Robust Proportional–Integral Sliding Mode Control for Induction Motors with Input Time Delay

Abstract

This paper proposes a control strategy applied to a three-phase induction motor (TIM) subject to parametric uncertainties, perturbations, and input time delay, whose primary objective is to achieve high-performance speed–torque control. The control design involves a predictive sliding mode observer (P-SMO) and a robust proportional–integral sliding mode control (PISM), aimed at reducing the detrimental effects of time delay and perturbations. The proposed control strategy’s effectiveness is investigated through computational simulations, carried out for different scenarios, whose distinctions focus on the consideration of delay in the feedback signals, the predictive character of the sliding mode observer, and the type of controller used. The presented results show the superior performance of the PISM controller compared with the classic PI controller for all tested scenarios. In the test scenario that considers the transport delay in the feedback signals, the sliding mode observer (SMO) without prediction does not stabilize the system, requiring the application of P-SMO to ensure stability and accurate tracking of the TIM speed reference.

1. Introduction

Induction motors are widely utilized in the industrial sector and electric vehicles due to their advantages, including physical robustness, low cost of construction, ease of maintenance, and safety during operation. A field-oriented control (FOC) strategy can be applied to three-phase induction motors (TIM) for improved speed–torque control [1,2,3]. This technique decouples magnetic flux control and torque control, enhancing the dynamic performance of the induction motor. Under this approach, the complex model of the TIM becomes equivalent to a simpler model of a separately excited direct current (DC) motor. Both linear and nonlinear control techniques can be utilized to control the decoupled structure.

Linear control laws, including proportional–integral (PI) control, are widely utilized in the industrial sector. The design of a robust PI controller involves adjusting its gains to effectively reject disturbances in the operating range, which necessitates an accurate understanding of the parameters of the induction motor and mechanical load. However, some of the parameters of the induction motor models are either poorly known or can vary based on the operating point, leading to a decline in the performance of PI controllers [4,5]. As a result, various nonlinear control techniques have been proposed in the literature to self-adjust PI gains [6,7,8].

A robust nonlinear control approach is the variable structure control (VSC). This control strategy is characterized by the use of a control law with a high switching frequency that keeps the trajectory of the system state variables within a designated region in the state space, referred to as the “sliding surface” [9,10,11,12]. When the trajectory of the state variables reaches the sliding surface and stays there, the system is said to be in sliding mode. VSC has the advantage that in sliding mode, the system is immune to a class of uncertainties and perturbations [10]. However, high-frequency switching may be degenerative for the system performance due to the “chattering effect” [11]. To avoid the chattering effect, a continuous sliding mode control law (SMC) can be used to smooth the switched VSC. SMC uses a continuous sigmoid function, such that the absolute robustness on the sliding surface of the system converges to a small domain referred to as the “boundary layer” [12].

The use of digital devices to implement PI and SMC control laws in applications is currently highly advantageous. Digital signal processing also provides the added advantage of enabling the implementation of wireless remote control structures and Networked Control Systems (NCS). In general, the application of NCS in industrial processes results in reduced costs and maintenance requirements [13,14,15]. However, there are numerous problems associated with NCS that have been investigated, including issues related to data loss failures or time delays in control signals, which have become increasingly significant in the context of controlling dynamic systems [15,16,17,18,19,20].

The high switching frequency in the control law makes the performance of the VSC highly sensitive to input time delay. As a result, SMC is more effective than VSC for controlling systems that are subject to input time delay [19,20]. Numerous designs of SMC applied to speed–torque control of three-phase induction motors have been documented in the literature [21,22,23,24,25,26,27]. These works mainly focus on analyzing the robustness of the control laws with respect to parametric uncertainties and input disturbances, ignoring the effect of input time delay. Some other studies on induction motor control consider both uncertainties and input time delay but do not employ SMC strategies [28,29].

The present work proposes a control design using proportional–integral with sliding mode (PISM) for controlling a three-phase induction motor. The main objective of the proposed PISM control strategy is to combine the advantages of proportional–integral control (PI), commonly used in industry, with sliding mode control (SMC) to provide robustness against a significant portion of the disturbances and uncertainties present in the system. Furthermore, a predictive control strategy (PISM-P) is proposed such that it guarantees robustness concerning input time delay.

This paper is organized as follows: Section 2 reviews the nonlinear model of a three-phase induction motor. Section 3 describes and analyzes the robustness of a sliding mode observer (SMO) applied to estimate the state variables of the induction motor model. Section 4 describes and examines the robustness of a proportional–integral sliding mode controller (PISM). Section 5 proposes a predictive sliding mode observer (P-SMO) that considers input time delay in the induction motor model. Section 6 analyzes the robustness when the P-SMO and PISM controllers are used for speed control. Simulation results are presented in Section 7 to corroborate the analyses. Finally, the conclusions are presented in Section 8.

2. Mathematical Model of the Three-Phase Induction Motor

A model of the three-phase induction motor consists of the coordinates’ transformation into a single coordinate. This single coordinate is defined by a variable representing the magnetic flux in the rotor. Thus, a model similar to the DC motor, called vector control [1,2,3], can be obtained, providing more effective control of the three-phase induction motor.

Consider the base values of the motor: current ($I_B$); angular speed (${\omega }_B$); and mechanical torque ($m_B$). A normalized mathematical model of the induction motor [1,2,3], including uncertainties and disturbances, is as follows:

$${\dot{x}}_1\left(t\right)=-\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)x_1\left(t\right)+\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left(u_1\left(t\right)+{\Delta }_{u1}\right)$$

(1)

$${\dot{x}}_2\left(t\right)={\omega }_B{x}_3\left(t\right)+\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\frac{\left(u_2\left(t\right)+{\Delta }_{u2}\right)}{x_1\left(t\right)}$$

(2)

$${\dot{x}}_3\left(t\right)=\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)x_1\left(t\right)\left(u_2\left(t\right)+{\Delta }_{u2}\right)-\frac{1}{{\tau }_m}\nu \left(t\right)$$

(3)

where:

• $u_1\left(t\right)=i_{Sd}\left(t\right)/\phantom{i_{Sd}\left(t\right)I_B}{I}_B$ and $u_2\left(t\right)=i_{Sq}\left(t\right)/\phantom{i_{Sq}\left(t\right)I_B}{I}_B$ are the normalized control inputs;
• $i_{sd}\left(t\right)$ and $i_{sq}\left(t\right)$ are direct and quadrature axes stator currents, respectively;
• $x_1\left(t\right)=i_{mR}\left(t\right)/\phantom{i_{mR}\left(t\right)I_B}{I}_B$ is the normalized magnitude of the rotor magnetizing current, with $x_1\left(t\right)\ne 0$;
• $x_2\left(t\right)=\rho \left(t\right)$ is the phase of the rotor magnetizing current vector;
• $x_3\left(t\right)=\omega \left(t\right)/\phantom{\omega \left(t\right){\omega }_B}{\omega }_B$ is the normalized rotor angular speed;
• $\nu \left(t\right)=m_L\left(t\right)/\phantom{m_L\left(t\right)m_B}{m}_B$ is the normalized load torque;
• $m_L$ is the load torque;
• ${\tau }_R=L_R/R_R$ is the nominal rotor time constant;
• $L_R$ and $R_R$ are the inductance and resistance of the rotor, respectively;
• $k_M=\left(2L_0{I}_B^2\right)/\phantom{\left(2L_0{I}_B^2\right)\left[3\left(1+{\sigma }_R\right)m_B\right]}\left[3\left(1+{\sigma }_R\right)m_B\right]$ is the nominal electric constant;
• $L_0$ is the mutual inductance;
• ${\sigma }_R$ is the rotor leakage factor;
• ${\tau }_m=\left(J{\omega }_B\right)/\phantom{\left(J{\omega }_B\right)m_B}{m}_B$ is the nominal mechanical constant of the induction motor;
• J is the inertia moment constant;
• ${\Delta }_{u1}\left(t\right)$, ${\Delta }_{u2}\left(t\right)$ are unknown but bounded input disturbances;
• ${\Delta }_{tr}\left(t\right)$, ${\Delta }_{km}\left(t\right)$ are unknown but bounded parametric uncertainties;
• The uncertain parameters $\left(1/{\tau }_R+{\Delta }_{tr}\right)$ and $\left(k_M/{\tau }_m+{\Delta }_{km}\right)$ have positive numerical values;
• ${\omega }_{ref}$ is the reference rotor speed in radians per second (rad/s);
• ${\omega }_B$ is the nominal rotor speed in radians per second (rad/s).

If input time delay is present, then the control laws $u_1\left(t\right)$ and $u_2\left(t\right)$ in (1)–(3) are replaced by $u_1(t-h)$ and $u_2(t-h)$, where $h>0$ is the time delay.

The main objective of the TIM control is to generate three-phase stator currents such that the rotor speed tracks a reference $x_{3ref}\left(t\right)=\omega {}_{\mathrm{re}f}\left(t\right)/\omega {}_B$. Also, like an independent DC motor with constant excitation current, a strategy used in field coordinates is to make the rotor magnetizing current track a constant value $x_{1\mathrm{Re}f}={i_{mR}}_{\mathrm{Re}f}/I_B$.

The available variables are the rotor speed $x_3\left(t\right)=\omega \left(t\right)/\phantom{\omega \left(t\right){\omega }_B}{\omega }_B$ and the three-phase stator currents $i_a\left(t\right)$,$i_b\left(t\right)$,$i_c\left(t\right)$. Thus, $x_1\left(t\right)=i_{mR}\left(t\right)/\phantom{i_{mR}\left(t\right)I_B}{I}_B$ and $x_2\left(t\right)=\rho \left(t\right)$ are internal variables and therefore must be estimated through a state variable observer. The estimated state variables are ${\tilde{x}}_1\left(t\right)={\tilde{i}}_{mR}\left(t\right)/\phantom{{\tilde{i}}_{mR}\left(t\right)I_B}{I}_B$ with ${\tilde{x}}_1\left(t\right)\ne 0$ for all time, and ${\tilde{x}}_2\left(t\right)=\tilde{\rho }\left(t\right)$. The control currents $u_1\left(t\right)=i_{Sd}\left(t\right)/\phantom{i_{Sd}\left(t\right)I_B}{I}_B$ and $u_2\left(t\right)=i_{Sq}\left(t\right)/\phantom{i_{Sq}\left(t\right)I_B}{I}_B$ generate the three-phase reference currents ${i_a}^{*}\left(t\right)$,${i_b}^{*}\left(t\right)$, and ${i_c}^{*}\left(t\right)$ for the internal control loop of the PWM inverter by computation of Equations (4)–(6):

$$\tilde{\rho }\left(t\right)=\int {\dot{\tilde{x}}}_2\left(t\right)dt$$

(4)

$$\left\{\begin{array}{c}{i}_{S1}\left(t\right)=i_{sd}\left(t\right)\cos \tilde{\rho }\left(t\right)-i_{sq}\left(t\right)\sin \tilde{\rho }\left(t\right)\hfill \\ i_{S2}\left(t\right)=i_{sd}\left(t\right)\sin \tilde{\rho }\left(t\right)+i_{sq}\left(t\right)\cos \tilde{\rho }\left(t\right)\hfill \end{array}\right\}$$

(5)

$$\left\{\begin{array}{c}{i_a}^{*}\left(t\right)=\frac{2}{3}{i}_{S1}\left(t\right)\hfill \\ {i_b}^{*}\left(t\right)=-\frac{1}{3}{\mathrm{i}}_{\mathrm{S}1}\left(t\right)+\frac{\sqrt{3}}{3}{i}_{S2}\left(t\right)\hfill \\ {i_c}^{*}\left(t\right)=-\frac{1}{3}{\mathrm{i}}_{\mathrm{S}1}\left(t\right)-\frac{\sqrt{3}}{3}{i}_{S2}\left(t\right)\hfill \end{array}\right\}$$

(6)

The control signals $u_1\left(t\right)+{\Delta }_{u1}\left(t\right)$ and $u_2\left(t\right)+{\Delta }_{u2}\left(t\right)$ can be obtained by the measured stator currents $i_a\left(t\right)$, $i_b\left(t\right)$, $i_c\left(t\right)$, and computation of (7) and (8):

$$\left\{\begin{array}{c}{i}_{S1}\left(t\right)=\frac{3}{2}{i}_a\left(t\right)\hfill \\ i_{S2}\left(t\right)=\frac{\sqrt{3}}{2}\left(i_b\left(t\right)-i_c\left(t\right)\right)\hfill \end{array}\right\}$$

(7)

$$\left\{\begin{array}{c}{u}_1\left(t\right)+{\Delta }_{u_1}\left(t\right)=i_{S1}\left(t\right)\cos \tilde{\rho }\left(t\right)+i_{S2}\left(t\right)\sin \tilde{\rho }\left(t\right)\hfill \\ u_2\left(t\right)+{\Delta }_{u_2}\left(t\right)=i_{S2}\left(t\right)\cos \tilde{\rho }\left(t\right)-i_{S1}\left(t\right)\sin \tilde{\rho }\left(t\right)\hfill \end{array}\right.$$

(8)

The estimated state variables ${\tilde{x}}_1\left(t\right)={\tilde{i}}_{mR}\left(t\right)/\phantom{{\tilde{i}}_{mR}\left(t\right)I_B}{I}_B$ and ${\tilde{x}}_2\left(t\right)=\tilde{\rho }\left(t\right)$ are computed in this work with a sliding mode observer, as presented by [30] and reviewed in the next section.

