𝐻_∞ Fuzzy Control of DC-DC Converters with Input Constraint

Abstract

This paper proposes a method for designing 𝐻_∞ fuzzy control of DC-DC converters under actuator saturation. Because linear control design methods do not take into account the nonlinearity of the system, a T-S fuzzy model and a controller design approach is used. The designed control not only handles the external disturbance but also the saturation of duty cycle. The input constraint is first transformed into a symmetric saturation which is represented by a polytopic model. Stabilization conditions for the 𝐻_∞ state feedback system of DC-DC converters under actuator saturation are established using the Lyapunov approach. The proposed method has been compared and verified with a simulation example.

1. Introduction

The main task of DC-DC converters is the adaptation of the voltage and current levels between sources and loads while maintaining a low power loss in the conversion [1–3]. With the extensive use of DC-DC converters in different industry applications (e.g. power supplies for personal computers, DC-motor drive, telecommunications equipment, etc.), improving their performances has become an interesting problem in recent years [1–17]. Recently, different converter circuits (buck converter, boost converter, buck-boost converter, Cuk converter, etc.) are known. According to each application purpose (increase or decrease the magnitude of the DC voltage and/or invert its polarity), the converter circuit was chosen. Among them, we consider here, the control of the basic Pulse-Width-Modulation (PWM) buck converters, but it could be easily adapted for other converters.

The DC-DC switching power converters are highly nonlinear systems [4, 5, 15, 16]. Consequently, the conventional linear controls based on averaging and linearization techniques [12, 14] will result in poor dynamic performance or system instability. In order to resolve this problem, the fuzzy logic approach has been proposed as an alternative solution [4, 5, 7, 8, 11, 13, 15, 16]. Specifically, in [4, 5], the authors have reported very significant results on the modelling and control of DC-DC converters with T-S fuzzy systems. In [4, 5, 8, 11, 15, 16], authors have proposed methods for designing fuzzy control of DC-DC converters. Nevertheless, in the aforementioned papers, they have not taken into account the inherent nonlinearity of actuator saturation (duty cycle) and the external disturbance. Motivated by this observation, our aim is to use T-S fuzzy systems in order to control a nonlinear DC-DC converter subject to external disturbances and actuator saturation.

The actuator saturation can degrade the performance of the closed-loop system and often make the stable closed-loop system unstable [1, 18–21]. The T-S fuzzy system in the presence of saturation has been receiving increasing attention for control of nonlinear systems [18–29]. In these works, the saturation effect is considered as a symmetric function. This is not the case in DC-DC converter application where the saturation function constrained is between 1 and 0. The solution of this problem proposed in linear control case [2] has been leading to complexity analysis with the nonlinearity of DC-DC converter model. Here, we proposed a simple mathematical transformation to obtain a symmetric saturation.

The 𝐻_∞ approach is used to analyze and to synthesize controllers/observers achieving an optimal level of disturbance attenuation and to guarantee the stability of the closed-loop system. To achieve this goal, the idea is to minimize the 𝐻_∞ norm which represents the maximum value ratio between the output signal energy (controlled output) and the input signal energy (disturbance input). Recently, many researchers have used this approach for control design of T-S fuzzy systems (see for example [20, 21, 30, 31] and their references). In most cases, the quadratic Lyapunov function and linear matrix inequality (LMI) techniques are used to analyze and synthesize of stabilization of T-S fuzzy systems [17–21, 30, 31]. Based on these works, we address the control problem of DC-DC converters via PDC controller with actuator saturation and external disturbance. In this paper, we will use T-S fuzzy systems to represent the DC-DC converters. The control design is based on the parallel distributed compensation (PDC) scheme [32]. The idea is that for each local linear model, a linear feedback control is designed. The control problem is formulated and solved as a LMI optimization problem [33].

This paper is organized as follows. Section 2 gives the averaged model of basic PWM buck converter. Section 3 presents the T-S fuzzy model of the DC-DC converter. Section 4 formulates the conditions for the 𝐻_∞ stabilization of fuzzy control with actuator saturation problem in terms of LMI. The simulation results to show the effectiveness of the proposed method are given in Section 4.

Notation 1. 𝐼_𝑟 denotes the set {1,2,…,𝑟}, ℜ denotes the set of real number and ℜ^𝑛×𝑚 the set of all 𝑛×𝑚 real matrix. 𝑀>(≥,,<,≤) is used to denote a symmetric positive definite (positive semidefinite, negative definite, negative semidefinite, resp.) matrix. * denotes the symmetric bloc matrix, 𝑋+(^∗) denotes 𝑋+𝑋^𝑇, × denotes the multiplication, co denotes the convex hull, and ⋂^𝑟_𝑗=1 is used to denote the intersection of 𝑟 sets.

