Boost Converters’ Proximate Constrained Time-Optimal Sliding Mode Control Based on Hybrid Switching Model

It is well known in the literature studies that the theoretical time-optimal control of boost converters can be achieved using switching surfaces based on the converter’s natural state trajectories. However, this method has two important drawbacks: First, the transient current peak of the time-optimal controller is far beyond the current limitations of related circuit elements in many practical cases. Second, switching based on the converter’s natural trajectories has high computational complexity and high dependence on circuit parameters. In this paper, based on the hybrid dynamical model of the converter and geometrical representation of its corresponding vector elds, a proximate constrained time-optimal sliding mode controller is proposed. e proposed method has a fast response that is near that of a time-optimal controller, with less computational complexity and sensitivity to parameter changes. e proposed method and its relevant theoretical framework are validated on an experimental setup with a boost converter prototype and an eZdsp TMS320F2812 processor board.


Introduction
Because of high demand for renewable-energy (RE) sources, power electronic systems as links between sources, storages, and loads play an increasingly important role in modern power systems [1]. In many RE applications, such as fuel cell and photovoltaic energy systems, the required load voltage is higher than the source voltage. In these cases, boost converters as a small-sized, low-cost, and power-e cient DC-DC converter are of special importance [2]. In addition, many of these RE applications need higher performance power converters with new control strategies. erefore, any improvement of boost converters' performance such as faster transient response or better response to source voltage and load variations would be bene cial and welcomed in the eld of RE systems. e common approaches to control the boost converter and other DC-DC power converters are based on linearized averaged models and standard frequency-domain design methods [3]. Nonlinear state space-averaged models are also used in many works to achieve better response [3]. An important assumption in the state space-averaging model is that the switching signal varies slowly compared with the changes in state variables [3]. is assumption causes the slow time response of the control method based on the model. Hybrid dynamical system modeling of converters, which directly takes into account the switching nature of the DC-DC converters, promises better transient response and large signal stability [4][5][6][7][8][9]. A useful outcome of considering the switching nature of DC-DC converters is the time-optimal control (TOC) of the converters [10][11][12]. Although there has been a vast body of work on TOC of boost converters [10,13,14], most of the focus has been on optimization of speci c controllers or optimization based on approximate ideal waveforms of the converter [15][16][17].
eoretical TOC of buck and boost converters has been done in recent studies [11,12] using optimal control theory methods like Pontryagin's maximum principle. ese researches show that the TOC of buck and boost converters is the minimum switching control based on the bang-bang property [11]. e ideal time-optimal (TO) controller can be implemented using the sliding mode controller (SMC) framework with switching surfaces based on natural state trajectories.
is TO controller has high computational complexity and high sensitivity to circuit parameters [12]. Some linear approximations for this optimal switching surface in the buck converter are proposed to relieve the computational complexity and circuit parameter dependence [7,13,16].
Up to date, SMC has been extensively studied [18][19][20][21][22]. Valuable studies are performed on novel sliding surfaces for some applications [23][24][25]. However, challenges of SMC may not be the same in different application areas. For example, DC-DC converters are switching systems in nature. ese systems are designed to achieve desired dynamics by rapidly switching between two or more subsystems. In this case, the control variable is a discrete signal that determines the ON/OFF state of the switches. Hence, the chattering issue in this application is different from the traditional one [23]. e key point to be addressed is finding the time-optimal sliding surface with a stable margin as to relieve the sensitivity to parameter changes [21,22].
Besides computational complexity and circuit parameter dependence, the traditional time-optimal control of the boost converter has two other important drawbacks. First, high transient current peak may exceed the current limit of the inductor element or the power source. Second, there are higher voltage and current fluctuations around the operating point.
In this paper, the boost converter is modeled as a switched affine (SA) system, and its behavior is studied using the phase portrait. We propose a proximate constrained time-optimal (PCTO) SMC to overcome the shortcomings of the traditional TOC using a piecewise linear switching surface. e proposed method satisfies the reaching and existence conditions of the sliding mode controller, which is visualized by the phase portrait and geometric analysis. e proposed method keeps the inductor current within the determined limits during the transient stage and, at the same time, on the switching surface in the other regions of the state space; the switching surface keeps the state vector on a linear approximation of the ideal switching surface to achieve fast and precise response. e speed response of the proposed method remains close to that of the TOC, while it needs less computational complexity and can be simply implemented using the digital signal processor (DSP). In addition, the performance of the proposed method has less sensitivity to circuit parameter changes.
An experimental prototype based on an eZdsp TMS320F2812 processor board with a sensor and signal conditioning circuit and a boost converter is built to verify our theoretical analysis and effectiveness of the proposed method.

