PWM-Based Finite-Time Tracking of Switched Buck Power Converters

In this paper, the problem of finite-time tracking is investigated for switched buck power converters based on the pulse width modulation (PWM) technique. For the continuous model, an equivalent continuous controller is solved by the backstepping technique, such that all signals are finite-time stable. PWM-based finite-time tracking with the equivalent control input is proposed for the switched buck converter, such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable. Simulation results are given to demonstrate the effectiveness of the proposed schemes.


Introduction
Electrical systems described by the multiple circuit topologies associated with the regulating switch are switched DC-DC power converters [1], which have widespread applications in power systems, photovoltaic systems, communication equipment, computers, and industrial electronics. e three basic topologies of switched DC-DC power converters are buck, boost, and buck-boost. e research of these power converters has drawn a great deal of attention for the development in both power electronics and control theory [2][3][4]. A good deal of results on point stabilization and trajectory tracking has been obtained under the switching method [5] and PWM technique [6]. Some switching methods have been investigated for stabilization of power converters, i.e., optimal switching instants based on a numerical optimization approach [7], state-dependent switching with the aid of variable-structure control [8], and switching law by combining sliding mode control with an equivalent control input [9][10][11][12]. e references mentioned above regard the switching signal as the control input directly based on switched systems theory. e output voltage regulation of PWM-based DC-DC power converters has already been an extremely active research. e output voltage tracking was discussed by selecting passivity-based control [13][14][15], energy shaping control [16,17], and state feedback indirect control with nonminimum phase [18,19]. In order to estimate the uncertain load resistances and enhance output performance, the adaptive backstepping controllers were proposed with better robustness and adaptability [20][21][22][23]. Most of the results focused on asymptotical convergence rate for DC-DC power converters. Compared with asymptotical stability, finite-time stability has a faster convergence performance in terms of time optimization. Finite-time control has aroused a great deal of interest in recent years. For DC-DC buck converters, finite-time tracking control via integral terminal sliding modes was presented in [24]. e output voltage regulation control is investigated to guarantee the finite-time convergence rate based on finite-time convergent observer [25,26] or via adaptive saturated finite-time control algorithm [27]. However, all these references above only considered point stabilization for continuous average models with the continuous control as a control input.
Little work has been done on PWM control to achieve the trajectory tracking problem of switched power converters so far. In this paper, we consider PWM-based finitetime tracking of switched buck power converters, where the system input is a digital control. e main work consists of the following aspects.
(1) Finite-time stability of a nonlinear system with continuous control is given as a priori information, and it is firstly proved that the closed-loop digital system is practically finite-time stable in the condition of the continuous control replaced by physical PWM control. (2) For the corresponding continuous system, an equivalent continuous controller is designed using the backstepping method, such that the tracking error tends to zero in a finite time, and the closed-loop error system is finite-time stable in terms of time optimization compared with [22,23]. (3) For the switched buck converter, PWM-based finitetime tracking with the equivalent control input is proposed such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable. e paper is organized as follows. Problem formulation is given in Section 2. PWM and its implementation are presented in Section 3. PWM-based finite-time tracking is researched in Section 4. Simulation results for buck converters are presented in Section 5. e paper is concluded in Section 6.
Notations: For a vector x, |x| denotes its usual Euclidean norm and x T denotes its transpose; |X| denotes the Frobenius norm of a matrix X defined by |X| � (Tr XX T ) (1/2) , where Tr(·) denotes the trace of a square matrix; R + denotes the set of all nonnegative real numbers; R n denotes the real n-dimensional space; R n×r denotes the real n × r matrix space; C i denotes the set of all functions with continuous i-th partial derivative; AB denotes the distance between A and B; S ABCD denotes the area of a rectangle ABCD.

Problem Formulation
Consider the switch-regulated buck converter circuit as shown in Figure 1. E is the voltage parameter of the supply voltage, R is the resistance of the resistor, C is the capacitance of the capacitor, L is the inductance of the inductor, VD is the diode, and VT is a PWM gate drive controlled switch. e control variable σ is introduced to denote the switch state, σ � 1 when VT � ON, and σ � 0 when VT � OFF.
at is, such a control input takes values in the discrete set 0, 1 { }. It is assumed that there is no noise in the circuit; Figure 2 is the equivalent circuit of Figure 1.
In order to establish the circuit model associated with the regulating switch position, we need to make an explanation for those physical symbols. e inductive current _ q L respects the derivative of the circulating electric charge q L , q C is the electrical charge stored in the capacitor, and (q C /C) is the capacitor voltage. e switch-regulated buck converter circuit is presented as Substituting the second equation into the first, it can be rewritten as (2) where x 1 and x 2 represent the capacitor voltage and the inductor current, respectively, and the variable σ is the digital control which represents time series produced by servo amplifiers and only take values from binary set 0, 1 { }. Let y � x 1 , that is, the output is the capacitor voltage, which is also the resistor voltage.
Given a smooth reference signal y r (t) ∈ C 2 (R), where y r (t), _ y r (t), € y r (t) are bounded with known constants. e aim of this paper is to design a switching signal σ for the switched buck converter (3), such that the capacitor voltage y � x 1 can be driven to track a given reference signal y r (t) in a finite time; simultaneously, all signals in the resulting system are required to be practically finite-time stable.
To study finite-time tracking, we introduce some basic concepts and lemmas that will serve as the basis for the development of our digital switching control.

