CONSENSUS CONTROL OF MULTI AGENT SYSTEM WITH CONSTRAINT

Consensus Control of Multi-agent System with Constraint Sun Chang (B.S.(Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN COMPUTATIONAL ENGINEERING (CE) SINGAPORE-MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE 2014 Acknowledgement This thesis could not have been completed if not for the assistance, patience, and support of many individuals. I would like to extend my gratitude first and foremost to my thesis advisor Professor Ong Chong Jin for his guidance throughout the graduate study. His insight leads to the original proposal of exploring the cooperative consensus problem. He has helped me through extremely difficult times over the problem analysis and the writing of the thesis. I sincerely thank him for his encouragement and guidance. I would also like to extend my appreciation to Professor Jacob K. White for his help while I was doing research in MIT. His rich knowledge about numerical computation and analysis helps a lot in understanding the matrix properties. I would additionally like to thank my friend Alexandra Lucas for his introduction to the vehicle to grid (V2G) problem which is an important application field of my research. This research would not have been possible without the assistance of SingaporeMIT Alliance and Mechanical Engineering department of National University of Singapore. They provided my the chance of doing graduate study and supported me until the completion. Finally I would like to extend my deepest gratitude to my parents Sun Rong and Shan Cheng Yan without whose love, support and understanding I could never have completed this doctoral degree. This research was supported by Singapore-MIT Alliance. Contents Introduction 1.1 13 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.1 Motivational Examples . . . . . . . . . . . . . . . . . . . . . . 14 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Consensus Under Switching Topology . . . . . . . . . . . . . . 17 1.2.2 Average Consensus . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Constrained Consensus . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Organization of The Thesis 23 1.2 . . . . . . . . . . . . . . . . . . . . . . . Review of Related Concepts and Theories 25 2.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Matrix Properties Related to Graphs . . . . . . . . . . . . . . . . . . 27 2.2.1 Irreducible Matrix and Connected Graph . . . . . . . . . . . . 27 2.2.2 Eigenvalue and Spectral Radius . . . . . . . . . . . . . . . . . 29 Mathematical Analysis and Convex Sets . . . . . . . . . . . . . . . . 32 2.3.1 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . 32 2.3.2 Convex Set and Its Properties . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Consensus Control on System with Constraint - The Scalar Case 35 2.3 2.4 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Preliminary and Problem Formulation . . . . . . . . . . . . . . . . . 36 3.3 The Update Law and Its Properties . . . . . . . . . . . . . . . . . . . 37 3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Consensus of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5.1 The special case of X1 = X2 = · · · = Xn . . . . . . . . . . . . 43 3.5.2 The special case of xi ∈ Rm and Xi is a box constraint . . . . 43 3.5.3 The special case when (A3-4) is not satisfied . . . . . . . . . . 44 3.6 Convergence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 Discussion and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Appendices 53 3.A Proof of Theorems and Lemmas . . . . . . . . . . . . . . . . . . . . . 53 3.B Derivations of Equations (3.20), (3.21), (3.23) and (3.24) . . . . . . . 67 Consensus Control on System with Constraint - The Multidimensional Case 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Motivational Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 The Full Approach and Its Properties . . . . . . . . . . . . . . . . . . 80 4.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 Consensus of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Simulation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Appendices 87 4.A Proof of Theorems and Lemmas . . . . . . . . . . . . . . . . . . . . . 87 4.B Derivations of Equations (4.27), (4.28), (4.30) and (4.31) . . . . . . . 92 Applications of Consensus Algorithm to V2G Problem 99 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.1 99 The V2G Model . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.2 Model Solving and Properties . . . . . . . . . . . . . . . . . . 102 5.2.3 Modifications of Algorithm 3-1 . . . . . . . . . . . . . . . . . 105 5.3 Simulation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Conclusion and Future Work 115 6.1 Summary of Main Contributions . . . . . . . . . . . . . . . . . . . . . 