Boost Converter as a Hybrid
Dynamical System 2.1. Switched Affine Model. Figure 1 shows a standard boost converter circuit considering the equivalent series resistances (ESRs) of the inductor and capacitor given by r L and r C , respectively. When the switch S is closed, the current of the inductor increases and energy gets stored in the inductor, and when it is open, the inductor current transfers the stored energy through the diode to the load side. A filter capacitor, C, regulates the output voltage to achieve a smooth voltage at the load side. e boost converter shown in Figure 1 has a linear dynamical model for each of the possible configurations of its power electronic switches. is converter has three operation states. In subsystem 1, the switch is ON and the diode is OFF. e dynamics of the system in this state is modeled by where In subsystem 2, the switch is OFF and the diode is ON. e boost converter is modeled as where

Complexity
Subsystem 3 corresponds to the case when both switches are OFF for which the state space model is as follows: where is model represents a hybrid dynamical system, and its discrete variable can be determined by the switch status and the switching condition of the diode, or simply by its operating state. Although the operating state is both state and input dependent, in many cases, the controller is designed so that mode 3 of operation does not occur corresponding to the continuous conduct mode (CCM) of the converter. In this case, the boost converter model can be simplified to an SA system with two input-dependent operating states: is SA system model is used in the following sections to study the boost converter's behavior.

Time Response.
Time response for a nonhomogeneous linear system like can be described by where e tA represents the matrix exponential. Each subsystem of the boost converter is a linear nonhomogeneous system similar to (8) with a constant g. erefore, the time response for the state q of the converter is given by Since A − 1 and A commute, using matrix exponential properties, we have Finally, in each subsystem, the boost converter's time response is represented as follows: For t � 0, the matrix exponential equals the identity matrix, so the initial condition is satisfied. If the eigenvalues of A q have negative real parts, the matrix exponential will converge to the zero matrix as t goes to infinity. In that case, the equilibrium of the boost converter is e state matrix A 1 has two real eigenvalues, implying that the equilibrium point is a simple node. e eigenvalues of A 2 are two complex numbers with negative real parts; hence, the equilibrium point is a spiral asymptotically stable point. e above equations are used for the phase portrait of the boost converter in its different operating modes in the following sections.

Phase Portrait Representation.
Phase portrait representations for a boost converter with the second circuit parameter set given in Table 1 are depicted in Figure 2. In this section, circuit 1 is used for better representation of the boost converter's state trajectories in its different operating subsystems. In the remaining parts of this paper, specifications of circuit 2 in Table 1 are used in all simulation and experimental tests.
As can be seen in Figure 2, in subsystem 1, each of the state trajectories converges to a point on the i L axis. Subsystem 2 has a spiral stable point in the middle of the continuous state space. e control input of a boost converter is a discrete switching signal which turns the switch ON or OFF. Fast switching between two phase portraits with varying average dwell time results in custom state trajectories in the continuous state space. Figure 2(c) shows a sample state trajectory for the fixed PWM input with 35% of duty cycle. To better illustrate the switching effect, the carrier frequency is kept low in this PWM signal. Figure 2(c) also depicts the possible equilibrium points of the converter state vector. To have closed-loop stability in a boost converter control system, the desired state values cannot be arbitrary points on the continuous state space. To hold the state trajectory in a neighborhood around a desired point, there should be a suitable switching sequence which moves the states in a limit cycle around the desired point. ese possible equilibrium points form a conic section in the continuous state space and can be calculated by simulating the PWM signals with different duty cycles. e result of this simulation is shown in Figure 2(c).

