# Distributed Control of Multiple Supercapacitors via Static Feedback

Authors: Zhikun Luo; Zhifeng Sun; Junmin Peng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

During recent years, many important distributed control approaches have been proposed to realize the stabilization of networked system including static feedback, which has been proven to be an efficient control strategy. Based on its traditional design method, for first- and second-order networked dynamical systems, we propose another kind of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain and a sum of nonlinear interaction functions so that all states will reach consensus at a value preset. By using integral Lyapunov functions, we can prove the stability of proposed control protocols. We also choose supercapacitors as simulation models to realize voltage equalization under our distributed control protocols.


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## Body

## 1. Introduction

In many large scale systems, decentralized control is the only feasible control strategy when sensing and actuation communication are limited. Recently, many important distributed control strategies have been proposed which are useful in dealing with stabilization of networked systems. For example, a cooperative distributed control paradigm was proposed for AC microgrids in [1]. In [2], an event-triggered communication based on distributed control scheme was proposed to help achieve stated goals of proportional current sharing along with average DC voltage regulation in a DC microgrid. In [3], authors claimed an efficient distributed control architecture for MMC system. Furthermore, distributed control for LPV systems over arbitrary graphs with communication latency was proposed in [4].To realize distributed control, many different design methods have been proposed by researchers like static feedback, integral action, and so on. There is no doubt that static feedback controllers have been widely used in a variety of distributed control problems. For example, they are commonly used in the consensus problem which has applications in supercapacitors [5], flocking [6], etc. Generally, for basic consensus problems, static controllers with linear form would be enough. As for some systems with inherent nonlinear dynamics, nonlinear consensus protocols seem to be necessary. Recently, many researchers have contributed a lot in this area, like [7], where authors introduced some perfect nonlinear consensus protocols for single and double-integrator dynamics. In [8], a nonlinear static state feedback control law via polynomial approach was proposed for stabilization and global asymptotic stabilization problems of saturated linear systems. In [9], implicit integrators coupled to a nonlinear static feedback were claimed for linear dynamics. In [10], distributed consensus tracking is addressed and well solved for multiagent systems with Lipschitz-type node dynamics.The distributed control algorithms mentioned in [7] are useful in dealing with consensus problems of first- and second-order networked dynamical systems. However, they cannot satisfy all kinds of requirements. For example, what if we want to set the consensus value of states as what we expect? Consider that this requirement is common and important in engineering design. However, the controller according to this theory cannot satisfy our requirements since the consensus value depends on initial values of states. Motivated by this, we intend to give out new distributed control protocols so that the nonlinear static feedback controller will be improved to meet more design requirements.The remainder of this paper is organized as follows. In Section2, the problem formulation is proposed. Then, in Section 3, distributed controllers are designed both for first-order and second-order systems. In Section 4, the controllers proposed above are applied to voltage equalization of multiple supercapacitors. Finally, conclusion and future work are given in Section 5.


## 2. Problem Formulation

In this section we introduce a unified mathematical notation and formalize our distributed control problems.


### 2.1. Notation

LetG be a graph. We define G=(ν,ε) where ν={1,…,n} is vertex set and ε={1,…,m} is edge set of graph G. Select some nodes as leaders which belong to vertex set νmajor and the rest of the nodes as slave nodes which belong to vertex set νslave. It is clear that νmajor∪νslave=ν. Let Ni informs the set of neighboring nodes of node i as Ni={j∣j∈ν,(i,j)∈ε}. We require that the undirected graphG is connected which means ∀i,j∈ν where there exists a path from node i to node j as {(i,k),(k,l),…,(m,j)}. Denote by B∈Rn×m the vertex-edge adjacency matrix of G and let L be its Laplacian matrix. For a undirected graph we have L=BBT. Define the scalar position of agent i as xi and its velocity as vi. We collect them into column vectors x=[x1,…,xn]T, v=[v1,…,vn]T. In denotes the identity matrix of n.


### 2.2. System Model

In this paper, we only consider agents endowed with either single-integrator dynamics as(1)x˙i=uior double-integrator dynamics as(2)x˙i=viv˙i=ui


### 2.3. Objective

The main objective of this paper is for agents to converge to a common state, i.e.,limt→∞xi(t)=x∗,∀i∈ν for single-integrator dynamics and limt→∞vi(t)=v∗,∀i∈ν for double-integrator dynamics. Theoretically consensus states x∗∈R and v∗∈R can be desired values set by us. Furthermore, for double-integrator dynamics, we also decide to realize limt→∞xi(t)=x∗,∀i∈ν where x∗∈R can be a desired value set by us.


## 2.1. Notation

LetG be a graph. We define G=(ν,ε) where ν={1,…,n} is vertex set and ε={1,…,m} is edge set of graph G. Select some nodes as leaders which belong to vertex set νmajor and the rest of the nodes as slave nodes which belong to vertex set νslave. It is clear that νmajor∪νslave=ν. Let Ni informs the set of neighboring nodes of node i as Ni={j∣j∈ν,(i,j)∈ε}. We require that the undirected graphG is connected which means ∀i,j∈ν where there exists a path from node i to node j as {(i,k),(k,l),…,(m,j)}. Denote by B∈Rn×m the vertex-edge adjacency matrix of G and let L be its Laplacian matrix. For a undirected graph we have L=BBT. Define the scalar position of agent i as xi and its velocity as vi. We collect them into column vectors x=[x1,…,xn]T, v=[v1,…,vn]T. In denotes the identity matrix of n.


