Predefined-time stabilization of permanent-magnet synchronous motor

Abstract

This paper designs a predefined-time convergent continuous control algorithm to stabilize a permanent-magnet synchronous motor (PMSM) system. Three cases have been considered: disturbance-free, in presence of a deterministic disturbance satisfying a Lipschitz condition, and in presence of both a stochastic white noise and a deterministic disturbance satisfying a Lipschitz condition. The designed control law is free from the restrictions of exponential control growth and exact initial conditions knowledge. This is the first predefined-time convergent continuous control algorithm applied to stabilizing a PMSM system with both deterministic and stochastic disturbances, which enables one to a priori set the predefined convergence time even in presence of various disturbances of different nature. Numerical simulations are provided for a PMSM system to validate the obtained theoretical results in each of the three considered cases. The simulation results demonstrate that the employed values of the predefined-time convergent control inputs are applicable in practice.

Keywords

Permanent-magnet synchronous motor predefined-time robust control

Introduction

Permanent-magnet synchronous motors (PMSMs) have received increasing attention from industry and household applications due to their characteristics of high power density, high efficiency, high power factor, compact structure, and excellent control performance. The PMSM has been applied in many areas such as aeronautics and astronautics, electric vehicle technology, domestic appliance, fan and pump applications (Kommuri et al., 2016; Kuang et al., 2017; Wang et al., 2020). The PMSM drive plays an important role in motion-control applications. On the other hand, the control performance of the PMSM servo drive is commonly influenced by uncertainties, such as unpredictable plant parameter variations, external load perturbations, or unmodeled dynamics of the controlled plant.

A major challenge for the control community is to design control laws that help to compensate for these uncertainties. The proportional-integral-derivative (PID) controller is conventionally applied to regulate the PMSM systems in industrial applications due to its simplicity, clarity, and functionality (Ang et al., 2005). However, a disadvantage of this controller is its sensitivity to the system uncertainties. An adaptive PI controller with an online tuning rule presents a method to improve the PID performance for PMSM (Hernandez-Guzman and Silva-Ortigoza, 2010).

Recently, many researchers have presented various advanced control strategies to efficiently control the PMSM systems such as linearization control (Zarchi et al., 2010), adaptive control (Ribeiro et al., 2007; Li and Liu, 2009), robust control (Baik et al., 2000; Nian and Deng, 2015; Wu and Zhang, 2018), sliding mode control (Qi et al., 2015; Wang and Wei, 2019; Xu et al., 2017), fixed-time convergent control (Basin et al., 2020), fractional-order control (Xie et al., 2019), fuzzy control (Mani et al., 2018), neural network control (Jon et al., 2017), and predictive control (Shengquan et al., 2020). Much attention has been paid to designing finite-time and fixed-time convergent control laws and estimating their convergence (settling) times. A continuous finite-time convergent algorithm for a second-order system was designed in Levant (1998). This algorithm, called super-twisting, drives a system state and its derivative at the origin in finite-time. The first upper estimates of the convergence times for twisting and super-twisting algorithms, based on the Lyapunov function approach, were obtained in Polyakov and Poznyak (2009a, 2009b). Fixed-time convergent algorithms, where the convergence times are independent of state initial conditions, are studied starting from Andrieu et al. (2008) and Polyakov (2012). Continuous fixed-time convergent scalar and multi-variable control laws are proposed in Basin et al. (2019) for a super-twisting system, whose state and disturbance initial conditions are unknown, and the upper estimates of their convergence times are calculated. A continuous fixed-time convergent control law driving the states of a stochastic super-twisting system at the origin for a fixed-time is designed in Guerra-Avellaneda and Basin (2020). A novel continuous controller ensuring an arbitrary predefined convergence time independent of initial conditions is proposed in Pal et al. (2020) for the $n$ -dimensional integrator chain, if no disturbances are present.

It can be observed that in most fixed-time approaches, the settling time cannot be assigned a priori. This drawback is corrected in predefined-time convergent algorithms, where the settling time can be assigned arbitrarily. However, the predefined-time convergent algorithm given in Pal et al. (2020) results in an exponentially growing control magnitude for negative state initial values and requires exact knowledge of state initial conditions in presence of disturbances.

