# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 1 Introduction
In this paper, we consider the fixed-time MSCO problem for a continuous-time multi-agent system with unknown external disturbances. Firstly, the local inequality constraints are removed using the penalty function method, resulting in a transformed problem without local constraints, and the solution gap is theoretically analyzed. Secondly, a three-stage fixed-time convergence strategy is designed: the first stage uses the integral sliding mode control to eliminate the external perturbation in fixed-time, the second stage uses the symmetry of the communication network to satisfy the global resource constraint in fixed-time, and the third stage uses the fixed-time gradient optimization algorithm to make all the agents converge to the minimum value of the transformed problem in fixed-time. Finally, the Lyapunov function analysis shows that the system can track the global resource demand and satisfy the local resource constraint in fixed-time. The main contribution of the paper can be summarized as follows:  
1. The penalty function method is used to remove the local constraints of the agents and the error bound of the approximate solution is analyzed. 2. A novel three-stage fixed-time distributed optimization algorithm is proposed to solve the MSCO problem with local inequality constraints, global equation constraints and unknown disturbances. 3. The effectiveness of the proposed algorithm is verified by the engineering case of a wind farm co-generation optimization problem considering the wake effect. The remainder of the paper is organized as follows: Section 2 gives some preliminary knowledge and formulates the MSCO problem. The main results are presented in Section 3, including the analysis of the optimal solution set and three-stage fixed-time MSCO algorithm. Simulation for wind farm co-generation is provided in Section 4 to validate all the results. Section 5 concludes the whole paper. **Notation 1**. : \(\mathbb{R}\) _is the set of real numbers, \(\mathbb{R}_{>0}\) is the set of positive real numbers, \(\mathbb{R}_{\geq 0}\) is the set of non-negative real numbers, and \(0_{n}\in\mathbb{R}^{n}\) is a full zero column vector. The \((\cdot)^{\top}\) represents the transpose, and let the second smallest eigenvalue of the matrix be \(\lambda_{2}(\cdot)\). \(f\in\mathbb{C}^{k}(U,V)\) represents the function \(f:U\to V\), \(U\subset\mathbb{R}^{n}\), \(V\subset\mathbb{R}^{m}\) are kth order continuous differentiable functions. \(\
abla f:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is the gradient of the function \(f\in C^{1}(\mathbb{R}^{n},\mathbb{R})\). \(\|\cdot\|_{p}\) is the \(l_{p}\) norm, and in particular, \(\|\cdot\|\) is the Euclidean norm._  
Suppose the system has \(N\) agents, and the communication topology \(\mathcal{G}=(A,\mathcal{V})\) between agents is an undirected graph with adjacency matrix \(A(t)=[a_{ij}(t)]\in\mathbb{R}^{N\times N}\), \(a_{ij}\in\{0,1\}\), and the set of nodes \(\mathcal{V}=\{1,2,\cdots,N\}\). The set of neighbor nodes of node \(i\) is denoted as \(\mathcal{N}_{i}(t)\), that is, \(\mathcal{N}_{i}(t)=\{j\in\mathcal{V}|a_{ij}(t)=1\}\). \(\mathrm{sgn}(\cdot)\) is the sign function. Given \(x=[x_{1},\cdots,x_{n}]^{\top}\in\mathbb{R}^{n}\), we define the function \(\mathrm{sign}^{\beta}:\mathbb{R}^{n}\to\mathbb{R}^{n},\beta\geq 0\) as  
\[\mathrm{sign}^{\beta}(x)=x\|x\|^{\beta-1}, \tag{1}\]  
and \(\mathrm{sign}^{\beta}(0_{n})=0_{n}\).