2. Averaged Model of Basic PWM Buck Converter

Figure 1 shows the basic circuit of the nonlinear PWM buck converter proposed in [4, 5] with an external disturbance 𝑖_load(𝑡) as proposed in [1, 2]. 𝑅_𝑚 is the on-state resistance of the MOSFET transistor, 𝑅_𝐿 is the winding resistance of inductor, 𝑉_𝑑 is the threshold voltage of the diode, and 𝑅_𝑐 is the equivalent series resistance of the filter capacitor.

Figure 1 

Schematic of a basic PWM buck converter.

By applying the Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) in on-state of the MOSFET transistor case, we obtain 𝑖_𝑙̇(𝑡)𝑣̇_𝑐(𝑡)=−1𝐿𝑅_𝑙+𝑅𝑅_𝑐𝑅+𝑅_𝑐−𝑅𝐿𝑅+𝑅_𝑐𝑅𝐶𝑅+𝑅_𝑐−1𝐶𝑅+𝑅_𝑐𝑖_𝑙(𝑡)𝑣_𝑐(𝑡)+1𝐿𝑉_𝑖𝑛+𝑅_𝑚𝑖_𝑙(𝑡0+)𝑅𝑅_𝑐𝐿𝑅+𝑅_𝑐−𝑅𝐶𝑅+𝑅_𝑐𝑖_load(𝑡). (2.1) Now, in off-state of the MOSFET transistor case and by applying of KVL and KCL, we get 𝑖_𝑙̇(𝑡)𝑣̇_𝑐(𝑡)=−1𝐿𝑅_𝑙+𝑅𝑅_𝑐𝑅+𝑅_𝑐−𝑅𝐿𝑅+𝑅_𝑐𝑅𝐶𝑅+𝑅_𝑐−1𝐶𝑅+𝑅_𝑐𝑖_𝑙(𝑡)𝑣_𝑐(𝑡)+−𝑉_𝑑𝐿0+𝑅𝑅_𝑐𝐿𝑅+𝑅_𝑐−𝑅𝐶𝑅+𝑅_𝑐𝑖_load(𝑡). (2.2) Using averaging method for one-time scale discontinuous system (AM-OTS-Ds) [9], the global dynamical behavior of the DC-DC converter is modeled as follows: 𝑖_𝑙̇(𝑡)𝑣̇_𝑐(𝑡)=−1𝐿𝑅_𝑙+𝑅𝑅_𝑐𝑅+𝑅_𝑐−𝑅𝐿𝑅+𝑅_𝑐𝑅𝐶𝑅+𝑅_𝑐−1𝐶𝑅+𝑅_𝑐𝑖_𝑙(𝑡)𝑣_𝑐(𝑡)+1𝐿𝑉_𝑖𝑛+𝑉_𝑑+𝑅_𝑚𝑖_𝑙(𝑡0𝑑(𝑡)+−𝑉)^𝑑𝐿0+𝑅𝑅_𝑐𝐿𝑅+𝑅_𝑐−𝑅𝐶𝑅+𝑅_𝑐𝑖_load(𝑡), (2.3) where 𝑑(𝑡) is the duty cycle.

3. T-S Fuzzy Model of DC-DC Converter

Our control approach is based on T-S fuzzy models and the robustness of this control depends on the ability of the fuzzy model to represent the real system. In this section our objective is to show the effectiveness of T-S fuzzy model to represent the DC-DC converter.

The T-S fuzzy model has been successfully used to approximate nonlinear systems by interpolation of numerous local linear models [34].

𝑅_𝑖: If 𝜉₁(𝑡) is about 𝑀_1𝑖 and 𝜉_𝑞(𝑡) is about 𝑀_𝑞𝑖 then 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_1𝑖𝑤(𝑡)+𝐵_2𝑖𝜎(𝑡)𝑧(𝑡)=𝐶_1𝑖𝑥(𝑡)+𝐷_1𝑖𝑤(𝑡)+𝐷_2𝑖𝜎(𝑡)For𝑖∈𝐼_𝑟 (3.1) in which 𝑀_𝑗𝑖 is the fuzzy set of 𝜉_𝑖 in rule 𝑅_𝑖, 𝑟 is the number of if-then fuzzy rules and 𝜉_𝑖(𝑡) are the decision variable assumed measurable, 𝑥(𝑡)∈ℜ^𝑛 is the system state vector, 𝜎(𝑡)∈ℜ^𝑚 is the saturate control input, 𝑦(𝑡)∈ℜ^𝑝 is the measurable output, 𝑧(𝑡)∈ℜ^𝑛𝑧 is the controlled output variable, and 𝑤(𝑡)∈ℑ₂ is the disturbance variable with ℑ₂={𝑤∈ℜ^𝑛𝑤∣‖𝑤‖₂≤𝑤,𝑤≻0}.