3. Sliding Mode Observer (SMO)

To estimate the variables, the following SMO [30] is used:

$${\dot{\tilde{x}}}_1\left(t\right)=-\frac{1}{{\tau }_R}{\tilde{x}}_1\left(t\right)+\frac{1}{{\tau }_R}\left(u_1\left(t\right)+\Delta u_1\right)$$

(9)

$${\dot{\tilde{x}}}_2\left(t\right)={\omega }_B{\tilde{x}}_3\left(t\right)+\frac{1}{{\tau }_R}\frac{\left(u_2\left(t\right)+\Delta u_2\right)}{{\tilde{x}}_1\left(t\right)}$$

(10)

$${\dot{\tilde{x}}}_3\left(t\right)=\frac{k_M}{{\tau }_m}{\tilde{x}}_1\left(t\right)\left(u_2\left(t\right)+\Delta u_2\right)-\frac{1}{{\tau }_m}\tilde{\nu }\left(t\right)+L_1\mathrm{sgm}\left(x_3\left(t\right)-{\tilde{x}}_3\left(t\right)\right)$$

(11)

$$\dot{\tilde{\nu }}\left(t\right)=L_2\mathrm{sgm}\left(x_3\left(t\right)-{\tilde{x}}_3\left(t\right)\right)$$

(12)

where $L_1$, $L_2>0$ are the gains, ${\tilde{x}}_1\left(t\right)\ne 0$ for all time, $\tilde{x}\left(t\right)=[{\tilde{x}}_1,{\tilde{x}}_2,{\tilde{x}}_3]$ is the estimated state vector, and $\tilde{\nu }\left(t\right)$ is the estimated mechanical load. The control signals $u_1\left(t\right)+\Delta u_1$ and $u_2\left(t\right)+\Delta u_2$ are obtained according to (4), (7) and (8). The continuous function $\mathrm{sgm}(·)$ is given by

$$\mathrm{sgm}\left(x\right)=\frac{x}{\left|x\right|+\delta };\delta \to 0^{+}$$

(13)

where $\delta$ is a small positive number and $\left|x\right|$ is the absolute value of x. The function $\mathrm{sgm}\left(x\right)$ is used in order to smooth the switched VSC to avoid the chattering phenomenon. The continuous function $\mathrm{sgm}\left(x\right)$ creates a boundary layer, proportional to $\delta$, around the sliding surface [11,12]. Thus, stability analyses must show the conditions for the trajectory of the state variables to reach the boundary layer and remain there. This is called the attractiveness condition.

Estimated errors are defined as

$$e_1\left(t\right)=x_1\left(t\right)-{\tilde{x}}_1\left(t\right)$$

(14)

$$e_2\left(t\right)=x_2\left(t\right)-{\tilde{x}}_2\left(t\right)$$

(15)

$$e_3\left(t\right)=x_3\left(t\right)-{\tilde{x}}_3\left(t\right)$$

(16)

$$e_4\left(t\right)=\nu \left(t\right)-\tilde{\nu }\left(t\right)$$

(17)

The sliding function of SMO is defined as

$${\sigma }_o\left(t\right)=e_3\left(t\right)$$

(18)

Using a Lyapunov function candidate as $V=(1/2){\left({\sigma }_o\left(t\right)\right)}^2$, the attractiveness condition of the state variables trajectory to the sliding surface is given by [12]

$${\sigma }_o\left(t\right){\dot{\sigma }}_o\left(t\right)<0$$

(19)

Substituting (3) and (11) into the derivative of (16), we have

$$e_3\left(t\right){\dot{e}}_3\left(t\right)=-L_1\underset{\le 0}{\underbrace{e_3\left(t\right)\mathrm{sgm}\left(e_3\left(t\right)\right)}}+e_3\left(t\right){\Delta }_1\left(t\right)$$

(20)

with

$${\Delta }_1\left(t\right)=\frac{k_M}{{\tau }_m}{e}_1\left(t\right)\left(u_2\left(t\right)+{\Delta }_{u2}\right)-\frac{1}{{\tau }_m}{e}_4\left(t\right)+{\Delta }_{km}{x}_1\left(t\right)\left(u_2\left(t\right)+{\Delta }_{u2}\right)$$

(21)

Taking into account the boundary layer, to satisfy the attractiveness condition (19), we have the conditions (22) and (23):

$$L_1>\sup \left|{\Delta }_1\left(t\right)\right|$$

(22)

$$\left|{\sigma }_o\left(t\right)\right|>\delta$$

(23)

where $\sup \left|{\Delta }_1\left(t\right)\right|$ is the supremum of $\left|{\Delta }_1\left(t\right)\right|$ and $\left|{\sigma }_o\left(t\right)\right|\le \delta$ is the boundary layer.

Figure 1 shows the scheme of a TIM control with proportional–integral sliding mode (PISM) as proposed in the next section.

4. Proportional–Integral with Sliding Mode Control (PISM)

This section proposes a control strategy which uses a proportional–integral control $u_{PI}\left(t\right)$ with sliding mode control $u_{SM}\left(t\right)$ according to (24):

$$\left\{\begin{array}{c}\left[\begin{array}{c}{u}_1\left(t\right)\\ u_2\left(t\right)\end{array}\right]=\left[\begin{array}{c}{u}_{PI1}\left(t\right)\\ u_{PI2}\left(t\right)\end{array}\right]+\left[\begin{array}{c}{u}_{SM1}\left(t\right)\\ u_{SM2}\left(t\right)\end{array}\right]\hfill \\ u\left(t\right)=u_{PI}\left(t\right)+u_{SM}\left(t\right)\hfill \end{array}\right.$$

(24)

where $u_{PI}\left(t\right)\in {\Re }^2$ establishes the dynamics of the sliding trajectory, and $u_{SM}\left(t\right)\in {\Re }^2$ is the control which keeps the state variables trajectory in the sling mode.

The proportional–integral control laws are given by (25) and (26):

$$u_{PI1}\left(t\right)=-\left[\left(k_{p1}\right){\overline{x}}_1\left(t\right)+\left(k_{i1}\right)\int {\overline{x}}_1\left(t\right)dt\right]$$

(25)

$$u_{PI2}\left(t\right)=-\frac{1}{{\tilde{x}}_1\left(t\right)}\left[\left(k_{p2}\right){\overline{x}}_3\left(t\right)+\left(k_{i2}\right)\int {\overline{x}}_3\left(t\right)dt\right]$$

(26)

where

$${\overline{x}}_1\left(t\right)={\tilde{x}}_1\left(t\right)-x_{1ref}\left(t\right)$$

(27)

$${\overline{x}}_3\left(t\right)=x_3\left(t\right)-x_{3ref}\left(t\right)$$

(28)

The sliding mode control laws are give by (29) and (30):

$$u_{SM1}\left(t\right)=-{\rho }_1\mathrm{sgm}\left({\sigma }_1\left(t\right)\right)$$

(29)

$$u_{SM2}\left(t\right)=-\frac{1}{{\tilde{x}}_1\left(t\right)}{\rho }_2\mathrm{sgm}\left({\sigma }_2\left(t\right)\right)$$

(30)

where the sliding functions are given by (31) and (32):

$${\sigma }_1\left(t\right)={\overline{x}}_1\left(t\right)$$

(31)

$${\sigma }_2\left(t\right)={\overline{x}}_3\left(t\right)$$

(32)

The parameters $k_{p1}$, $k_{p2}$, $k_{i1}$, $k_{i2}$, ${\rho }_1$, and ${\rho }_2$ are positive constants, $x_{1ref}$ and $x_{3ref}$ are the magnetizing current and rotor speed references, respectively. The $\mathrm{sgm}(·)$ function is given by (13). The estimated variable ${\tilde{x}}_1\left(t\right)$ is obtained by SMO.

