$ whose columns are unit vectors
uniformly spaced on the unit circle $S^{1}$. Then, the spectral radius of
$BB^{T}$ is $m/2$.

###### Proof.

The proof is by direct calculation. There is no loss of generality in assuming
the $m$ uniformly distributed vectors are
$(\cos\frac{2k\pi}{m},\sin\frac{2k\pi}{m})$. $BB^{T}$ is then given explicitly
by

$BB^{T}=\left(\begin{array}[]{ll}\sum_{k=1}^{m}\cos^{2}\frac{2k\pi}{m}&\sum_{k=1}^{m}\sin\frac{2k\pi}{m}\cos\frac{2k\pi}{m}\\\\[3.61371pt]
\sum_{k=1}^{m}\sin\frac{2k\pi}{m}\cos\frac{2k\pi}{m}&\sum_{k=1}^{m}\sin^{2}\frac{2k\pi}{m}\end{array}\right).$

Standard trig identities show the off-diagonal matrix entries are zero and the
diagonal entries are $m/2$, proving the proposition. ∎

For $n>2$, the conclusion is similar but slightly more complex. We consider
$n\times m$ matrices $B$ ($m>n$) whose columns are unit vectors in
$\mathbb{R}^{n}$. Following standard constructions of Euler angles (e.g.
http://www.baillieul.org/Robotics/740_L9.pdf), we define spherical coordinates
(or generalized Euler angles) on the $n-1$-sphere $S^{n-1}$ whereby points
$(x_{1},\dots,x_{n})\in S^{n-1}$ are given by

$\begin{array}[]{lll}x_{1}&=&\sin\theta_{1}\sin\theta_{2}\cdots\
\sin\theta_{n-1}\\\ x_{2}&=&\sin\theta_{1}\sin\theta_{2}\cdots\
\cos\theta_{n-1}\\\
x_{3}&=&\sin\theta_{1}\sin\theta_{2}\cdots\cos\theta_{n-2}\\\ &\vdots&\\\
x_{n-1}&=&\sin\theta_{1}\cos\theta_{2}\\\ x_{n}&=&\cos\theta_{1},\end{array}$

where $0\leq\theta_{1}\leq\pi,\ 0\leq\theta_{j}<2\pi$ for $j=2,\dots,n-1$. We
shall call a distribution of points on $S^{n-1}$ parametrically regular if it
is given by $\theta_{1}=\frac{j\pi}{N_{1}},\ (j=0,\dots,N_{1})$, and
$\theta_{k}=\frac{2j\pi}{N_{k}},\ (j=1,\dots,N_{k})$ for $2\leq k\leq n-1$.
The following extends Proposition 2 to $n\times m$ matrices where $n>2$.

######  Theorem 4.

Let the $n\times m$ matrix $B$ comprise columns consisting of all unit
$n$-vectors associated with the parametrically regular distribution
$(N_{1},\dots,N_{n-1})$ with all $N_{j}>2$. $B$ then has
$m=(N_{1}+1)N_{2}\cdots N_{n-1}$ columns; $BB^{T}$ is diagonal, and the
largest diagonal entry (eigenvalue) is $\frac{N_{1}+2}{2}N_{2}\cdots N_{n-1}$.

###### Proof.

The matrices $B$ of the theorem have the form

$\left(\begin{array}[]{ccc}&\sin(\frac{j_{1}\pi}{N_{1}})\sin(\frac{j_{2}2\pi}{N_{2}})\cdots\sin(\frac{j_{n-1}2\pi}{N_{n-1}})&\\\
\cdots&\sin(\frac{j_{1}\pi}{N_{1}})\sin(\frac{j_{2}2\pi}{N_{2}})\cdots\cos(\frac{j_{n-1}2\pi}{N_{n-1}})&\cdots\\\
\cdots&\vdots&\cdots\\\ &\cos(\frac{j_{1}\pi}{N_{1}})&\end{array}\right).$

The entries in the product $BB^{T}$ may be factored into products of sums of
the form

$\begin{array}[]{cccc}\
\sum_{k=1}^{N_{1}}\cos^{2}(\frac{k\pi}{N_{1}})=\frac{N_{1}+2}{2},&\sum_{k=1}^{N_{j}}\sin^{2}(\frac{k2\pi}{N_{j}})=\frac{N_{j}}{2},\\\
\sum_{k=1}^{N_{j}}\cos^{2}(\frac{k2\pi}{N_{j}})=\frac{N_{j}}{2},&\ \ \ \ \ \ \
\sum_{k=1}^{N_{j}}\sin(\frac{k2\pi}{N_{j}})\cos(\frac{k2\pi}{N_{j}})=0,(j=2,\dots,n-1),&\end{array}$

and from this the result follows. ∎

######  Remark 2.

Simulation experiments show that for matrices $B$ comprised of columns of
random unit vectors in $\mathbb{R}^{n}$ that are approximately parametrically
regular in their distribution, the matrix norm (=largest singular value) of
$BB^{T}$ is ${\cal O}(m)$, in agreement with the theorem.