The global dynamic system is inferred as follows: 𝑥̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝜉(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_1𝑖𝑤(𝑡)+𝐵_2𝑖𝜎(𝑡,𝑧(𝑡)=_𝑟𝑖=1)𝜇_𝑖(𝜉(𝑡))𝐶_1𝑖𝑥(𝑡)+𝐷_1𝑖𝑤(𝑡)+𝐷_2𝑖𝜎(𝑡), (3.2) where 𝜇_𝑖(𝜉(𝑡=∏))^𝑞𝑗=1𝑀_𝑗𝑖𝜉_𝑗(𝑡)∑^𝑟_𝑖=1∏^𝑞_𝑗=1𝑀_𝑗𝑖𝜉_𝑗(𝑡), (3.3)𝑀_𝑗𝑖(𝜉_𝑗(𝑡)) is the grade of membership of 𝜉_𝑗(𝑡) in 𝑀_𝑗𝑖. The normalized activation function 𝜇_𝑖(𝜉(𝑡)) in relation with the 𝑖th submodel is such that ^𝑟_𝑖=1𝜇_𝑖(𝜉(𝑡))=10≤𝜇_𝑖(𝜉(𝑡))≤1∀𝑖∈𝐼_𝑟, (3.4) Among different methods (identification method, linearization around different operating points, or by transformation in nonlinear sector), the method by transformation in nonlinear sector is based on bounded function and gives a minimum number of local models. Here we will use this method to represent the DC-DC converter.

The following Lemma will be used in the sequel of the paper.

Lemma 3.1. Let 𝑓(𝑥(𝑡))∶𝑅→𝑅 is a bounded function, it always exist tow functions, 𝜂₁(𝑥(𝑡)) and 𝜂₂(𝑥(𝑡)) and two scalars 𝛼 and 𝛽 such that: 𝑓(𝑥(𝑡))=𝛼×𝜂₁(𝑥(𝑡))+𝛽×𝜂₂(𝑥(𝑡)) (3.5) with 𝜂₁(𝑥(𝑡))+𝜂₂(𝑥(𝑡))=1,𝜂₁(𝑥(𝑡))≥0,𝜂₂(𝑥(𝑡))≥0. (3.6)

In this case, the obtained T-S fuzzy model represents exactly the nonlinear system for 𝑥∈𝑅^𝑛 with 2^nl locals models where nl represents the number of local models.

Assuming that, 𝐼_𝑙min≤𝑖_𝑙(𝑡)≤I_𝑙max and by Lemma 3.1, the system described by (2.3) is modeled with a two-rules T-S fuzzy system as follows: Rule 1: if 𝑖_𝑙(𝑡) is about 𝐼_min then: 𝑥̇(𝑡)=𝐴_𝑚𝑥(𝑡)+𝐵_𝑚1𝑖_load(𝑡)+𝐵_𝑚21𝑑(𝑡)+𝐷_𝑚. Rule 2: if 𝑖_𝑙(𝑡) is about 𝐼_max then: 𝑥̇(𝑡)=𝐴_𝑚𝑥(𝑡)+𝐵_𝑚1𝑖_load(𝑡)+𝐵_𝑚22𝑑(𝑡)+𝐷_𝑚.

The overall model of T-S fuzzy system can be given by the following: 𝑥̇(𝑡)=𝐴_𝑚𝑥(𝑡)+𝐵_𝑚1𝑖_load(𝑡)+𝐵_𝑚2𝜇𝑑(𝑡)+𝐷_𝑚, (3.7) with 𝑥₁(𝑡)=𝑖_𝑙(𝑡),𝑥₂(𝑡)=𝑣_𝑐(𝑡),𝐵_𝑚2𝜇=∑²_𝑖=1𝜇_𝑖(𝑖_𝑙(𝑡))𝐵_𝑚2𝑖 and 𝐴_𝑚=−1𝐿𝑅_𝑙+𝑅𝑅_𝑐𝑅+𝑅_𝑐−𝑅𝐿𝑅+𝑅_𝑐𝑅𝐶𝑅+𝑅_𝑐−1𝐶𝑅+𝑅_𝑐,𝐵_𝑚21=1𝐿𝑉_𝑖𝑛+𝑉_𝑑+𝑅_𝑚𝐼_𝑙min0,𝐵_𝑚1=𝑅𝑅_𝑐𝐿𝑅+𝑅_𝑐−𝑅𝐶𝑅+𝑅_𝑐,𝐵_𝑚22=1𝐿𝑉_𝑖𝑛+𝑉_𝑑+𝑅_𝑚𝐼_𝑙max0,𝐷_𝑚=−𝑉_𝑑𝐿0. (3.8) The membership function is such that 𝜇₁𝑖_𝑙(𝑡)=−𝑖_𝑙(𝑡)+𝐼_𝑙max𝐼_𝑙max−𝐼_𝑙min,𝜇₂𝑖_𝑙(𝑡)=𝑖_𝑙(𝑡)−𝐼_𝑙min𝐼_𝑙max−𝐼_𝑙min. (3.9) To show the effectiveness of the fuzzy model to represent the nonlinear system, we simulate the fuzzy model and the nonlinear system for the same inputs with different initial conditions. The simulation parameters used in this work are as follows [4]: 𝑅=6Ω,𝑅_𝑙=48.5mΩ,𝑅_𝑐=0.16Ω,𝑅_𝑚=0.27Ω,𝑉_𝑖𝑛=30V,𝐿=98.58mH,𝐶=202.5𝜇F,𝑓=1KH,𝐼_𝑙min=0A,𝐼_𝑙max=10A, (3.10) with 𝑑(𝑡)=0.5 and the input current 𝑖_load(𝑡)=0.25sin(5000𝑡).

