Pinning control
| Mario di Bernardo and Pietro DeLellis (2014), Scholarpedia, 9(8):29958. | doi:10.4249/scholarpedia.29958 | revision #143295 [link to/cite this article] |
Pinning control is a feedback control strategy for synchronization and consensus of complex dynamical networks. Specifically, a virtual leader (the pinner) is added to the network and defines its desired trajectory. The pinner directly controls only a small fraction of the network nodes (the pinned nodes), by exerting a control action that is a function of the pinning error vector, whose $i$-th component is given by the difference between the output of the pinner and the output of the $i$-the node, see the schematic in Figure 1.
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Overview
In the last decades, much research effort has been devoted to collecting data from real networks and characterize their main features. An important area of investigation is the study of emergent properties and, in particular, the emergence of coordinated motion. Indeed, synchronization and consensus are common features in complex networks of dynamical systems and are frequently encountered in nature and technology. In general, the network dynamics are described in terms of those of each node given by: \[\tag{1} \dot{x}_i=f(x_i,t)-c\sum_{j=1}^N \mathcal{L}_{ij}h(x_j), \] where $x_i\in\mathbb{R}^n$ is the state of the $i$-th node, $f:\mathbb{R}^n\times\mathbb{R}^+\rightarrow\mathbb{R}^n$ is the vector field describing the individual dynamics of each node, $\mathcal{L}_{ij}$ is the element $(i,j)$ of the weighted Laplacian matrix $\mathcal{L}$ describing the network topology, $h(x_j):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is the output function of the network nodes, and $c>0$ is the overall coupling strength. When all the nodes' trajectories converge towards a common unknown evolution, we say that the network achieves synchronization, that is, \[ \lim_{t\rightarrow\infty}(x_i(t)-x_j(t))=0,\qquad i,j=1,\ldots,N. \] When all the nodes converge towards a point in the state space, then we say that the network achieves consensus.
Recently, pinning control has been proposed a viable approach for steering the synchronization/consensus dynamics towards a desired common trajectory/point in the state space. The term pinning control was firstly introduced in the field of partial differential equations (PDEs) [Grigoriev, Cross and Schuster, 1997], and was then adopted in complex networks to describe a feedback control action exerted on a fraction of the network nodes [Wang and Chen, 2002]. Namely, pinning control of a complex networks of $N$ nodes can be modeled as \[ \dot{x}_i=f(x_i,t)-c\sum_{j=1}^N \mathcal{L}_{ij}h(x_j)+u_i,\qquad i=1,\ldots,N,\tag{2} \] where $u_i$ is the pinning control input, defined as \[\tag{3} u_i=\delta_i \left(g_i(x_i)-g_i(s)\right),\qquad i=1,\ldots,N. \] where $g_i:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is the pinning function determining the control input of node $i$, for $i=1,\ldots,N$; $\delta_i$ is $1$ if node $i$ is pinned, and $0$ otherwise; $s(t)$ is the desired network trajectory (also called pinner's trajectory ), defined as \[\tag{4} \dot{s}=f(s,t). \] Here, we remark that the number $M=\displaystyle \sum_{i=1}^N \delta_i$ of pinned nodes is typically much smaller than the size of the network, i.e. $M \ll N$.
The aim of pinning control is to make the trajectories of every node of the network converge onto the desired trajectory $s(t)$ described by the pinner. Namely, \[ \lim_{t\rightarrow\infty}(x_i(t)-s(t))=0,\qquad i=1,\ldots,N. \]
Since its proposal in 2002, a wide variety of different approaches to pinning control have been proposed in the literature to tame the behaviour of complex networks of nonlinear systems coupled in a plethora of different ways (e.g. with and without delays, via partial or full state coupling etc). Here we chose to summarise only some of the existing results with a focus on those the authors were directly involved with. The interested reader is referred to the existing literature on synchronisation and control of networks for further approaches and results.