Motivated by the foregoing discussion, this paper focuses on designing novel predefined-time convergent control laws for stabilizing PMSM systems affected by deterministic disturbances and stochastic noises, which are free from the mentioned restrictions of exponential control growth and exact initial conditions knowledge. To the best of our knowledge, this is the first attempt to design a predefined-time convergent control law for a super-twisting system with stochastic noises, although stochastic noises are broadly used for modelling independent unknown disturbances in technical systems. Another motivation point is to demonstrate that the employed values of the predefined-time convergent control inputs are applicable in practical cases of PMSM stabilization.

The contributions of this paper are as follows:

Designing predefined-time convergent continuous control laws driving the states of a super-twisting system to the origin for an a priori assigned time in three cases: disturbance-free, in presence of a deterministic disturbance satisfying a Lipschitz condition, and in presence of both a stochastic white noise and a deterministic disturbance satisfying a Lipschitz condition.

Achieving that the designed predefined-time convergent continuous control laws are free from the restrictions of exponential control growth and exact initial conditions knowledge.

Applying for the first time the developed control algorithm to stabilizing a PMSM system with both deterministic and stochastic disturbances, which enables one to a priori set the predefined convergence time even in presence of various disturbances of different nature.

Validating efficiency of the developed control algorithm for a PMSM system via numerical simulations in each of the considered three cases.

The paper is organized as follows. The mathematical model of the PMSM and the stabilization problem statement are given in Section 2. Sections 3, 4, and 5 present predefined-time convergent continuous control laws for a PMSM in three cases: without disturbances, in presence of deterministic disturbances, and in presence of deterministic disturbances and stochastic noises, respectively. Numerical simulations to validate the obtained theoretical results are given in each section. Section 6 concludes this study.

The standard notation is used: $| x (t) |$ denotes the absolute value of a scalar $x (t)$ , $∥ x (t) ∥$ denotes the Euclidean norm of a vector $x (t)$ , $a^{p}$ corresponds to a value $a$ raised to a power $p$ . The function $sign (x)$ is introduced according to Filippov’s definition (1998).

Problem statement

The mathematical model of a PMSM (Figure 1) can be represented by the following equations (Wang et al., 2020)

\begin{matrix} \frac{d ω}{dt} = \frac{K_{t}}{J} i_{q} - \frac{B}{J} ω - \frac{F}{J} \\ \frac{d i_{d}}{dt} = - \frac{R_{s}}{L_{d}} i_{d} + n_{p} ω i_{q} + \frac{u_{d}}{L_{d}} \\ \frac{d i_{q}}{dt} = - \frac{R_{s}}{L_{q}} i_{q} + n_{p} ω i_{d} - \frac{n_{p} ϕ_{v}}{L_{q}} ω + \frac{u_{q}}{L_{q}} \end{matrix}

(1)

Here, $L_{d}$ and $L_{q}$ are the inductances of $d - q$ axes satisfying $L_{d} = L_{q} = L$ , $u_{d}$ and $u_{q}$ the stator voltages of $d - q$ axes, $i_{d}$ and $i_{q}$ the stator currents of $d - q$ axes, $R_{s}$ the stator resistance, $ω$ the rotor velocity, $n_{p}$ the number of pole pairs, $ϕ_{v}$ the rotor flux linkage, $K_{t} = \frac{3 n_{p} ϕ_{v}}{2}$ , $F$ the load torque, $J$ the moment of inertia, and $B$ the viscous friction coefficient.

Figure 1.

PMSM electromechanical diagram.

In this paper, the strategy of $i_{d} = 0$ is adopted (Wang et al., 2020). Then, the first equation in (1) can be rewritten as

\frac{d ω}{dt} = \frac{K_{t}}{J} i_{q} - \frac{B}{J} ω + ζ, ω (t_{0}) = ω_{0},

(2)

where $i_{q}$ is the control input and $ζ = - \frac{F}{J}$ is a disturbance.

The control problem is to design a continuous control law driving the states of the resulting closed-loop system at the origin for a predefined time in the sense of the following definitions.