115 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Summary This thesis studies the consensus control of a group of agents connected via a dynamically changing communication network where the states of the agents lie within individually-defined constraints. A new algorithm is proposed in Algorithm 3-1 to solve the average constrained consensus problem when the state variables are scalars. The proofs of convergence, consensus and convergence rate under appropriate assumptions are provided and they are original. Another algorithm (Algorithm 4-1) is proposed for the average constrained consensus problem when the state variables are vectors and the constraints are general closed convex sets. The proofs of convergence and consensus under appropriate assumptions are provided. The proposed algorithm for the scalar case is also adapted to solve a real world vehicle to grid (V2G) problem. Simulation results are provided to verify the application of the proposed algorithm to the V2G problem. List of Figures 1-1 Illustration of projection method for constrained consensus . . . . . . 21 2-1 Graph associated to different matrices. . . . . . . . . . . . . . . . . . 29 3-1 Plots of xi (k), cij (k) and Ri (k) versus k for Example I. . . . . . . . . 47 3-2 Plots of xi (k) and cij (k) versus k for Example 1. . . . . . . . . . . . . 48 3-3 Plots of xi (k) and Ri (k) versus k for Example II. . . . . . . . . . . . 49 3-4 Plots of xi (k) versus k in different network for Example II. . . . . . . 49 3-5 Plots of xi (k) against k for i = 5, 15, 25, 35 . . . . . . . . . . . . . . . 51 3-6 Depiction of the θi and the state trajectories for the unicycles example. 52 4-1 Network connections for Examples and 2. Brown lines indicate communication links, Black lines for δij . Shaded regions are individual feasible domains identified by the Xi . . . . . . . . . . . . . . . . . . 75 4-2 Examples and for Discussion . . . . . . . . . . . . . . . . . . . . . 78 4-1 Network switching alternatively between A1 and A2 . . . . . . . . . . 85 4-2 Trajectory of state variables in 6-agent system . . . . . . . . . . . . . 85 4.A.1Illustration of some concepts in the proof . . . . . . . . . . . . . . . . 90 5-1 Illustration of V2G Model . . . . . . . . . . . . . . . . . . . . . . . . 101 5-1 Plot of xi (k) against k under different initial conditions . . . . . . . . 108 5-2 Convergence under different network connectivity . . . . . . . . . . . 108 5-3 Dynamics of vehicles entering and leaving the system . . . . . . . . . 109 5-4 Power demanded by the grid . . . . . . . . . . . . . . . . . . . . . . . 109 5-5 Consensus value for all the vehicles during the hours . . . . . . . . 110 5-6 Consensus value of all the vehicles during the first 15 minutes . . . . 111 5-7 State variable changes with constraint-free algorithm during the first 10s112 5-8 State variable changes with constrained consensus control during the first 10s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5-9 Energy sold by each vehicle in group 5-9(a) and 5-9(b) . . . . . . 113 List of Tables 1.1 Summarize of existing results . . . . . . . . . . . . . . . . . . . . . . 22 3.A.1Table of cjl (k)δlj (k) − cil (k)δli (k) in different cases . . . . . . . . . . . 61 Figure 5-7: State variable changes with constraint-free algorithm during the first 10s Algorithm 5-1 and Algorithm 3-1 keep the power demand on each vehicle within its restrictive constraint. The proposed constrained consensus method is proven to be a suitable control law to this system. In this problem, the communication rate is taken to be 1ms = 0.001s, as a result, the consensus value can be achieved in about seconds. The time to achieve consensus is inverse proportional to the communication rate. If communication rate is changed to 0.01s, the it will take around 60 seconds to achieve consensus. Similarly, if the communication rate is 0.1s, 600 seconds is needed to achieve consensus which is more than an hour. In those cases, the amount of energy sold to the grid may vary greatly for two vehicles plugging in and off at the same time. Therefore, 1ms is a preferred communication rate for this model. While the consensus variable is the power, it is interesting to find out the total energy supplied by the groups of EVs. Figure 5-9 shows the result. Both groups sell essentially the same amount of energy subject to a maximal relative errors of 0.02% and 0.04% respectively. The maximal relative error for each group is defined as Emax −Emin , Emax where Emax and Emin are the maximal and minimal energy supplied in the group respectively. This simulation verifies that applying Algorithm 5-1, for any two vehicles which enter and leave the grid at the same time, they sell essentially the same amount of energy with a maximal relative error that is less than 0.04%. 112 Power Vs Time with Algorithm 3-1 kW 0 10 10 t(s) Power Vs Time with Algorithm 5-1 kW 0 t(s) Figure 5-8: State variable changes with constrained consensus control during the first 10s (a) Group (b) Group Figure 5-9: Energy sold by each vehicle in group 5-9(a) and 5-9(b) 5.4 Conclusion The aim of this chapter is to test the usage of constrained consensus control in a V2G service. The V2G problem is modeled and solved by a modified Algorithm 3-1 to manage the amount of energy supplied by a fleet of vehicles to the grid. The consensus property is proven under appropriate assumptions and a heuristic method (Algorithm 5-1) is proposed to accelerate the convergence speed. A simulation is conducted to verify the usage of the proposed algorithm. Results show that the use of constrained consensus control makes the system reach consensus in about seconds while the limitation of battery are not exceeded. Moreover, the simulation shows 113 that the two groups of vehicles sold essentially the same amount of energy. With the assessed conditions, each agent of the first group sells 