A nullcline-based control strategy for PWL-shaped oscillators

Abstract

Starting from the piecewise-linear framework applied to networked FitzHugh-Nagumo oscillators, this paper aims at presenting a control strategy relying on phase portrait reshaping through the manipulation of their nullclines in order to fulfil both phase and time requirements. This has been achieved by relating the slopes of piecewise linearly approximated model’s nullclines in its limit cycle to its oscillation period. Additionally, the targeted issue addressed by this paper has been assessed by combining the former framework with an event-driven control strategy aimed at reducing its effects to specific time instants instead of continuous, instantaneous phases, which is much more computationally expensive. The strategy is therefore motivated by its simplicity and is supported by key applications to bioinspired locomotion control in legged robots, which suggest how a dynamics-preserving approximation of the phase portrait combined with a sampled control action can produce pre-defined phase topologies in directed, non-diffusive tree graphs.

Appendices

Appendix

P-FHNs and stability issues

Because of the nature of P-FHNs, we want to prove that asymptotic stability of this kind of systems does not depend on the particular fixed point as in the case of linear systems, but before doing this, we have to prove the following statement.

Lemma 1

Given a PWL function as described by Eq. 7, let $s_j\doteq \text {sgn}\left( x-E_j\right)$, $\bar{{\mathbf {m}}}\doteq \left[ m_1,\ldots ,m_{n-1}\right]$ and $\bar{{\mathbf {s}}}\doteq \begin{bmatrix} s_1-s_2\\ \vdots \\ s_{n-1}-s_{n} \end{bmatrix}$, then its derivative with respect to x is:

$$\begin{aligned} \frac{\text {d}\varPi (x)}{\text {d}x}=0.5m_0(1-s_1)+0.5m_n(1+s_n)+0.5\bar{{\mathbf {m}}}\bar{{\mathbf {s}}}\nonumber \\ \end{aligned}$$

(26)

Proof

Every PWL function in the form of Eq. 7 admits the following derivative:

$$\begin{aligned} \frac{\text {d}\varPi (x)}{\text {d}x}=a_1+\sum _{j=1}^{n}b_js_j \end{aligned}$$

If ${\mathbf {s}}\doteq \begin{bmatrix} s_1\\ \vdots \\ s_{n} \end{bmatrix}$ and ${\mathbf {m}}\doteq \left[ m_0,\ldots ,m_n\right]$, then the summation can be rewritten as follows:

$$\begin{aligned} \begin{aligned} \sum _{j=1}^{n}b_js_j =&\, 0.5\sum _{j=1}^{n}(m_j-m_{j-1})s_j \\ =&\, 0.5\sum _{j=1}^{n}m_js_j-0.5\sum _{j=1}^{n}m_{j-1}s_j \\ =&\, 0.5\left[ \bar{{\mathbf {m}}}, m_n\right] {\mathbf {s}} - 0.5\left[ m_0,\bar{{\mathbf {m}}}\right] {\mathbf {s}} \end{aligned} \end{aligned}$$

because every $b_j$ depends on both $m_j$ and $m_{j-1}$. Thus:

$$\begin{aligned} \begin{aligned} \frac{\text {d}\varPi (x)}{\text {d}x} =&\, 0.5m_0+0.5m_n+0.5\left[ \bar{{\mathbf {m}}}, m_n\right] {\mathbf {s}}\\&-0.5\left[ m_0,\bar{{\mathbf {m}}}\right] {\mathbf {s}} \\ =&\, 0.5m_0+0.5m_n+0.5\bar{{\mathbf {m}}}\left[ s_1,\ldots ,s_{n-1}\right] ^{\mathrm{T}} \\&+ 0.5m_ns_n-0.5m_0s_1-0.5\bar{{\mathbf {m}}}\left[ s_2,\ldots ,s_n\right] ^{\mathrm{T}} \end{aligned} \end{aligned}$$

By grouping together all the common terms:

$$\begin{aligned} \frac{\text {d}\varPi (x)}{\text {d}x}=0.5m_0(1-s_1)+0.5m_n(1+s_n)+0.5\bar{{\mathbf {m}}}\bar{{\mathbf {s}}} \end{aligned}$$

$\square$

This lemma is valid $\forall E_j$, therefore it still holds if $E_{j-1}<E_j<E_{j+1}$. However, if the former ordering criterion holds too, the aforementioned expression given by the previous lemma simply states that the derivative of the PWL function is given by the slope of the branch where $x_1$ is located in. For example, if $s_1=-1$ (therefore, $s_j=-1$, with $j\ge 2$) then the selected branch is the leftmost one, thus the derivative is equal to $m_0$. Similarly, if $s_n=1$ (therefore, $s_j=1$, with $j<n$) then the selected branch is the rightmost one, thus the derivative is equal to $m_n$. In all the other cases, $\exists !j:s_j=-s_{j+1}$: in this region, the derivative is equal to $m_j$. We can summarise these statements as follows:

Corollary 1

If the hypotheses of Lemma 1 hold, then:

$$\begin{aligned} \frac{\text {d}\varPi (x_1)}{\text {d}x_1}= \left\{ \begin{array}{ll} m_0 &{} x_1\in \left( -\infty ,E_1\right) \\ m_j &{} x_1\in \left[ E_j,E_{j+1}\right) \\ m_n &{} x_1\in \left[ E_n,+\infty \right) \end{array} \right. \end{aligned}$$

(27)

Now, we are ready to enunciate a simple stability condition which employs the previous results:

Lemma 2

(Stability of fixed points) Given the system 8 with n break-points, asymptotic stability within the $j+1$th sub-region is achieved iff $b>\frac{m_j}{\varepsilon }$ and $bm_j<1, j\in \left\{ 0,1,\ldots ,n\right\}$.