Consider a $n$ -dimensional system

\begin{matrix} \overset{\cdot}{x} (t) = u (t) + ξ (t) + σ (t, x (t)) dW (t), \\ x (t_{0}) = x_{0} \in R^{n}, t > 0, \end{matrix}

(3)

where $x (t) \in R^{n}$ is a state, $u (t) \in R^{n}$ is a control input, $ξ \in R^{n}$ is a deterministic disturbance satisfying the Lipschitz condition

∥ ξ (t_{1}) - ξ (t_{2}) ∥ \leq L | t_{1} - t_{2} |

for any $t_{1}$ , $t_{2}$ $\leq t_{0}$ , with a certain constant $L$ . $W (t)$ is a Wiener process defined on the complete probability space $(Ω, F, P)$ , where $Ω$ is the sample space, $F$ is a $σ$ -field with a filtration ${F_{t}}_{t \geq 0}$ , and $P$ a probability measure. The condition $σ (t, 0) = 0$ is satisfied for all $t \geq 0$ .

Consider two cases:

(a) $σ (t, x (t)) = 0$ . The system (3) reduces to a deterministic system.

Definition 1. (Predefined-time convergence for a deterministic system.) The system (3) is called predefined-time convergent to the origin, if

(1) it is fixed-time convergent to the origin, for any initial state $x_{0}$ , $\in R^{n}$ , there exists a positive constant $T_{\max} > 0$ independent of $x_{0}$ , such that $x (t) = 0$ for all $t \geq T_{\max}$ ,

(2) $T_{\max}$ is independent of initial conditions and disturbances and can be arbitrarily chosen in advance, and

(3) $T_{\max} \geq T_{f}$ , where $T_{f}$ is the true convergence time.

(b) $σ (t, x (t)) \neq 0$ . The system (3) is a stochastic system.

Definition 2. (Predefined-time convergence for a stochastic system.) The system (3) is called predefined-time convergent to the origin in $ρ$ -mean, if

(1) it is fixed time convergent to the origin in $ρ$ -mean, i.e., for any initial state $x_{0}$ , $\in R^{n}$ , there exists a positive constant $T_{\max} > 0$ independent of $x_{0}$ , such that $E [x (t)]^{ρ} = 0$ for all $t \geq T_{\max}$ ,

(2) $T_{\max}$ is independent of initial conditions and disturbances and can be arbitrarily chosen in advance, and

(3) $T_{\max} \geq T_{f}$ , where $T_{f}$ is the true convergence time.

Predefined-time stabilization without disturbances

Control design

Consider the system (2), where $ζ = 0$ . Then, equation (2) reduces to

\frac{d ω}{dt} = \frac{K_{t}}{J} i_{q} - \frac{B}{J} ω, ω (t_{0}) = ω_{0},

(4)

where the control law is designed as

i_{q} = \frac{J}{K_{t}} (u_{1} (t) + u_{2} (t)) .

(5)

Here, $u_{1} (t) = \frac{B}{J} ω$ is the compensation term and $u_{2} (t)$ is defined as

u_{2} (t) = {\begin{matrix} - η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)), & t_{0} \leq t < t_{f}, \\ 0, & t_{f} \leq t, \end{matrix}

(6)

where $η > 1$ .

Theorem 1. The control law (5) drives the state $ω (t)$ of the system (4) and its derivative $\overset{\cdot}{ω} (t)$ to the origin for an a priori assigned time $t_{f}$ and stays there afterwards for any $t \geq t_{f}$ . In other words, the closed-loop system (4), (5) is predefined-time convergent to the origin.

Proof. Substituting (5) into (4) yields the equation

\overset{\cdot}{ω} (t) = u_{2} (t), ω (t_{0}) = ω_{0},

(7)

or, equivalently

\begin{matrix} \overset{\cdot}{ω} (t) = \frac{- η (e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)), t_{0} \leq t < t_{f}, \\ \overset{\cdot}{ω} (t) = 0, t \geq t_{f}, \\ ω (t_{0}) = ω_{0}, \end{matrix}

(8)

whose solution is given by

\begin{matrix} ω (t) = \ln (C_{1} (t_{f} - t)^{η} + 1) sign (ω_{0}), t_{0} \leq t < t_{f} \\ ω (t) = 0, t \geq t_{f} \end{matrix}

(9)

where $C_{1} = \frac{e^{| ω_{0} |} - 1}{{(t_{f} - t_{0})}^{η}}$ . Hence, $ω (t)$ is smooth function, equal to zero for all $t \geq t_{f}$ .

