## 2 Main Results
### Hybrid PID Controller with Smooth Transition to the Setpoint
#### 2.3.2 The Exponential Trajectory to the Setpoint  
As mentioned before, the proposed control scheme considers an exponential function, which is reset by the state of the hybrid system described earlier. The structure of such and exponential function is  
\[\big{(}x_{h}-x_{sp}\big{)}\exp(-\alpha t_{h}). \tag{6}\]  
Here, \(-\alpha\) represents the decay rate, where \(\alpha\) is the decay factor, which is a positive real number and a design parameter. Additionally, \(x_{h}\) and \(t_{h}\) are values provided by the hybrid system, and \(x_{sp}\) denotes the setpoint. Figure 2: Scheme of the closed-loop system with the hybrid PID controller and exponential trajectories for a smooth transition to the setpoint. According to Figure 2, the error \(e(t)\) is expressed as follows:  
\[e(t)=x_{sp}+\big{(}x_{h}-x_{sp}\big{)}\exp(-at_{h})-x(t), \tag{7}\]  
but at every instant the hybrid system is reset, i.e., when \(t=t_{r}\), the expression (7) simplifies to  
\[e(t) = x_{sp}+\big{(}x_{h}-x_{sp}\big{)}-x(t),\] \[e(t) = x_{h}-x(t), \tag{8}\]  
because at the reset instant \(t=t_{r}\), \(t_{h}\) is set to zero. Additionally, from (7), it follows that  
\[\lim_{t_{h}\to\infty}e(t)=x_{sp}-x(t), \tag{9}\]  
where \(t_{r}\) are the instants at which the hybrid system is reset. At this point, the following results naturally arise. **Theorem 1**. : _Suppose the following assumptions hold:_  
* _The value_ \(x(t)\) _is available at all_ \(t\)_._
* _The design parameter_ \(\alpha\in\mathbb{R}\) _satisfies_ \(\alpha>0\) _and the envelope generated by_ \(\exp(-\alpha t_{h})\) _is bounded by the system response under the influence of the mistuned PID controller._
* _The hybrid system (_4_)-(_5_) is reset at instant_ \(t=t_{r}\)_._
* _There exist gains_ \(K_{P}\)_,_ \(K_{I}\)_, and_ \(K_{D}\) _(and possibly_ \(N\)_) such that the PID controller given by (_1_) (or correspondingly (_2_)) is capable of stabilizing the process variable_ \(x(t)\) _in a neighborhood of_ \(x_{sp}\)_._ _Then, the exponential function (_6_) starts from_ \(x_{h}=x(t)\) _with_ \(t_{h}=0\) _at_ \(t=t_{r}\)_. Furthermore, the exponential function (_6_) tends asymptotically to zero as_ \(t\to\infty\)_. Consequently, the error defined by (_7_) is reset at_ \(t=t_{r}\) _starting at_ \(x_{h}-x(t)\) _and tending to_ \(x_{sp}-x(t)\) _as_ \(t\to\infty\)_, and accordingly, the PID controller (_1_) (or correspondingly (_2_)) takes the process variable_ \(x(t)\) _to a neighborhood of_ \(x_{sp}\) _following an exponential trajectory._  
Proof. : The proof follows directly from the previous discussion. **Remark 1**. : _\"Exponential trajectory to the setpoint\": Notice that at the instants when the hybrid system is reset, namely when \(t=t_{r}\), the error \(e(t)\) is initially zero rather than \(x_{sp}-x(t)\). This characteristic often leads to a significant reduction in the magnitude of the PID controller's output. The reason for this behavior lies in the exponential function, which is reset at \(t=t_{r}\) to start from the current value of \(x(t)\), i.e., \(x_{h}=x(t)\) and \(t_{h}=0\). Subsequently, the exponential function (6) asymptotically approaches zero as \(t_{h}\to\infty\). As a result, the error \(e(t)\) gradually converges to \(e(t)=x_{sp}-x(t)\), leading to a smooth and progressive reduction in \(e(t)\)._  
**Remark 2**. : _Note that the selection of the decay rate \(\alpha\) depends on the behavior induced by the mistuned PID in the sense that \(\alpha\) should be selected to produce a slower behavior than that induced by the classical PID, thus maintaining the stability originally provided by the PID controller. Furthermore, the exponential function, which is restored by the hybrid system, ensures that the error remains bounded. In other words, the envelope generated by \(\exp(-\alpha t_{h})\) must be bounded by the system response under the influence of the mistuned PID controller._  
**Remark 3**.