## 2.2. System Model

In this paper, we only consider agents endowed with either single-integrator dynamics as(1)x˙i=uior double-integrator dynamics as(2)x˙i=viv˙i=ui


## 2.3. Objective

The main objective of this paper is for agents to converge to a common state, i.e.,limt→∞xi(t)=x∗,∀i∈ν for single-integrator dynamics and limt→∞vi(t)=v∗,∀i∈ν for double-integrator dynamics. Theoretically consensus states x∗∈R and v∗∈R can be desired values set by us. Furthermore, for double-integrator dynamics, we also decide to realize limt→∞xi(t)=x∗,∀i∈ν where x∗∈R can be a desired value set by us.


## 3. Distributed Control with Static Nonlinear Feedback

In this section, we define a class of nonlinear consensus algorithms where the input of each agent can be decoupled into a product of a nonlinear gain function and a nonlinear interaction function to solve consensus problems. In Section3.1 we first study consensus for single-integrator dynamics by nonlinear protocols. In Section 3.2 we consider a nonlinear consensus protocol for agents with double-integrator dynamics. Finally, in Section 3.3, we give a nonlinear consensus protocol with nonlinear, state-dependent damping for agents with double-integrator dynamics.


### 3.1. Consensus for Single-Integrator Dynamics

Considering agents with dynamics (1), for the major node i, i∈νmajor, we define ui as what is given by(3)ui=-μixi∑j∈Niαijxi-xj+γixi-x∗where x∗∈R is an arbitrary value given by us. For the slave node i, i∈νslave, we define ui as what is given by(4)ui=-μixi∑j∈Niαijxi-xjAssumption 1.
The nonlinear gainμi is continuous and 0<μmin≤μi(x)≤μmax,∀i∈ν,∀x∈R.Assumption 2.
The interaction functionαij(·) is continuous ∀(i,j)∈ε, and(1)
αij(-y)=-αij(y),∀(i,j)∈ε,∀y∈R.(2)
αij(-y)=-αji(y),∀(i,j)∈ε,∀y∈R.(3)
αij(y)>0,∀(i,j)∈ε,∀y>0.Assumption 3.
The functionγi(·) is continuous ∀i∈νmajor, and(1)
γi(-y)=-γi(y),∀i∈νmajor,∀y∈R.(2)
γi(y)>0,∀i∈νmajor,∀y>0.Remark 4.
Assumptions2 and 3 guarantee that leader agents move in the direction of both their neighbors and a given value x∗ while other agents just move in the direction of their neighbors. A consequence of Assumption 2 is αij(0)=0,∀(i,j)∈ε, and a consequence of Assumption 3 is γi(0)=0,∀i∈νmajor.Now we state the main results of this section.Theorem 5.
Givenn agents with dynamics (1) and ui is given by (3) and (4), where μi(·) satisfies Assumption 1 for all i∈ν, αij(·) satisfies Assumption 2 for (i,j)∈ε, and γi(·) satisfies Assumption 3 for all i∈νmajor, then agents converge asymptotically to an agreement point as limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us under any initial condition x(0).Proof.
The proof is Lyapunov based. Consider the following candidate Lyapunov function:(5)Vx=∑i∈ν∫x∗xiy-x∗μiydy
It can be easily verified thatV(x∗1n×1)=0. Note that V(x)>0 for x≠x∗1n×1. To prove it, first we consider the case when xi>x∗; let z=y-x∗; then ∫x∗xi‍y-x∗/μi(y)dy=∫0xi-x∗z/μi(z+x∗)dz>0. The case when xi<x∗ is treated analogously. Let z=x∗-y; then ∫x∗xiy-x∗/μi(y)dy=∫0x∗-xiz/μi(x∗-z)dz>0. Thus we have V(x)>0, if and only if x=x∗1n×1 and then V(x)=0. Now we consider V˙(x).(6)V˙x=-∑i∈νmajorxi-x∗μixiμixi∑j∈Niαijxi-xj+γixi-x∗-∑i∈νslavexi-x∗μixiμixi∑j∈Niαijxi-xj=-∑i∈νxi-x∗∑j∈Niαijxi-xj-∑i∈νmajorxi-x∗γixi-x∗=-∑i∈νxi∑j∈Niαijxi-xj+∑i∈νx∗∑j∈Niαijxi-xj-∑i∈νmajorxi-x∗γixi-x∗
Due to Assumption2, the first term of (6) can be written as ∑i∈νxi∑j∈Ni‍αij(xi-xj)=1/2∑i∈ν‍∑j∈Ni‍(xi-xj)αij(xi-xj)≥0. Well according to Assumption 2 it can be shown that ∑i∈ν‍x∗∑j∈Ni‍αij(xi-xj)=0 and clearly ∑i∈νmajor‍(xi-x∗)γi(xi-x∗)≥0 due to Assumption 3. Thus V˙(x) can be written as V˙(x)=-∑i∈ν‍xi∑j∈Ni‍αij(xi-xj)-∑i∈νmajor‍(xi-x∗)γi(xi-x∗)<0, unless xi=xj,∀(i,j)∈ε and xi=x∗,∀i∈νmajor, and then V˙(x)=0. Note that G is a connected graph; thus xi=xj=x∗,∀i,j∈ν. Therefore agents can converge to xi=x∗,∀i∈ν when t→∞ where x∗∈R is an arbitrary value and the proof is completed.


