1. Introduction

2. Mathematical Model of a WT

3. Design of the Dynamic Controller for the Tracking of the Angular Velocity Reference

3.2. Design of the Dynamic Controller for the Tracking of the Angular Velocity Reference

Given the model (1), and considering the nominal values $R_s^{\circ }$, $L^{\circ }$, ${\varphi }_m^{\circ }$, $k_m^{\circ }$, $f^{\circ }$, $k_w^{\circ }$, one obtains the following model

$$\begin{array}{cc}\hfill \frac{d{i}_d}{dt}& =-\frac{R_s^{\circ }}{L^{\circ }}{i}_d+p\omega i_q+\frac{1}{L^{\circ }}{v}_d+{\Delta }_1\hfill \\ \hfill \frac{d{i}_q}{dt}& =-\frac{R_s^{\circ }}{L^{\circ }}{i}_q-p\omega i_d-p\frac{{\varphi }_m^{\circ }}{L^{\circ }}\omega +\frac{1}{L^{\circ }}{v}_q+{\delta }_2\hfill \\ \hfill \dot{\omega }& =-\frac{k_m^{\circ }}{J^{\circ }}{i}_q-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2+{\Delta }_3\hfill \end{array}$$

(4)

with the perturbation terms

$$\begin{array}{cc}\hfill {\Delta }_1& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)i_d+\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right)v_d\hfill \\ \hfill {\delta }_2& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)i_q-p\left(\frac{{\varphi }_m}{L}-\frac{{\varphi }_m^{\circ }}{L^{\circ }}\right)\omega +\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right)v_q\hfill \\ \hfill {\Delta }_3& =-\left(\frac{k_m}{J}-\frac{k_m^{\circ }}{J^{\circ }}\right)i_q-\left(\frac{f}{J}-\frac{f^{\circ }}{J^{\circ }}\right)\omega +\left(\frac{k_w}{J}-\frac{k_w^{\circ }}{J^{\circ }}\right){\omega }^2.\hfill \end{array}$$

The final perturbation term ${\Delta }_2$ for the dynamics of $i_q$ will be given by the sum of ${\delta }_2$ and additional terms arising from the derivative of the reference imposed on $i_q$. This will become clearer in the subsequent computations.

The following result solves the posed control problem for the system (4).

Theorem 1. 

If the signals $i_d$, $i_q$, ω are measurable, the dynamic controller

$$\begin{array}{cc}\hfill {\dot{\xi }}_{1,1}& =-\frac{R^{\circ }}{L^{\circ }}{i}_d+p\omega i_q+\frac{1}{L^{\circ }}{v}_d-{\alpha }_{1,1}{⌊{\xi }_{1,1}-(i_d-i_{d,\mathrm{ref}})⌉}^{1/2}+{\xi }_{1,2}\hfill \\ \hfill {\dot{\xi }}_{1,2}& =-{\alpha }_{1,2}{⌊{\xi }_{1,1}-(i_d-i_{d,\mathrm{ref}})⌉}^0\hfill \\ \hfill {\dot{\xi }}_{2,1}& =-\frac{R^{\circ }}{L^{\circ }}{i}_q-p\omega i_d-p\frac{{\varphi }_m^{\circ }}{L^{\circ }}\omega +\frac{1}{L^{\circ }}{v}_q-D_{i_{q,\mathrm{ref}}}^{\circ }-{\alpha }_{2,1}{⌊{\xi }_{2,1}-(i_q-i_{q,\mathrm{ref}})⌉}^{1/2}+{\xi }_{2,2}\hfill \\ \hfill {\dot{\xi }}_{2,2}& =-{\alpha }_{2,2}{⌊{\xi }_{2,1}-(i_q-i_{q,\mathrm{ref}})⌉}^0\hfill \\ \hfill {\dot{\xi }}_{3,1}& =c\left(-\frac{k_m^{\circ }}{J^{\circ }}{i}_q-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2\right)-{\alpha }_{3,1}{⌊{\xi }_{3,1}-(\omega -{\omega }_{\mathrm{ref}})⌉}^{1/2}+c{\xi }_{3,2}\hfill \\ \hfill {\dot{\xi }}_{3,2}& =-\frac{{\alpha }_{3,2}}{c}{⌊{\xi }_{3,1}-(\omega -{\omega }_{\mathrm{ref}})⌉}^0\hfill \\ \hfill {\dot{I}}_{e_{i_d}}& =i_d-i_{d,\mathrm{ref}}\hfill \\ \hfill {\dot{I}}_{e_{i_q}}& =i_q-i_{q,\mathrm{ref}}\hfill \\ \hfill {\dot{I}}_{e_{\omega }}& =\omega -{\omega }_{\mathrm{ref}}\hfill \\ \hfill v_d& =L^{\circ }\left(-k_{1,p}(i_d-i_{d,\mathrm{ref}})-k_{1,i}{I}_{e_{i_d}}+\frac{R_s^{\circ }}{L^{\circ }}{i}_d-p\omega i_q+\frac{d}{dt}{i}_{d,\mathrm{ref}}-{\xi }_{1,2}\right)\hfill \\ \hfill v_q& =L^{\circ }\left(-k_{2,p}(i_q-i_{q,\mathrm{ref}})-k_{2,i}{I}_{e_{i_q}}+\frac{R_s^{\circ }}{L^{\circ }}{i}_q+p\omega i_d+p\frac{{\varphi }_m^{\circ }}{L^{\circ }}\omega +D_{i_{q,\mathrm{ref}}}^{\circ }-{\xi }_{2,2}\right)\hfill \end{array}$$

