## 4 Control Strategies
### The Third Strategy  
The adaptive controller \(G_{c}^{a}\) shown in Figure 4 is a dynamic system with on-line parameter estimation. It updates an uncertain plant or controller's parameters, while running with the use of the measured signal or even computed system response. The objective of such an approach is to obtain a consistent performance of a control system during operation in an inconsistent environment. Indirect control with an adaptation estimates plant parameters to update the controller's parameters at the end after application of some updating rules. The procedure aims at a time convergence of the parameter values to the true unknowns. The direct adaptation estimates the controller's parameters only since, in this work, we apply the second one.  
In the third strategy below, we apply an adaptation represented by the block diagram shown in Figure 4.  
In the problem of \(\dot{\theta}_{lr}(t)\) reference trajectory to be followed by \(\dot{\theta}_{ln}(t)\) of the actual model of our multi-degree of freedom system, the model reference control engages a subsystem in which the input signal, being a difference \(e_{l}(t))\)-error of angular position of the driven load, is delivered to two adaptation blocks. The first one is responsible for estimation of the parameters of the controller operating on \(e_{l}(t)\), while the second one uses the error of the driving motor position:  
\[e_{m}(t)=\theta_{mr}(t)-\theta_{ma}(t),\quad\tilde{e}_{m}(t)=\beta_{2}e_{m}(t), \tag{23}\]  
both having a PID form.  
Figure 4: Block diagram of the closed-loop model reference adaptive PID (fractional PID, as well) control system with the additional time-dependent disturbances of the damping coefficient \(b_{la}(t)\), and a PI compensation of the resulting second-stage error dynamics.  
Before the worked out signal of the difference of the two adaptive PID controller outputs will reach the actual model input \(u_{a}(t)\), it is subject to the third dynamical action \(G_{Pl}\); see Figure 3. It computes a sum of a proportional and integral transformation of the error signal. The final step in the control subsystem is used to cancel the remaining non-compensated error dynamics. According to the procedure, the following adaptive control law (see adaptive control design methods and analysis in Reference [23]) is used in a modified form:  
\[\begin{split} u_{l}(t)=& a^{\left\lvert b\alpha_{l}(t) \right\rvert}+c\dot{\epsilon}_{l}(t)+I_{l}(t)\int_{t_{j}}^{t_{j+1}}\dot{ \epsilon}_{l}(\tau)d\tau\\ u_{m}(t)=& a^{\left\lvert b\alpha_{m}(t)\right\rvert }+c\dot{\epsilon}_{m}(t)+I_{m}(t)\int_{t_{j}}^{t_{j+1}}\dot{\epsilon}_{m}(\tau )d\tau\\ u_{a}(t)=& K_{pa}\Bigg{(}(u_{l}(t)-u_{m}(t))+\frac{1}{T_{ ia}}\int_{t_{j}}^{t_{j+1}}(u_{l}(\tau)-u_{m}(\tau))d\tau\Bigg{)},\end{split} \tag{24}\]  
for \(j=0,1,\ldots,j_{s}+j_{u}\).  
For adaptation of the time-dependent parameter of integral action \(I.(t)\), where \({}^{\omega},{}^{\omega}\) is to be replaced by \({}^{\omega}\)\({}^{\omega}\) or \({}^{\omega}\)\({}^{\omega}\)\({}^{\omega}\), the following adaptation for this part is applied:  
\[I.(t)=\begin{cases}\frac{1}{2},&\text{for}\;D>f\\ d,&\text{for}\;D\leq f,\end{cases} \tag{25}\]  
where \(D=c|\dot{\epsilon}.(t)|\). It is assumed that the constants: \(a\), \(b\), \(c\), \(d\), \(f\) are the same for both adaptation laws \(u_{l}(t)\) and \(u_{m}(t)\). In the numerical simulation the parameters of the regulator for both considered types of disturbance will be valued as follows: \(a=250\), \(b=8\), \(c=8\), \(d=0.001\), \(f=0.02\), \(K_{pa}=1\), \(T_{ia}=1\), \(\beta_{i}=1\).