17.2kW h to the grid during all hours with a maximal relative error of 0.02%. Each agent of the second group sells 7.19kW h during the second hour reporting a maximal relative error of 0.04%. The conclusion is that the constrained consensus algorithm has a high potential of application in V2G problems due to its decentralized nature, fast convergence speed and capability of handling high number of agents. Broadening the consensus theory to other V2G services such as regulation is an opportunity for future work. 114 Chapter Conclusion and Future Work 6.1 Summary of Main Contributions This section summarizes the main contributions of this thesis. The scope of the study in this thesis is the consensus problem where the agents are confined within individually-defined constraints. The objective is to preserve the average consensus property while achieving consensus. In addition, the consensus value should depend on the initial states and not on the sequence of topological changes. The first contribution is a new algorithm proposed in Algorithm 3-1 to solve the above mentioned problem when the state variables are scalars. The idea of Algorithm 3-1 is to use a user-defined weight to control the step size of each movement. From the update law (3.7) of cij (k), one can see that the weight depends on the relative position of the state variables xi (k) and xj (k) and it may become zero as the state variables approach the boundaries. This may cause virtual disconnection from agent i to j even if they are physically connected. This makes the mathematical structure of our problem different from the previous works and the consensus conditions of the existing results are not applicable. As a result, the consensus of Algorithm 31 has to be proven differently. The proof consists of three parts. The first is the convergence of Algorithm 3-1 under mild assumptions. This is done by showing that all subsequences converge to the same limit. The second is the consensus result which requires additional assumptions. The third is the convergence rate for a fully 115 connected network. The results are all self-contained and can be of independent interest. Algorithm 3-1 can also be applied to achieve consensus for some special cases with slight modifications, namely when all constraints are identical; when (A3-4) is not satisfied; when state variables are vectors with box constraints. Three simulation examples are provided to illustrate Algorithm 3-1. The second contribution is a new algorithm (Algorithm 4-1) for the average constrained consensus problem when the state variables are vectors and the constraints are general closed convex sets. Several motivational examples are used to show that a naive extension of Algorithm 3-1 fails. Algorithm 4-1 is motivated by these examples and like Algorithm 3-1 uses the weight as control variable but with a different update law. The proofs of the convergence and consensus under proper assumptions are self-contained and original. An example that illustrates Algorithm 4-1 is also provided. The last contribution is the adaptation of Algorithm 3-1 to a real world V2G problem. V2G service is a potential solution to reduce the cost of energy generation. The battery of each vehicle is used to store the energy during non-peak hours and supply back to the grid during peak hours. The objective of the control is to ensure that every vehicle in the grid sells the same power to the grid subjected to a maximal value. Moreover the total amount of power supplied must equal to the power demand by the grid. This condition is achieved by average consensus property. When the power demand and number of vehicles not change in a sufficiently long time Algorithm 3-1 ensures that power supplied by all EVs reach consensus. Moreover, a modified algorithm (Algorithm 5-1) is proposed to accelerate the rate of convergence. Simulation results are provided for this problem. The power demand of the simulation makes use of real world data. 6.2 Future Work There are several open problems in this thesis. These are given below: In the proof of convergence (Theorem 3.3), the number of subsequences is as116 sumed to be finite or countably infinite. If the number of subsequences is uncountably infinite, i.e., the limit set of subsequences is dense, the proof may not work. From our experience the number of subsequences cannot be uncountably infinite, but a rigorous proof is needed. In the proof of convergence rate (Theorem 3.17), it is assumed that the asymptotic rate of convergence for agent i Ri exists for all i ∈ Zn . This, from our experience, is true under a fully connected network, but a rigorous proof is needed. Step (2) of the Algorithm 4-1 requires the search of a subset Si (k) that satisfies (P1) and (P2a). This is a mixed integer programming problem which can be computational intensive. In cases where the number of neighbors is not too large, the computational time is acceptable. Methods to speed up this search, heuristic or otherwise is a future work. The assumption (A4-1) is not easy to check. Approaches that can relax (A4-1) is useful. The convergence rate of Algorithm 4-1 has not been investigated. The algorithms developed in this thesis apply to undirected network. A good direction of future work is to extend this result for directed networks. The communication delays are not considered in this thesis, each agent is assumed to be a single integrator and the constraints for each agent are independent from each other. In real world applications, communication delay always exists, the dynamics of each agent is more complex and the constraints of the agents sometimes are coupled together.It would be more practically meaningful to take these into consideration. Also, it would be interesting to think about alternatives to represent communication failure rather than switching network. In the application to V2G problem, the state variable is the power sold to the grid. 