(5)

${⌊·⌉}^{\alpha }={|·|}^{\alpha }\mathrm{sign}(·)$, with $c=1-{\lambda }_{max}^{\circ }/\widehat{\lambda }$, $k_{j,p},k_{j,i}>0$, $j=1,2,3$, and

$$\begin{array}{cccc}\hfill {\alpha }_{1,1}& >0,\hfill & \hfill {\alpha }_{1,2}& >2\frac{{\theta }_1^2}{{\alpha }_{1,1}^2}+3{\theta }_1\hfill \\ \hfill {\alpha }_{2,1}& >0,\hfill & \hfill {\alpha }_{2,2}& >2\frac{{\theta }_2^2}{{\alpha }_{2,1}^2}+3{\theta }_2\hfill \\ \hfill {\alpha }_{3,1}& >0,\hfill & \hfill {\alpha }_{3,2}& >2\frac{c^2{\theta }_3^2}{{\alpha }_{3,1}^2}+3c{\theta }_3\hfill \end{array}$$

(6)

for some finite positive reals ${\theta }_i$, $i=1,2,3$, ${\omega }_{\mathrm{ref}}$ given by (3),

$$\begin{array}{cc}\hfill i_{d,\mathrm{ref}}& =\phi \left(t\right)\hfill \\ \hfill i_{q,\mathrm{ref}}& =\frac{J^{\circ }}{k_m^{\circ }}\left(\frac{k_{3,p}}{c}(\omega -{\omega }_{\mathrm{ref}})+\frac{k_{3,i}}{c}{I}_{e_{\omega }}+{\xi }_{3,2}-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2\right)\hfill \end{array}$$

(7)

$\phi \left(t\right)$ bounded function with bounded derivative, and with

$$\begin{array}{cc}\hfill D_{i_{q,\mathrm{ref}}}^{\circ }& =\frac{J^{\circ }}{k_m^{\circ }}[\frac{k_{3,p}}{c}\left(-\frac{k_m^{\circ }}{J^{\circ }}c{e}_{i_q}-k_{3,p}{e}_{\omega }-k_{3,i}{I}_{e_{\omega }}-c{\xi }_{3,2}\right)\hfill \\ \hfill & +\frac{k_{3,i}}{c}{e}_{\omega }-\frac{{\alpha }_{3,2}}{c}{⌊e_{3,1}⌉}^0+\left(2\frac{k_w^{\circ }}{J^{\circ }}\omega -\frac{f^{\circ }}{J^{\circ }}\right)\left(-\frac{k_m^{\circ }}{J^{\circ }}{i}_q-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2\right)]\hfill \\ \hfill D_{i_{q,\mathrm{ref}}}& =\frac{J^{\circ }}{k_m^{\circ }}\left(\frac{k_{3,p}}{c}c+2\frac{k_w^{\circ }}{J^{\circ }}\omega -\frac{f^{\circ }}{J^{\circ }}\right){\Delta }_3\hfill \end{array}$$

(8)

ensures that the tracking error $\omega -{\omega }_{\mathrm{ref}}$, converges to zero asymptotically. Moreover, the errors $i_d-i_{d,\mathrm{ref}}$, $i_q-i_{q,\mathrm{ref}}$ converge asymptotically to zero. ⋄

Proof. 