The derivative of (9) is given by

\overset{\cdot}{ω} (t) = \frac{- η C_{1} {(t_{f} - t)}^{η - 1}}{C_{1} {(t_{f} - t)}^{η} + 1} sign (ω (t)) .

(10)

It follows from (10) that when $\overset{\cdot}{ω} (t) = 0$ for $t_{f} = t$ and $t_{f} > t$ as well. Thus, $\overset{\cdot}{ω} (t)$ is continuous function, equal to zero for all $t \geq t_{f}$ . $\overset{\cdot}{ω} (t)$ is also smooth, if $η > 2$ . ■

PMSM simulations

To demonstrate efficiency of the proposed control law, numerical simulations have been performed for the PMSM system (4), (5). The PMSM parameters used in the simulation are assigned as $J = 1.74 \times 10^{- 4} kg \times m^{2}$ , $B = 7.403 \times 10^{- 5} N$ , $n_{p} = 4$ , $ϕ_{v} = 0.1167 wb$ . The control parameters are $η = 30$ , $t_{f} = 20$ . The treated initial conditions and the corresponding settling times obtained by simulation are given in Table 1. Figures 2 –5 present the time histories of $ω (t)$ for initial conditions $ω_{0} = 100$ and $ω_{0} = - 100$ , as well as their zoomed time histories, which demonstrate that the system (4) is predefined-time convergent to the origin for the desired time $t_{f}$ , as stated in Theorem 1. Finally, Figure 6 shows the control input (5) for the initial condition of $ω_{0} = 100$ .

Table 1.

Convergence times vs initial conditions.

$ω_{0}$	10	100	1000	$10^{4}$	$10^{5}$
$t_{sim}$	15.5	19.65	19.9	20	20

Figure 2.

Time histories of $ω (t)$ with initial condition $ω_{0} = 100$ .

Figure 3.

Time histories of $ω (t)$ with initial condition $ω_{0} = 100$ (zoomed).

Figure 4.

Time histories of $ω (t)$ with initial condition $ω_{0} = - 100$ .

Figure 5.

Time histories of $ω (t)$ with initial condition $ω_{0} = - 100$ (zoomed).

Figure 6.

Control input (5).

To provide comparisons with the predefined-time control law given in Pal et al. (2020), numerical simulations are conducted for the system (7) with initial condition $ω_{0} = - 100$ and the system (1) in Pal et al. (2020) with initial condition $x_{0} = - 100$ . The corresponding graphs of the control inputs are shown in Figures 7 –9. The simulation results demonstrate that the control law of Pal et al. (2020) is not realizable in view of its too large magnitude, whereas the control law (6) remains in the range of practically acceptable values.

Figure 7.

Control input (6) with initial condition $ω_{0} = - 100$ .

Figure 8.

Control input of Pal et al. (2020) with initial condition $x_{0} = - 100$ .

Figure 9.

Control input of Pal et al. (2020) with initial condition $x_{0} = - 100$ (zoomed).

Note that the control law (6) depends only on the absolute values of the state $ω (t)$ . Therefore, the state trajectories and control inputs corresponding to positive and negative initial values of the same magnitude are symmetrically mirrored with respect to the time axis, so the exponential growth phenomenon does not take place even for negative state initial values.

Predefined-time stabilization with deterministic disturbances

In this section, two predefined-time convergent control laws are designed for a PMSM system in presence of a deterministic disturbance $ζ (t)$ satisfying the Lipschitz condition with a constant $L$ .

Control design

Consider equation (2), where $ζ \neq 0$ and the control law is designed as

i_{q} = \frac{J}{K_{t}} (u_{1} (t) + u_{2} (t)),

(11)

where $u_{1} (t)$ is defined in Section 3 and $u_{2} (t)$ is to be assigned. Substituting $i_{q} (t)$ into (2) yields

\overset{\cdot}{ω} (t) = u_{2} (t) + ζ (t),

where $u_{2} (t)$ is given by

u_{2} (t) = {\begin{matrix} - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - α \int_{0}^{t} sign (ω (s)) ds \\ - η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)), t_{0} \leq t < t_{f}, \\ - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - α \int_{0}^{t} sign (ω (s)) ds, t \geq t_{f} . \end{matrix}

(12)

Here, $λ_{1}, α > 0$ , $λ_{2} \geq 0$ , $η > 1$ , and $p > 1$ . The gain $λ_{2}$ can be selected as $λ_{2} = 0$ or $λ_{2} > 0$ .