To keep the focus on channel intermittency and pursue the emulation problems
of the following section, we shall assume the parameter $\alpha$ appearing in
Theorem 2 is inversely related to the size, $m$, of the matrix $B$, and for
the case of quantized inputs considered next, other bounds on input magnitude
may apply as well.

We conclude by considering discrete-time descriptions of (1) in which the
control inputs to each channel are selected from finite sets having two or
more elements. To start, suppose (1) is steered by inputs that take continuous
values in $\mathbb{R}^{m}$ but are piecewise constant between uniformly spaced
instants of time: $t_{0}<t_{1}<\cdots;\ t_{k+1}-t_{k}=h$, and $u(t)=u(t_{k})$,
for $t_{k}\leq t<t_{k+1}$. Then the state transition between sampling instants
is given by

$x(t_{k+1})=F_{h}x(t_{k})+\Gamma_{h}u(t_{k})$ (10)

where

$F_{h}=e^{Ah}\ \ {\rm and}\ \ \Gamma_{h}=\int_{0}^{h}e^{A(h-s)}\,ds\cdot B.$

When $B$ (and hence $\Gamma_{h}$) has more columns than rows, there is a
parameter lifting $\hat{\Gamma}:\mathbb{R}^{m\times
n}\rightarrow\mathbb{R}^{n\times n}$, and it is possible to find $\hat{F}$
satisfying $\hat{\Gamma}(\hat{F})=F_{h}$. The relationship between $\hat{B}$
and $\hat{\Gamma}$ and between the lifted parameters $\hat{A}$ and $\hat{F}$
is highly nonlinear, and will not be explored further here. Rather, we
consider first order approximations to $F_{h}$ and $\Gamma_{h}$ and rewrite
(10) as

$\begin{array}[]{ccl}x(t_{k+1})&=&(I+Ah)x(t_{k})+hBu(t_{k})+o(h)\\\\[3.61371pt]
&=&x(t_{k})+h(Ax(t_{k})+Bu(t_{k}))+o(h).\end{array}$

We may use lifted parameters of the first order terms and write the first
order, discrete time approximation to (1) as

$x(k+1)=(I+h\hat{\rm B}(\hat{A}))x(k)+hBu(k).$ (11)

Having approximated the system in this way, we consider the problem of
steering (11) by input vectors $u(k)$ having each entry selected from a finite
set, say $\\{-1,1\\}$ in the simplest case. A variety of MEMS devices that
operate utilizing very large arrays of binary actuators can be modeled in this
way, and successful control designs in adaptive optics applications have been
either open-loop or hybrid open-loop combined with modulation from optical
wavefront sensors ([15],[20]). The approach being pursued in the present work
aims to close feedback loops using real-time measurements of the state. For
any state $x(k)$, binary choices must be made to determine the value of the
input to each of the $m$ channels. Control design is thus equivalent to
implementing a selection function along the lines of [16]. Some details of
research on such designs are described in the next section, but complete
results will appear elsewhere.

##  VI Conclusions and work in progress

It is easy to show that if inputs $u(k)$ to (11) can take a continuum of
values in an open neighborhood of the origin in $\mathbb{R}^{m}$, then
feedback laws based on sample-and-hold versions of (6) can be designed to
asymptotically steer the state of (11) to the origin of the state space
$\mathbb{R}^{n}$. Current work is aimed at extending the ideas we have
described in the above sections of the paper to a theory of feedback control
designs for (11) in which control inputs at each time step are selected from a
finite set ${\cal U}$ of admissible control values. One goal is design of both
selection functions and modulation strategies whereby systems of the form (11)
with finitely quantized feedback can emulate the motions of continuous time
systems like the ones treated in the preceding sections. In the quest to find
control theoretic abstractions of widely studied problems in neuroscience, the
work is organized around three themes: neural coding (characterizing the brain
responses to stimuli by collective electrical activity—spiking and bursting—of
groups of neurons and the relationships among the electrical activities of the
neurons in the group), neural modulation (real-time transformations of neural
circuits induced by various means including chemically or through neural input
from projection neurons) wherein connections within the anatomical connectome
are made to specify the functional brain circuits that give rise to behavior
[23], and neural emulation (finding patterns of neural activity that associate
potential actions to the outcomes those actions are expected to produce,
[21]).

