# A Robust Asynchronous Sampled-Data Control Design for Nonlinear Systems with Actuator Failures
## 2 Actuator Fault Observer and Asynchronous Sampled-Data Control Designs
### Event-Triggered Control Design  
The main objective of this sub-section is to complement the stated actuator fault observer design granted in Theorem 1 by using a discrete-time event-triggered rule. To begin with, let us assume that there exists a constant gain matrix \(K\) such that the state feedback control law  
\[u(t)=Kx(t), \tag{10}\]  
the system \((A+BK)\) is a Hurwitz matrix; i.e., the closed-loop system \(\dot{x}(t)=(A+BK)x(t)\), its origin equilibrium point is exponentially stable. Therefore, the system \((A,B)\) is assumed controllable too.  
Going further, let us invoke the discrete-time version of (10):  
\[u(t_{k})=Kx(t_{k});\quad t_{k}\leq t<t_{k+1},\quad k\in N, \tag{11}\]  
where \(x(t_{k})\) is the sampled of \(x(t)\) at \(t=t_{k}\), and the sequence \(t_{1},t_{2},..,t_{k}\). is an increasing one. The sampled-data control means that the value of \(u(t_{k})\) is hold, by employing Zero-Order-Hold (ZOH) devices (for instance, Microchip Technology Inc., Chandler, AZ, USA) embedded in the digital controller (for instance, Microchip Technology Inc., Chandler, AZ, USA), until the next event happens at \(t=t_{k+1}\). Then, the event-triggered control signal design consists of finding a sampling-rule to grant the next sample-time \(t_{k+1}\). In our approach, we first consider:  
\[\ddot{x}(t)=x(t_{k})-x(t),\quad k\in N. \tag{12}\]  
From the above equation, the sampled-data closed-loop system yields:  
\[\dot{x}(t) = Ax(t)+BKx(t_{k}) \tag{13}\] \[= (A+BK)x(t)+BK\ddot{x}(t).\]  
And because \(K\) is designed such that the system \((A+BK)\) is a Hurwitz matrix, the above system is then Input-to-State Stable in the Lyapunov's sense. Moreover, if \(\ddot{x}(t)\) is a bounded term, then the above dynamic will be a BIBO-stable system. So, from (13), it is clear that an option to design a triggered sampling rule is by employing the _internal_ signal given by:  
\[u(t_{k})-u(t)=K[x(t_{k})-x(t)]. \tag{14}\]Based on the previous evidences, next is our main result.  
**Theorem 2**.: _Given the linear model (2) to the nonlinear system (1), the next dynamic system locally stabilizes its origin equilibrium point by using self-triggered sampled-data information:_  
\[\hat{x} = Ax+B(u-\hat{u}_{a})+L(x-\hat{x}),\quad\text{with a given $L=L^{T}>0$,} \tag{15}\] \[\hat{u}_{a} = -\alpha B^{T}(x-\hat{x});\quad\text{with a given $\alpha\in R^{+}$}\] (16) \[u(t_{k}) = Kx(t_{k})+\hat{u}_{a}(t_{k});\quad t_{k}\leq t<t_{k+1},\quad k \in N, \tag{17}\]  
_where \(K\) is selected such that \((A+BK)\) is a Hurwitz matrix, and by using the next sampling-rate rule_  
\[t_{k+1}=\text{inf}\{t>t_{k}|\quad||u(t_{k})-u(t)||>\delta\},\quad\forall k\in N, \tag{18}\]  
_for some user given small positive constant \(\delta\)._  
**Remark 1**.: To observe, the controller in Theorem 2 is a non-uniform sampled-data system whereas the faulty estimation signal \(\hat{u}_{a}(t)\) is expressed in continuous-time domain (See Figure 1). But, obviously, this signal can be translate into a discrete-time domain if preferable for digital-monitoring propose.