Robust H ∞ Switching Rule Design for Boost Converters with Uncertain Parameters and Disturbances

and Applied Analysis 3 The uncertainties can be written as ΔA 1 (t) = M 1 F 1 (t)N 1 , ΔA 2 (t) = M 2 F 2 (t)N 2 , (6) where F i T (t)F i (t) ≤ I, M 1 = M 2 = [ 1 0


Introduction
The last years have witnessed the crescent interest of the scientific community in the study of hybrid systems because of their wide applications in many fields, such as chemical processing, communication networks, traffic control, automotive engine control, and aircraft control [1][2][3][4].The stochastic hybrid systems, also called Markov jump systems, represent an important class of hybrid systems that are popular in modeling many practical systems, such as manufacturing systems, power systems, aerospace systems, and networked control systems that may experience random abrupt changes in their structures and parameters [5][6][7][8].Recently, the problem of the adaptive tracking for a class of stochastic nonlinear systems with stationary Markovian switching was considered in [9].The problem of robust mode-dependent delayed state feedback  ∞ control for a class of uncertain time-delay systems with Markovian switching parameters and mixed discrete, neutral, and distributed delays was investigated in [10].
As a specific type of hybrid systems, switched systems have received a great attention in the last decades.Many examples of such systems can be found in real world, such as power electronics, networked control systems [11].The switching rule design is one of the key issues in the study of switched systems.Based on the switching rule design, many important problems, such as stability, model reduction,  ∞ control or filtering problem, and tracking control, have been extensively investigated in [12][13][14][15][16][17][18][19][20][21][22].For example, the problems of stability and stabilization of impulsive switched systems with time delays were considered in [12].The problems of finite-time stability analysis and stabilization for switched nonlinear discrete-time systems were addressed in [13].The model reduction of switched systems was investigated in [14,15].In [16][17][18][19][20][21][22], the  ∞ control and filtering problems of switched systems were studied.In [23], the tracking control for switched linear systems with time delay was investigated, and a tracking control law was designed such that the  ∞ model reference tracking performance is satisfied.There are also many papers discussing the switching rule design for switched affine systems; see, for instance, [24][25][26] and the references therein.
On the other hand, the switch-mode DC-DC converters are widely applied in DC motor drives and regulated DC power supplies where the object is to convert the unregulated DC input into a desired DC output voltage level [27].Due to the fact that the DC-DC converters operate in a switch mode where an interaction between continuous and discrete dynamics exists, they can be modeled as hybrid systems.Recently, a unified method for fast modeling of DC-DC switching converters in the continuous conduction mode and discontinuous conduction mode was proposed in [28].A tristate converter was modeled as a switched affine system, and a controller was designed to ensure that the corresponding closed-loop system is always stable in the safe set [29].In [27], a hybrid feedback switching rule based on the common Lyapunov function approach was applied to a Buck converter.
The method dealing with switched affine systems was used to study DC-DC converters, and a set of equilibrium points and a class of switching rules were determined in [30,31].In order to improve the switching strategy, the clock delay control was discussed in [32].In addition, the method of state feedback  ∞ controller design for Monrovian switching systems with mixed delays was applied to DC-DC converters in [33].However, DC-DC converters with uncertain parameters and disturbances have not yet gained sufficient research attention, which motivates the present study.
In this paper, we are interested in investigating the robust  ∞ switching rule design for Boost converters with uncertain parameters and disturbances.Firstly, the Boost converter under consideration is modeled as a switched affine system.Then the common quadratic Lyapunov function approach is utilized for the design of switching rule.By determining a set of attainable equilibrium points, a class of switching rules and the activation region of each subsystem are developed.A sufficient condition for the existence of such switching rule guaranteeing the  ∞ model reference tracking performance is formulated in terms of linear matrix inequality (LMI).
The rest of this paper is organized as follows.Section 2 presents the model of switched affine linear system for a Boost converter with uncertain parameters and disturbances.In Section 3, a switching rule is designed such that the  ∞ model reference tracking performance is satisfied.The simulation result is illustrated through a numerical example in Section 4. Brief conclusion is discussed in Section 5.
Notations.  denotes the -dimensional Euclidean space, and  × is the set of  ×  real matrices.For real matrices or vectors, the superscript "" stands for the transpose.The convex combination is given by   = ∑  =1     , where  belongs to the set of Λ composed by all nonnegative vectors such that ∑  =1   = 1.The space of square integrable functions on [0, ∞) is denoted by  2 [0, ∞).

Model of Boost Converter
A Boost converter with the inductor equivalent series resistance (ESR), uncertain parameters, and disturbance is shown in Figure 1.
Suppose that the converter operates in a continuous current conduction mode.Let the inductor current  and the capacitance voltage   be the state variables, and denote that From Figure 1, it can be seen that the converter operates between two modes.Considering that the output load resistance  and its variation are bounded, when the switch  is closed, the system equation is when the switch  is open, the system equation is where  is the disturbance of the input voltage and Δ 1 and Δ 2 are the variations of the load resistance.
In order to separate the uncertainties from the system equation, we can use the first-order Taylor series expansion, and the system equation can be written as when the switch  is closed, when the switch  is open, where Abstract and Applied Analysis 3 The uncertainties can be written as where and Δ max is the upper bound of the variation of the load resistance.