Figure 2 shows the membership functions of the fuzzy model. The DC-DC converter responses of nonlinear system (2.3) and T-S fuzzy system (3.7) with initial conditions (𝑣_𝑐(0)=0V,𝑖_𝑙(0)=5A) and (𝑣_𝑐(0)=5V,𝑖_𝑙(0)=0A) are shown in Figures 3 and 4, respectively.

Figure 2 

Membership functions of the T-S fuzzy model.

(a)

(b)

Figure 3 

The nonlinear DC-DC converter responses and its T-S fuzzy representation with initial conditions (𝑣𝑐(0)=0V,𝑖𝑙(0)=5A).

(a)

(b)

Figure 4 

The nonlinear DC-DC converter responses and its T-S fuzzy representation with initial conditions (𝑣𝑐(0)=5V,𝑖𝑙(0)=0A).

These figures demonstrate that, with different initial conditions, the T-S fuzzy system (2.3) has the same behaviour as the nonlinear system (3.7). This means the satisfactory approximation ability of the fuzzy model.

4. 𝐻_∞ Fuzzy Control of DC-DC Converter

In this section, we present an 𝐻_∞ fuzzy approach to the control design of DC-DC converter.

4.1. Saturated Control Analysis

In order to control the output voltage of the DC-DC converter, we define the following variables: 𝑒₁(𝑡)=𝑣_𝑐(𝑡)−𝑉_ref,𝑒₂(𝑡)=𝑒₁̇(𝑡), (4.1) where 𝑉_ref is the reference voltage of 𝑣_𝑐(𝑡).

The time-derivative of 𝑒₂(𝑡) is as follows: 𝑒₂̇(𝑡)=𝑣̈_𝑐(𝑡)=𝑅𝐶𝑅+𝑅_𝑐𝑖_𝑙̇(𝑡)−1𝐶𝑅+𝑅_𝑐𝑣̇_𝑐(𝑡)−𝑅𝐶𝑅+𝑅_𝑐𝑖̇_load(𝑡). (4.2) Using the converter’s model defined in (2.3), we have 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵_𝑤𝑤₁(𝑡)+𝐵₂(𝑥)𝑑(𝑡)𝑥+𝐷,₁(𝑡)=𝑒₁(𝑡),𝑥₂(𝑡)=𝑒₂(𝑡),𝑤₁(𝑡)=𝑖_load(𝑡)𝑖̇_load(𝑡),−𝐴=01𝑅+𝑅_𝑙𝐿𝐶𝑅+𝑅_𝑐−𝐿+𝑅_𝑙𝐶𝑅+𝑅_𝑐+𝐶𝑅𝑅_𝑐𝐿𝐶𝑅+𝑅_𝑐,𝐵₂(𝑥)=0𝑅𝐿𝐶𝑅+𝑅_𝑐𝑉_𝑖𝑛+𝑉_𝑑−𝑅_𝑚𝑖_𝑙(𝑡,𝐵_𝑤)=−𝑅00_𝑙𝑅𝐿𝐶𝑅+𝑅_𝑐−𝑅𝐶𝑅+𝑅_𝑐0−,𝐷=𝑅𝑉_𝑑+𝑉_ref+𝑅_𝑙𝑉_ref𝐿𝐶𝑅+𝑅_𝑐. (4.3) By Lemma 3.1, this system can be modeled with a two-rule T-S fuzzy system as follows: Rule 1: if 𝑖_𝑙(𝑡) is about 𝐼_min then: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵_𝑤𝑤₁(𝑡)+𝐵₂₁𝑑(𝑡)+𝐷. Rule 2: if 𝑖_𝑙(𝑡) is about 𝐼_max then: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵_𝑤𝑤₁(𝑡)+𝐵₂₂𝑑(𝑡)+𝐷.