Theorem 2. The control law (12) drives the state $ω (t)$ of the system (4) and its derivative $\overset{\cdot}{ω} (t)$ to the origin for an a priori assigned time $t_{f}$ and stays there afterwards for any $t \geq t_{f}$ . In other words, the closed-loop system (4), (12) is predefined-time convergent to the origin in the presence of a deterministic disturbance $ζ (t)$ satisfying the Lipschitz condition with a constant $L$ , if the following conditions hold: $η > 1$ , $α > L$ , $λ_{1} > \sqrt{2 α}$ , $λ_{2} \geq 0$ , and $p > 1$ .

The resulting closed-loop control system can be represented in the conventional super-twisting form

\begin{matrix} \overset{\cdot}{ω} (t) = {\begin{matrix} - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) \\ - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)) + y (t), t_{0} \leq t < t_{f}, \\ - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) \\ - λ_{2} | ω (t) |^{p} sign (ω (t)) + y (t), t \geq t_{f}, \end{matrix} \\ \overset{\cdot}{y} (t) = - α sign (ω (t)) + \overset{\cdot}{ζ} (t), ω (t_{0}) = ω_{0}, y (t_{0}) = y_{0} . \end{matrix}

(13)

Then, both states $ω (t)$ and $y (t)$ converge to the origin for the predefined time $t_{f}$ .

Proof: As follows from Theorem 4.5 in Shtessel et al. (2014), the conditions $α > L$ and $λ_{1} > \sqrt{2 α}$ provide finite-time convergence to the origin of both states of the super-twisting system (13), if $η = 0$ , where the term $- λ_{1} | ω (t) |^{1 / 2} sign (ω (t))$ is responsible for finite-time convergence of an undisturbed system and the term $- α sign (ω (t))$ is responsible for suppressing the disturbance $\overset{\cdot}{ζ} (t)$ . Thus, the first and third terms in the control law (12), without the fourth exponential term (6), already yield finite-time convergence of system states (13) to the origin and suppression of the disturbance $\overset{\cdot}{ζ} (t)$ . As follows from Theorem 1, adding the fourth exponential term (6) in the control law (12), $- η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t))$ , makes this finite-time convergence predefined for a time no greater than $t_{f}$ . Finally, as follows from Theorem 1 in Basin et al. (2019), adding the second term in the control law (12), $- λ_{2} | ω (t) |^{p} sign (ω (t))$ with $λ_{2} > 0$ , yields fixed-time convergence of system states (13) to the origin, if $η = 0$ , and suppression of the disturbance $\overset{\cdot}{ζ} (t)$ . Hence, the combination of all four terms in the control law (12) with $λ_{2} > 0$ makes this fixed-time convergence predefined for a time no greater than $t_{f}$ and may only increase the convergence rate in comparison to the case $λ_{2} = 0$ . ■.

PMSM simulations

To demonstrate efficiency of the proposed control law, numerical simulations have been performed for the PMSM system (2), (12). First, the case $λ_{2} = 0$ is considered. The PMSM parameters used in the simulation are assigned as $J = 1.74 \times 10^{- 4} kg \times m^{2}$ , $B = 7.403 \times 10^{- 5} N$ , $n_{p} = 4$ , $ϕ_{v} = 0.1167 wb$ . The deterministic disturbance is selected as $ζ (t) = \frac{1}{J} (0.1 t + 0.01 \cos (t))$ ; accordingly, the Lipschitz constant is set to $L = 633$ . The control parameters are $η = 30$ , $t_{f} = 20$ , $λ_{1} = 50$ , $α = 635$ , which are selected to satisfy the conditions of Theorem 2. The treated initial conditions and the corresponding settling times obtained by simulation are given in Table 2. Figures 10 and 11 present the time histories of $ω (t)$ and $y (t)$ for initial conditions $ω_{0} = 100$ , $y_{0} = - 50$ and $ω_{0} = - 100$ , $y_{0} = - 50$ , as well as their zoomed time histories, which demonstrate predefined-time convergence of the PMSM states to the origin for the desired time $t_{f}$ , as stated in Theorem 2. Figure 12 shows the control input (11) for the initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Table 2.