If ${\rho }_1=0.0$ and ${\rho }_2=0.0$, then (24)–(26) represents the conventional proportional–integral control law (PI).

If ${\rho }_1>0.0$ and ${\rho }_2>0.0$, then (24)–(32) represents the proportional–integral with sliding mode control law (PISM).

4.1. Robustness Analysis of PISM

For stability, we have the following conditions:

$${\overline{x}}_1\left(t\right){\dot{\overline{x}}}_1\left(t\right)<0$$

(33)

$${\overline{x}}_3\left(t\right){\dot{\overline{x}}}_3\left(t\right)<0$$

(34)

If the sliding mode condition (22) of SMO is satisfied, then using (25), (27), (29), (31), and (1), we have

$${\overline{x}}_1\left(t\right){\dot{\overline{x}}}_1\left(t\right)=-\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left[\begin{array}{c}\left(1+k_{p1}\right){\left({\overline{x}}_1\left(t\right)\right)}^2\hfill \\ +\left({\rho }_1\right){\overline{x}}_1\left(t\right)\mathrm{sgm}\left({\overline{x}}_1\left(t\right)\right)\hfill \end{array}\right]-{\Delta }_2\left(t\right)$$

(35)

where

$${\Delta }_2\left(t\right)={\overline{x}}_1\left(t\right)\left\{{\dot{x}}_{1ref}\left(t\right)+\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left[\begin{array}{c}\left(k_{i1}\right)\int {\overline{x}}_1\left(t\right)dt\hfill \\ +x_{1ref}-{\Delta }_{u1}\hfill \end{array}\right]\right\}$$

(36)

From (35), to satisfy (33), we have conditions (37) and (38):

$$\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left[\begin{array}{c}\left(1+k_{p1}\right){\left({\overline{x}}_1\left(t\right)\right)}^2\hfill \\ +\left({\rho }_1\right){\overline{x}}_1\left(t\right)\mathrm{sgm}\left({\overline{x}}_1\left(t\right)\right)\hfill \end{array}\right]>\left|{\Delta }_2\left(t\right)\right|;\forall t\ge t_0$$

(37)

$$\left|{\sigma }_1\left(t\right)\right|>\delta$$

(38)

Using (3), (26), (28), (30), and (32), we have

$${\overline{x}}_3\left(t\right){\dot{\overline{x}}}_3\left(t\right)=-\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)\left[\begin{array}{c}\left(k_{p2}\right){\left({\overline{x}}_3\left(t\right)\right)}^2\hfill \\ +\left({\rho }_2\right){\overline{x}}_3\left(t\right)\mathrm{sgm}\left({\overline{x}}_3\left(t\right)\right)\hfill \end{array}\right]-{\Delta }_3\left(t\right)$$

(39)

where

$${\Delta }_3\left(t\right)={\overline{x}}_3\left(t\right)\left[\begin{array}{c}\frac{1}{{\tau }_m}\nu \left(t\right)+{\dot{x}}_{3ref}\left(t\right)\hfill \\ +\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)\left(k_{i2}\int {\overline{x}}_3\left(t\right)dt-{\Delta }_{u2}{x}_1\left(t\right)\right)\hfill \end{array}\right]$$

(40)

From (39), to satisfy (34), we have conditions (41) and (42):

$$\left(\begin{array}{c}\frac{k_M}{{\tau }_m}+{\Delta }_{km}\hfill \end{array}\right)\left[\begin{array}{c}\left(k_{p2}\right){\left({\overline{x}}_3\left(t\right)\right)}^2+\left({\rho }_2\right){\overline{x}}_3\left(t\right)\mathrm{sgm}\left({\overline{x}}_3\left(t\right)\right)\hfill \end{array}\right]>\left|{\Delta }_3\left(t\right)\right|;\forall t\ge t_0$$

(41)

$$\left|{\sigma }_2\left(t\right)\right|>\delta$$

(42)

With bounded uncertainties and perturbations, then $\left|{\Delta }_2\left(t\right)\right|$ and $\left|{\Delta }_3\left(t\right)\right|$ are also bounded. Therefore, limited numerical values exist for the parameters $k_{p1}$, $k_{p2}$, ${\rho }_1$, and ${\rho }_2$ such that (37) and (41) can be satisfied. Thus, we have ${\overline{x}}_1\left(t\right)\approx 0$ and ${\overline{x}}_3\left(t\right)\approx 0$ in steady state, confined within the boundary layer.

4.2. Robustness Analysis of PISM with Input Time Delay

If input time delay $h>0$ is present in (1), then using (25), (27), (29), and (31), we have

$${\overline{x}}_1\left(t\right){\dot{\overline{x}}}_1\left(t\right)=-\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left[\begin{array}{c}{\left({{\overline{x}}^{}}_1\left(t\right)\right)}^2+\left(k_{p1}\right){\overline{x}}_1\left(t\right){\overline{x}}_1(t-h)\hfill \\ +\left({\rho }_1\right){\overline{x}}_1\left(t\right)\mathrm{sgm}\left({\overline{x}}_1(t-h)\right)\hfill \end{array}\right]-{\overline{\Delta }}_2\left(t\right)$$

(43)

where

$${\overline{\Delta }}_2\left(t\right)={\overline{x}}_1\left(t\right)\left[{\dot{x}}_{1ref}\left(t\right)+\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left(\begin{array}{c}{k}_{i1}\int {\overline{x}}_1(t-h)dt\hfill \\ +x_{1ref}\left(t\right)-{\Delta }_{u1}\hfill \end{array}\right)\right]$$

(44)

From (43), to satisfy (33), we have conditions (45)–(47):

$$\mathrm{sgn}\left({\overline{x}}_1\left(t\right)\right)=\mathrm{sgn}\left({\overline{x}}_1(t-h)\right);\forall t\ge t_0$$

(45)

$$\left(\frac{1}{{\tau }_R}+{\Delta }_{tr}\right)\left[\begin{array}{c}{\left({{\overline{x}}^{}}_1\left(t\right)\right)}^2+\left(k_{p1}\right){\overline{x}}_1\left(t\right){\overline{x}}_1(t-h)\hfill \\ +\left({\rho }_1\right){\overline{x}}_1\mathrm{sgm}\left({\overline{x}}_1(t-h)\right)\hfill \end{array}\right]>\left|{\overline{\Delta }}_2\left(t\right)\right|;\forall t\ge t_0$$

(46)

$$\left|{\sigma }_1(t-h)\right|>\delta$$

(47)

Using (3), (26), (28), (30), and (32) with time delay, we have

$$\begin{array}{c}{\overline{x}}_3\left(t\right){\dot{\overline{x}}}_3\left(t\right)=-\left(\begin{array}{c}\frac{k_M}{{\tau }_m}\hfill \\ +{\Delta }_{km}\hfill \end{array}\right)\left(\frac{x_1\left(t\right)}{x_1(t-h)}\right)\left[\begin{array}{c}\left(k_{p2}\right){\overline{x}}_3\left(t\right){\overline{x}}_3(t-h)\hfill \\ +\left({\rho }_2\right){\overline{x}}_3\left(t\right)\mathrm{sgm}\left({\overline{x}}_3(t-h)\right)\hfill \end{array}\right]-{\overline{\Delta }}_3\left(t\right)\hfill \end{array}$$