### 3.2. Consensus for Double-Integrator Dynamics

Considering agents with double-integrator dynamics (2), for the major node i, i∈νmajor, we define ui as what is given by(7)ui=-μivi∑j∈Niαijxi-xj+βijvi-vj+γivi-v∗where v∗∈R is an arbitrary value given by us. And for the slave node i, i∈νslave, ui is given by(8)ui=-μivi∑j∈Niαijxi-xj+βijvi-vjBy using an integral Lyapunov function, we are thus able to prove that agents reach consensus under the nonlinear consensus protocol also in case of double-integrator dynamics.Theorem 6.
Givenn agents with dynamics (2) and ui is given by (7) and (8), where μi(·) satisfies Assumption 1 for all i∈ν, αij(·) and βij(·) satisfy Assumption 2 for all (i,j)∈ε and γi(·) satisfies Assumption 3 for all i∈νmajor, then agents asymptotically achieve consensus, i.e., |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞ under any initial conditions (x(0),v(0)). What’s more, the velocities converge to limt→∞vi(t)=v∗,∀i∈ν where v∗∈R is an arbitrary value given by us.Proof.
This proof relies on Lyapunov techniques. We assume there existsξ,1≤ξ≤n nodes in νmajor. First we define v-0∈Rn×1, where the ith element of v-0 is vi-v∗ if i∈νmajor and 0 otherwise. v-=[v1-v∗,…,vn-v∗]T, x-=BTx, and Γv=diag([μ1(v1),…,μn(vn)]). Now we combine (7) and (8) in vector form as:(9)x˙=vv˙=-ΓvBαx-+BβBTv+γv-0where α(·),β(·) and γ(·) are componentwise.
Now we define another Lyapunov function as(10)Vx-,v=∑i∈ν∫v∗viy-x∗μiydy+∑i,j∈ε∫0x-ijαijydywhere x-ij=xi-xj is an element in x-.
It is clear thatV([01×m,v∗11×n]T)=0. Now we need to demonstrate V(x-,v)>0 for x-≠0m×1,v≠v∗1n×1. To prove this, first we consider the situation when v≠v∗1n×1, and thus ∑i∈ν‍∫x∗xi‍y-x∗/μi(y)dy>0 as we have proven in Theorem 5. As for ∫0x-ij‍αij(y)dy, if x-ij>0, obviously ∫0x-ij‍αij(y)dy>0. And if x-ij<0, while ∫0x-ij‍αij(y)dy=∫x-ij0‍αij(-y)dy>0, then for x-≠0m×1,∑(i,j)∈ε‍∫0x-ij‍αij(y)dy>0. We can conclude that V(x-,v)>0 unless x-=0m×1,v=v∗1n×1; then V(x-,v)=0. Differentiating V(x-,v) with respect to time yields(11)V˙x-,v=∂V∂vdvdt+∂V∂x-dx-dt=∑i∈νvi-v∗μiviui+∑i∈ν∑j∈Niαijxi-xjvi=-v-TBαx-+BβBTv+γv-0+vTBαx-=-vTBβBTv-v-0Tγv-0≤0
Due to Assumptions2 and 3, we can conclude that if and only if BTv=0,v-0=0 then V˙(x-,v)=0. Since graph G is connected, BTv=0,v-0=0 equates to v=v∗1n×1. Thus V˙(x-,v)=0 is given by T1={(x-,v)∣v=v∗1n×1}. Hence, v˙=0 and it can be shown that x-=0 in T1. To testify it, we assume x-≠0 in T1. Then we can define xi-=minxj,j∈ν which means ∃κ∈Ni-:xκ>xi-. Thus(12)v˙i-=-μi-vi-∑j∈Ni-αi-jxi--xj+βi-jvi--vj+γi-vi--v∗=-μi-vi-∑j∈Ni-αi-jxi--xj>-μi-vi-αi-κxi--xκ>0
The result contrasts with our conclusionv˙=0. Thus T1 can be rewritten as T1={(x-,v)∣v=v∗1n×1,x-=0} which implies consensus can be accomplished as |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞ and limt→∞vi(t)=v∗,∀i∈ν where v∗∈R is an arbitrary value given by us. So proof is completed.