The simplest coding and emulation problems in this context involve finding
state-dependent binary vectors $u[k]\in\\{-1,1\\}^{m}$ that elicit the desired
state transitions of (11). For finite dimensional linear systems, it is
natural to consider using inputs that are both spatially and temporally
discrete to emulate continuous vector fields as follows. Let $H$ be an
$n\times n$ Hurwitz matrix. A discretized Euler approximation to the solution
of $\dot{x}=Hx;\ x(0)=x_{0}$ is

$x(k+1)=(I+hH)x(k);x(0)=x_{0}.$

Two specific problems of emulating this system by (11) with binary inputs are:

  * •

Find a partition of the state space $\\{U_{i}\,:\,\cup\ U_{i}=\mathbb{R}^{n};\
\ U_{i}^{o}\cap U_{j}^{o}=\emptyset;\ U_{i}^{o}={\rm interior}\ U_{i}\\}$ and
a rule for assigning coordinates of $u(k)$ to be either +1 or -1, so that for
each $x\in U_{i}$, $Ax+Bu(k)$ is as close as possible to $Hx$ (in an
appropriate metric).

  * •

Find a partition and a rule that for each $U_{i}$ both assigns $\pm 1$
coordinate entries of $u(k)$ together with a coordinate projection operation
$P(k)$ determining which channels are operative in the partition cell $U_{i}$
with the property the $Ax+BP(k)u(k)$ is as close as possible to $Hx$ (in an
appropriate metric).

Figure 4: (a) Linear vectors fields scale in magnitude with the distance from
the origin. (b) The choices of vector directions for the matrix $B$ with
binary $\pm 1$ inputs. (c) Choices of vector directions as shown in (b) and
specified by $\pm 1$ inputs to the channels of $B$ to provide plausible
approximations to the continuous vector field of (a), and (d) is typical paths
followed by applying the $\pm 1$ inputs as prescribed in (c).

######  Example 4.

Returning to the two-state, three-input examples considered in Section II,
suppose $A=0$ and $B=\left(\begin{array}[]{ccc}0&1&1/\sqrt{2}\\\
1&0&1/\sqrt{2}\end{array}\right)$. We have normalized the third column of $B$
in order that all three channels are equally influential in steering with
binary inputs chosen from $\\{-1,1\\}$. Consider
$H=\left(\begin{array}[]{cc}0&1\\\ -1&-2\end{array}\right)$, a Hurwitz matrix
with both eigenvalues equal to -1. A sketch of the vector field and flow is
shown in Fig. 4(a). It is noted that length scales of linear vector fields
increase in proportion to their distance from the origin. This needs to be
accounted for in interpolations using binary affine maps in a way that may be
similar to the way that neurons in the entorhinal cortex—grid cells in
particular—encode information about length scales. In mammalian brains, grid
cells show stable firing patterns that are correlated with hexagonal grids of
locations that an animal visits as it moves around a range space. Grid cells
appear to be arranged in discrete modules anatomically arranged in order of
increasing scale along the dorso-ventral axis of the medial entorhinal cortex
[26, 22], with each module’s neurons firing at module-specific scales.

The mechanisms of neural coding and modulation are ares of active research in
brain science, and it is in part for this reason that we have avoided being
precise in specifying approximation metrics in this emulation problem. Both
binary codings depicted in Fig. 4 steer the quantized system (11) toward the
origin—but along qualitatively different paths. In observing animal movements
in field and laboratory settings, it is found that they can exhibit a good
deal of individuality in choosing paths from a common start point to a common
goal ([28]). There is also evidence that among regions of the brain that guide
movement, some exhibit neural activity that is more varied than others such as
the grid cell modules in the entorhinal cortex [27], which tend to be similar
in neuroanatomy from one animal to the next. As future work will be focused on
systems with large numbers of simple inputs along with learning strategies, we
will be aiming to understand the emergence of multiple and diverse solutions
that meet similar control objectives.

In treating neuro-inspired approaches to input modulation along the lines
suggested by the second bullet above, we note that with enough control
channels available, it is possible to pursue quantized control input designs
in which a large enough and appropriately chosen group of binary inputs can
simulate certain amplitude-modulated analog signals. How this approach scales
with control task complexity and numbers of neurons needed for satisfactory
execution remains ongoing work.

Finally, we note that our work exploring neuromimetic control has focused on
linear systems only because they are the most widely studied and best
understood. There is no reason that exploration of nonlinear control systems
that model land and air vehicles cannot be approached in ways that are similar
to what we have presented. For such models, connections between task geometry
and time-to-execute can be studied. Deeper connections to neurobiological
aspects of sensing and control in and of themselves will also call for
nonlinear models. This is foreshadowed in the piecewise selection functions
that define the quantized control actions in this section. Connections with
established theories of threshold-linear networks ([29]) and with other models
competitive neural dynamics ([30]) remain under investigation. The
neurobiology literature on the control of physical movement is very rich, and
there appears to be much to explore.

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