and the global T-S fuzzy model can be given by the following: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵_𝑤𝑤₁(𝑡)+𝐵_2𝜇𝑑(𝑡)+𝐷, (4.4) with: 𝑤₁(𝑡)=𝑖_load(𝑡)𝑖̇_load(𝑡),𝐵_2𝜇=²_𝑖=1𝜇_𝑖𝑖_𝑙(𝑡)𝐵_2𝑖,𝐵₂₁=0𝑅𝐿𝐶𝑅+𝑅_𝑐𝑉_𝑖𝑛+𝑉_𝑑−𝑅_𝑚𝐼_𝑙min,𝐵₂₂=0𝑅𝐿𝐶𝑅+𝑅_𝑐𝑉_𝑖𝑛+𝑉_𝑑−𝑅_𝑚𝐼_𝑙max, (4.5) The output voltage of the DC-DC converter can be controlled by means of variation of duty cycle (Figure 1). The duty cycle is defined by the ratio 𝑇_on/𝑇𝑠, where 𝑇𝑠 is the frequency of the PWM circuit and 𝑇_on is the time when the MOSFET is on (Figure 5). A transistor gate driver (Figure 1) switches the MOSFET between the conducting (on) and blocking (off) states using a binary signal 𝑢_𝑏(𝑡). This signal is produced by the modulator. In this operation, the duty cycle is compared with a sawtooth signal 𝑉_𝑠(𝑡) of amplitude equals to 1. Consequently, the duty-cycle is constrained in amplitude between 0 and 1.

Figure 5 

Waveforms of the PWM process.

Thus, in this application, the control input is subject to actuator saturation with the following saturation: 0≤𝑑(𝑡)≤1, (4.6) which is not a symmetric saturation. However, most results reported in open literature have treated the controller design analysis with symmetric saturation [18–21]. To transform this saturation into a symmetric saturation, authors in [2] have proposed a change of variables to the linear converter model. This idea cannot be used in our case (nonlinear converter model) because it increases the complexity of the control analysis. Here, we will propose another idea that allows a simple analysis thereafter.

The system described in (4.4) can be written as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵_𝑤𝑤₁(𝑡)+𝐵₂(𝑥𝑑(𝑡)−1)2+12𝐵₂(𝑥)𝑥̇(𝑡)(𝑡)+𝐷,=𝐴𝑥+𝐵₁𝑤(𝑡)+𝐵₂(𝑥)𝛿(𝑡), (4.7) with 𝑤(𝑡)=12𝑅𝐿𝐶𝑅+𝑅_𝑐𝑉_𝑖𝑛+𝑉_𝑑−𝑅_𝑚𝑖_𝑙(𝑡−𝑅)^𝑙𝑅𝐿𝐶𝑅+𝑅_𝑐𝑖_load(𝑡)−𝑅𝐶𝑅+𝑅_𝑐𝑖̇_load(𝑡)−𝑅𝑉_𝑑+𝑉_ref+𝑅_𝑙𝑉_ref𝐿𝐶𝑅+𝑅_𝑐 (4.8) is the external disturbance, 𝐵₁=[01]^𝑇, and 𝛿(𝑡)=𝑑(𝑡)−1/2 is the new control input.

This last system can be represented with the following T-S fuzzy model: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵₁𝑤(𝑡)+𝐵_2𝜇𝛿(𝑡). (4.9) Thus, the new control input 𝛿(𝑡) is constrained as follows: −12(𝑡)≤1≤𝛿2. (4.10)

In this work, the controller is a nonlinear state feedback which shares the same activation functions as the T-S fuzzy model (3.2) of the following form [32]: 𝑢(𝑡)=𝐾_𝜇𝑥(𝑡),𝐾_𝜇=^𝑟_𝑗=1𝜇_𝑗(𝜉(𝑡))𝐾_𝑗, (4.11) where and 𝐾_𝑗∈ℜ^𝑚×𝑛 is the local controller matrix to be determined.

The control input is subject to actuator saturation the saturation function means: 𝜎(𝑡)=sat(𝑢(𝑡),𝑢)=𝑠₁,𝑠₂,…,𝑠_𝑚^𝑇,𝑠_𝑖=sign𝑢_𝑖min𝑢_𝑖,𝑢_𝑖, (4.12)𝑢∈𝑅^𝑚 denotes the saturation level and 𝑢_𝑖 and 𝑢_𝑖 denote the 𝑖th element of 𝑢 and 𝑢(𝑡). In our case: 𝑢_𝑖=𝑢=0.5. (4.13) The saturation obtained in (3.6) is a symmetric saturation and the following lemma can be used to the stability analysis.