### 3.3. Consensus for Double-Integrator Dynamics with State-Dependent Damping

Considering agents with double-integrator dynamics (2), for the major node i, i∈νmajor, ui is given by(13)ui=-σixivi-∑j∈Niαijxi-xj-γixi-x∗and for the slave node i, i∈νslave, ui is given by(14)ui=-σixivi-∑j∈Niαijxi-xjWith this framework, we are able to generalize the simple consensus to a much broader class of consensus functions.Theorem 7.
Consider agents with dynamics (2) and ui is given by (13) and (14), where σi(·) satisfies Assumption 1 for all i∈ν, αij(·) satisfies Assumption 2 for all (i,j)∈ε, and γi(·) satisfies Assumption 3 for all i∈νmajor. Then agents asymptotically achieve consensus, i.e., |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞. What’s more, the scalar positions converge to limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us. And velocities converge to limt→∞vi(t)=0,∀i∈ν.Proof.
The proof is also Lyapunov based. In this section we define the candidate Lyapunov function as:(15)Vx,v=∑i∈νvi22+∑i,j∈ε∫0xi-xjαijydy+∑i∈νmajor∫x∗xiγiy-x∗dy
First, let us prove∫x∗xi‍γi(y-x∗)dy>0 for xi≠x∗,i∈νmajor. To show this, if xi>x∗, obviously ∫x∗xi‍γi(y-x∗)dy>0, and if xi<x∗, ∫x∗xi‍γi(y-x∗)dy=∫xix∗‍γi(x∗-y)dy>0. Therefore V(x,v)>0 unless v=0, xi=xj,∀(i,j)∈ε and xi=x∗,∀i∈νmajor and then V(x,v)=0. Differentiating V(x,v) with respect to time yields(16)V˙x,v=∑i∈νmajorvi-σixivi-∑j∈Niαijxi-xj-γixi-x∗+∑i∈νslavevi-σixivi-∑j∈Niαijxi-xj+∑i∈νvi∑j∈Niαijxi-xj+∑i∈νmajorγixi-x∗vi=∑i∈ν-σixivi2≤0
If and only ifv=0 then V˙(x,v)=0. Thus V˙(x,v)=0 is given by T2={(x,v)∣v=0}. Therefore v˙=0 and it can be proven that x-=0 in T2. To show it, we assume x-≠0 in T2 and define xi-=minxj,j∈ν which means ∃κ∈Ni-:xκ>xi-.
Ifi- is a slave node then(17)v˙i-=-∑j∈Niαi-jxi--xj>-αi-κxi--xκ>0It contrasts with our conclusion v˙=0.
Otherwise ifi- is a major node then first we have(18)∑i∈νui=-∑i∈ν∑j∈Niαijxi-xj-∑i∈νmajorγixi-x∗=-∑i∈νmajorγixi-x∗=0Note that(19)v˙i-=-∑j∈Niαi-jxi--xj-γi-xi--x∗=0Due to Assumption 2 we have(20)-∑j∈Niαi-jxi--xj>-αi-κxi--xκ>0
Thusγi-(xi--x∗)>0 which means xi->x∗ and(21)-∑i∈νmajorγixi-x∗<-γi-xi--x∗<0
We find (21) contrasts with our conclusion (18). Thus x-=0 in T2 and xi=x∗,∀i∈νmajor. Because G is a connected graph, we can say x=x∗1n×1 and T2 can be rewritten as T2={(x,v)∣x=x∗1n×1,v=0} which implies consensus has been realized as |xi-xj|→0,vi-vj→0,∀i,j∈ν as t→∞ and the scalar positions converge to limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us. Well velocities converge to limt→∞vi(t)=0,∀i∈ν. The proof is completed.