Pinning controllability
Observing equations (2)-(4), it is apparent that, in pinning control design, it is crucial to appropriately tailor the function $g$, select the nodes to be pinned, that is, decide the value of $\delta_i$ for $i=1,\ldots,N$, and select the coupling gain $c$. This problem is known in the literature as pinning controllability.
Local pinning controllability was investigated by Sorrentino et al. (2007), by extending the so-called Master Stability Function (MSF) approach. Basically, the MSF approach is applied to an extended network of $N+1$ dynamical systems $y_i=x_i$ for $i=1,\ldots,N$, and $y_{N+1}=s$. The desired common evolution is given by the evolution of an extra virtual vertex, which is added to the original network. Specifically, if we take $g_i(z)=\kappa_i h(z)$ in (3), with $\kappa_i>0$, the control input becomes $u_i=-\delta_i\kappa_i(h(x_i)-h(s))$. Then, equations (2)-(4) can be rewritten as \[ \dot{y}_i=f(x_i,t)-c\sum_{j=1}^{N+1} \mathcal{M}_{ij}h(x_j),\qquad i=1,\ldots,N+1,\tag{5} \] where $\mathcal{M}=\left\{\mathcal{M}_{ij}\right\}$ is an $(N+1)$-dimensional square matrix with \[{{\mathcal M}_{ij}} = \left[ {\begin{array}{*{20}{c}} {{{\mathcal L}_{11}} + {\delta _1}{\kappa _1}}& \cdots &{{{\mathcal L}_{1N}}}&{ - {\delta _{11}}{\kappa _1}}\\ \vdots & \ddots & \vdots & \vdots \\ {{{\mathcal L}_{N1}}}& \cdots &{{{\mathcal L}_{NN}} + {\delta _N}{\kappa _N}}&{ - {\delta _N}{\kappa _N}}\\ 0& \cdots &0&0 \end{array}} \right].\] Equation (5) can be studied using the classical MSF approach. Once the functional form of $f$ and $h$ are assigned, the local stability of the reference evolution can be investigated in term of the eigenvalues of the matrix $\mathcal{M}$. Specifically, it can be shown that if the eigenvalues lie in a (possibly empty) area of the complex plane, then local stability is guaranteed.
Lyapunov-based approaches were proposed to investigate global pinning controllability in linear pinning control schemes. Specifically, the global stability properties of the error vector $e=\left[e_1^T\ldots e_N^T\right]^T$ were assessed under the hypothesis that the functions $f$ and $h$ fulfill some specific assumptions. For instance, Porfiri and Bernardo (2008) assumed $h(x)=x$ and that $f$ is such that, for any $z_1, z_2\in\mathbb{R}^n$, we can write \[\nonumber f(z_1)-f(z_2)=F_{z_1,z_2}(z_1-z_2), \] where $F_{z_1,z_2}\in\mathbb{R}^{n\times n}$ is bounded, that is, $\left\|F_{z_1,z_2}\right\|\le \alpha$. The case of networks of QUAD dynamical systems (see work by DeLellis, Bernardo and Russo (2011), for details) was investigated by Chen, Liu and Lu (2007), Yu, Chen and Lü (2009), and Song and Cao (2010), instead, where the function $h$ was assumed to be linear, namely, $h(x)=\Gamma x$, where $\Gamma\in\mathbb{R}^{n\times n}$ is called the inner coupling matrix. Global stability results for specific node dynamics were reported by Duan and Chen (2009), while Chen, Liu and Lu (2007) give sufficient conditions for stability when only a single node is pinned.
Adaptive strategies for pinning control
The classical linear pinning control approach presented above requires a centralized knowledge of the network topology to detect whether there are possible coupling and control gains ensuring pinning controllability. Alternative decentralized approaches have been recently proposed and discussed in the literature. Specifically, adaptive strategies have been designed to tailor the coupling and control gains of the network. Also, recent works propose to evolve the network topology to better cope with the cases in which the initial topology makes pinning controllability not feasible with a static approach.
