# RobustH∞ Switching Rule Design for Boost Converters with Uncertain Parameters and Disturbances

Authors: Yang Yang; Hamid Reza Karimi; Zhengrong Xiang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the design problem of robustH∞ switching rule for Boost converters with uncertain parameters and disturbances. Firstly, the Boost converter is modeled as a switched affine linear system with uncertain parameters and disturbances. Then, using common Lyapunov function approach and linear matrix inequality (LMI) technique, a novel switching rule is proposed such that the H∞ model reference tracking performance is satisfied. Finally, a simulation result is provided to show the validity of the proposed method.


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## Body

## 1. 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 [1–4]. 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 [5–8]. 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 H∞ 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, H∞ control or filtering problem, and tracking control, have been extensively investigated in [12–22]. 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 [16–22], the H∞ 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 H∞ model reference tracking performance is satisfied. There are also many papers discussing the switching rule design for switched affine systems; see, for instance, [24–26] 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 H∞ 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 robustH∞ 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 H∞ model reference tracking performance is formulated in terms of linear matrix inequality (LMI).The rest of this paper is organized as follows. Section2 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 H∞ 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. Rn denotes the n-dimensional Euclidean space, and Rn×n is the set of n×n real matrices. For real matrices or vectors, the superscript “T” stands for the transpose. The convex combination is given by Aλ=∑i=1mλiAi, where λ belongs to the set of Λ composed by all nonnegative vectors such that ∑i=1mλi=1. The space of square integrable functions on [0,∞) is denoted by L2[0,∞).


## 2. Model of Boost Converter

A Boost converter with the inductor equivalent series resistance (ESR), uncertain parameters, and disturbance is shown in Figure1.Figure 1
A Boost converter with inductor ESR.Suppose that the converter operates in a continuous current conduction mode. Let the inductor currenti and the capacitance voltage uc be the state variables, and denote that x=[x1x2]T=[iuc]T. From Figure 1, it can be seen that the converter operates between two modes. Considering that the output load resistance R and its variation are bounded, when the switch S is closed, the system equation is
(1)[i˙u˙c]=[-R0L00-1(R+ΔR1)C][iuc]+[EL0]+[1L0]ω,u0=[01][iuc],
when the switch S is open, the system equation is
(2)[i˙u˙c]=[-R0L-1L1C-1(R+ΔR2)C][iuc]+[EL0]+[1L0]ω,u0=[01][iuc],
where ω is the disturbance of the input voltage and ΔR1 and ΔR2 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 switchS is closed,
(3)x˙(t)=(A1+ΔA1(t))x(t)+b1+D1ω(t),y(t)=C1x(t),
when the switch S is open,
(4)x˙(t)=(A2+ΔA2(t))x(t)+b2+D2ω(t),y(t)=C2x(t),
where
(5)A1=[-R0L00-1RC],b1=[EL0],A2=[-R0L-1L1C-1RC],b2=[EL0],D1=[1L0],D2=[1L0],C1=[01],C2=[01].
The uncertainties can be written as
(6)ΔA1(t)=M1F1(t)N1,ΔA2(t)=M2F2(t)N2,
where FiT(t)Fi(t)≤I,M1=M2=[1001],N1=N2=[000ΔRmax/R2C], and ΔRmax is the upper bound of the variation of the load resistance.