## 3.1. Consensus for Single-Integrator Dynamics

Considering agents with dynamics (1), for the major node i, i∈νmajor, we define ui as what is given by(3)ui=-μixi∑j∈Niαijxi-xj+γixi-x∗where x∗∈R is an arbitrary value given by us. For the slave node i, i∈νslave, we define ui as what is given by(4)ui=-μixi∑j∈Niαijxi-xjAssumption 1.
The nonlinear gainμi is continuous and 0<μmin≤μi(x)≤μmax,∀i∈ν,∀x∈R.Assumption 2.
The interaction functionαij(·) is continuous ∀(i,j)∈ε, and(1)
αij(-y)=-αij(y),∀(i,j)∈ε,∀y∈R.(2)
αij(-y)=-αji(y),∀(i,j)∈ε,∀y∈R.(3)
αij(y)>0,∀(i,j)∈ε,∀y>0.Assumption 3.
The functionγi(·) is continuous ∀i∈νmajor, and(1)
γi(-y)=-γi(y),∀i∈νmajor,∀y∈R.(2)
γi(y)>0,∀i∈νmajor,∀y>0.Remark 4.
Assumptions2 and 3 guarantee that leader agents move in the direction of both their neighbors and a given value x∗ while other agents just move in the direction of their neighbors. A consequence of Assumption 2 is αij(0)=0,∀(i,j)∈ε, and a consequence of Assumption 3 is γi(0)=0,∀i∈νmajor.Now we state the main results of this section.Theorem 5.
Givenn agents with dynamics (1) and ui is given by (3) and (4), where μi(·) satisfies Assumption 1 for all i∈ν, αij(·) satisfies Assumption 2 for (i,j)∈ε, and γi(·) satisfies Assumption 3 for all i∈νmajor, then agents converge asymptotically to an agreement point as limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us under any initial condition x(0).Proof.
The proof is Lyapunov based. Consider the following candidate Lyapunov function:(5)Vx=∑i∈ν∫x∗xiy-x∗μiydy
It can be easily verified thatV(x∗1n×1)=0. Note that V(x)>0 for x≠x∗1n×1. To prove it, first we consider the case when xi>x∗; let z=y-x∗; then ∫x∗xi‍y-x∗/μi(y)dy=∫0xi-x∗z/μi(z+x∗)dz>0. The case when xi<x∗ is treated analogously. Let z=x∗-y; then ∫x∗xiy-x∗/μi(y)dy=∫0x∗-xiz/μi(x∗-z)dz>0. Thus we have V(x)>0, if and only if x=x∗1n×1 and then V(x)=0. Now we consider V˙(x).(6)V˙x=-∑i∈νmajorxi-x∗μixiμixi∑j∈Niαijxi-xj+γixi-x∗-∑i∈νslavexi-x∗μixiμixi∑j∈Niαijxi-xj=-∑i∈νxi-x∗∑j∈Niαijxi-xj-∑i∈νmajorxi-x∗γixi-x∗=-∑i∈νxi∑j∈Niαijxi-xj+∑i∈νx∗∑j∈Niαijxi-xj-∑i∈νmajorxi-x∗γixi-x∗
Due to Assumption2, the first term of (6) can be written as ∑i∈νxi∑j∈Ni‍αij(xi-xj)=1/2∑i∈ν‍∑j∈Ni‍(xi-xj)αij(xi-xj)≥0. Well according to Assumption 2 it can be shown that ∑i∈ν‍x∗∑j∈Ni‍αij(xi-xj)=0 and clearly ∑i∈νmajor‍(xi-x∗)γi(xi-x∗)≥0 due to Assumption 3. Thus V˙(x) can be written as V˙(x)=-∑i∈ν‍xi∑j∈Ni‍αij(xi-xj)-∑i∈νmajor‍(xi-x∗)γi(xi-x∗)<0, unless xi=xj,∀(i,j)∈ε and xi=x∗,∀i∈νmajor, and then V˙(x)=0. Note that G is a connected graph; thus xi=xj=x∗,∀i,j∈ν. Therefore agents can converge to xi=x∗,∀i∈ν when t→∞ where x∗∈R is an arbitrary value and the proof is completed.


## 3.2. Consensus for Double-Integrator Dynamics

Considering agents with double-integrator dynamics (2), for the major node i, i∈νmajor, we define ui as what is given by(7)ui=-μivi∑j∈Niαijxi-xj+βijvi-vj+γivi-v∗where v∗∈R is an arbitrary value given by us. And for the slave node i, i∈νslave, ui is given by(8)ui=-μivi∑j∈Niαijxi-xj+βijvi-vjBy using an integral Lyapunov function, we are thus able to prove that agents reach consensus under the nonlinear consensus protocol also in case of double-integrator dynamics.Theorem 6.
Givenn agents with dynamics (2) and ui is given by (7) and (8), where μi(·) satisfies Assumption 1 for all i∈ν, αij(·) and βij(·) satisfy Assumption 2 for all (i,j)∈ε and γi(·) satisfies Assumption 3 for all i∈νmajor, then agents asymptotically achieve consensus, i.e., |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞ under any initial conditions (x(0),v(0)). What’s more, the velocities converge to limt→∞vi(t)=v∗,∀i∈ν where v∗∈R is an arbitrary value given by us.Proof.
This proof relies on Lyapunov techniques. We assume there existsξ,1≤ξ≤n nodes in νmajor. First we define v-0∈Rn×1, where the ith element of v-0 is vi-v∗ if i∈νmajor and 0 otherwise. v-=[v1-v∗,…,vn-v∗]T, x-=BTx, and Γv=diag([μ1(v1),…,μn(vn)]). Now we combine (7) and (8) in vector form as:(9)x˙=vv˙=-ΓvBαx-+BβBTv+γv-0where α(·),β(·) and γ(·) are componentwise.
Now we define another Lyapunov function as(10)Vx-,v=∑i∈ν∫v∗viy-x∗μiydy+∑i,j∈ε∫0x-ijαijydywhere x-ij=xi-xj is an element in x-.
It is clear thatV([01×m,v∗11×n]T)=0. Now we need to demonstrate V(x-,v)>0 for x-≠0m×1,v≠v∗1n×1. To prove this, first we consider the situation when v≠v∗1n×1, and thus ∑i∈ν‍∫x∗xi‍y-x∗/μi(y)dy>0 as we have proven in Theorem 5. As for ∫0x-ij‍αij(y)dy, if x-ij>0, obviously ∫0x-ij‍αij(y)dy>0. And if x-ij<0, while ∫0x-ij‍αij(y)dy=∫x-ij0‍αij(-y)dy>0, then for x-≠0m×1,∑(i,j)∈ε‍∫0x-ij‍αij(y)dy>0. We can conclude that V(x-,v)>0 unless x-=0m×1,v=v∗1n×1; then V(x-,v)=0. Differentiating V(x-,v) with respect to time yields(11)V˙x-,v=∂V∂vdvdt+∂V∂x-dx-dt=∑i∈νvi-v∗μiviui+∑i∈ν∑j∈Niαijxi-xjvi=-v-TBαx-+BβBTv+γv-0+vTBαx-=-vTBβBTv-v-0Tγv-0≤0
Due to Assumptions2 and 3, we can conclude that if and only if BTv=0,v-0=0 then V˙(x-,v)=0. Since graph G is connected, BTv=0,v-0=0 equates to v=v∗1n×1. Thus V˙(x-,v)=0 is given by T1={(x-,v)∣v=v∗1n×1}. Hence, v˙=0 and it can be shown that x-=0 in T1. To testify it, we assume x-≠0 in T1. Then we can define xi-=minxj,j∈ν which means ∃κ∈Ni-:xκ>xi-. Thus(12)v˙i-=-μi-vi-∑j∈Ni-αi-jxi--xj+βi-jvi--vj+γi-vi--v∗=-μi-vi-∑j∈Ni-αi-jxi--xj>-μi-vi-αi-κxi--xκ>0
The result contrasts with our conclusionv˙=0. Thus T1 can be rewritten as T1={(x-,v)∣v=v∗1n×1,x-=0} which implies consensus can be accomplished as |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞ and limt→∞vi(t)=v∗,∀i∈ν where v∗∈R is an arbitrary value given by us. So proof is completed.