Lemma 4.1 (see [18]). Let 𝐸 be the set of 𝑚×𝑚 diagonal matrices whose diagonal elements are 1 or 0. Suppose that |𝑣_𝑖|≤𝑢_𝑖 for all 𝑖∈𝐼_𝑚, where 𝑣_𝑖 and 𝑢_𝑖 denote the 𝑖th element of 𝑣∈ℜ^𝑚 and 𝑢∈ℜ^𝑚, respectively. If 𝑥∈⋂^𝑟_𝑗=1ℑ(𝐻_𝑗) for 𝑥∈ℜ^𝑛, then: sat(𝑢,𝑢)=^2𝑚_𝑠=1𝛼_𝑠𝐸_𝑠𝑢+𝐸_𝑠𝜈,_2𝑚𝑠=1𝛼_𝑠=1,0≤𝛼_𝑠≤1,𝜈=^𝑟_𝑗=1𝜇_𝑗𝐻_𝑗ℑ𝐻𝑥,_𝑗=𝑥∈ℜ^𝑛∣ℎ^𝑗_𝑖𝑥≤𝑢_𝑖, (4.14) where 𝐸_𝑠 denotes all elements of 𝐸,𝐸_𝑠=𝐼−𝐸_𝑠, 𝐻_𝑗 is 𝑚×𝑛 matrix and ℎ^𝑗_𝑖 is the 𝑖th row of the matrix 𝐻_𝑗.

In our case 𝑚=1,𝐸_𝑠∈co{0,1}, 𝑢₁ is the saturation level (𝑢₁=1/2) and 𝑟(𝑟=2) is the number of local models. Consequently, the saturation function in (4.10) can be rewritten as follows: sat𝐾_𝜇𝑥(𝑡)=𝐸_𝛼𝐾_𝜇+𝐸_𝛼𝐻_𝜇𝑥(𝑡), (4.15) with ℎ^𝑗_𝑖𝑥≤12,∀𝑖∈𝐼_𝑚,𝑗∈𝐼_𝑟, (4.16) and 𝐻_𝜇=∑²_𝑖=1𝜇_𝑖𝐻_𝑖,𝐸_𝛼=∑²_𝑠=1𝛼_𝑠𝐸_𝑠 and 𝐸_𝛼=∑²_𝑠=1𝛼_s𝐸_𝑠.

4.2. Quadratic Lyapunov Stability

For a constant 𝜌>0 and a symmetric positive matrix 𝑃, define an ellipsoid as follows: 𝜀𝑃,𝜌=𝑥∈ℜ^𝑛∣𝑥^𝑇𝑃𝑥≤𝜌. (4.17) This ellipsoid can be rewritten as follows: 𝜀𝑃,𝜌=𝑥∈ℜ^𝑛∣𝑥^𝑇𝐏𝑥≤1 (4.18)𝜌≻0, this implied that 𝐏=𝑃/𝜌∈ℜ^𝑛×𝑛 is a symmetric positive matrix.

We define the Lyapunov function as follows: 𝑉(𝑥)=𝑥(𝑡)^𝑇𝐏𝑥(𝑡). (4.19) An ellipsoid 𝜀(𝑝,𝜌) is said to be contractively invariant set if 𝑉̇(𝑥)<0, for  all 𝑥∈𝜀(𝑃,𝜌)∣{0}. Therefore, if an ellipsoid is contractively invariant, it is inside the domain of attraction.

An ellipsoid 𝜀(𝑃,𝜌)={𝑥∈𝑅^𝑛∣𝑥^𝑇𝐏𝑥≤1} is inside ⋂^𝑟_𝑗=1ℑ(𝐻_𝑗) if and only if for all 𝑖∈𝐼_𝑚,𝑗∈𝐼_𝑟: ℎ^𝑗_𝑖^𝑇𝐏ℎ^𝑗_𝑖≤𝑢²_𝑖𝐼. (4.20) In order to design a DC-DC converter to perform adequately in the presence of external disturbances, the 𝐻_∞ attenuation performance is chosen as the performance measure, which is defined as follows: ^𝑇₀𝑧(𝜏)^𝑇𝑧(𝜏)𝑑𝜏<𝛾^2𝑇₀𝑤(𝜏)^𝑇𝑤(𝜏)𝑑𝜏, (4.21) where 𝛾 is a positive scalar and small as possible, 𝑤(𝑡)∈𝐿₂[01] and 𝑥(0)=0.