Switching Rule Design
Because the hybrid system model of the converter described in Section 2 involves uncertain parameters and disturbance, the model can be described by the following switched affine systems: where () ∈   is the system state, () ∈   is the output,  ∈   is the bounded exogenous disturbance of the system, () : [0, ∞) →  = {1, 2, . . ., } is a piecewise constant function of time, so-called switching rule, and  is the number of system modes.The th system mode (  ,   ,   ,   ) is active at time  when () = .  =   + Δ  , Δ  represents the uncertainty, and   ,   ,   , and   are constant matrices of appropriate dimensions.
The problem concerned here is to design a switching rule () and determine an equilibrium point   ∈   that is attainable under such switching rule, that is, () →   as  → ∞.Furthermore, the switching rule should guarantee the  ∞ model reference tracking performance.
Given the desired equilibrium point   and system (7), the tracking error system is ė  () =  ()   () +  () +  ()  () , Consider the measurement vector () of system (7), and define the output tracking error as follows: Given a performance index then the problem under consideration can be transformed to design a switching rule () such that error system (8) is stabilizable with an  ∞ disturbance attenuation .
Assumption 1.The uncertainty Δ  is given as follow: where   and   are known constant matrices of the appropriate dimensions and   () is an unknown bounded function which satisfies Definition 2. Given a constant  > 0, error switched system (8) is said to be stabilizable with an  ∞ disturbance attenuation  if there exists a switching rule () such that the following conditions are satisfied: (1) system (8) with () ≡ 0 is asymptotically stable for all admissible uncertainties; (2) under zero-initial condition, that is, holds for all admissible uncertainties and all nonzero () ∈  2 [0, ∞).
Lemma 4 (see [24]).Consider switched affine linear system (7) with  = 2; the point   = 0 is a quadratic stabilizable switched equilibrium if and only if there exists  ∈ (0, 1) such that Lemma 5 (see [34]).If  and  are constant matrices of appropriate dimensions, then where  is a positive scalar.
The following theorem provides an LMI condition for designing the switching rule () such that error system ( 8) is stabilizable with an  ∞ disturbance attenuation .Theorem 6.Let   be a given constant vector representing the desired equilibrium point of the system (7), and consider error switched system (8).Given constants  > 0 and  > 0, if there exist  ∈ Λ and a symmetric positive definite matrix  ∈  × such that for all  ∈ , then the following switching rule can make the error switched system (8) stabilizable with an  ∞ disturbance attenuation .
From Theorem 6, it can be seen that ( 18) is a LMI; thus we can first get the matrix  through LMI toolbox of MATLAB [35].Then the desired switching rule can be obtained from (20) for any selected equilibrium point   .
According to the switched affine system model of the Boost converter described in Section 3, if the converter operates in a continuous current conduction mode, we can get  = 2.In the case of two subsystems, if we ignore the uncertainties in the switching point average system, then we can obtain the convex combination as in Lemma 4 and in [8], and the convex combination is given by where If the dwell time of the th subsystem is proportional to   , as the concept of "duty ratio" in DC-DC converters, then ( 27) is an average model of switched system (7).However, unlike time average control, the quadratic stability of the switched equilibrium takes the state feedback form.A necessary and sufficient condition for the quadratic stability of the switched equilibrium is shown in Lemma 4. In addition, the proposed switching rule depends on state variables; hence rapid switching can be avoided.
Condition (19) is an expression of the desired equilibrium point   for DC-DC converters.Notice that the inequality (18) imposes that   = ∑ 2 =1     is asymptotically stable, where   can be regard as the concept of "duty ratio" in DC-DC converters.Then we can calculate the desired equilibrium point   for DC-DC converters by the following equation: where The switching rule (20) can be expressed as where From (18), we can easily obtain Denote Then the activation regions of subsystems 1 and 2 can be defined as follow: Therefore, the switching rule (20) in the case of two subsystems can be rewritten as where with   ,   ,   ,   , and   ,  = 1, 2, have been given in ( 3), (4), and (6).
Remark 7. In Theorem 6, a method of robust  ∞ switching rule design is proposed for Boost converters.It is worth pointing out that the proposed design method can also be applied to other types of converters that can be described by switched affine system models, such as Buck converters and Buck-Boost converters.
Remark 8.It can be seen from ( 34) that the matrix  is necessary to the implementation of the switching rule (34); because the matrix  defined in Theorem 6 does not depend on each  ∈ Λ associated with   .Hence, for any equilibrium point   = [    ]  selected by the designer, the matrix  can be derived from (18), and consequently, the switching rule (34) can be implemented in practice.
In addition, for this example, when the value of  is decreased to 393, the feasible solution of (18) cannot be found.Thus, it can be obtained that a larger  is favorable for the feasibility of matrix inequality (18).On the other hand, we can get from Remark 8 that the solution of (18) does not dependent on the equilibrium point   .But the proposed switching rule depends on the equilibrium point   , which can be seen from (34).

Conclusions
The problem of robust  ∞ switching rule design for Boost converters with uncertain parameters and disturbance has been discussed in this paper.Firstly, a switched linear affine system model for the Boost converter under consideration is built.Then, the state-dependent switching rule is designed for the switched linear affine systems by using common Lyapunov function technique.The simulation result is given to demonstrate that the proposed switching rule can guarantee the  ∞ model reference tracking performance.Our further work will focus on extending the proposed design method to other types of converters.We will also consider the robust  ∞ switching rule design based on min-projection strategy.

Figure 1 :
Figure 1: A Boost converter with inductor ESR.
Figures 2-4 shows the response of the previous Boost converter, where the initial state is given by  0 = [0 0]  .

Figure 4 :
Figure 4: Response of output voltage in steady state.