## 3. Switching Rule Design

Because the hybrid system model of the converter described in Section2 involves uncertain parameters and disturbance, the model can be described by the following switched affine systems:
(7)x˙(t)=A-σ(t)x(t)+bσ(t)+Dσ(t)ω(t),y(t)=Cσ(t)x(t),
where x(t)∈Rn is the system state, y(t)∈Rq is the output, ω∈Rp is the bounded exogenous disturbance of the system, σ(t):[0,∞)→m_={1,2,…,m} is a piecewise constant function of time, so-called switching rule, and m is the number of system modes. The ith system mode (Ai,bi,Di,Ci) is active at time t when σ(t)=i. A-i=Ai+ΔAi, ΔAi represents the uncertainty, and Ai, bi, Di, and Ci are constant matrices of appropriate dimensions.The problem concerned here is to design a switching ruleσ(t) and determine an equilibrium point xr∈Rn that is attainable under such switching rule, that is, x(t)→xr as t→∞. Furthermore, the switching rule should guarantee the H∞ model reference tracking performance.Given the desired equilibrium pointxr and system (7), the tracking error system is
(8)e˙r(t)=A-σ(t)er(t)+k-σ(t)+Dσ(t)ω(t),k-i=bi+A-ixr,er(t)=x(t)-xr.
Consider the measurement vector y(t) of system (7), and define the output tracking error as follows:
(9)ε(t)=y(t)-Cσ(t)xr=Cσ(t)er(t).
Given a performance index
(10)J=∫t0∞εT(t)ε(t)dt-γ2∫t0∞ωT(t)ω(t)dt,
then the problem under consideration can be transformed to design a switching rule σ(t) such that error system (8) is stabilizable with an H∞ disturbance attenuation γ.Assumption 1.
The uncertaintyΔAi is given as follow:
(11)ΔAi(t)=MiFi(t)Ni,
where Mi and Ni are known constant matrices of the appropriate dimensions and Fi(t) is an unknown bounded function which satisfies
(12)FiT(t)Fi(t)≤I.Definition 2.
Given a constantγ>0, error switched system (8) is said to be stabilizable with an H∞ disturbance attenuation γ if there exists a switching rule σ(t) such that the following conditions are satisfied:(1)
system (8) with ω(t)≡0 is asymptotically stable for all admissible uncertainties;(2)
under zero-initial condition, that is,x(0)=0,
(13)J=∫t0∞εT(t)ε(t)dt-γ2∫t0∞ωT(t)ω(t)dt<0holds for all admissible uncertainties and all nonzero ω(t)∈L2[0,∞).Lemma 3 (see [31]).
Consider switched system (7) with bσ(t)≡0, if there exist λ∈Λ and a symmetric positive-definite matrix P∈Rn×n such that
(14)AλTP+PAλ<0.
Then the switching rule
(15)σ(x)=argmini∈m_xTPAix
makes the equilibrium point xr=0 globally asymptotically stable.Lemma 4 (see [24]).
Consider switched affine linear system (7) with m=2; the point xr=0 is a quadratic stabilizable switched equilibrium if and only if there exists α∈(0,1) such that
(16)Aλ=αA1+(1-α)A2isHurwitz,bλ=αb1+(1-α)b2=0.Lemma 5 (see [34]).
IfX and Y are constant matrices of appropriate dimensions, then
(17)XTY+YTX≤ξ-1XTX+ξYTY,
where ξ is a positive scalar.The following theorem provides an LMI condition for designing the switching ruleσ(t) such that error system (8) is stabilizable with an H∞ disturbance attenuation γ.Theorem 6.
Letxr be a given constant vector representing the desired equilibrium point of the system (7), and consider error switched system (8). Given constants γ>0 and ε>0, if there exist λ∈Λ and a symmetric positive definite matrix P∈Rn×n such that
(18)[AiTP+PAiε-1NiTεPMiγ-1PDiCiTε-1Ni-I000εMiTP0-I00γ-1DiTP00-I0Ci000-I]<0,(19)Aλxr+bλ=0,
for all i∈m_, then the following switching rule
(20)σ(t)=argmini∈m_{erT(t)(AiTP+PAi+ε-2NiTNi+ε2PMiMiTPmmmmmmmmmmm+γ-2PDiDiTP+CiTCi)er(t)}
can make the error switched system (8) stabilizable with an H∞ disturbance attenuation γ.Proof.
Choose the following Lyapunov function:(21)V(er)=erT(t)Per(t).
Along the trajectory of system (8), we obtain from (20) that