Let us define the angular velocity tracking error $e_{\omega }=\omega -{\omega }_{\mathrm{ref}}=c\omega$. The dynamics of $e_{\omega }$ are

$$\begin{array}{cc}\hfill {\dot{e}}_{\omega }& =c\dot{\omega }=c\left(-\frac{k_m^{\circ }}{J^{\circ }}{i}_q-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2+{\Delta }_3\right)\hfill \\ \hfill & =c\left(-\frac{k_m^{\circ }}{J^{\circ }}{e}_{i_q}-\frac{k_m^{\circ }}{J^{\circ }}{i}_{q,\mathrm{ref}}-\frac{f^{\circ }}{J^{\circ }}\omega +\frac{k_w^{\circ }}{J^{\circ }}{\omega }^2+{\Delta }_3\right)\hfill \\ \hfill & =-\frac{k_m^{\circ }}{J^{\circ }}c{e}_{i_q}-k_{3,p}{e}_{\omega }-k_{3,i}{I}_{e_{\omega }}-e_{3,2}\hfill \end{array}$$

(9)

where $e_{i_q}=i_q-i_{q,\mathrm{ref}}$, with $i_{q,\mathrm{ref}}$ as in (7), and $e_{3,2}=c({\xi }_{3,2}-{\Delta }_3)$. The dynamics of $e_{i_q}$ are

$$\begin{array}{cc}\hfill {\dot{e}}_{i_q}& =-\frac{R_s}{L}{i}_q-p\omega i_d-p\frac{{\varphi }_m}{L}\omega +\frac{1}{L}{v}_q-\frac{d{i}_{q,\mathrm{ref}}}{dt}\hfill \\ & =-\frac{R_s^{\circ }}{L^{\circ }}{i}_q-p\omega i_d-p\frac{{\varphi }_m^{\circ }}{L^{\circ }}\omega +\frac{1}{L^{\circ }}{v}_q+{\delta }_2-D_{i_{q,\mathrm{ref}}}^{\circ }-D_{i_{q,\mathrm{ref}}}\hfill \\ & =-\frac{R_s^{\circ }}{L^{\circ }}{i}_q-p\omega i_d-p\frac{{\varphi }_m^{\circ }}{L^{\circ }}\omega +\frac{1}{L^{\circ }}{v}_q-D_{i_{q,\mathrm{ref}}}^{\circ }+{\Delta }_2\hfill \end{array}$$

with the derivative of $i_{q,\mathrm{ref}}$ given by

$$\begin{array}{cc}\hfill \frac{d}{dt}{i}_{q,\mathrm{ref}}& =\frac{J^{\circ }}{k_m^{\circ }}\left(\frac{k_{3,p}}{c}{\dot{e}}_{\omega }+\frac{k_{3,i}}{c}{e}_{\omega }+{\dot{\xi }}_{3,2}+\left(2\frac{k_w^{\circ }}{J^{\circ }}\omega -\frac{f^{\circ }}{J^{\circ }}\right)\dot{\omega }\right)=D_{i_{q,\mathrm{ref}}}^{\circ }+D_{i_{q,\mathrm{ref}}}\hfill \end{array}$$

$D_{i_{q,\mathrm{ref}}}^{\circ }$, $D_{i_{q,\mathrm{ref}}}$ as in (8), and with the (unknown) perturbation term ${\Delta }_2$ given by

$$\begin{array}{cc}\hfill {\Delta }_2& ={\delta }_2-D_{i_{q,\mathrm{ref}}}=-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)i_q-p\left(\frac{{\varphi }_m}{L}-\frac{{\varphi }_m^{\circ }}{L^{\circ }}\right)\omega +\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right)v_q\hfill \\ & +\frac{J^{\circ }}{k_m^{\circ }}\left(\frac{k_{3,p}}{c}c+2\frac{k_w^{\circ }}{J^{\circ }}\omega -\frac{f^{\circ }}{J^{\circ }}\right)\left[\left(\frac{k_m}{J}-\frac{k_m^{\circ }}{J^{\circ }}\right)i_q+\left(\frac{f}{J}-\frac{f^{\circ }}{J^{\circ }}\right)\omega -\left(\frac{k_w}{J}-\frac{k_w^{\circ }}{J^{\circ }}\right){\omega }^2\right].\hfill \end{array}$$