(48)

where

$${\overline{\Delta }}_3\left(t\right)={\overline{x}}_3\left(t\right)\left[\begin{array}{c}\frac{1}{{\tau }_m}\nu \left(t\right)+{\dot{x}}_{3ref}\left(t\right)-{\Delta }_{u2}{x}_1\left(t\right)\hfill \\ +\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)\left(\frac{x_1\left(t\right)}{x_1(t-h)}\right)\left(k_{i2}\int {\overline{x}}_3(t-h)dt\right)\hfill \end{array}\right]$$

(49)

From (48), to satisfy (34), we have conditions (50)–(52):

$$\mathrm{sgn}\left({\overline{x}}_3\left(t\right)\right)=\mathrm{sgn}\left({\overline{x}}_3(t-h)\right);\forall t\ge t_0$$

(50)

$$\left(\begin{array}{c}\frac{k_M}{{\tau }_m}\hfill \\ +{\Delta }_{km}\hfill \end{array}\right)\left(\frac{x_1\left(t\right)}{x_1(t-h)}\right)\left[\begin{array}{c}\left(k_{p2}\right){\overline{x}}_3\left(t\right){\overline{x}}_3(t-h)\hfill \\ +\left({\rho }_2\right){\overline{x}}_3\left(t\right)\mathrm{sgm}\left({\overline{x}}_3(t-h)\right)\hfill \end{array}\right]>\left|{\overline{\Delta }}_3\left(t\right)\right|;\forall t\ge t_0$$

(51)

$$\left|{\sigma }_2(t-h)\right|>\delta$$

(52)

Conditions (45) and (50) make it more difficult to ensure stability when fast transient is present. To improve the robustness of stability with respect to input time delay, a predictive control strategy is proposed in the next section. Therefore, this section shows that PISM is not robust concerning control delay.

5. Predictive Sliding Mode Observer (P-SMO)

We propose the predictive estimated state variable ${\tilde{x}}_{1p}\left(t\right)={\tilde{i}}_{mRp}\left(t\right)/I_B$ as [19,20]

$${\tilde{x}}_{1p}\left(t\right)=e^{\left(-\frac{1}{{\tau }_R}\right)h_d}{\tilde{x}}_1\left(t\right)+\underset{-h_d}{\overset{0}{\int }}{e}^{\left(\frac{1}{{\tau }_R}\right)\theta }\frac{1}{{\tau }_R}{u}_{1p}(t+\theta )d\theta$$

(53)

where the estimated state variable ${\tilde{x}}_1\left(t\right)$ of (53) is obtained by (9), and $h_d>0$ is the design input time delay. With a nominal system, the derivative of ${\tilde{x}}_{1p}\left(t\right)$ is

$${\dot{\tilde{x}}}_{1p}\left(t\right)=-\frac{1}{{\tau }_R}\tilde{x}{}_{1p}\left(t\right)+\frac{1}{{\tau }_R}\left(u_{1p}\left(t\right)+{\overline{\Delta }}_{u1}\right)$$

(54)

where ${\overline{\Delta }}_{u1}=e^{\left(-\frac{1}{{\tau }_R}\right)h_d}{\Delta }_{u1}(t-h_d)$.

From (54), we note that it is free of input time delay. The other estimated predictive state variables are computed as

$${\dot{\tilde{x}}}_{2p}\left(t\right)={\omega }_B{\tilde{x}}_{3p}\left(t\right)+\frac{1}{{\tau }_R}\frac{u_{2p}\left(t\right)}{{\tilde{x}}_{1p}\left(t\right)}$$

(55)

$$\begin{array}{c}{\dot{\tilde{x}}}_{3p}\left(t\right)=\frac{1}{{\tau }_m}\left(k_M{\tilde{x}}_{1p}\left(t\right)u_{2p}\left(t\right)-{\tilde{\nu }}_p\left(t\right)\right)+L_1\mathrm{sgm}\left(x_3\left(t\right)-{\tilde{x}}_{3p}\left(t\right)\right)\hfill \end{array}$$

(56)

$${\dot{\tilde{\nu }}}_p\left(t\right)=L_2\mathrm{sgm}\left(x_3\left(t\right)-{\tilde{x}}_{3p}\left(t\right)\right)$$

(57)

The P-SMO is performed by (53) and (55)–(57), and it is free of input time delay. P-SMO is called the predictive system of the delayed TIM model.

The estimated error of predictive magnetizing current is defined as

$$e_{1p}\left(t\right)=x_1\left(t\right)-{\tilde{x}}_{1p}(t-h)$$

(58)

Substituting (1) and (54) into the derivative of (58), replacing $u_1\left(t\right)$ with $u_{1p}(t-h)$, and using ${\tilde{x}}_{1p}(t-h)$, we have

$$\begin{array}{c}{\dot{e}}_{1p}\left(t\right)=-\left(\frac{1}{{\tau }_R}\right)e_{1p}\left(t\right)+\left(\frac{1}{{\tau }_R}\right)\left(1-e^{\left(-\frac{1}{{\tau }_R}\right)h}\right){\Delta }_{u1}\hfill \\ -{\Delta }_{tr}\left(x_1\left(t\right)-u_{1p}(t-h)-{\Delta }_{u1}\right)\hfill \end{array}$$

(59)

The sliding function of P-SMO is defined as

$${\sigma }_{op}\left(t\right)=e_{3pp}^{}\left(t\right)$$

(60)

where

$$e_{3pp}^{}\left(t\right)=x_3\left(t\right)-{\tilde{x}}_{3p}\left(t\right)$$

(61)

The attractiveness condition to the sliding mode is given by

$${\sigma }_{op}\left(t\right){\dot{\sigma }}_{op}\left(t\right)<0$$

(62)

Substituting (3), (56), and (61) into (62) and replacing $u_2\left(t\right)$ for $u_{2p}(t-h)$, we obtain

$$e_{3pp}\left(t\right){\dot{e}}_{3pp}\left(t\right)=-\left(\begin{array}{c}{L}_1{e}_{3pp}\left(t\right)\mathrm{sgm}\left(e_{3pp}\left(t\right)\right)-e_{3pp}\left(t\right){\Delta }_{1p}\left(t\right)\hfill \end{array}\right)$$

(63)

$$e_{4pp}\left(t\right)=\nu \left(t\right)-{\tilde{\nu }}_p\left(t\right)$$

(64)

where

$${\Delta }_{1p}\left(t\right)=\left[\begin{array}{c}\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)x_1\left(t\right)\left(u_{2p}(t-h)+{\Delta }_{u2}\right)-\frac{k_M}{{\tau }_m}{\tilde{x}}_{1p}\left(t\right)u_{2p}\left(t\right)-\frac{1}{{\tau }_m}{e}_{4pp}\left(t\right)\hfill \end{array}\right]$$

(65)

From (63), to satisfy the attractiveness condition (62) and considering the boundary layer, we have conditions (66) and (67):

$$L_1>\sup \left|{\Delta }_{1p}\left(t\right)\right|$$

(66)

$$\left|{\sigma }_{op}\left(t\right)\right|>\delta$$

(67)

With bounded uncertainties, bounded perturbations, and stabilizing control, $\left|{\Delta }_{1p}\left(t\right)\right|$ has limited numerical values for all time. Thus, a limited value for $L_1$ exists such that (66) is satisfied.