## 3.3. Consensus for Double-Integrator Dynamics with State-Dependent Damping

Considering agents with double-integrator dynamics (2), for the major node i, i∈νmajor, ui is given by(13)ui=-σixivi-∑j∈Niαijxi-xj-γixi-x∗and for the slave node i, i∈νslave, ui is given by(14)ui=-σixivi-∑j∈Niαijxi-xjWith this framework, we are able to generalize the simple consensus to a much broader class of consensus functions.Theorem 7.
Consider agents with dynamics (2) and ui is given by (13) and (14), where σi(·) satisfies Assumption 1 for all i∈ν, αij(·) satisfies Assumption 2 for all (i,j)∈ε, and γi(·) satisfies Assumption 3 for all i∈νmajor. Then agents asymptotically achieve consensus, i.e., |xi-xj|→0,|vi-vj|→0,∀i,j∈ν as t→∞. What’s more, the scalar positions converge to limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us. And velocities converge to limt→∞vi(t)=0,∀i∈ν.Proof.
The proof is also Lyapunov based. In this section we define the candidate Lyapunov function as:(15)Vx,v=∑i∈νvi22+∑i,j∈ε∫0xi-xjαijydy+∑i∈νmajor∫x∗xiγiy-x∗dy
First, let us prove∫x∗xi‍γi(y-x∗)dy>0 for xi≠x∗,i∈νmajor. To show this, if xi>x∗, obviously ∫x∗xi‍γi(y-x∗)dy>0, and if xi<x∗, ∫x∗xi‍γi(y-x∗)dy=∫xix∗‍γi(x∗-y)dy>0. Therefore V(x,v)>0 unless v=0, xi=xj,∀(i,j)∈ε and xi=x∗,∀i∈νmajor and then V(x,v)=0. Differentiating V(x,v) with respect to time yields(16)V˙x,v=∑i∈νmajorvi-σixivi-∑j∈Niαijxi-xj-γixi-x∗+∑i∈νslavevi-σixivi-∑j∈Niαijxi-xj+∑i∈νvi∑j∈Niαijxi-xj+∑i∈νmajorγixi-x∗vi=∑i∈ν-σixivi2≤0
If and only ifv=0 then V˙(x,v)=0. Thus V˙(x,v)=0 is given by T2={(x,v)∣v=0}. Therefore v˙=0 and it can be proven that x-=0 in T2. To show it, we assume x-≠0 in T2 and define xi-=minxj,j∈ν which means ∃κ∈Ni-:xκ>xi-.
Ifi- is a slave node then(17)v˙i-=-∑j∈Niαi-jxi--xj>-αi-κxi--xκ>0It contrasts with our conclusion v˙=0.
Otherwise ifi- is a major node then first we have(18)∑i∈νui=-∑i∈ν∑j∈Niαijxi-xj-∑i∈νmajorγixi-x∗=-∑i∈νmajorγixi-x∗=0Note that(19)v˙i-=-∑j∈Niαi-jxi--xj-γi-xi--x∗=0Due to Assumption 2 we have(20)-∑j∈Niαi-jxi--xj>-αi-κxi--xκ>0
Thusγi-(xi--x∗)>0 which means xi->x∗ and(21)-∑i∈νmajorγixi-x∗<-γi-xi--x∗<0
We find (21) contrasts with our conclusion (18). Thus x-=0 in T2 and xi=x∗,∀i∈νmajor. Because G is a connected graph, we can say x=x∗1n×1 and T2 can be rewritten as T2={(x,v)∣x=x∗1n×1,v=0} which implies consensus has been realized as |xi-xj|→0,vi-vj→0,∀i,j∈ν as t→∞ and the scalar positions converge to limt→∞xi(t)=x∗,∀i∈ν where x∗∈R is an arbitrary value given by us. Well velocities converge to limt→∞vi(t)=0,∀i∈ν. The proof is completed.