Using the controller (5), one works out

$$\begin{array}{cc}\hfill {\dot{e}}_{i_q}& =-k_{2,p}{e}_{i_q}-k_{2,i}{I}_{e_{i_d}}-e_{2,2}\hfill \end{array}$$

(10)

with $e_{2,2}={\xi }_{2,2}-{\Delta }_2$. Considering (9), (10), one finally gets the coupled dynamics

(11)

where the dynamics of the errors $e_{3,1}$, $e_{3,2}$, $e_{2,1}$, $e_{2,2}$, are

$$\begin{array}{cc}\hfill {\dot{e}}_{3,1}& =-{\alpha }_{3,1}{⌊e_{3,1}⌉}^{1/2}+e_{3,2}\hfill \\ \hfill {\dot{e}}_{3,2}& =-{\alpha }_{3,2}{⌊e_{3,1}⌉}^0-c{\dot{\Delta }}_3\hfill \\ \hfill {\dot{e}}_{2,1}& =-{\alpha }_{2,1}{⌊e_{2,1}⌉}^{1/2}+e_{2,2}\hfill \\ \hfill {\dot{e}}_{2,2}& =-{\alpha }_{2,2}{⌊e_{2,1}⌉}^0-{\dot{\Delta }}_2.\hfill \end{array}$$

(12)

The errors $e_{3,1}$, $e_{3,2}$ tend to zero in finite time. The proof of the global convergence to the origin in finite time of $e_{3,1}$, $e_{3,2}$ can be demonstrated with Lyapunov arguments [39,40], considering the Lyapunov candidate

This candidate is continuous but not differentiable. As explained in [39], the use of the classical version of Lyapunov’s theory [41], instead of nonsmooth versions [42,43], is allowed since $V_3$ is differentiable almost everywhere, i.e., except when $e_{3,1}=0$. The fact that the trajectories of (12) may pass through $e_{3,1}=0$ for time instants constituting a set of null measure before reaching the origin from a certain finite-time instant on allows using the classical version of Lyapunov’s theory. Therefore, considering the necessary modifications with respect to the proof given in [39]

$${\dot{V}}_3=-\frac{{\alpha }_{3,1}}{2|e_{3,1}{|}^{1/2}}{E}_3^T{Q}_{3,1}{E}_3-\frac{c{\dot{\Delta }}_3}{2|e_{3,1}{|}^{1/2}}{⌊e_{3,1}⌉}^0{E}_3^T{Q}_{3,0}{E}_3$$

where

We will proof later that, with the proposed controller, ${\dot{\Delta }}_3$ remains bounded, with $|{\dot{\Delta }}_3|\le {\theta }_3$ for all time instants. To take into account the perturbation ${\dot{\Delta }}_3$, ${\dot{V}}_3$ can be bounded as follows

with α_3,1>0, Q₃>0 under the conditions (6), and Also here, slight modifications with respect to the proof of [39] were necessary. This proves the finite−time convergence of E₃ to the origin.

With the same arguments, also the errors $e_{2,1}$, $e_{2,2}$ can be proved to tend to zero in finite time, considering the Lyapunov candidate

(13)

Also in this case it will be shown that ${\dot{\Delta }}_2$ remains bounded, with $|{\dot{\Delta }}_2|\le {\theta }_2$.

Since $e_{2,1}$, $e_{2,2}$, $e_{3,1}$, $e_{3,2}$ tend to zero in finite time, the tracking error dynamics (11) has the origin globally asymptotically stable, namely $e_{\omega }$, $I_{e_{\omega }}$, $e_{i_q}$, $I_{e_{i_q}}$ tend to zero asymptotically.