Figure 2 shows the complete scheme of a TIM networked control taking into account the input time delay, with P-SMO and predictive proportional–integral sliding mode control (PISM-P) as proposed in the next section.

6. Predictive Proportional–Integral Sliding Mode Control (PISM-P)

The predictive controller PISM-P $u_{1p}\left(t\right)$ and $u_{2p}\left(t\right)$ are given by the same equations as PISM (24)–(32) but using the predictive state variables obtained by P-SMO, that is, replacing ${\overline{x}}_1\left(t\right)$ and ${\overline{x}}_3\left(t\right)$ with ${\overline{x}}_{1p}\left(t\right)$ and ${\overline{x}}_{3p}\left(t\right)$, respectively. Thus, we obtain

$${\overline{x}}_{1p}\left(t\right)={\tilde{x}}_{1p}\left(t\right)-x_{1ref}\left(t\right)$$

(68)

$${\overline{x}}_{3p}\left(t\right)={\tilde{x}}_{3p}\left(t\right)-x_{3ref}\left(t\right)$$

(69)

Using (68) and (69), if ${\rho }_1=0.0$ and ${\rho }_2=0.0$, then (24)–(26) will be called the PI-P control law.

Remark 1.

If a control law $u_p\left(t\right)$ stabilizes the predictive system (54)–(57), then the delayed uncertain model (1)–(3) with $u\left(t\right)=u_p(t-h)$ is also stabilized [31].

Proposition 1.

The predictive controller PISM-P stabilizes the predictive system (53)–(57) if conditions (70) and (71) are satisfied:

$$\left[\begin{array}{c}{\left({\overline{x}}_{1p}\left(t\right)\right)}^2\left(1+k_{p1}\right)+\left({\rho }_1\right){\overline{x}}_{1p}\left(t\right)\mathrm{sgm}\left({\overline{x}}_{1p}\left(t\right)\right)\hfill \end{array}\right]>\left|{\Delta }_{2p}\left(t\right)\right|,\forall t\ge t_0$$

(70)

$$\left[\begin{array}{c}\left(k_{p2}\right){\left({\overline{x}}_{3p}\left(t\right)\right)}^2+\left({\rho }_2\right){\overline{x}}_{3p}\left(t\right)\mathrm{sgm}\left({\overline{x}}_{3p}\left(t\right)\right)\hfill \end{array}\right]>\left|{\Delta }_{3p}\left(t\right)\right|,\forall t\ge t_0$$

(71)

where

$${\Delta }_{2p}\left(t\right)=\left|{\overline{x}}_{1p}\left(t\right)\left[\begin{array}{c}\left(\frac{1}{{\tau }_R}\right)\left(x_{1ref}\left(t\right)-e^{\left(-\frac{1}{{\tau }_R}\right)h}{\Delta }_{u1}\right)+\left(k_{i1}\right)\int {\overline{x}}_{1p}\left(t\right)dt+{\dot{x}}_{1ref}\left(t\right)\hfill \end{array}\right]\right|$$

(72)

$${\Delta }_{3p}\left(t\right)=\left|\left({\overline{x}}_{3p}\left(t\right)\right)\left[\begin{array}{c}\frac{k_M}{{\tau }_m}\left(k_{i2}\right)\int \left({\overline{x}}_{3p}\left(t\right)\right)dt+\frac{1}{{\tau }_m}{\tilde{\nu }}_p\left(t\right)-L_1\mathrm{sgm}\left(e_{3pp}\left(t\right)\right)+{\dot{x}}_{3ref}\left(t\right)\hfill \end{array}\right]\right|$$

(73)

Proof. 

For stability, we have the following conditions:

$${\overline{x}}_{1p}\left(t\right){\dot{\overline{x}}}_{1p}\left(t\right)<0$$

(74)

$${\overline{x}}_{3p}\left(t\right){\dot{\overline{x}}}_{3p}\left(t\right)<0$$

(75)

Using PISM-P into (74) and (75), we have (76) and (77):

$$\begin{array}{c}{\overline{x}}_{1p}\left(t\right){\dot{\overline{x}}}_{1p}\left(t\right)=-\left(\frac{1}{{\tau }_R}\right)\underset{>0}{\underbrace{\left[\begin{array}{c}\left(1+k_{p1}\right){\left({{\overline{x}}_{1p}}^{}\left(t\right)\right)}^2+\left({\rho }_1\right){\overline{x}}_{1p}\left(t\right)\mathrm{sgm}\left({\overline{x}}_{1p}\left(t\right)\right)\hfill \end{array}\right]}}-{\Delta }_{2p}\left(t\right)\hfill \end{array}$$

(76)

$$\begin{array}{c}{\overline{x}}_{3p}\left(t\right){\dot{\overline{x}}}_{3p}\left(t\right)=-\frac{k_M}{{\tau }_m}\underset{>0}{\underbrace{\left[\begin{array}{c}\left(k_{p2}\right){\left({\overline{x}}_{3p}\left(t\right)\right)}^2+\left({\rho }_2\right){\overline{x}}_{3p}\left(t\right)\mathrm{sgm}\left({\overline{x}}_{3p}\left(t\right)\right)\hfill \end{array}\right]}}-{\Delta }_{3p}\left(t\right)\hfill \end{array}$$

(77)

 ☐

From (76) and (77), the stability conditions (74) and (75) are satisfied if the inequalities (70) and (71) are also satisfied.

With bounded uncertainties and perturbations, then $\left|{\Delta }_{2p}\left(t\right)\right|$ and $\left|{\Delta }_{3p}\left(t\right)\right|$ are also bounded. Thus, limited numerical values exist for the parameters $k_{p1}$, $k_{p2}$, ${\rho }_1$, and ${\rho }_2$ such that (70) and (71) can be satisfied.

If conditions (70) and (71) are satisfied, then the control law PISM-P stabilizes the predictive model (54)–(57). According to Remark 1, the nonlinear uncertain model (1)–(3) will also be stabilized by PISM-P, even when input time delay is present.

7. Computational Simulations

To corroborate the previous analyses, computational simulations were performed on a 25 CV three-phase induction motor model.

7.1. Numerical Values

The uncertain models in (1)–(3) were computed using MatLab/Simulink^®. The nominal values are shown in

Table 1. Nominal values.