Convergence times vs initial conditions.

$y_{0}$ \ $ω_{0}$	10	100	1000	$10^{4}$	$10^{5}$
−100	0.35	0.55	1.37	6.49	13.67
−50	0.23	0.47	1.33	6.40	13.61
−10	0.15	0.42	1.31	6.31	13.55
−5	0.14	0.42	1.31	6.30	13.54
−1	0.13	0.42	1.31	6.30	13.54
1	0.11	0.41	1.31	6.29	13.54
5	0.12	0.41	1.31	6.28	13.53
10	0.15	0.41	1.30	6.27	13.52
50	0.64	0.64	1.30	6.18	13.47
100	1.12	1.12	1.33	6.06	13.40

Figure 10.

Time histories of $ω (t)$ and $y (t)$ when $λ_{2} = 0$ with initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Figure 11.

Time histories of $ω (t)$ and $y (t)$ when $λ_{2} = 0$ with initial conditions $ω_{0} = - 100$ and $y_{0} = - 50$ .

Figure 12.

Control input with $λ_{2} = 0$ and initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Then, the case $λ_{2} > 0$ is considered, where the values $λ_{2} = 1$ , $p = 1.5$ are selected to satisfy the conditions of Theorem 2 and the other control parameters are the same as in the preceding set of simulations. The treated initial conditions and the corresponding settling times obtained by simulation are given in Table 3. Figures 13 and 14 present the time histories of $ω (t)$ and $y (t)$ for initial conditions $ω_{0} = 100$ , $y_{0} = - 50$ and $ω_{0} = - 100$ , $y_{0} = - 50$ , as well as their zoomed time histories, which demonstrate predefined-time convergence of the PMSM states to the origin for the desired time $t_{f}$ , as stated in Theorem 2. Figure 15 shows the control input (11) for the initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ . Note that in both cases the magnitude of the control input $i_{q}$ remains less than $5 A$ , which is acceptable in practice. Thus, both control laws provide reliable stabilization of the PMSM states to the origin in presence of a deterministic disturbance within the a priori assigned time of $20$ sec. The second control law with $λ_{2} \neq 0$ yields considerably lesser convergence times for large state initial values, although its magnitude is only slightly higher than that in the first case.

Table 3.

Convergence times vs initial conditions.

$y_{0}$ \ $ω_{0}$	10	100	1000	$10^{4}$	$10^{5}$
−100	0.34	0.46	0.57	0.62	0.63
−50	0.23	0.37	0.48	0.53	0.54
−10	0.14	0.30	0.42	0.46	0.48
−5	0.13	0.29	0.41	0.46	0.47
−1	0.11	0.28	0.40	0.45	0.46
1	0.11	0.28	0.40	0.45	0.46
5	0.11	0.25	0.40	0.44	0.46
10	0.15	0.26	0.39	0.44	0.45
50	0.64	0.64	0.64	0.64	0.64
100	1.12	1.12	1.12	1.12	1.12

Figure 13.

Time histories of $ω (t)$ and $y (t)$ with $λ_{2} \neq 0$ and initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Figure 14.

Time histories of $ω (t)$ and $y (t)$ with $λ_{2} \neq 0$ and initial conditions $ω_{0} = - 100$ and $y_{0} = - 50$ .

Figure 15.

Control input with $λ_{2} \neq 0$ and initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Remark 1. The presented simulations were run in MatLab with discretization step 0.001. This discretization is the reason why the state trajectory does not reach zero exactly but forms a limit cycle (chattering) with a certain amplitude. In this case, the finite-time convergence is determined according to the Levant’s test (Shtessel et al., 2014), which establishes that the finite-time convergence takes place if the magnitude of the limit cycle for the state variable of relative degree one with respect to a discontinuous control input is of the same order as the discretization step, the magnitude of the limit cycle for the state variable of relative degree two is of the same order as the discretization step in square, the magnitude of the limit cycle for the state variable of relative degree three is of the same order as the discretization step in cube, and so forth. In the considered case, the state variable of relative degree one, $y (t)$ , with respect to a discontinuous control input $sign (ω (t))$ reaches the limit cycle of the magnitude $10^{- 3}$ , which is equal to that of the discretization step, and the state variable of relative degree two, $ω (t)$ , reaches the limit cycle of the magnitude $10^{- 9}$ , which is even less than that of the discretization step in square $10^{- 6}$ . Thus, the finite-time convergence of both state variables to zero is established according to the Levant’s test.