Time-Optimal Control
In [15], it is shown using Pontryagin's maximum principle that time-optimal control of a boost converter is a bang-bang control, and it is proved that this time-optimal control is also minimum switching control [7]. In a physical sense, when the load resistance steps up, the controller should turn the switch ON to store sufficient energy in the inductor. e controller will turn the switch OFF when the state variables reach a special trajectory of the switch OFF mode. is trajectory will reach the desired state variable point. en, the controller will keep the state variables near the desired point. is time-optimal control can be implemented as a sliding mode controller whose switching surface is defined Complexity 3 by combining two natural state trajectories of subsystem 1 and subsystem 2 of the boost converter [6]. Figure 3 shows an ideal state trajectory for time-optimal control of a boost converter with specifications of circuits given in Table 1. Voltage and current waveforms of the ideal time-optimal controller are depicted in Figure 3. Despite the fast response time, this time-optimal controller has the following four major drawbacks: (1) High computational complexity  Although the simulated waveforms show good steadystate operation, in experimental tests, high current and voltage fluctuations will occur because of the aligned direction of the switching surface and trajectories.
Linear approximations of the optimal state trajectories can be used to overcome the first issue highlighted above and to some extent the second issue. In this paper, a conventional sliding mode controller is tuned to reduce the current overshoot within the determined limits. en, to improve the response time, we investigate piecewise linear switching surfaces to maintain the inductor current within its allowed maximum designed value.

Sliding Mode Control Principle.
In this section, a conventional SMC is tuned to keep the boost converter's response time low while maintaining the current limit. In the SMC, the system state trajectory first moves toward a surface in the state space, namely, the sliding surface. Second, the system state slides near the surface and converges to a desired equilibrium point (8) [14]. e first and second parts of the SMC state trajectory are called the reaching mode and sliding mode, respectively (8). e design procedure of an SMC has two major steps. In the first step, a switching surface is defined which has the desired convergence properties and provides the required asymptotic behavior. In the second step, a switching law is designed so that the reaching condition is guaranteed.

Sliding Surface Definition.
e sliding surface of a conventional SMC for the boost converter can be defined as where e corresponding switching law for the defined sliding is where q is the index for subsystems defined in Section 2.

Existence Condition.
e existence condition for the sliding surface guarantees that there exists a region near the sliding surface, in which the switching law can steer each initial state inside it to hit the sliding surface. is condition is formulated as is condition guarantees that, in some regions around the surface, the switching law steers state variables toward them. According to (1), (3), (15), and (16), the time derivative S ′ is Since state transition functions of both modes are continuous, S 1 ′ and S 2 ′ are continuous. Hence, for the existence condition, it is sufficient that where ∧ represents intersection and Condition S � 0 in (18) makes i L and v C dependent so that the inequalities are functions of i L and k. For an ideal boost converter without parasitic elements, the existence of inequalities (17) can be solved to derive limitations for the parameter k factor. However, when parasitic elements exist, these inequalities are expanded as follows.

Inequality 1
where Inequality 2 where It can be shown analytically that inequality 1 holds for positive values of i L and k. But S 2 ′ in inequality 2 has a thirdorder two-variable function, and it is better to solve it numerically. Figure 4 depicts a contour plot for S 1 ′ and S 2 ′ 6 Complexity with respect to k and i L for a sample boost converter with circuit 1 parameters given in Table 1.
According to (18), S 1 ′ should be greater than zero. Figure 4 shows that there are no limitations when i L and k are positive. According to (18), S 2 ′ should be smaller than zero. For every positive value of k, there is a minimum value of i L for S 2 ′ to be positive. e zero level of the contour determines this limit.