Parameters	Symbol	Value
Voltage	$U_B$	220 V
Stator current	$I_B$	59.7 A
Rotor speed	${\omega }_B$	122.5 rad/s
Torque	$m_B$	15.3 kgf
Rotor resistance	$R_R$	1.14 $\mathsf{\Omega }$
Rotor inductance	$L_R$	100 mH
Mutual inductance	$L_0$	92 mH
Inertia moment	J	1.4135 kg m${}^2$
Leakage factor	${\sigma }_R$	0.08
Rotor time constant	${\tau }_R=\raisebox{1ex}{$L_R$}\!\left/ \!\raisebox{-1ex}{$R_R$}\right.$	87.7ms
Mechanical time constant	${\tau }_m=\left({Jw}_B\right)/m_B$	1.155 s
Electric constant	$k_M=\left(2L_0{I}_B^2\right)/\left[3\right(1+{\sigma }_R\left)m_B\right]$	1.3499

.

Parametric uncertainties ${\Delta }_{tr}$ and ${\Delta }_{km}$ directly affect the observer performance. We define measures for the parametric variations for these uncertainties according to (78):

$$\left\{dtr=\frac{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_R$}\right.+{\Delta }_{tr}\right)}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_R$}\right.}; dkt=\frac{\left(\raisebox{1ex}{$k_M$}\!\left/ \!\raisebox{-1ex}{${\tau }_m$}\right.+{\Delta }_{km}\right)}{\raisebox{1ex}{$k_M$}\!\left/ \!\raisebox{-1ex}{${\tau }_m$}\right.}\right.$$

(78)

The disturbances in the control signals directly affect the dynamics of the system in sliding mode due to the boundary layer [11,12]. We define measures for these perturbations of both control input signals according to (79):

$$d{u}_i\left(t\right)=\frac{\left(u_i\left(t\right)+{\Delta }_{ui}\left(t\right)\right)}{u_i\left(t\right)};i=1,2;u_i\left(t\right)\ne 0$$

(79)

Table 2. Parameters variation and control disturbance.

Symbol	Time Interval (s)	Value (p.u.)
dtr	$\left\{\begin{array}{c}0\le t<50\hfill \\ 50\le t\le 160\hfill \end{array}\right.$	$\left\{\begin{array}{c}1.0: \mathrm{undisturbed}\hfill \\ 1.6+\left(0.6\right)sen\left(\pi t\right)\hfill \end{array}\right.$
dkt	$\left\{\begin{array}{c}0\le t<50\hfill \\ 50\le t\le 160\hfill \end{array}\right.$	$\left\{\begin{array}{c}1: \mathrm{undisturbed}\hfill \\ 1.3+\left(0.3\right)sen\left(\pi t\right)\hfill \end{array}\right.$
$\left.\begin{array}{c}{du}_1\hfill \\ {du}_2\hfill \end{array}\right\}$	$\left\{\begin{array}{c}0\le t<40\hfill \\ 40\le t<60\hfill \\ 60\le t\le 160\hfill \end{array}\right.$	$\left\{\begin{array}{c}1.0: \mathrm{undisturbed}\hfill \\ 1+0.3\ast \mathrm{s}\mathrm{i}\mathrm{n}(10\ast (t\left)\right)\hfill \\ 1.0: \mathrm{undisturbed}\hfill \end{array}\right.$

presents the adopted numerical values of parametric uncertainties (78) and control perturbations (79). Sinusoidal disturbances were used to improve graphical visualization of the influences on induction motor dynamics. The constant gains are in

Table 3. Observer and controller gains.

Parameter	Value	Parameter	Value
$L_1$	10	$k_{i1}$	15
$L_2$	7	$k_{p2}$	15
$\delta$	0.01	$k_{i2}$	15
$h_d$	10	${\rho }_1$	0 or 15
$k_{p1}$	15	${\rho }_2$	0 or 15

.

7.2. Operating Conditions

The operating conditions with respect to the rotor speed reference and load torque are summarized in

Table 4. Speed rotor reference and load torque.

Parameter	Time Interval (s)	Value (p.u.)
${\omega }_{\mathrm{re}f}=x_{3ref}$	$\left\{\begin{array}{c}0\le t<5\hfill \\ 5\le t<70\hfill \\ 70\le t<75\hfill \\ 75\le t<130\hfill \\ t=130\hfill \\ 130<t\le 160\hfill \end{array}\right.$	$\left\{\begin{array}{c}\mathrm{ramp}(0\mathrm{to}0.8)\hfill \\ 0.8\hfill \\ \mathrm{ramp}(0.8\mathrm{to}0.2)\hfill \\ 0.2\hfill \\ \mathrm{step}(0.2\mathrm{to}0.3)\hfill \\ 0.3\hfill \end{array}\right.$
$m_L=v$	$0\le t\le 160$	0.9

.

7.2.1. TIM Uncertain Model without Input Time Delay, Controlled by (i) PI and (ii) PISM

Two simulations are performed and the results are compared: in the first one, (i) the PI controller is used, and in the second, (ii) the PISM controller is used. In these simulations, the input time delay is not present, that is $h=0$ during all simulation times. Only parametric uncertainties and control perturbations are present, according to Table 2.

7.2.2. TIM Uncertain Model with Input Time Delay, Controlled by (i) PI and (ii) PISM

Two simulations are performed and the results are compared: in the first one, (i) the PI controller is used, and in the second, (ii) the PISM controller is used. In these simulations, the input time delay is present in some time intervals, as shown in

Table 5. Input time delay.

Parameter	Time Interval (Seconds)	Value (ms)
h	$\left\{\begin{array}{c}0\le t<15\hfill \\ 15\le t<35\hfill \\ 35\le t<65\hfill \\ 65\le t<95\hfill \\ 95\le t<120\hfill \\ 120\le t<140\hfill \\ 140\le t\le 160\hfill \end{array}\right.$	$\left\{\begin{array}{c}0\hfill \\ 10\hfill \\ 0\hfill \\ 10\hfill \\ 0\hfill \\ 13\hfill \\ 0\hfill \end{array}\right.$

. Note that the actual time delay h has different numerical values, which are 0 ms, 10 ms, and 13 ms.

7.2.3. TIM Uncertain Model with Input Time Delay, Controlled by (i) PI-P and (ii) PISM-P

Two simulations are performed and the results are compared: in the first one, (i) the PI-P controller is used, and in the second, (ii) the PISM-P controller is used. In these simulations, the input time delay is present in some time intervals, as shown in Table 5. The purpose of these simulations is to analyze the performance of PI-P and PISM-P acting with different numerical values of input time delay with a predictive observer P-SMO. Note that the input time delay $h_d$ used in theP-SMO design has a constant numerical value of 10 milliseconds (ms).

7.3. Obtained Results

The digital control was performed by zero-order block with a sampling time of 1 millisecond. The results are presented in graphical form. The variables of rotor speed, torque, and currents are per unit (p.u.). The time interval of simulation is 0 at 160 s. With the purpose of evaluating the performance of each controller, we defined the following numerical performance indices:

Speed performance:

$$\mathrm{SP}=\underset{0}{\overset{160}{\int }}\left|\left(x_3\left(t\right)-x_{3ref}\left(t\right)\right)\right|dt$$

(80)

Torque performance:

$$\mathrm{TP}=\underset{0}{\overset{160}{\int }}\left|\left(v\left(t\right)-m_d\left(t\right)\right)\right|dt$$

(81)

Magnetizing current performance:

$$\mathrm{MP}=\underset{0}{\overset{160}{\int }}\left|\left(x_1\left(t\right)-x_{1ref}\left(t\right)\right)\right|dt$$

(82)

The variable $m_d\left(t\right)$ is the actual electric torque computed by

$$m_d\left(t\right)={\tau }_m\left(\frac{k_M}{{\tau }_m}+{\Delta }_{km}\right)x_1\left(t\right)\left(u_2\left(t\right)+{\Delta }_{u2}\right)$$

(83)

The ideal values for the performance indices are zero.