Predefined-time stabilization with deterministic disturbances and stochastic noises

Control design

In this section, a predefined-time convergent control law is designed for a PMSM system in presence of a deterministic disturbance $ζ (t)$ satisfying the Lipschitz condition with a constant $L$ and a stochastic white noise.

Consider the PMSM system (2) in presence of a deterministic disturbance $ζ (t)$ satisfying the Lipschitz condition and a stochastic white noise

\begin{matrix} d ω = (\frac{K_{t}}{J} i_{q} - \frac{B}{J} ω + ζ) dt + σ (t, ω) dW (t), \\ ω (0) = ω_{0}, \end{matrix}

(14)

where $W (t)$ is a Wiener process defined in $(3)$ .

The control law is represented as

i_{q} = \frac{J}{K_{t}} (u_{1} (t) + u_{2} (t)),

(15)

where $u_{1} (t)$ is defined in Section 3 and $u_{2} (t)$ is to be assigned. Substituting $i_{q} (t)$ into (2) yields

d ω (t) = (u_{2} (t) + ζ (t)) dt + σ (t, ω) dW (t),

where $u_{2} (t)$ is given by

u_{2} (t) = {\begin{matrix} - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - α \int_{0}^{t} sign (ω (s)) ds \\ - η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)), t_{0} \leq t < t_{f}, \\ - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - α \int_{0}^{t} sign (ω (s)) ds, t \geq t_{f} . \end{matrix}

(16)

Here, $λ_{1}, α > 0$ , $λ_{2} > 0$ , $η > 1$ , and $p > 1$ .

Theorem 3. The control law (15) drives the $ρ$ -th initial moment $E [ω (t)]^{ρ}$ of the state $ω (t)$ of the system (14) to the origin for an a priori assigned time $t_{f}$ and stays there afterwards for any $t \geq t_{f}$ . In other words, the closed-loop system (14), (15) is predefined-time convergent to the origin in $ρ$ -mean in the presence of a deterministic disturbance $ζ (t)$ satisfying the Lipschitz condition with a constant $L$ and a stochastic white noise with diffusion $σ (t, ω) = | ω |^{r}$ , if the following conditions hold: $ρ > 1$ , $η > 1$ , $α > L$ , $λ_{1} > \sqrt{2 α}$ , $λ_{2} \geq 0$ , $p > 1$ , $2 λ_{1} > ρ - 1 > 0$ , $2 λ_{2} > ρ - 1 > 0$ , and $\frac{3}{2} \leq 2 r \leq (1 + p)$ .

The resulting closed-loop control system can be represented in the conventional super-twisting form

\begin{matrix} \overset{\cdot}{ω} (t) = {\begin{matrix} - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) \\ - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ - η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t)) \\ + y (t) + σ (t, ω) dW (t), t_{0} \leq t < t_{f}, \\ - λ_{1} | ω (t) |^{1 / 2} sign (ω (t)) \\ - λ_{2} | ω (t) |^{p} sign (ω (t)) \\ + y (t) + σ (t, ω) dW (t), t \geq t_{f}, \end{matrix} \\ \overset{\cdot}{y} (t) = - α sign (ω (t)) + \overset{\cdot}{ζ} (t), ω (t_{0}) = ω_{0}, y (t_{0}) = y_{0} . \end{matrix}

(17)

Then, both states $ω (t)$ and $y (t)$ converge to the origin for the predefined-time $t_{f}$ .

Note that the predefined-time convergence of $E | x (t) |^{ρ}$ to zero means that the expectation and the absolute value in power $ρ$ of the centralized difference of values of a stochastic process $x (t)$ converge to zero as time tends to an a priori predefined value. Conventionally, the value of $ρ$ is set to 2, which corresponds to the mean-square convergence of a stochastic process $x (t)$ to zero.

