Neuromimetic Control — A Linear Model Paradigm

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.

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.

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.

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

This function lifting is depicted graphically by

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

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.

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.

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.

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 $n>2$.

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.

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).

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.