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.
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 Section 2, 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
Let be a graph. We define where is vertex set and is edge set of graph . Select some nodes as leaders which belong to vertex set and the rest of the nodes as slave nodes which belong to vertex set . It is clear that . Let informs the set of neighboring nodes of node as . We require that the undirected graph G is connected which means where there exists a path from node to node as . Denote by the vertex-edge adjacency matrix of and let be its Laplacian matrix. For a undirected graph we have . Define the scalar position of agent as and its velocity as . We collect them into column vectors , . denotes the identity matrix of .
2.2. System Model
In this paper, we only consider agents endowed with either single-integrator dynamics asor double-integrator dynamics as
2.3. Objective
The main objective of this paper is for agents to converge to a common state, i.e., for single-integrator dynamics and for double-integrator dynamics. Theoretically consensus states and can be desired values set by us. Furthermore, for double-integrator dynamics, we also decide to realize where 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 Section 3.1 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 , , we define as what is given bywhere is an arbitrary value given by us. For the slave node , , we define as what is given by
Assumption 1. The nonlinear gain is continuous and .
Assumption 2. The interaction function is continuous , and(1).(2).(3).
Assumption 3. The function is continuous , and(1).(2).
Remark 4. Assumptions 2 and 3 guarantee that leader agents move in the direction of both their neighbors and a given value while other agents just move in the direction of their neighbors. A consequence of Assumption 2 is , and a consequence of Assumption 3 is .
Now we state the main results of this section.
Theorem 5. Given agents with dynamics (1) and is given by (3) and (4), where satisfies Assumption 1 for all , satisfies Assumption 2 for , and satisfies Assumption 3 for all , then agents converge asymptotically to an agreement point as where is an arbitrary value given by us under any initial condition .
Proof. The proof is Lyapunov based. Consider the following candidate Lyapunov function:It can be easily verified that . Note that for . To prove it, first we consider the case when ; let ; then . The case when is treated analogously. Let ; then . Thus we have , if and only if and then . Now we consider .Due to Assumption 2, the first term of (6) can be written as . Well according to Assumption 2 it can be shown that and clearly due to Assumption 3. Thus can be written as , unless and , and then . Note that is a connected graph; thus . Therefore agents can converge to when where 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 , , we define as what is given bywhere is an arbitrary value given by us. And for the slave node , , is given byBy 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. Given agents with dynamics (2) and is given by (7) and (8), where satisfies Assumption 1 for all , and satisfy Assumption 2 for all and satisfies Assumption 3 for all , then agents asymptotically achieve consensus, i.e., as under any initial conditions . What’s more, the velocities converge to where is an arbitrary value given by us.
Proof. This proof relies on Lyapunov techniques. We assume there exists nodes in . First we define , where the th element of is if and 0 otherwise. , , and . Now we combine (7) and (8) in vector form as:where and are componentwise.
Now we define another Lyapunov function aswhere is an element in .
It is clear that . Now we need to demonstrate for . To prove this, first we consider the situation when , and thus as we have proven in Theorem 5. As for , if , obviously . And if , while , then for . We can conclude that unless ; then . Differentiating with respect to time yieldsDue to Assumptions 2 and 3, we can conclude that if and only if then . Since graph is connected, equates to . Thus is given by . Hence, and it can be shown that in . To testify it, we assume in . Then we can define which means . ThusThe result contrasts with our conclusion . Thus can be rewritten as which implies consensus can be accomplished as as and where 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 , , is given byand for the slave node , , is given byWith 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 is given by (13) and (14), where satisfies Assumption 1 for all , satisfies Assumption 2 for all , and satisfies Assumption 3 for all . Then agents asymptotically achieve consensus, i.e., as . What’s more, the scalar positions converge to where is an arbitrary value given by us. And velocities converge to .
Proof. The proof is also Lyapunov based. In this section we define the candidate Lyapunov function as:First, let us prove for . To show this, if , obviously , and if , . Therefore unless , and and then . Differentiating with respect to time yieldsIf and only if then . Thus is given by . Therefore and it can be proven that in . To show it, we assume in and define which means .
If is a slave node thenIt contrasts with our conclusion .
Otherwise if is a major node then first we haveNote thatDue to Assumption 2 we haveThus which means andWe find (21) contrasts with our conclusion (18). Thus in and . Because is a connected graph, we can say and can be rewritten as which implies consensus has been realized as as and the scalar positions converge to where is an arbitrary value given by us. Well velocities converge to . 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.
We need to construct a mathematical model for this circuit. Considering each supercapacitor is an agent in this networked system and the voltage of agent is
is voltage of supercapacitor . is charging or discharging current. are capacity and resistance, respectively. is duty ratio of switch 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.
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 Figure 2.
We construct its topological structure as each agent only contacts with its neighbors. Thus the Laplacian matrix will be
We choose agents 2 and 3 as leaders and assume relevant parameters are appropriate so that can satisfy all control inputs. Now we give out three kinds of controller designs to realize voltage equalization.
4.1. Simulation A
Let
Therefore 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 , , is given byFor slave node , , is given by
We defineas nonlinear gain where . The interaction function is defined aswhere . is defined with the same structure as (29) for .
Consulting that the rated voltage of Maxwell supercapacitor generally is around , we set initial conditions for as . We set consensus value and . The following are simulation results [see Figures 3 and 4].
Our results show that consensus between four agents can be realized and the consensus voltage is what is given by us as . What’s more, our design can make sure that control inputs will be strictly limited in the area which is in our simulation.
4.2. Simulation B
Now we add four integrators to the controllers and let
We define as rate of change of . 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 , , is given byFor slave node , , is given by
has the structure (28) for all , have the structure (29) for all , and has the structure (29) for . Given initial conditions for as and . We set consensus value . The following are simulation results [see Figures 5 and 6].
From results we can see that voltages of four supercapacitors can reach consensus and they change together at a rate 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 , , is given byFor slave node , , is given by
is defined with the structure (28) for all , is defined with the structure (29) for all , and is defined with the structure (29) for . Given initial conditions for as and . We set consensus value . The following are simulation results [see Figures 7 and 8].
It can be concluded from results that consensus can be realized for voltages of supercapacitors at a value and the rate of change of each voltage reaches consensus at a value 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.
Data Availability
All data included in this study are available upon request by contact with the corresponding author.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61741315, Hunan Provincial Natural Science Foundation of China under Grant 2017JJ4056, and Commonweal Technology Research Project of Zhejiang Province under Grant LGG18F030005.