## 4. Voltage Equalization for Supercapacitors

In this section, we are going to apply our theory to realize voltage equalization for supercapacitor which is an efficient energy storage element with some significant advantages, like high power density, extremely high cycling capability, and being environment friendly as introduced by [11–13].In application, multiple supercapacitors are series-connected or parallel-connected together to satisfy voltage requirement which brought one key issue, voltage equalization. Recently many researchers have contributed a lot in this area. Like [14], authors proposed a PI observer to realize distributed voltage equalization. In [15], a MLLMS control algorithm for supercapacitors was provided to mitigate power quality problems in grid. In [16], a novel fuzzy logic strategy was proposed for supercapacitors.Motivated by [17], we first give out a voltage equalization circuit for supercapacitors as Figure 1.Figure 1
Power source together with voltage equalization circuit [17].We need to construct a mathematical model for this circuit. Considering each supercapacitor is an agent in this networked system and the voltage of agenti is(22)
v
˙
i
=
1
C
i
I
-
v
i
R
i
,
t
∈
d
i
T
1
C
i
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1
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d
i
Tv i ∈ R is voltage of supercapacitor i. I is charging or discharging current. Ci,Ri are capacity and resistance, respectively. 0≤di≤1 is duty ratio of switch Si at each period.Remark 8.
In application, the voltage dynamic across each supercapacitor can be formulated as follows with aid of state space average modeling method.(23)v˙i=1CiI-viRidi+1CiI1-di=1CiI-viRidid i is the final control signal to be carried out depending on the control input we give. Generally, we can design different controllers for different control purpose.Based on the mathematical model of supercapacitors circuit, now we apply our theory to realize voltage equalization for supercapacitors by simulation. Consider a voltage equalization circuit with four supercapacitors as Figure2.Figure 2
Example system.We construct its topological structure as each agent only contacts with its neighbors. Thus the Laplacian matrix will be(24)L=1-100-12-100-12-100-11We choose agents 2 and 3 as leaders and assume relevant parameters are appropriate so thatdi can satisfy all control inputs. Now we give out three kinds of controller designs to realize voltage equalization.


### 4.1. Simulation A

Let(25)v˙i=1CiI-viRidi=ui,1≤i≤4Therefore this system can be regarded as a networked system with dynamics (1). Theorem 5 can be used to solve its consensus problem. For major node i, i=2,3, ui is given by(26)ui=-μivi∑j∈Niαijvi-vj+γivi-v∗For slave node i, i=1,4, ui is given by(27)ui=-μivi∑j∈Niαijvi-vjWe define(28)μix=a-e-x,x≥0ex,x<0as nonlinear gain where 1≤i≤4,a>0. The interaction function αij(·) is defined as(29)αijx=τ,x>τx,x≤τ-τ,x<-τwhere (i,j)∈ε,τ>0. γi(·) is defined with the same structure as (29) for i=2,3.Consulting that the rated voltage of Maxwell supercapacitor generally is around2.70V, we set initial conditions for vi,1≤i≤4 as v1(0)=2.66V,v2(0)=2.57V,v3(0)=2.43V,v4(0)=2.32V. We set consensus value v∗=2.50V and a=τ=2. The following are simulation results [see Figures 3 and 4].Figure 3
Voltage for each supercapacitor.Figure 4
Control input for each supercapacitor.Our results show that consensus between four agents can be realized and the consensus voltage is what is given by us as2.50V. What’s more, our design can make sure that control inputs ui,1≤i≤4 will be strictly limited in the area (-3aτ,+3aτ) which is (-12,+12) in our simulation.


### 4.2. Simulation B

Now we add four integrators to the controllers and let(30)v˙i=1CiI-viRidi=ωiω˙i=ui,1≤i≤4We defineωi as rate of change of vi. Therefore this system can be regarded as a networked system with dynamics (2). Theorem 6 can be used to solve its consensus problem. For major node i, i=2,3, ui is given by(31)ui=-μiωi∑j∈Niαijvi-vj+βijωi-ωj+γiωi-ω∗For slave node i, i=1,4, ui is given by(32)ui=-μiωi∑j∈Niαijvi-vj+βijωi-ωjμ i ( · ) has the structure (28) for all i∈ν, αij(·),βij(·) have the structure (29) for all (i,j)∈ε, and γi(·) has the structure (29) for i=2,3. Given initial conditions for vi,ωi,1≤i≤4 as v1(0)=1.52V,v2(0)=1.33V,v3(0)=1.02V,v4(0)=0.96V and ω1(0)=0.05,ω2(0)=0.00,ω3(0)=-0.03,ω4(0)=0.01. We set consensus value ω∗=0.04. The following are simulation results [see Figures 5 and 6].Figure 5
Voltage for each supercapacitor.Figure 6
Rate of change of voltage for each supercapacitor.From results we can see that voltages of four supercapacitors can reach consensus and they change together at a rateω∗=0.04 in the end. We can also find that it takes more time to reach consensus compared with results in Simulation A.