Reaching Condition.
e existence condition guarantees sliding and remaining on the surface when the state is near the defined sliding surface, but the reaching condition guarantees that when the initial state is far from the sliding surface, it will reach the surface in finite time. e reaching condition of states to the surface is simply proved according to the time response of the converter in (11). From this equation, it can be concluded that as long as the eigenvalues of state matrices have negative real parts, the trajectory will converge to x e for different subsystems: According to the switching rule in (15), the switch will be ON when S < 0 and OFF when S > 0; then, all trajectories will reach the sliding surface if the following conditions are met: is reaching condition can be properly interpreted in the phase portrait.
It can be proved that if the equilibrium point of one subsystem is located in the opposite region where the other subsystem works, then the reaching condition is satisfied. Figure 5 depicts a representation of this condition.

Sliding Dynamics.
e equivalent dynamics of the system on the sliding surface will steer the state toward the desired state. Suppose that the system state is on the surface and slides on it, then the equivalent time derivative of S should be zero. From (19), we have where m is the slope and r is a vector which plays a role similar to the duty cycle in PWM switching. A candidate solution to r will be defined as Sliding dynamics of the system is the dynamics of the state variable on the sliding surface. A measure of the state variable position on the sliding surface is defined by where n is a vector perpendicular to m and parallel to the surface. erefore, the sliding dynamics will be the time derivative of this defined position:  x i1 x i2 Switching surface x r Complexity (29) But the above determinant is positive below the conic section of possible equilibrium points and negative above it. Since R is positive above x r and above the conic section for positive k values, we have D × D ′ < 0, for all x on S � 0.
(30) erefore, the system state converges to x r for any initial state.

Simulation of Tuned SMC.
e performance of the tuned conventional SMC is simulated using the MATLAB software. Figure 6 shows the state trajectory and waveforms for the sample boost converter. is SMC reduces the transient current overshoot and can be simply implemented. However, voltage and current waveforms show that the time response is slow compared with the time-optimal control. Another drawback of this controller is the output voltage dependent on circuit parameters.

Principle of Operation.
Although the tuned conventional SMC takes care of the inductor current limit and has an acceptable response speed, the conventional controller may not be the fastest controller with this determined constraint. To achieve the fastest possible response, it is necessary to hold the converter's current in its maximum allowed value during the transient. is idea leads to a constrained TOC. But TOC has a high computational complexity again. In this section, a modified piecewise linear switching surface will be defined. is modified surface keeps the inductor current high enough in the transient state, and then the system state converges to its desired value in a suitable manner near the time-optimal trajectory. is method keeps the response speed as fast as possible and the steady-state voltage oscillations as low as possible.

Piecewise Sliding Surface.
e new piecewise switching surface of the proposed SMC is defined as where i max represents the maximum allowed inductor current. In (31), k is a positive factor but has small values to keep the time response fast.

Existence Condition.
e existence condition for the first part of the new sliding surface is similar to that of the conventional SMC, but for the second part corresponding to a constant-current region, it is investigated as follows. e derivative of the surface function for the constant-current region is Condition S � 0 in (32) makes i L and v C dependent so that the inequalities are functions of i L and k. erefore, the inequalities can be expanded as follows.

Inequality 3
Since r L ≪ 1, this inequality is true in most practical cases. e second inequality can be derived as follows.

Inequality 4
is inequality guarantees that if the initial voltage is greater than the source voltage, the output current can slide on the constant-current switching surface, and it will not exceed it. But when the initial voltage is much lower than the source voltage, the converter may exceed the maximum current limits. It can be seen in the phase portrait of the vector fields that this is a natural property of the boost converter and does not belong to the proposed controller. To overcome this overcurrent problem for small initial voltages, it is suitable to start the converter with a startup higher resistance load and then switch to the full load.

Reaching Condition.
Reaching condition can be proved similarly to the conventional SMC case. Since the equilibrium point of subsystem 1 is in the opposite side of the switching surface, any trajectory starting from the subsystem 1 side will hit the sliding surface. is is true for trajectories starting from the subsystem 2 side, similarly.

