Distributed optimization for a class of uncertain MIMO nonlinear multi-agent systems with arbitrary relative degree

doi:10.1016/j.ins.2019.08.010

Information Sciences

Volume 506, January 2020, Pages 58-77

https://doi.org/10.1016/j.ins.2019.08.010 Get rights and content

Abstract

The paper is concerned with the distributed optimization problem (DOP) for a class of uncertain multi-input-multi-output (MIMO) nonlinear multi-agent systems with arbitrary relative degree. The target is to design distributed control laws such that the outputs of the agent systems converge to a consensus value and on which the sum of the local cost functions is minimized. By introducing pseudo gradient technique, internal model technique and adaptive control technique, a novel state based distributed control law is firstly constructed. An incremental type Lyapunov function based approach is presented to show that the proposed DOP is solved by the state based control law without requiring the eigenvalue information of Laplacian matrix. By further introducing distributed high-gain observer technique, an output based distributed control law is constructed and by which the DOP is solved under some mild assumption. The proposed control laws are validated on a group of Euler-Lagrange systems, a group of robot manipulators with flexible joints and a group of Chua circuit systems. The simulation results illustrate the effectiveness of the proposed methods.

Introduction

The distributed optimization problem was firstly introduced to the control community by the pioneering work [16]. The last decade has witnessed the flourishing research activities on the distributed optimization problem. This is because the distributed optimization has many applications in the modern industry, such as cyber-physical systems [27], wireless resource allocation [19] and machine learning [2].

The distributed computation problem or model was firstly studied in [26] and has received renewed interest in the recent years. In the seminal work [16], the focus is extended to a distributed optimization problem over the so-called multi-agent system. Each agent has a locally known, probably different, and convex cost function. The target is to minimize the sum of the cost functions through information exchange among neighboring agents. Build on this framework, many algorithms are proposed to solve the distributed optimization problem (DOP), just name a few [6], [9], [14], [15], [18], [21], [22], [30], [31], [37] and the references therein. It is worth to note that a control perspective for DOP is presented in a pioneering work [28]. As pointed out in [28], the distributed optimization problem can be reformulated as the controller design problem such that the multi-agent system can reach consensus on an optimal value, i.e., the sum of the local cost functions is minimized on the value. Works along this line have recently appeared in the literature such as [10], [12], [13], [20], [24], [25], [29], [35], [36]. The linear multi-agent systems are considered in [10], [13], [20], [36] and nonlinear systems in [12], [24], [25], [29], [35].

Although some progress has been made on the DOP for multi-agent systems, the systems considered in the existing works such as [12], [24], [25], [29], [35] have some special forms and the control laws are designed by taking advantage of the special forms. In this paper we will consider the multi-agent system that consists of n agents. Each agent takes on the following form: $\begin{matrix} {\dot{z}}_{i} & = & q_{i} (z_{i}, x_{i 1}, w_{i}), \\ {\dot{x}}_{i 1} & = & x_{i 2}, \\ {\dot{x}}_{i 2} & = & x_{i 3}, \\ ⋮ & = & ⋮ \\ {\dot{x}}_{i, ρ - 1} & = & x_{i ρ}, \\ {\dot{x}}_{i ρ} & = & p_{i} (z_{i}, x_{i 1}, \dots, x_{i ρ}, w_{i}) + u_{i}, i = 1, \dots, n; \end{matrix}$ where $z_{i} \in R^{h_{i}}$ is the internal state, $x_{i j} \in R^{m}$ is the external state and $u_{i} \in R^{m}$ is the control input, x_i1 is the output that is to be synchronized to an optimal consensus value, ρ ≥ 1 is the relative degree, w_i is an uncertain constant parameter that belongs to a known compact set, q_i and p_i are continuous functions. It is noted that the system (1) is in a normal form that is defined in [8]. Roughly speaking, any affine nonlinear system with certain relative degree in some region can be transformed to the normal form system and the details can be found in Chapter 13 of [8]. The system (1) can include the systems considered in [12], [13], [24], [29], [35] as special cases. It is worth to note that [12], [29] consider the normal form nonlinear systems with unitary and second relative degrees, respectively. However, the methods in [12], [29] cannot be extended to the higher relative degree case. To circumvent the difficulty introduced by higher relative degree, a pseudo gradient technique is originally proposed in this paper. Along with internal model technique and adaptive control technique, a novel state based distributed control law is constructed such that the proposed DOP is solved for the multi-agent system (1). By further introducing distributed high-gain observer technique, an output based distributed control law is constructed and proved to solve the DOP under some mild assumption.

In [12], [29], [35], the selection of the parameters of the proposed control laws is dependent on the eigenvalue information of the Laplacian matrix. The eigenvalue information is in fact a global information that is hard to obtain for large-scale network. On the other hand, the eigenvalues of the Laplacian matrix may be changed when the network is subject to link failures and reconstructions. Therefore, the requirement for the eigenvalue information of the Laplacian matrix is restrictive and unrealistic. In this paper we present a novel incremental type Lyapunov function based approach such that the selection of the parameters of the proposed state based control law is independent of the Laplacian matrix. The current approach is inspired by the characteristic of the DOP that the outputs of the multi-agent systems “forget” the initial conditions and converge to the optimal consensus value, that is uniquely determined by the local cost functions. This initial condition insensitivity characteristic can be also found in the incrementally stable systems [1].