Proof

The Jacobian matrix of such systems is given by:

$$\begin{aligned} {\mathbf {J}}= \begin{bmatrix} \frac{1}{\varepsilon }\frac{d\varPi (x_1)}{dx_1}&\quad -\,\frac{1}{\varepsilon } \\ 1&\quad -\,b \end{bmatrix} \end{aligned}$$

(28)

where $\frac{\text {d}\varPi (x_1)}{\text {d}x_1}$ is given by Eq. 27. Due to the nature of the system, the characteristic polynomial is simply given by:

$$\begin{aligned} p(\lambda )=\lambda ^2-\text {Tr}({\mathbf {J}})\lambda +\text {det}({\mathbf {J}}) \end{aligned}$$

(29)

By applying the Descartes’ rule of signs, we can infer that:

$$\begin{aligned} \text {Spec}({\mathbf {J}})\subseteq \text {LHS}\Leftrightarrow \left\{ \begin{array}{ll} b> \frac{1}{\varepsilon }\frac{d\varPi (x_1)}{dx_1}=\frac{m_j}{\varepsilon } \\ 1 > b\frac{d\varPi (x_1)}{dx_1}=bm_j \end{array} \right. \end{aligned}$$

(30)

where $\text {LHS}\doteq \left\{ \mu \in {\mathbb {C}}|\text {Re}\left\{ \mu \right\} <0\right\}$ stands for Left-Half Semiplane of the Argand–Gauss plane and $0\le j\le n$. $\square$

To apply the previous lemma, it is sufficient to check if Condition 30 is verified within the region of interest, according to the desired branch of the phase portrait. It derives that P-FHNs constitute a simpler case study with $n=2$ and therefore we have only 3 slopes. Thus, if these slopes were chosen according to Table 2, then the central region of the phase portrait would be always unstable.

Real and virtual equilibrium points in P-FHNs

The calculation of fixed points of a P-FHN consists on finding those points which nullify both the nullclines. In order to calculate P-FHN’s fixed points, we can compute the intersection points between the $x_2$-nullcline and each straight line the phase portrait is made of. As stated by Eq. 18, we can write the generic line with pre-defined slope $m\in \left\{ m_0,m_1,m_2\right\}$ and selected break-point $E\in \left\{ E_1,E_2\right\}$ alternatively as $x_2=mx_1+(m_1-m)E+a_0$. Thus, the intersection between the former line and the $x_2$-nullcline gives equilibrium points in the form $\left[ x_1^{\mathrm{(eq)}},\frac{1}{b}x_1^{\mathrm{(eq)}}+\frac{a}{b}\right]$, where:

$$\begin{aligned} x_1^{\mathrm{(eq)}}=\frac{(m_1-m)bE+ba_0-a}{1-bm} \end{aligned}$$

(31)

Equation 31 gives the abscissa of several fixed points, depending on the number of slopes. However, some of them may not belong to the PWL function. To make it clearer, suppose a P-FHN system with $m_0=-m_1=m_2=-1$, $E_1=-E_2=-1$, $a=a_0=0$ and $b=0.5$. Thus, fixed points are $\left[ 0,0\right]$, $\left[ -2/3,-4/3\right]$ and $\left[ 2/3,4/3\right]$. However, the last two points do not belong to the N-shaped nullcline, even though they are drawn from the computation we have introduced before. This can be easily proven by calculating $\varPi (\pm 2/3)=\pm 2/3\ne \pm 4/3$, according to Eq. 5. Despite they can be considered fixed points, they are virtual fixed points, whilst the first one is effectively a real fixed point.

The presence of virtual fixed points can be expressed differently as follows. The PWL framework implies a state space fragmentation where each single “slice” is adequately characterised, as stated by Lemma 2. However, the former lemma does not consider the existence of fixed points belonging to regions they are not assigned to. This fact has a great relevance when demonstrating the emergence of limit cycles. Thereby, what follows is an immediate consequence of the previous results which guarantee the existence of limit cycles in P-FHNs:

Lemma 3

(Stable limit cycles) Given a P-FHN system as stated by Eq. 8, if Eq. 7 are verified and the fixed points associated to the most external slopes of its phase portrait are virtual and asymptotically stable, then the system admits a limit cycle.