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Automatica, 47(7):1366–1378, 2011. 127 [...]... discrete time system with undirected time-varying networks The rest of this chapter gives an overview of the consensus problem 1.1 Background Cooperative control of multi- agent system is a decentralized control scheme Agents are connected via a communication network to other agents and each agent follows its own control law Collectively, the agents achieve some desirable outcome The word consensus means... concept of reaching consensus and asymptotic consensus is defined first 15 Definition 1.1 In a multi- agent system of n agents, let xi ∈ Rm i = 1, · · · , n be the state variable of agent i The system is said to reach consensus if and only if x1 = x2 = · · · = xn Definition 1.2 In the case when the state variables xi (k), i = 1, · · · , n changes with time k = 1, 2, · · · , the system reaches asymptotic consensus. .. Average Consensus The following theorem is shown in [50], Theorem 1.5 Consider a multi- agent system with undirected network topology G and its associated consensus algorithm of (1.3) Let G be connected and cij aij = cji aji , then the system reaches consensus with consensus value x∞ being the average of the initial states, or, x∞ = 1 n n i=1 xi (0) This type of consensus problem where the consensus. .. element denoted by xi Xi Constraint set of agent i Ni The set of neighbors of agent i G(V, E) A graph with vertex set V and edge set E A = [aij ] Adjacency matrix of a graph G(V, E) cij The weight associated with aij (k) V2G Vehicle to grid service SoC State of charge EV Electric vehicle 11 12 Chapter 1 Introduction This thesis studies the cooperative consensus problem for a multi- agent system operating in... to the partial differentiation of gi (xi ) with respect to each element of xi The first part of (1.6) is the attractive part which guarantees the consensus while the second part 21 is the repulsive part when agent states are close to the boundaries of its respective constraint The work of [35] considers a multi- agent system with an undirected static network However the consensus under switching network... for discrete time multi- agent system that reaches consensus with switching topologies and bounded time delay under proper assumptions In [83], the author solves the consensus problem for a second order dynamical system Reference [17] further considers 18 consensus of heterogeneous agents with both switching topologies and time delay and gives a sufficient consensus condition in the form of linear matrix... motivation of this thesis Problem considered Average consensus Constrained consensus Under switching topology What to do? consensus value is the Can we incorporate average of initial value the feature of consensus value depends average consensus into on switching sequence constrained consensus? Table 1.1: Summarize of existing results From the above table, the consensus value of the existing constrained consensus. .. to the system breaks the symmetry of the updating law and the average consensus property cannot be guaranteed Some other works on constrained consensus includes Moore et al [43] who studies the consensus of system when the states of agents are partially constrained and J.Lee et al in [34] using a model predictive control (MPC) framework for the constrained consensus problem when the incremental of states... problem in the presence of state constraints and switching networks when the state variables are scalars Most of the consensus literature [44, 55, 48, 3, 63, 57, 79, 34] is for constraint- free systems with the exception of [4, 35, 43] The original work of [4] relies on the projection operation to satisfy the state constraints However, the consensus value is dependent on the sequence of network switching... According to [48], one of the main approaches is to use a vector representation of the relative position of the nearby UAVs and apply a consensus- based controller with an input bias The problem of formation control can be formulated as a local optimization problem Each agent i minimizes the local cost function Ui = 1 2 j∈Ni ||xi − xj − rij ||2 , where xi is the position vector of agent i, rij is the desired . the discrete time system with undirected t ime-varying networ ks. The rest of this chapter gives an overview of the consensus problem. 1.1 Background Cooperative control of multi- agent system is a. Consensus Control of Multi- agent System with Constr aint Sun Chang (B.S.(Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN COMPUTATIONAL. concept of reaching consensus and asymptotic con- sensus is defined first 15 Definition 1.1. In a multi- age nt system of n agents, let x i ∈ R m i = 1, · · · , n be the state variable of agent i. The system