The main contributions of the paper are summarized as follows:

•
A general framework on the DOP for a class of uncertain multi-input-multi-output nonlinear multi-agent systems with arbitrary relative degree is presented. To circumvent the difficulty introduced by higher relative degree, pseudo gradient technique is originally proposed. Along with internal model technique and adaptive control technique, a novel state based distributed control law is constructed such that the DOP is solved. Compared to the existing results on the DOP for nonlinear multi-agent systems, the proposed state based feedback control law does not require the eigenvalue information of Laplacian matrix and moreover the gradients of the local cost functions are not required to be globally Lipschitz.
•
By further introducing the distributed high-gain observer technique, an output based distributed control law is proposed and proved to solve the DOP under some mild assumption. The singular perturbation based method is used to show the stability of the system.
•
Last but not least, incremental stability property of the DOP is exploited in the paper. Inspired by the property, an incremental type Lyapunov function based approach is presented to derive the stability of the closed-loop system under state feedback control law. The approach may shed light on the time-varying distributed optimization problem.

The remaining contents are organized as follows. In Section 2, we give the distributed optimization problem formulation on the multi-agent system (1). Section 3 proposes the state based distributed control law design for the DOP and also presents the convergence analysis of the closed-loop system. The results for the output feedback control law are given in Section 4. Section 5 gives the numerical simulations that illustrate the effectiveness of the methods. Finally, some concluding remarks and future work are given in Section 6.

Section snippets

Notations and definitions

Throughout the paper, $R^{n}$ and $R^{n \times m}$ denote the n-dimensional real vector space and real matrix space of size n × m, respectively. $R^{+}$ means [0, ∞). ‖ · ‖ is the Euclidean norm. The symbol ⊗ denotes the Kronecker product. σ_min(A) and σ_max(A) denote the smallest and biggest eigenvalue of the matrix A, respectively. The notation $1_{n}$ is defined as $1_{n} = {[1 1 \dots, 1]}^{T}$ . diag[x₁, x₂, . . . , x_n] means the n × n diagonal matrix with its diagonal elements as x₁, x₂, . . . , x_n. ∇f( · ) means the gradient of a

Global convergence

The state based distributed control law is given as follows: $\begin{matrix} u_{i} & = & - α \nabla f_{i} (y_{i}) - \sum_{j = 1}^{n} a_{i j} (y_{i} - y_{j}) - \sum_{j = 1}^{n} a_{i j} (η_{i} - η_{j}) - ϕ_{i} \\ {\dot{ϕ}}_{i} & = & α \nabla f_{i} (y_{i}) + \sum_{j = 1}^{n} a_{i j} (y_{i} - y_{j}) + \sum_{j = 1}^{n} a_{i j} (η_{i} - η_{j}) \\ {\dot{η}}_{i} & = & \sum_{j = 1}^{n} a_{i j} (y_{i} - y_{j}), i = 1, \dots, n; \end{matrix}$ where α > 0 is a parameter to be verified shortly, $ϕ_{i}, η_{i} \in R^{m}$ and $y_{i} = \sum_{j = 1}^{ρ - 1} k_{j} x_{i j} + x_{i ρ}$ where $k_{j}, j = 1, \dots, ρ - 1$ are chosen such that the polynomial $s^{ρ - 1} + k_{ρ - 1} s^{ρ - 2} + \dots + k_{2} s + k_{1}$ is Hurwitz and $k_{1} = 1$ . In practice, we can choose $k_{ρ - 1}, \dots, k_{1}$ such that $s^{ρ - 1} + k_{ρ - 1} s^{ρ - 2} + \dots + k_{2} s + k_{1} = {(s + 1)}^{ρ - 1}$ .

Remark 3.1

It is noted that the so-called pseudo gradient ∇f_i(y_i)

Output feedback

In the last section, the state based feedback control law (6) is considered. However, the state may not be available for feedback control design under some scenarios. In this section an output based feedback control law is proposed by introducing distributed high-gain observer.

Example 1

In the first example, we consider the DOP on the multi-agent system where each agent takes the Euler-Lagrange form [23]: $M_{i} (q_{i}) {\ddot{q}}_{i} + C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + G_{i} (q_{i}) = τ_{i}, i = 1, \dots, 5;$ where $q_{i} = {[q_{i 1}, q_{i 2}]}^{T} \in R^{2}$ is the general position and $τ_{i} \in R^{2}$ is the control input. The details on $M_{i} (q_{i}), C_{i} (q_{i}, {\dot{q}}_{i})$ and G_i(q_i) can be found in [23], [35]. The local cost functions are $f_{1} (q_{1}) = {(q_{11}^{2} + 1)}^{2} + q_{12}^{2} + {(q_{11} + q_{12})}^{2}, f_{2} (q_{2}) = q_{21}^{2} + {(q_{22}^{2} - 1)}^{2}, f_{3} (q_{3}) = 4 {(q_{31} + 2)}^{2} + q_{32}^{2},$ $f_{4} (q_{4}) = 2 q_{41}^{2} + q_{41} + q_{42}^{2} + 1, f_{5} (q_{5}) = q_{51}^{4} + q_{51}^{2} + 3 q_{52}^{2} + 1 .$ In [35], distributed control

Conclusion

In this paper the distributed optimization problem (DOP) for a class of uncertain minimum-phase multi-input-multi-output nonlinear multi-agent systems with arbitrary relative degree is considered. Both the state and output based feedback control laws are proposed. When the state based control law is applied, its parameter can be selected independent of the Laplacian matrix and the local cost functions can be any strongly convex functions. The local cost functions have to be in quadratic forms

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under grant numbers 61603084, 61621004, 61420106016, and in part by the Fundamental Research Funds for the Central Universities under Grant N170404017 and in part by the State Key Laboratory of Synthetical Automation for Process Industries under Grant 2018ZCX03.

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Distributed optimization for a class of uncertain MIMO nonlinear multi-agent systems with arbitrary relative degree

Abstract

Introduction

Section snippets

Notations and definitions

Global convergence

Output feedback

Example 1

Conclusion

Declaration of Competing Interest

Acknowledgement

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