Finally, considering the expression of the reference $i_{d,\mathrm{ref}}$ as in (7) and the control $v_d$ as in (5), for the error $e_{i_d}=i_d-i_{d,\mathrm{ref}}$ one obtains

$$\begin{array}{cc}\hfill {\dot{e}}_{i_d}& =-\frac{R_s}{L}{i}_d+p\omega i_q+\frac{1}{L}{v}_d-\frac{d}{dt}{i}_{d,\mathrm{ref}}\hfill \\ \hfill & =-\frac{R_s^{\circ }}{L^{\circ }}{i}_d+p\omega i_q+\frac{1}{L^{\circ }}{v}_d-\frac{d}{dt}{i}_{d,\mathrm{ref}}+{\Delta }_1\hfill \\ \hfill & =-k_{1,p}{e}_{i_d}-k_{1,i}{I}_{e_{i_d}}-e_{1,2}\hfill \end{array}$$

(14)

where

$$\begin{array}{cc}\hfill {\dot{e}}_{1,1}& =-{\alpha }_{1,1}{⌊e_{1,1}⌉}^{1/2}+e_{1,2}\hfill \\ \hfill {\dot{e}}_{1,2}& =-{\alpha }_{1,2}{⌊e_{1,1}⌉}^0-{\dot{\Delta }}_1.\hfill \end{array}$$

(15)

With the same arguments used before, $e_{1,1}$, $e_{1,2}$ tend to zero in finite time, since $|{\dot{\Delta }}_1|\le {\theta }_1$. As a consequence, the closed–loop dynamics

has the origin globally asymptotically stable, namely $e_{i_d},I_{e_{i_d}}$ tend to zero asymptotically.

We can now show what stated before, namely, that ${\dot{\Delta }}_1,{\dot{\Delta }}_2,{\dot{\Delta }}_3$ remain bounded for all time instants. In fact, since the initial tracking errors $e_{i_d}\left(0\right)$, $e_{i_q}\left(0\right)$, $e_{\omega }\left(0\right)$ are finite, so are

$$\begin{array}{cc}\hfill {\Delta }_1\left(0\right)& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)i_d\left(0\right)+\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right)v_d\left(0\right)\hfill \\ \hfill {\Delta }_2\left(0\right)& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)i_q\left(0\right)-p\left(\frac{{\varphi }_m}{L}-\frac{{\varphi }_m^{\circ }}{L^{\circ }}\right)\omega +\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right)v_q\left(0\right)-D_{i_{q,\mathrm{ref}}}\left(0\right)\hfill \\ \hfill {\Delta }_3\left(0\right)& =-\left(\frac{k_m}{J}-\frac{k_m^{\circ }}{J^{\circ }}\right)i_q\left(0\right)-\left(\frac{f}{J}-\frac{f^{\circ }}{J^{\circ }}\right)\omega \left(0\right)+\left(\frac{k_w}{J}-\frac{k_w^{\circ }}{J^{\circ }}\right){\omega }^2\left(0\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\dot{\Delta }}_1\left(0\right)& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)\frac{d{i}_d}{dt}{|}_{t=0}+\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right){\dot{v}}_d{|}_{t=0}\hfill \\ \hfill {\dot{\Delta }}_2\left(0\right)& =-\left(\frac{R_s}{L}-\frac{R_s^{\circ }}{L^{\circ }}\right)\frac{d{i}_q}{dt}{|}_{t=0}-p\left(\frac{{\varphi }_m}{L}-\frac{{\varphi }_m^{\circ }}{L^{\circ }}\right)\dot{\omega }{|}_{t=0}+\left(\frac{1}{L}-\frac{1}{L^{\circ }}\right){\dot{v}}_q{{|}_{t=0}-{\dot{D}}_{i_{q,\mathrm{ref}}}|}_{t=0}\hfill \\ \hfill {\dot{\Delta }}_3\left(0\right)& =-\left(\frac{k_m}{J}-\frac{k_m^{\circ }}{J^{\circ }}\right)\frac{d{i}_q}{dt}{|}_{t=0}-\left(\frac{f}{J}-\frac{f^{\circ }}{J^{\circ }}\right)\dot{\omega }{{|}_{t=0}+\left(\frac{k_w}{J}-\frac{k_w^{\circ }}{J^{\circ }}\right)2\omega \left(0\right)\dot{\omega }|}_{t=0}.\hfill \end{array}$$

and the right sides of the equations (4) for $t=0$. Since the functions involved are also locally Lipschitz, this remains true for finite time intervals $[0,\delta )$, $\delta >0$. This implies that the dynamics (12), (15) converge to zero for $t\in [0,\delta )$. This reasoning can be iterated, so that the local Lipschitzianity implies that ${\dot{\Delta }}_1,{\dot{\Delta }}_2,{\dot{\Delta }}_3$ remain bounded for all time instants.

This concludes the proof. □

4. Simulation results

5. Conclusions

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