Complexity
Definition 1 (see [29,28]). Consider a nonlinear system where f: D ⟶ R n is continuous, f(0) � 0, and D is an open neighborhood of the origin. e origin of system (4) is finite-time stable (FS), if it is Lyapunov stable and finite-time convergent in a neighborhood D 0 ⊆ D of the origin. By "finite-time convergence," we mean that there is a settling When D 0 � D � R n , the origin is a globally finite-time stable equilibrium.
Definition 2 (see [30]). e origin of system (4) is said to be practical finite-time stable (PFS), if for all initial conditions x 0 , there exist a constant ε T > 0 and a settling time Remark 1. For PFS, if ε T depends on T and T ⟶ 0, ε T ⟶ 0, then the effect of PFS is the same as FS. As T gets smaller, the bound ε T of PFS here becomes smaller and even tends to zero compared with boundedness of [30].
Lemma 1 (see [31]). For any real number x i , i � 1, . . . , n, and 0 < p ≤ 1, the following inequality holds: Lemma 2 (see [28]). For system (4), suppose there exist constants a > 0, 0 < β < 1, and a positive definite function then, the origin of system (4) is finite-time stable, and the settling time T * satisfies If D � R n and V(x) is radially unbounded, the origin of system (4) is globally finite-time stable.

PWM and Its Implementation
For a nonlinear system where x ∈ R n , f ∈ R n×1 is a locally Lipschitz continuous function, B ∈ R n is a constant vector, and u ∈ R is the digital control produced by servo amplifiers; it can only take values from binary set 0, 1 { }. In order to design such digital control, we first give a priori hypothesis for system (10).
ere exists a continuous static control such that the origin of the closed-loop system (10) with continuous static control (11) is finite-time stable. To achieve digitization, regarding μ(t): � μ(x(t)) as a duty ratio, from the viewpoint of mathematics, a PWM control is a strategy defined as [32] where t k is a sampling instant and T represents the sampling period, where T is parameter-tuning. However, it is very difficult to implement the PWM control (12). e one is how to get exact μ(t k ) at t k with variable-step resolution, the other one comes from the fact that PWM control should be a physical signal produced by a servo circuit instead of numerical signal directly.
Remark 2. By [1], systems (10) and (11) are the average models of the PWM controlled system (10) with (12). As T ⟶ 0, the duty ratio μ(t) as the equivalent control replaces the PWM control σ T (t) in (12). e duty ratio μ(t) may be designed in the form of a state feedback function, and thus it is easy to achieve some performance index of the closed-loop PWM switched system. In practice, the PWM control is generated as a discrete gate pulse signal [33], comparing a desired analog control signal with a fixedfrequency ramp. One approximate implementation of PWM control (12) is presented as where μ(t) is the function as (11) and u 0 (t) is a triangular wave with magnitude A � 1 and same period T as (12) (see Figure 3). If continuous control (11) is changed to PWM implementation (13), the stability analysis of system (10) will be given as follows.
For PWM implementation (13), we will discuss the relationship between σ T (t) and μ(t) on the interval [0, t].
e solution of system (10) with (11) is denoted as x μ , and the solution of system (10) and PWM implementation (13) is denoted as x σ T ; by formula (19), the following relationship holds which implies that According to Assumption 1, the origin of system (10) with (11) is finite-time stable. By Definition 1, there exists a settling time T * ∈ (0, ∞), such that

Complexity
From (22) and 21, we have that As a consequence, there exists e T , such that |D T | ≤ e T , and According to Definition 2, the origin of closed-loop digital system (10) and (13) is practically finite-time stable.
From the above analysis, we can get the result.
Remark 3. According to eorem 1, system (10) with (13) is practically finite-time stable. From (24), as T ⟶ 0, e T ⟶ 0. is implies that the sampling period T can be tuned smaller, and system (10) with (13) is closer to be finitetime stable. Moreover, all signals can be made arbitrarily small for small enough T.