In our problem, the controlled output is the error 𝑒₁(𝑡), that is: 𝑧(𝑡)=𝐶₁𝑥(𝑡), (4.22) where 𝐶₁=[01]. The closed-loop system composed of (4.9), (4.22), and (4.15) is given by the following: 𝑥̇(𝑡)=𝐴+𝐵_2𝜇𝐸_𝛼𝐾_𝜇+𝐸_𝛼𝐻_𝜇𝑥(𝑡)+𝐵_𝑚1𝑤(𝑡),𝑧(𝑡)=𝐶₁𝑥(𝑡). (4.23)

Theorem 4.2. The ellipsoid 𝜀(𝑃,𝜌) is contractively invariant set of the closed-loop system (4.15) and achieves in a disturbance rejection level 𝛾, if there exist a symmetric positive definite matrix 𝑄 and matrices 𝐹_𝑗∈ℜ^𝑚×𝑛, solutions of the following LMI problem: min^𝛾_{𝑄,𝐹𝑗,𝑍𝑗},14𝑧^𝑗_𝑖𝑧^𝑗_𝑖^𝑇𝑄≥0,∀𝑖∈𝐼_𝑚,𝑗∈𝐼_𝑟𝐴𝑄+𝐵_2𝑖𝐸_𝑠𝐹_𝑗+𝐵_2𝑖𝐸_𝑠𝑍_𝑗+^∗∗∗𝐵^𝑇₁−𝛾𝐼^∗𝐶₁𝑄0−𝛾𝐼<0,∀𝑖∈𝐼_𝑟,𝑗∈𝐼_𝑟. (4.24)(4.25)(4.26) Then one gets 𝐾_𝑗=𝐹_𝑗𝑄⁻¹,𝐻_𝑗=𝑍_𝑗𝑄⁻¹. (4.27) Using Lemma 3.1 for inequality (4.20), one has: ℎ^𝑗_𝑖^𝑇𝐏ℎ^𝑗_𝑖≤14,∀𝑖∈𝐼_𝑚,𝑗∈𝐼_𝑟. (4.28) Let the following change of variables: 𝑄=𝐏⁻¹,𝑍_𝑗=𝐻_𝑗𝑄. (4.29)(4.30) Consequently, the inequality (4.28) can be rewritten as follows 𝑄4−𝑧^𝑗_𝑖^𝑇𝑧^𝑗_𝑖≥0, (4.31) where 𝑧^𝑗_𝑖 is the 𝑖th row of the matrix 𝑍_𝑗. By Schur complement [33], this last inequality can be transformed as LMI (4.25).

Proposition 4.3 (see [30]). If the Lyapunov function defined in (4.19) satisfies the following Hamilton-Jacobi-Bellman inequality: 𝑉̇(𝑥(𝑡))+𝛾⁻¹𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾𝑤(𝑡)^𝑇𝑤(𝑡)<0, (4.32) along (4.23) for all 𝑥(𝑡)≠0 and 𝑤(𝑡)∈𝐿₂[01]≜{𝑤∈ℜ^𝑛𝑤∣∫^𝑇₀‖𝑤‖²𝑑𝑡≤𝑤,𝑤>0}, then (4.21) is verified. Using (4.23), one has: 𝑉̇(𝑥(𝑡))−𝛾𝑤(𝑡)^𝑇𝑤(𝑡)+𝛾⁻¹𝑧(𝑡)^𝑇𝑧(𝑡)=𝑥(𝑡)𝑤(𝑡)^𝑇𝐴+𝐵_2𝜇𝐸_𝑠𝐾_𝜇+𝐵_2𝜇𝐸_𝑠𝐻_𝜇^𝑇𝐏+^∗∗𝐵^𝑇₁𝐏−𝛾𝐼+𝛾⁻¹𝐶^𝑇₁0𝐶₁0𝑥(𝑡)𝑤(𝑡). (4.33) By Schur complement [13], the condition (4.32) holds if Σ_𝜇=𝐏^𝑇𝐴+𝐵_2𝜇𝐸_𝑠𝐾_𝜇+𝐵_2𝜇𝐸_𝑠𝐻_𝜇+^∗∗∗𝐵^𝑇₁𝐏−𝛾𝐼^∗𝐶₁0−𝛾𝐼<0. (4.34) Pre- and postmultiplying this last inequality by Γ=diag(𝑄,𝐼,𝐼) one has ΓΣ_𝜇𝐴Γ=^𝑖𝑄+𝐵_2𝑖𝐸_𝑠𝐾_𝑗𝑄+𝐵_2𝑖𝐸_𝑠𝐻_𝑗𝑄+^∗∗∗𝐵^𝑇₁−𝛾𝐼^∗𝐶₁𝑄0−𝛾𝐼<0. (4.35) Taking account (4.27) and (4.30), one gets the following LMI: ΓΣ_𝜇𝐴Γ=^𝜇𝑄+𝐵_2𝜇𝐹_𝜇+^∗∗∗𝐵^𝑇₁−𝛾𝐼^∗𝐶₁𝑄0−𝛾𝐼<0, (4.36) which ends the proof.

