The block diagram of the variable-structure sliding mode control of an asynchronous motor (SM-DTC) is illustrated in Fig. 1. The quantities that are required are the stator flux, the torque and the motor speed. It is possible, in certain cases, to suppress the speed control loop and to control the motor using only its torque and flux [19]. The combination of the SMC with the DTC-SPWM provides a robust control which keeps the parameters to be adjusted within a well-defined range (Fig. 1) [20].

The complete flowchart of the proposed algorithm is illustrated in Fig. 2.

### Stator flux and torque control

The main task of the variable structure controller, illustrated in Fig. 3, is to obtain a fast and reliable control of the torque and the stator flux. For this reason, two sliding mode controllers with PI regulators are designed, and the direct and quadrature reference voltages are obtained at the output of the controller to generate the SPWM. In the following illustration, the reference and the estimated values are designated respectively by (*) and (^).

Based on (1)–(2) and under the assumed orientation where the d component is aligned to the stator flux vector direction, i.e., the quadrature stator flux is zero ‘φ_{qs} = 0’, the developed equations can be written as:

$$ \left\{\begin{array}{l}{V}_{ds}={R}_s{I}_{ds}+\frac{d{\varphi}_{ds}}{dt}\\ {}{\mathrm{V}}_{qs}={R}_s{I}_{qs}+{\omega}_s{\varphi}_{ds}\\ {}{T}_e=\frac{3P}{4}\frac{\varphi_{ds}}{R_s}\left({\mathrm{V}}_{qs}-{\omega}_s{\varphi}_{ds}\right)\end{array}\right. $$

(4)

The above equations indicate that the direct and quadrature stator voltage components V_{ds} and V_{qs} can be employed for flux and torque control, respectively.

From the same perspective as the PI-DTC-SPWM control strategy, two sliding surfaces (S_{1},S_{2}) are selected according to (4), in which S_{1} is defined from the error of the stator flux to control the direct voltage component, while the surface S_{2} represents the error of the electromagnetic torque to allow the determination of the quadrature voltage component. Since defining a sliding surface based only on the error will not allow the imposition of the dynamics for the error correction [17], these two surfaces S_{1} and S_{2} are designed so as to enforce sliding-mode operation with first-order dynamics of \( {\mathrm{S}}_1={\dot{\mathrm{S}}}_1=0 \) and \( {\mathrm{S}}_2={\dot{\mathrm{S}}}_2=0 \) as:

$$ \left[\begin{array}{c}{S}_1\\ {}{S}_2\end{array}\right]=\left[\begin{array}{c}{e}_{\varphi s}+{c}_{\varphi s}\frac{de_{\varphi s}}{dt}\\ {}{e}_{Te}+{c}_{Te}\frac{de_{Te}}{dt}\end{array}\right] $$

(5)

where c_{φs} and c_{Te} are constant gains to be defined according to the desired dynamics. e_{φs} and e_{Te} are theerror functions that must be minimized:

$$ \left\{\begin{array}{l}{e}_{\varphi s}=\left|{\varphi}_s^{\ast}\right|-\left|{\hat{\varphi}}_s\right|\\ {}\ {e}_{Te}={T}_e^{\ast }-{\hat{T}}_e\end{array}\right. $$

(6)

In sliding mode, the control laws limit the state of the system to the surfaces (S_{1} and S_{2}) and their behavior is exclusively governed by (S_{1} = S_{2} = 0). First-order linear torque and flux error dynamics resulting from (5) are:

$$ \left\{\begin{array}{c}{e}_{\varphi s}=-{c}_{\varphi s}\frac{de_{\varphi s}}{dt}\\ {}{e}_{Te}=-{c}_{Te}\frac{de_{Te}}{dt}\end{array}\right. $$

(7)

Then, the control law can be proposed in a similar way as:

$$ \left\{\begin{array}{l}{V}_{ds}=\left({K}_{p\varphi s}+\frac{K_{I\varphi s}}{S}\right) sat\left({S}_1\right)\\ {}{V}_{qs}=\left({K}_{p Te}+\frac{K_{I Te}}{S}\right) sat\left({S}_2\right)+{\omega}_s{\hat{\varphi}}_s\end{array}\right. $$

(8)

where K_{pφs}, K_{pTe} are the proportional gains of the PI regulators which allow the convergence of the errors, and must be chosen to satisfy the condition of stability \( {S}_1\frac{dS_1}{dt}<0\ \mathrm{et}\ {S}_2\frac{dS_2}{dt}<0 \) using the Lyapunov criterion [21]. K_{Iφs}, K_{ITe} are the integral gains which also ensure convergence of the errors and decoupling between the torque and the flux. In order to reduce chattering phenomena, the traditional sign function of the switching control is replaced by a more flexible saturation function “sat(S_{1}) and sat(S_{2})”.

### Speed control

Figure 4 shows the speed loop using sliding mode control. The controller is designed so that the regulation loop generates the reference of the electromagnetic torque with a rapid dynamic response. The speed sliding surface is defined by:

$$ \left\{\begin{array}{c}{S}_{\varOmega }={\Omega}^{\ast }-\hat{\Omega}\\ {}{\dot{S}}_{\Omega}={\dot{\Omega}}^{\ast }-\hat{\dot{\Omega}}\end{array}\right. $$

(9)

The mechanical equation of the asynchronous motor is given by:

$$ J\dot{\varOmega}={T}_e-{T}_L-f\hat{\varOmega} $$

(10)

By substituting (9) into (10), the surface derivative is:

$$ {\dot{S}}_{\varOmega }=\dot{\Omega}\ast -\frac{1}{J}\left({T}_e-{T}_L-f\hat{\Omega}\right) $$

(11)

Based on the sliding mode theory, there is:

$$ {T}_e={T}_{eeq}+{T}_{en} $$

(12)

The equivalent command part (T_{eeq}) is defined during the sliding mode state with Ṡ_{Ω} = 0, T_{en} = 0 and \( {\dot{\varOmega}}^{\ast } \) =0, and is:

$$ {T}_{eeq}={T}_L+f\Omega $$

(13)

The non-linear part (Ten) is defined as:

$$ {T}_{en}={c}_{\varOmega } sat\left({S}_{\varOmega}\right) $$

(14)

From (13) and (14) the torque control equation in sliding mode is given by:

$$ {T}_e={T}_L+f\hat{\varOmega}+{c}_{\varOmega } sat\left({S}_{\varOmega}\right) $$

(15)

where c_{Ω} is a positive gain.