Adaptation of the coupling and control gains
The first adaptive mechanism introduced in the literature on pinning control was the adaptation of the control gains, see Zhou, Lu and Lü (2008) and Wang et al. (2008). Specifically, the control input is defined as \[ u_i=-\delta_i\kappa_i(x_i-s),\nonumber \] where the control gains are adapted as \[\tag{6} \dot{\kappa}_i=q_i\left\|e_i\right\|. \] The global stability of this control scheme was proved under the assumption that the vector field $f$ describing the individual dynamics satisfies the Lipschitz assumption. Later, to enhance the performance of the control strategies, the coupling gains were adapted as well by DeLellis, Bernardo and Turci (2010). Specifically, the off-diagonal elements of the Laplacian matrix are selected as \[ \mathcal{L}_{ij}=-a_{ij}\sigma_{ij}, \] where $a_{ij}$ is the element $(i,j)$ of the adjacency matrix of the network, that is equal to 1 if there is an edge between $i$ and $j$, while it is zero otherwise; the coupling gains $\sigma_{ij}$ defines the mutual coupling strength between each couple $(i,j)$ of nodes, and is adapted as \[\tag{7} \dot{\sigma}_{ij}=\eta_{ij}\left\|e_i-e_j\right\|^2, \qquad\qquad i,j=1,\ldots,N,\:i\ne j. \] This control scheme was shown to be effective in controlling networks of QUAD dynamical systems. Adaptive schemes to compensate for communications delays in the network have been proposed by Wang et al. (2010).
Pinning control via network evolution
The design of adaptive strategies based on the local error between neighboring nodes avoids the need for an offline tuning of the gains based on the knowledge of the network topology. Nonetheless, the selection of the nodes to be pinned is still decided offline. Specifically, $\delta_i$ in (4) is set to one if node $i$ is pinned, and to zero otherwise. A different approach was proposed by DeLellis, Bernardo and Porfiri (2011). The selection of the pinned nodes is not made a priori, but the link activation is related to the network evolution. Specifically, $\delta_i$ is not fixed and binary, and its evolution is described by \[ \delta_i=b_i^2,\nonumber \] where \[\tag{8} \ddot{b}_i+\zeta\dot{b}_i+\frac{\mathrm{d}}{\mathrm{d} b_i}U(b_i)=g(\left\| e_i\right\|), \quad i=1,\ldots,N. \] Notice that each $b_i$ updates according to the edge-snapping mechanism. Namely, its dynamics are those of a unitary mass in a double-well potential $U$ subject to an external force $g$ which is function of the pinning error $e_i$ and a linear damping described by $\zeta$.
A manageable selection for the potential is the following: \[\tag{9} U(z)=kz^2(z-1)^2, \] where $k$ is a parameter defining the height of the barrier between the two wells. With this simple choice of the potential, the dynamical system (8) has only two stable equilibria, namely, $0$ and $1$. These states correspond to the node $i$ being either pinned or not, respectively.
The steady-state selection of the pinned sites, that is, the asymptotic values of $\delta_i$, $i=1,\ldots,N$, is highly dependent on the initial state of the networked dynamical systems. Such initial conditions may indeed lead to a broad range of emerging pinning configurations, spanning from networks where every possible node is pinned to networks where pinning control is only applied to a single node. Therefore, a variety of coexisting equilibrium configurations are possible. DeLellis, Bernardo and Porfiri (2011) give conditions for the local stability of these configurations. Furthermore, sufficient conditions for network (2) to be controlled to the desired trajectory and for the edge snapping mechanism (8) to drive the network to a steady-state pinning configuration are also given. The global stability of the error dynamics is addressed by assuming the vector fields of the agents are QUAD.