Sliding Dynamics.
Sliding dynamics for the first part of the surface is exactly the same as that of the conventional SMC. For the constant-current part, the following definitions lead to a similar proof:  10 Complexity where x s is the intersection of the constant-current line with the conic section. Using these definitions, it can be shown that if the system state reaches the constant-current surface, it will slide up to the corner of the two switching surfaces and then toward the desired state.

Simulation of the Proposed Controller.
Simulation results of the proposed PCTO-SMC for circuit 2 parameters given in Table 1  e k parameter determines the slope of the near-optimal part of the switching surface. In practice, since r l , r c << 1, the existence condition, (20) and (22), can be simplified as k > k min where k min � L · v ref /(R · C · v s ) is the slope of the time-optimal switching surface and has a small positive value. For example, Complexity for circuit 2 parameters given in Table 1, k min � 0.007. Hence, as long as k is near k min , the controller's response speed remains close to that of the time-optimal controller. On the contrary, when k is very close to k min , the controller has a marginal behavior in sliding existence and stability.

State Trajectory and Startup.
Simulation results in the previous sections show faster time response for the proximate constrained time-optimal SMC. But the other capabilities of this controller, which are proved in previous sections, are better illustrated in experimental results. e proposed controller's performance is validated in the laboratory using a prototype boost converter with the parameters of circuit 2 in Table 1. An eZdsp F2812 board is used to implement the controller.
Because of switching frequency limitations, the sample time of the eZdsp board is set to 10 μs. is sample time causes higher current fluctuations in the state trajectory and current waveforms. erefore, for comparison with the experimental waveforms, simulation results with the practical sampling frequency will be compared here. Figure 8 shows the state trajectory and waveforms of the PCTO-SMC with 10 μs sample time and 5 V source voltage and 20 V desired output voltage starting from an initial voltage of 5 V. It is shown in these simulations that, with larger sample time, the converter's current ripple will be higher. e corresponding experimental results for voltage and current waveforms are shown in Figure 9. e state trajectory is exported to the personal computer using the Oscilloscope software tool and then plotted in Figure 9. It is depicted in this figure that the experimental validation has similar results to the previous simulations.
e reminder of this section shows the experimental results related to changes in load, source voltage, and output voltage reference.

Response to Load, Source Voltage, and Reference Voltage
Changes. Several experimental tests were carried out to investigate the controller's response to the load resistance, source voltage, and reference output changes. Possible changes of load between R � 112 Ω and R � 56 Ω and between R � 112 Ω and R � 232 Ω have been tested. Figure 10 shows some of these results. Output voltage varies about 1% of its nominal voltage in response to load changes. Note that sensitivity to the load changes can be further reduced by tuning the switching surface slope k. Reducing the k parameter leads to smaller output voltage variations but more current fluctuations. Current fluctuations can be reduced again by increasing the switching speed (corresponding to shortening the sampling time).
is is not possible in the present experimental setup. Source voltage changes between 1.5 V, 3.3 V, 5 V, and 10 V have been tested. Figure 10(c) shows output voltage response when the source voltage changes from 10 V to 5 V.
e results show less than 0.5% of nominal value changes in the output voltage. Several experimental tests for step changes in the reference voltage were also performed.

Conclusion
A hybrid switching system model of the boost converter and phase portrait geometric representation of the corresponding vector fields lead to better description of its behavior. en, a new proximate constrained time-optimal sliding mode controller is proposed which directly controls the converter switch.
is controller has faster response time compared to conventional SMC. e proposed method has a current limitation capability by designing a piecewise sliding surface to maintain the inductor current constraint. Experiment results show good response to changes in load resistance, source voltage, and output voltage reference. Limited current and low output variation in load and source variations are beneficial in many RE applications. Compared to the time-optimal control, the proposed controllers achieve much lower current peak and steady-state voltage fluctuations and also lower computational complexity.

Data Availability
e MATLAB file data used to support the findings of this study are available from the corresponding author upon request.  Complexity 13