(22)V˙(er(t))=x˙T(t)Per(t)+erT(t)Px˙(t)=2erT(t)Px˙=2erT(t)P(Aσ(t)+ΔAσ(t)(t))x(t)+2erT(t)Pbσ(t)+2erT(t)PDσ(t)ω(t)=mini∈m_(2erT(t)P(Aix(t)+ΔAi(t)x(t)mmmmmmmmmm+bi+Diω(t)))=mini∈m_(erT(t)(AiTP+PAi+ΔAiT(t)P+PΔAi(t))er(t)+2erT(t)P((Ai+ΔAi(t))xr+bi+Diω(t)))=mini∈m_(erT(t)(AiTP+PAi+NiTFiT(t)MiTPmmmmm+PMiFi(t)Ni)er(t)+2erT(t)PDiω(t))+minλ∈Λ(2erT(t)P((Aλ+ΔAλ(t))xr+bλ)).
By Lemma 5, one obtains
(23)NiTFiT(t)MiTP+PMiFi(t)Ni≤ε-2NiTFiT(t)Fi(t)Ni+ε2PMiMiTP≤ε-2NiTNi+ε2PMiMiTP,2erT(t)PDiω(t)≤γ-2erT(t)PDiDiTPer(t)+γ2ωT(t)ω(t).
It follows from (19) and (23) that
(24)V˙(er(t))≤mini∈m_(erT(t)(AiTP+PAi+ε-2NiTNi+ε2PMiMiTP+γ-2PDiDiTP)er(t)+γ2ωT(t)ω(t)).
For anyt≥0, the feasibility of (18) implies V˙(er(t))<0, which ensures the asymptotic stability of system (8) with ω(t)≡0.
On the other hand, denote that{(tk,σ(tk))∣k=0,1,…,s;0=t0≤t1≤⋯≤ts<tf} as the switching sequence of σ(t) on the interval [0,tf). Under zero condition, we have
(25)∫t0tf[εT(t)ε(t)-γ2ωT(t)ω(t)]dt=∑k=0s-1{∫tktk+![εT(t)ε(t)-γ2ωT(t)ω(t)+V˙(er(t))]dt-[V(er(tk+1))-V(er(tk))]∫tktk+![εT(t)ε(t)-γ2ωT(t)ω(t)+V˙(er(t))]dt}+∫tstf[εT(t)ε(t)-γ2ωT(t)ω(t)+V˙(er(t))]dt-[V(er(tf))-V(er(ts))]=∑k=0s-1∫tktk+![erT(t)CiTCier(t)-γ2ωT(t)ω(t)+V˙(er(t))]dt+∫tstf[εT(t)ε(t)-γ2ωT(t)ω(t)+V˙(er(t))]dt-V(er(tf))≤∑k=0s-1∫tktk+![erT(t)(AiTP+PAi+ε-2NiTNi+ε2PMiMiTPmmmmmmmmmm+γ-2PDiDiTP+CiTCi)er(t)]dt+∫tstf[erT(t)(AiTP+PAi+ε-2NiTNi+ε2PMiMiTP+γ-2PDiDiTP+CiTCi)er(t)]dt.
According to Schur’s complement and the inequality (25), it is easy to get
(26)∫t0tfεT(t)ε(t)dt≤γ2∫t0tfωT(t)ω(t)dt.
Letting tf=∞, we get J<0.
This completes the proof.From Theorem6, it can be seen that (18) is a LMI; thus we can first get the matrix P through LMI toolbox of MATLAB [35]. Then the desired switching rule can be obtained from (20) for any selected equilibrium point xr.According to the switched affine system model of the Boost converter described in Section3, if the converter operates in a continuous current conduction mode, we can get m=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
(27)x˙(t)=Aλx(t)+bλ,
where Aλ=∑i=12λiAi, bλ=∑i=12λibi with 0<λi<1 and ∑i=12λi=1. If the dwell time of the ith subsystem is proportional to λi, 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 xr for DC-DC converters. Notice that the inequality (18) imposes that Aλ=∑i=12λiAi is asymptotically stable, where λi can be regard as the concept of “duty ratio” in DC-DC converters. Then we can calculate the desired equilibrium point xr for DC-DC converters by the following equation:
(28)xr=-Aλ-1bλ,
where Aλ=∑i=12λiAi, bλ=∑i=12λibi, 0<λi<1, and ∑i=12λi=1.The switching rule (20) can be expressed as
(29)σ(t)=argmini∈m_{erT(t)Tier(t)},
where
(30)Ti=AiTP+PAi+ε-2NiTNi+ε2PMiMiTP+γ-2PDiDiTP+CiTCi.
From (18), we can easily obtain
(31)Ti=AiTP+PAi+ε-2NiTNi+ε2PMiMiTP+γ-2PDiDiTP+CiTCi≤0.
Denote
(32)S={x∈Rn∣(x-xr)T(T1-T2)(x-xr)=0}.
Then the activation regions of subsystems 1 and 2 can be defined as follow:
(33)Ψ1={x∈Rn∣(x-xr)TT1(x-xr)<(x-xr)TT2(x-xr)},Ψ2={x∈Rn∣(x-xr)TT1(x-xr)>(x-xr)TT2(x-xr)}.
Therefore, the switching rule (20) in the case of two subsystems can be rewritten as
(34)σ(x)={1,if(x-xr)T(T1-T2)(x-xr)<0,2,if(x-xr)T(T1-T2)(x-xr)>0,
where
(35)T1=A1TP+PA1+ε-2N1TN1+ε2PM1M1TP+γ-2PD1D1TP+C1TC1,T2=A2TP+PA2+ε-2N2TN2+ε2PM2M2TP+γ-2PD2D2TP+C2TC2,
with Ai, Mi, Ni, Ci, and Di, i=1,2, have been given in (3), (4), and (6).Remark 7.
In Theorem6, a method of robust H∞ 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 P is necessary to the implementation of the switching rule (34); because the matrix P defined in Theorem 6 does not depend on each λ∈Λ associated with xr. Hence, for any equilibrium point xr=[irucr]T selected by the designer, the matrix P can be derived from (18), and consequently, the switching rule (34) can be implemented in practice.


