# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 1 Introduction  
The coordinated control of multi-agent systems has emerged as a pivotal area of contemporary research, driven by the proliferation of distributed network resources and the imperative for collaboration among various mechanical devices [1; 2; 3]. Key aspects of such systems encompass group interaction and communication, coordination and cooperation, and conflict resolution. A prominent example of these concepts is the multi-flight simulator system [4]. Additionally, multi-agent systems have notable applications in the healthcare sector [5]. Exploration of multi-agent systems has yielded numerous methodologies and control strategies aimed at ensuring consistent stability. For instance, dynamic surface control combined with a first-order low-pass filter has been utilized to address the inherent \"differential explosion\" issue in traditional recursive methods [6; 7; 8; 9]. While this method simplifies computational complexity, it also introduces errors associated with filter computation and increases the number of control parameters, potentially compromising transient performance management. Recent research [10] has attempted to resolve the algebraic loop problem intrinsic to the backstepping method, thereby ensuring transient tracking performance, albeit at the cost of increased control complexity due to the introduction of new variables. In addressing the problem of unknown states in high-order multi-agent systems, observers are often employed. Among these, the use of high-gain observers is a common research approach [11; 12]. Through the application of such observers, disturbances within the system, including external input disturbances, system coupling, and other factors, can be effectively managed [13; 14; 15; 16; 17]. These mechanisms ensure the stability of the entire closed-loop system while optimizing the tracking performance of each subsystem. Various types of observers employ different methodologies; for example, interval observers utilize neural networks, resilient control techniques, and fuzzy equations to estimate unmeasurable states of the system [18]. By narrowing the interval width, this method ensures that the closed-loop system remains semi-globally ultimately bounded, allowing for a convergence of tracking and observation errors within a small neighborhood near the origin [19]. Similar developments have been documented in discrete systems [20]. Optimizations of high-gainobservers involve dimensionality reduction, as seen in the current literature [21; 22] where the observer model's complexity is optimized within manageable limits while ensuring that all closed-loop signals are ultimately bounded under the influence of a continuously differentiable switching function. Therefore, building upon the research foundation of high-gain observers, this study considers the addition of compensatory terms to achieve error reduction and structural simplification. Furthermore, with the expansion of material science applications, particularly in multi-agent systems equipped with intelligent materials, significant challenges arise. For example, piezoelectric ceramics and ionic polymer metal composites have gained attention due to their exceptional performance and desirable physical characteristics [23]. However, these materials also exhibit performance deficiencies, such as non-smoothness, nonlinearity, and hysteresis, which can significantly impact the precision and stability of control systems, particularly due to the effects of hysteresis signals and coupling characteristics [24]. To address nonlinear hysteresis inputs, two primary strategies are typically employed. The first strategy involves mitigating hysteresis effects through the development of an adaptive hysteresis inverse controller [25]. The second strategy, similar to the methods outlined in [26], employs an adaptive algorithm to reduce the impacts of hysteresis by modeling it with both linear and nonlinear components.