4.3. The Initial Conditions Constraint

The main purpose of the DC-DC converter control is to maintain the voltage level 𝑣_𝑐(𝑡) equal to the desired level 𝑉_ref. The initial condition for the state vector is as follows: 𝑥₀=𝑥₁(0)𝑥₂(0)^𝑇=𝑒₁(0)𝑒₂(0)^𝑇=𝑉_ref0^𝑇. (4.37) However, it is not possible to assure that any initial condition will belong to the ellipsoid of stability given in Theorem 4.2. In this section, the objective is to guarantee the start up stability with maximum elimination of external disturbances in the presence of saturation.

The point 𝑥₀∈𝜀(𝑃,𝜌) is equivalent to the following: 𝑥^𝑇₀𝐏𝑥₀≤1. (4.38) Or in LMI form by the following: 1𝑥^𝑇₀𝑥₀𝑄≥0. (4.39) The optimization problem proposed in Theorem 4.2 can be formulated as follows: min^𝛾_{𝑄,𝐹𝑗,𝑍𝑗}.LMI(4.27),LMI(4.28),LMI(4.41) (4.40)

4.4. Simulation Results

In order to demonstrate the effectiveness of the proposed method, the DC-DC converter is controlled by the proposed 𝐻_∞-based state feedback and the PDC controller proposed in [4]. In this simulation, the current perturbation step equals to 1 A and the PWM frequency equals to 5 kHz.

Solving the optimization problem described in (4.40) for initial condition 𝑥₀=[12𝑉0]^𝑇, we have: 𝑄=817.3975−6.1238×10⁵−6.1238×10⁵5.5691×10⁵,𝐻₁=𝐻₂=[−0.03540],𝛾=0.7255,𝐾₁=[−1.4681−0.0016],𝐾₂=[−1.5030−0.0017]. (4.41) The control is defined by 𝑑(𝑡)=sat(𝐾_𝜇𝑥(𝑡))+1/2.

Figures 6, 7, and 8 show the DC-DC converter responses (output voltage and inductance current, resp.). Figure 9 shows the trajectory of the saturated control input signal (duty cycle 𝑑(𝑡)). Figure 8 shows, with the presence of saturation, the 𝐻_∞ control can guarantee the stability of all initial conditions in the interval 𝑒₁(0)∈[−12𝑉12𝑉] in presence of actuator saturation and external disturbances (see initial time and 0.1 s time).

Figure 6 

Response of the output voltage 𝑣𝑐(𝑡) of PWM buck converter.

Figure 7 

Response of the inductance current 𝑖𝑙(𝑡) of PWM buck converter.

Figure 8 

The trajectory of the error 𝑒1(𝑡).

Figure 9 

The trajectory of the saturated control (duty cycle 𝑑(𝑡)).

Simulation results (Figures 6–8) demonstrate that our controller guarantees better stabilization performance, better perturbation rejection (see initial time in Figure 6) and better time response (see Figures 6–8) Moreover, the proposed controller is robust with respect to frequency change. Figure 10 shows the simulation results for a frequency of 1 kHz. It shows that our controller achieves better performance even with different frequency values.

Figure 10 

Response of the output voltage 𝑣𝑐(𝑡) of PWM buck converter for 1 kHz.

Despite that we take into account not only the actuators saturation but also the external disturbance in the designing control, our controller gives better stabilization performance (perturbation rejection and better time response) of the system as reported in the open literature [4]. Moreover, with different frequency values, our saturated control gives robust control signal (guarantees the same performances despite the change of frequency). These results demonstrate the effectiveness of proposed method.

5. Conclusion

This work has presented an 𝐻_∞ controller design for DC-DC converters via state feedback under actuator saturation and external disturbances. The T-S fuzzy system is first used to describe the DC-DC converter. Then, the state controller is designed to guarantee the stability of the closed loop system with 𝐻_∞ performance. The saturation effect is represented by a polytopic model. Based on Lyapunov approach, the problem of 𝐻_∞ stabilization in the presence of actuator saturation was formulated as an LMI optimization and solved easily by using existing numerical tools. Finally, the simulation results on a DC-DC converter have demonstrated the effeteness of the proposed control.

The analysis given here is applied to the buck converter; we can extend this work to other converters (boost converter, buck-boost converter, Cuk converter, etc.). Finally, the results developed in the paper can be extended to the case that the underlying systems are involved with any switching dynamics such as a nondeterministic or stochastic switching system (see [35–37]).