Proof

If Eq. 7 are satisfied, then Lemma 2 assures the existence of two asymptotically stable sub-regions (i.e., the most external ones) and one unstable sub-region between them. For the sake of simplicity, we refer to them as ${\mathcal {R}}^{\mathrm{Left}}$, ${\mathcal {R}}^{\mathrm{Centre}}$ and ${\mathcal {R}}^{\mathrm{Right}}$. On account of the aforementioned results, each region comprises a fixed point, which could be either real or virtual. We refer to these solutions as ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Left}}$, ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Centre}}$ and ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Right}}$. Let $\varsigma ({\mathbf {x}})$ be the trajectory of a P-FHN starting from ${\mathbf {x}}$. Because of the stable and virtual nature of both ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Left}}$ and ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Right}}$, $\forall \bar{{\mathbf {x}}}\in \left\{ {\mathcal {R}}^{\mathrm{Left}},{\mathcal {R}}^{\mathrm{Right}}\right\}$$\varsigma (\bar{{\mathbf {x}}})$ will start to converge to virtual points, which are fixed solutions placed in a region they do not actually belong to. In fact, ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Left}}\notin {\mathcal {R}}^{\mathrm{Left}}$ and ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Right}}\notin {\mathcal {R}}^{\mathrm{Right}}$ by definition. Conversely, $\forall \bar{{\mathbf {x}}}\in {\mathcal {R}}^{\mathrm{Centre}}$ the trajectory will escape from ${\mathbf {x}}^{\mathrm{(eq)}}_{\mathrm{Centre}}$ naturally, reaching either ${\mathcal {R}}^{\mathrm{Left}}$ or ${\mathcal {R}}^{\mathrm{Right}}$. Once the trajectory has reached a stable region having a virtual fixed point, it will be directed towards it, but having placed it within a different region the trajectory will not be able to reach it. Thus, this moving trajectory will create a limit cycle due to the unreachability of its stable fixed points. That having been said, we have to prove that if $x_1\ll E_1$ ($x_1\gg E_2$) the system reduces to a stable one. This is true because in these cases it results taht $\varPi (x_1)\approx a_1x_1$, where $a_1<0$ by construction. Therefore, the asymptotic state matrix is given by:

$$\begin{aligned} {\mathbf {A}}_\infty =\begin{bmatrix} \frac{a_1}{\varepsilon }&\quad -\,\frac{1}{\varepsilon } \\ 1&\quad -\,b \end{bmatrix} \end{aligned}$$

which satisfies Lemma 2, thus it is a Hurwitz matrix and therefore the Poincaré–Bendixson theorem [22] about stability of limit cycles in continuous dynamical systems is verified. $\square$

Lemma 3 has an important consequence for further applications and considerations. In fact, it is possible to switch from an oscillatory regime to a steady stable by turning a virtual but stable fixed point into a real one.

P-FHNs as switching systems

When $\varepsilon \ll 1$, systems in a PWL framework can be considered examples of switching systems, whose dynamics can be linearised through the employment of switching functions which allow to change among diverse working conditions, or modalities.

Let ${\mathbb {M}}\doteq \left\{ M_1,M_2,M_3\right\}$ be the set of working modalities; each modality determines a proper linearised version of the original system through some matrices ${\mathbf {A}}(\text {M})$ and ${\mathbf {B}}(\text {M})$, with $M\in {\mathbb {M}}$, such that the overall dynamics can be expressed as

$$\begin{aligned} \dot{{\mathbf {x}}}={\mathbf {A}}(\text {M}){\mathbf {x}}+ {\mathbf {B}}(\text {M}) \end{aligned}$$

(32)

In particular, $\exists {\mathcal {S}}:\varOmega \rightarrow {\mathbb {M}}$ which gives the next operating modality according to the current values of the state variables. For P-FHNs, this switching function is given by:

$$\begin{aligned} {\mathcal {S}}(x_1)= \left\{ \begin{array}{ll} \text {M}_1 &{} x_1\in \left( -\infty ,E_1\right) \\ \text {M}_2 &{} x_1\in \left( E_1,E_2\right) \\ \text {M}_3 &{} x_1\in \left( E_2,+\infty \right) \end{array}\right. \end{aligned}$$

(33)

which in turns give:

$$\begin{aligned} \begin{array}{lll} \text {M}_j &{} \mapsto &{} m_{j-1}\\ {\mathbf {A}}({\mathrm{M}}_j) &{} \mapsto &{} {\mathbf {J}} \end{array} \end{aligned}$$

(34)

where $j\in \left\{ 1,2,3\right\}$, ${\mathbf {J}}$ is the Jacobian matrix. This notation is perfectly coherent to the works in [66] about the convergence of hybrid systems to limit cycles. We have reported Fig. 12 to show how modalities are linked together and how a P-FHN switches among them, according to Table 9.

Fig. 12

Allowed modalities P-FHNs can switch among, depending on the position of $x_1$ over the $x_1$-$x_2$ plane. This FSM is founded on the legend shown in Table 9

Table 9 Legend that explains the FSM depicted in Fig. 12