## 4. Simulation Results

To illustrate the obtained result, consider the Boost converter described in Section2. The parameters of the Boost converter are E=12 V, L=10 mH, C=100 μF, R=30 Ω, R0=0.1 Ω, ΔR1=sin(θt)Ω, ΔR2=cos(θt)Ω, θ=100, and ω=0.001e-100t, and the steady-state duty ratio is 0.5. From (3), (4), and (6), we can obtain
(36)A1=[-1000-331.3],A2=[-10-10010000-331.3],b1=[12000],b2=[12000],ΔA1=[1001]sin(100t)[00011.11],ΔA2=[1001]cos(100t)[00011.11],C1=C2=[01],D1=D2=[1000].Letλ1=0.5, λ2=0.5, ε=1, and γ=1000; then we can get xr=[1.578823.684] by (28). Solving (18), the matrix P can be obtained as follows:
(37)P=[19.2121-0.0031-0.00310.1922].From (34), the desired switching rule can be derived. Figures 2–4 shows the response of the previous Boost converter, where the initial state is given by x0=[00]T.Figure 2
Response of inductor current.Figure 3
Response of output voltage.Figure 4
Response of output voltage in steady state.From Figures2 and 3, it can be observed that the inductor current and capacitance voltage uc can track the reference current and the reference voltage rapidly, respectively. Therefore, the converter with the switching rule (34) has an H∞ model reference tracking performance. This demonstrates the effectiveness of the proposed result.In addition, for this example, when the value ofγ is decreased to 393, the feasible solution of (18) cannot be found. Thus, it can be obtained that a larger γ 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 xr. But the proposed switching rule depends on the equilibrium point xr, which can be seen from (34).


## 5. Conclusions

The problem of robustH∞ 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 H∞ 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 H∞ switching rule design based on min-projection strategy.



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