Proof. As follows from Theorem 2 in Guerra-Avellaneda and Basin (2020), the conditions $2 λ_{1} > ρ - 1 > 0$ , $2 λ_{2} > ρ - 1 > 0$ and $\frac{3}{2} \leq 2 r \leq (1 + p)$ provide fixed-time convergence to the origin in $ρ$ -mean of both states of the stochastic super-twisting system (17), if $η = 0$ , where the terms $- λ_{1} | ω (t) |^{1 / 2} sign (ω (t))$ and $- λ_{2} | ω (t) |^{p} sign (ω (t))$ with $λ_{2} > 0$ are responsible for fixed-time convergence in $ρ$ -mean of an undisturbed system and the term $- α sign (ω (t))$ is responsible for suppressing the disturbance $\overset{\cdot}{ζ} (t)$ . Thus, the first three terms in the control law (16), without the fourth exponential term (6), already yield fixed-time convergence in $ρ$ -mean of system states (17) to the origin and suppression of the disturbance $\overset{\cdot}{ζ} (t)$ . As follows from Theorem 1 of this paper, adding the fourth exponential term (6) in the control law (16), $- η \frac{(e^{| ω (t) |} - 1)}{e^{| ω (t) |} (t_{f} - t)} sign (ω (t))$ , makes this fixed-time convergence in $ρ$ -mean predefined for a time no greater than $t_{f}$ . ■.

PMSM simulations

To demonstrate efficiency of the proposed control law, numerical simulations have been performed for the PMSM system (14), (15). The PMSM parameters, deterministic disturbance, Lipschitz constant, and control parameters are the same as in Section 4. The stochastic noise parameter is given by $r = 0.75$ . The stochastic convergence is regarded in mean-square sense, $ρ = 2$ , to satisfy the conditions of Theorem 3. The treated initial conditions and the corresponding settling times obtained by simulation are given in Table 4. Figures 16 and 17 present the time histories of $ω (t)$ and $y (t)$ for initial conditions $ω_{0} = 100$ , $y_{0} = - 50$ and $ω_{0} = - 100$ , $y_{0} = - 50$ , as well as their zoomed time histories, which demonstrate predefined-time convergence of the PMSM states to the origin in $ρ$ -mean for the desired time $t_{f}$ , as stated in Theorem 3. Figure 18 shows the control input (15) for the initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ . The magnitude of the control input $i_{q}$ also remains less than $5 A$ , which is acceptable in practice. Thus, both control laws provide reliable stabilization of the PMSM states to the origin in presence of a deterministic disturbance and a stochastic noise within the a priori assigned time of $20$ sec.

Table 4.

Convergence times vs initial conditions.

$y_{0}$ \ $ω_{0}$	10	100	1000	$10^{4}$	$10^{5}$
−100	0.32	0.49	0.62	0.64	0.65
−50	0.22	0.38	0.51	0.54	0.54
−10	0.14	0.30	0.43	0.46	0.46
−5	0.13	0.30	0.43	0.45	0.45
−1	0.13	0.29	0.42	0.44	0.45
1	0.12	0.28	0.41	0.44	0.44
5	0.12	0.27	0.40	0.43	0.44
10	0.15	0.27	0.39	0.42	0.44
50	0.64	0.64	0.64	0.64	0.64
100	1.12	1.12	1.12	1.12	1.12

Figure 16.

Time histories of stochastic $ω (t)$ and $y (t)$ with initial conditions $ω_{0} = - 100$ and $y_{0} = - 50$ .

Figure 17.

Time histories of stochastic $ω (t)$ and $y (t)$ with initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Figure 18.

Stochastic control input with initial conditions $ω_{0} = 100$ and $y_{0} = - 50$ .

Conclusions

This paper has presented a predefined-time convergent continuous control algorithm for a PMSM which is free from the restrictions of exponential control growth and exact initial conditions knowledge. Three cases have been considered:

disturbance-free,

in presence of a deterministic disturbance satisfying a Lipschitz condition, and

in presence of a deterministic disturbance satisfying a Lipschitz condition and a stochastic white noise.

The performance of the developed algorithm is verified with numerical simulations, which demonstrate reliable predefined-time convergence of the PMSM system states to the origin in all three cases. The employed values of the predefined-time convergent control inputs are applicable in practical cases of PMSM stabilization. The ongoing research focuses on designing adaptive predefined-time control algorithms for scalar and multi-variable systems with deterministic disturbances and stochastic white noises.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Mexican National Science and Technology Council (CONACyT) under Grant 250611.

ORCID iD

Michael Basin

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