7.4. Results for the TIM Uncertain Model without Input Time Delay, Controlled by (i) PI and (ii) PISM

(i)

Figure 3 shows the obtained responses of rotor speed and electric torque when PI is used to control the TIM uncertain model without input time delay.

(ii)

Figure 4 shows the obtained responses of rotor speed and electric torque when PISM is used to control the TIM uncertain model without input time delay.

It may be noted by Figure 3a and Figure 4a that both controllers maintained the rotor speed tracked to the reference. The computed numerical value of the rotor speed performance index for PI is SP = 0.7461, and SP = 0.2389 for PISM.

Figure 3b and Figure 4b show the electric torque and load torque. The computed numerical value of the performance index for the torque of PI is TP = 5.3427, and TP = 2.8412 for PISM. At the 40–60 s interval, there is a degenerative influence of the control disturbance in both the electric torque and the rotor speed.

The SP and TP performance indices indicate that the PISM controller performed better than the PI in the simulated case.

(i) Figure 5 shows the obtained responses of three-phase currents for the TIM when PI is used to control the TIM uncertain model without input time delay. (ii) Figure 6 shows the obtained responses of three-phase currents for the TIM when PISM is used to control the TIM uncertain model without input time delay.

Figure 5a and Figure 6a show the currents during the entire simulated period. Figure 5b and Figure 6b illustrate the currents when the induction motor is starting. Figure 5c and Figure 6c show the currents when the reference abruptly changes from 0.2 p.u. to 0.3 p.u.

(i) Figure 7 shows the motor magnetizing current and direct and quadrature control currents when PI is used to control the TIM uncertain model without input time delay. (ii) Figure 8 shows the motor magnetizing current and direct and quadrature control currents when PISM is used to control the TIM uncertain model without input time delay.

Figure 7a and Figure 8a show the motor magnetizing current $\left(i_{mR}\right)$. The reference signal for this current is a constant value $x_{1ref}={\mathrm{I}}_{mRref}=1.0$. The computed numerical value of the performance index for the magnetizing current of PI is MP = 0.0332, and MP = 0.0117 for PISM, which indicates that the PISM controller performed better than the PI in the simulated case.

Direct control currents $\left(i_{sd}\right)$ and quadrature control currents $\left(i_{sq}\right)$ are shown in Figure 7b and Figure 8b of PI and PISM, respectively. The figures clearly illustrate the degenerative influence caused by the disturbance in the control signal in both cases.

7.4.1. Results for the TIM Uncertain Model with Input Time Delay, Controlled by (i) PI and (ii) PISM

The input time delay is present in the nonlinear model, as shown in Table 5.

(i)

Figure 9 shows the rotor speed, the electric torque, and load torque when PI is used to control the TIM uncertain model with input time delay.

(ii)

Figure 10 shows the rotor speed, the electric torque, and load torque when PISM is used to control the TIM uncertain model with input time delay.

Figure 9a and Figure 10a show the rotor speed. Figure 9b and Figure 10b show the electric torque and load torque.

According to Figure 9, it can be noted that with the PI controller, after 19 s, the computation was aborted because of instability caused by the input time delay of 10 ms, as analyzed in item 4.2.

Figure 10 shows that with the PISM controller, after 15 s, the computation was aborted because of instability caused by the input time delay of 10 ms, indicating greater sensitivity to input time delay when compared with the PI controller, corroborating the analysis performed in item 4.2.

7.4.2. Results for the TIM Uncertain Model with Input Time Delay, Controlled by (i) PI-P and (ii) PISM-P

In this case, the P-SMO predictive observer is used to provide the feedback variables for the PI-P and PISM-P controllers. The actual input time delay is listed in Table 5. (i) Figure 11 shows the obtained responses of rotor speed and torque when PI-P is used to control the TIM delayed uncertain model. (ii) Figure 12 shows the obtained responses of rotor speed and torque when PISM-P is used to control the TIM delayed uncertain model.

Figure 11a and Figure 12a illustrate that both controllers maintained the rotor speed tracked to the reference. The computed numerical value of rotor speed performance index for PI-P is SP = 0.7779, and SP = 0.2604 for PISM-P. Figure 11b and Figure 12b show the electric torque and load torque. The computed numerical value of the performance index for torque of PI-P is TP = 5.6252, and TP = 3.1148 for PISM-P. Because of the predictor P-SMO, it was possible to have a good robustness in respect to input time delay. As can be seen in

Table 6. Performance of each controller.

Simulate Condition	Speed Performance (SP)	Torque Performance (TP)
TIM uncertain model without input time delay, controlled by PI	0.7461	5.3427
TIM uncertain model without input time delay, controlled by PISM	0.2389	2.8412
TIM uncertain model with input time delay, controlled by PI	INIFINITE	INIFINITE
TIM uncertain model with input time delay, controlled by PISM	INIFINITE	INIFINITE
TIM uncertain model with input time delay, controlled by PI-P	0.7779	5.6252
TIM uncertain model with input time delay, controlled PISM-P	0.2604	3.1148

, the SP and TP performance indices indicate that the PISM-P controller performed better than the PI-P in the present simulated case when input time delay and perturbations are present.

Analyzing Table 6, the PISM controller performed better than the PI controller in the test without input time delay. In the condition with input time delay, the PISM-P controller also performed better than the PI-P controller, with values close to the PISM results in the without-input time delay test. In addition, the SMO-P guaranteed the stability of the two controllers in the scenario that considers the input time delay.

8. Conclusions

This paper proposed a robust predictive proportional–integral sliding mode controller (PISM-P) for high-performance induction motors subject to uncertainties, disturbances, and input time delay.

The comparative presented results show that the PISM controller obtained better performance than the PI controller for all tested scenarios, either in the joint operation with SMO, when the feedback occurs without delay (Figure 3 and Figure 4), or in operation with P-SMO, when the transport delay is caused by the digital communication network (Figure 11 and Figure 12).

Furthermore, it can also be stated that PI and PISM controllers operating with the SMO are not able to stabilize the system against significant delays in the feedback signals (Figure 9 and Figure 10). In these cases, the proposed P-SMO system, associated with the PI-P and PISM-P controllers, ensured the stability and accurate tracking of the TIM speed reference, even against load disturbances and significant delays in the feedback signals (Figure 11 and Figure 12).

Finally, it is noteworthy that even in the face of the complexity of the processes, the PISM-P controller algorithm can be executed quickly by modern digital signal processors (DSPs) or field-programmable gate arrays (FPGAs), which operate at high frequencies and are capable of reducing the computational cost of its digital implementation. Studies and analyses on time-varying control delay and experimental results should be performed.