### 4.3. Simulation C

Consider the model (24) as we have informed in Simulation B. Another controller design can realize consensus with state-dependent damping. For major node i, i=2,3, ui is given by(33)ui=-σiviωi-∑j∈Niαijvi-vj-γivi-v∗For slave node i, i=1,4, ui is given by(34)ui=-σiviωi-∑j∈Niαijvi-vjσ i ( · ) is defined with the structure (28) for all i∈ν, αij(·) is defined with the structure (29) for all (i,j)∈ε, and γi(·) is defined with the structure (29) for i=2,3. Given initial conditions for vi,ωi,1≤i≤4 as v1(0)=2.60V,v2(0)=2.53V,v3(0)=2.43V,v4(0)=2.32V and ω1(0)=0.33,ω2(0)=0.00,ω3(0)=-0.52,ω4(0)=0.12. We set consensus value v∗=2.50V. The following are simulation results [see Figures 7 and 8].Figure 7
Voltage for each supercapacitor.Figure 8
Rate of change of voltage for each supercapacitor.It can be concluded from results that consensus can be realized for voltages of supercapacitors at a valuev∗=2.50V and the rate of change of each voltage reaches consensus at a value ωi=0,1≤i≤4 in the end.


## 4.1. Simulation A

Let(25)v˙i=1CiI-viRidi=ui,1≤i≤4Therefore this system can be regarded as a networked system with dynamics (1). Theorem 5 can be used to solve its consensus problem. For major node i, i=2,3, ui is given by(26)ui=-μivi∑j∈Niαijvi-vj+γivi-v∗For slave node i, i=1,4, ui is given by(27)ui=-μivi∑j∈Niαijvi-vjWe define(28)μix=a-e-x,x≥0ex,x<0as nonlinear gain where 1≤i≤4,a>0. The interaction function αij(·) is defined as(29)αijx=τ,x>τx,x≤τ-τ,x<-τwhere (i,j)∈ε,τ>0. γi(·) is defined with the same structure as (29) for i=2,3.Consulting that the rated voltage of Maxwell supercapacitor generally is around2.70V, we set initial conditions for vi,1≤i≤4 as v1(0)=2.66V,v2(0)=2.57V,v3(0)=2.43V,v4(0)=2.32V. We set consensus value v∗=2.50V and a=τ=2. The following are simulation results [see Figures 3 and 4].Figure 3
Voltage for each supercapacitor.Figure 4
Control input for each supercapacitor.Our results show that consensus between four agents can be realized and the consensus voltage is what is given by us as2.50V. What’s more, our design can make sure that control inputs ui,1≤i≤4 will be strictly limited in the area (-3aτ,+3aτ) which is (-12,+12) in our simulation.


## 4.2. Simulation B

Now we add four integrators to the controllers and let(30)v˙i=1CiI-viRidi=ωiω˙i=ui,1≤i≤4We defineωi as rate of change of vi. Therefore this system can be regarded as a networked system with dynamics (2). Theorem 6 can be used to solve its consensus problem. For major node i, i=2,3, ui is given by(31)ui=-μiωi∑j∈Niαijvi-vj+βijωi-ωj+γiωi-ω∗For slave node i, i=1,4, ui is given by(32)ui=-μiωi∑j∈Niαijvi-vj+βijωi-ωjμ i ( · ) has the structure (28) for all i∈ν, αij(·),βij(·) have the structure (29) for all (i,j)∈ε, and γi(·) has the structure (29) for i=2,3. Given initial conditions for vi,ωi,1≤i≤4 as v1(0)=1.52V,v2(0)=1.33V,v3(0)=1.02V,v4(0)=0.96V and ω1(0)=0.05,ω2(0)=0.00,ω3(0)=-0.03,ω4(0)=0.01. We set consensus value ω∗=0.04. The following are simulation results [see Figures 5 and 6].Figure 5
Voltage for each supercapacitor.Figure 6
Rate of change of voltage for each supercapacitor.From results we can see that voltages of four supercapacitors can reach consensus and they change together at a rateω∗=0.04 in the end. We can also find that it takes more time to reach consensus compared with results in Simulation A.


## 4.3. Simulation C

Consider the model (24) as we have informed in Simulation B. Another controller design can realize consensus with state-dependent damping. For major node i, i=2,3, ui is given by(33)ui=-σiviωi-∑j∈Niαijvi-vj-γivi-v∗For slave node i, i=1,4, ui is given by(34)ui=-σiviωi-∑j∈Niαijvi-vjσ i ( · ) is defined with the structure (28) for all i∈ν, αij(·) is defined with the structure (29) for all (i,j)∈ε, and γi(·) is defined with the structure (29) for i=2,3. Given initial conditions for vi,ωi,1≤i≤4 as v1(0)=2.60V,v2(0)=2.53V,v3(0)=2.43V,v4(0)=2.32V and ω1(0)=0.33,ω2(0)=0.00,ω3(0)=-0.52,ω4(0)=0.12. We set consensus value v∗=2.50V. The following are simulation results [see Figures 7 and 8].Figure 7
Voltage for each supercapacitor.Figure 8
Rate of change of voltage for each supercapacitor.It can be concluded from results that consensus can be realized for voltages of supercapacitors at a valuev∗=2.50V and the rate of change of each voltage reaches consensus at a value ωi=0,1≤i≤4 in the end.


## 5. Conclusion

Motivated by traditional design approaches of nonlinear static feedback for distributed control, in this paper, we propose another kind of control algorithm based on static feedback so that states can reach consensus at the value preset. Moreover, we support our theory by giving its theoretical proof and using supercapacitors as example models to realize voltage equalization. Our future work will extend the method to output feedback controller.



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*Source: 7171623-2019-03-14.xml*