PWM-Based Finite-Time Tracking
For the switched buck converter (3), in order to achieve the tracking target, we give the ideas for design as follows: (1) Design the equivalent continuous control by the backstepping method (2) e digital control is produced using the equivalent continuous control based on the PWM technique

Design of a Finite-Time State Feedback Controller.
System (3) with equivalent continuous control μ is presented as For the continuous system (25), the backstepping method is adopted to design a finite-time state feedback controller μ, such that the capacitor voltage x 1 tracks the given reference voltage y r (t) in a finite time.
Introduce the track error Step 1. From (25) and (26), the derivative of z 1 with respect to time turns out to be Choosing the Lyapunov function V 1 � (1/2)z 2 1 , the derivative of V 1 along the system (27) is given by As (1/C)x 2 is just a variable and not an effective control input, (27) cannot be enforced for all t ≥ 0. Nevertheless, it shows that the desired value for the variable (1/C)x 2 is where c 1 > 0 and β ∈ ((1/2), 1) are design parameters. Substituting (29) into (28), we have Step 2. Indeed, design the error From (25) and (31), deriving z 2 with respect to time yields For the second Lyapunov-like function V � V 2 � V 1 + (1/2)z 2 2 , the derivative of V along the system (32) satisfies where c 2 > 0 is a design parameter. e finite-time state feedback controller is designed as

Substituting (34) into (33) results in
In view of the above control analysis, we give the stability result of the continuous system in the following. (27), (32), and (34) are finite-time stable, and the tracking error converges to the origin in a finite time. Moreover, the equivalent continuous control (34) is bounded, and by choosing design parameters appropriately, μ can be obtained such that

Theorem 2. For system (3) with continuous control μ (34), all the signals of the closed-loop error systems
Proof. By defining z � (z T 1 , z T 2 ) T and V(z) � (1/2)z 2 1 + (1/2)z 2 2 , V(z) is a positive definite function. From (35) and Lemma 1, where c � min 2c 1 , 2c 2 and β ∈ ((1/2), 1). According to Lemma 2, the closed-loop error system is finite-time stable. Besides, based on Definition 1, we can obtain that By the definition of z, that is, the tracking error converges to the origin in a finite time. Furthermore, from (38), this implies that z is bounded.
For the boundedness of y r , _ y r , and € y r , it can be obtained that α 1 (34), that is, by choosing positive parameters E, c 1 , c 2 appropriately. Based on PWM implementation (13) and continuous signal (40), the binary PWM control is designed as where σ is the digital switch in the buck converter (3) and u 0 (t) is the triangular wave with period T. In order to indicate the proposed PWM-based finitetime tracking control approach more clearly, the control block diagram is shown in Figure 5, where the switching signal σ(t) is generated by the PWM technique comparing equivalent continuous control μ(t) with fixed-frequency triangle wave by comparator.  Proof. According to eorem 2, the closed-loop error systems (27) and (32) with (40) are finite-time stable. Based on eorem 1, the closed-loop error systems (27) and (32) with (41) are practically finite-time stable.
Furthermore, z 1 � x 1 − y r is a component of states z of the closed-loop error system; based on Remark 3, when T is small enough, the tracking error x 1 − y r can be made arbitrarily small. □ Remark 4. Finite-time point regulation problem was discussed for continuous average models in [25][26][27] with the continuous control as a control input. Compared with [25][26][27], PWM control based on the equivalent control input (i.e., the duty ratio function) is considered for finite-time trajectory tracking of switched power converters in this paper, and here the digital control is the system input.

Simulation Results
Example 1. To investigate the effectiveness of the proposed control method, we give the simulation results as follows.
To make comparisons of simulation results, the two system cases are discussed keeping other parameters intact, respectively. Figures 6 and 7 show the responses of the corresponding closed-loop system with the different cases. Figure 6 implies that the tracking error tends to the origin in a finite time. From Figure 7, it can be learned that the tracking error converges to a small neighborhood of the origin in a finite time. Compared with Figure 6, Figure 7 illustrates the validity of PWMbased finite-time tracking control of the closed-loop system. Complexity tracking control with the equivalent control input is proposed for the switched buck converter, such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable. Simulation results are given to demonstrate the effectiveness of the proposed schemes. It is an interesting and challenging direction under current research. e finite-time control method can be further generalized to solve other converters, such as boost converter, buck-boost converter, andĆuk converters; another future research is regarding stochastic PWM control of systems based on random theory [34,35].

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 8 Complexity