To take advantage of the possible evolution of the network topology DeLellis, Bernardo and Porfiri (2011) used the edge snapping mechanism to control the network topology, so that each couple of controlled nodes can mutually negotiate the activation of the corresponding link. Namely, each off-diagonal element of the Laplacian matrix of the network is set as \[ \mathcal{L}_{ij}=-\sigma_{ij}\alpha_{ij}^2, \qquad i,j=1,\ldots,N,\:i\ne j,\tag{10} \] where $\sigma_{ij}$ is the mutual coupling strength between each couple $(i,j)$ of nodes, that is adapted as in (7), and where the weight $\alpha_{ij}$ is associated to every undirected edge of the target pinning edge and is adapted using the following second order dynamics \[ \dot{\alpha}_{ij}+\nu \dot{\alpha}_{ij}+\frac{\mathrm{d}}{\mathrm{d}\alpha_{ij}}U(\alpha_{ij})=c(\Vert e_{ij}\Vert),\quad i,j=1,\ldots,N,\:i\ne j,\tag{11} \] where $e_{ij}=e_j-e_i$, and the potential is the same as in (9). The target network topology evolves in a decentralized way; the local mismatch of the trajectories acts as an external forcing on the edge dynamics, inducing the activation of the corresponding link.
Examples
Decentralized adaptive pinning control
Here, we illustrate the decentralized pinning control strategy firstly proposed by DeLellis, Bernardo and Turci (2010). Adopting this scheme, both the control and coupling gains are independently and adaptively estimated on-line. Specifically, we show the effectiveness of this approach in controlling a complex network of 30 Chua's circuits coupled through an Erdös-Rènyi random network. The individual dynamics of the Chua's circuits are described by \begin{align*}\tag{12} \dot{x}_{i1}&=\upsilon_1\left(-x_{i1}+x_{i2}-g(x_{i1})\right),\\ \dot{x}_{i2}&=x_{i1}-x_{i2}+x_{i3},\\ \dot{x}_{i3}&=-\upsilon_2 x_{i2}, \end{align*} where $g(x_{i1})$ is the piecewise linear characteristic of the Chua's diode, described by \[\tag{13} g(x_{i1})=\beta_1 x_{i1}+\frac{1}{2}(\beta_2-\beta_1)(|x_{i1}+1|-|x_{i1}-1|). \]
In this example, we consider the case in which only one node is directly pinned. Moreover, the initial conditions are taken randomly from a uniform distribution between zero and ten. As illustrated in Figures 2 and 3, the coupling and control gains are adapted so that the network nodes converge towards the pinner's trajectory, as clearly depicted in Figure 4. Differently from the classical approaches, in which the coupling and control gains need to be decided offline, here they are evolved online on the basis of the local error between neighboring nodes, without requiring a global knowledge of the network topology.
Pinning edge snapping
Here, we illustrate the fully decentralized snapping control presented by DeLellis, Bernardo and Porfiri (2011), where the coupling and control gains are adapted as in equations (6) and (7), and the network and control edges are activated according to the edge snapping mechanisms described in (8) and (11) . Specifically, we refer to a network of 30 Lorenz oscillators, whose individual dynamics are described by \begin{align*} \dot x_{i1}&=a_1(x_{i2}-x_{i1}),\\ \dot x_{i2}&=a_2x_{i1}-x_{i2}-x_{i1}x_{i3},\\ \dot x_{i3}&=x_{i1}x_{i2}-a_3x_{i3}, \end{align*} for $i=1,\ldots,30$, where $a_1=10$, $a_2=28$, and $a_3=8/3$. The initial conditions on the target network nodes are taken from a uniform distribution between $0.5$ and $3$, while the initial conditions for the pinner are $x_s(0)=[0,1,2]^T$. Moreover, $\sigma_{ij}(0)$ with $(i,j)\in\mathcal E$ and $q_i(0)$ with $i=1,\ldots,N$ are selected to be zero.
In this example Figure 6 demonstrates that all the coupling and control gains converge to steady-state values. Figure 7 illustrates the adaptive selection of the network edges and the pinned nodes.
References
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