# Global Harmonic Current Rejection of Nonlinear Backstepping Control with Multivariable Adaptive Internal Model Principle for Grid-Connected Inverter under Distorted Grid Voltage

Authors: Yang Yu; Zengqiang Mi; Yiming Che; Tong Zhao; Yikun Xu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Based on a brief review on current harmonics generation mechanism for grid-connected inverter under distorted grid voltage, the harmonic disturbances and uncertain items are immersed into the original state-space differential equation of grid-connected inverter. A new algorithm of global current harmonic rejection based on nonlinear backstepping control with multivariable internal model principle is proposed for grid-connected inverter with exogenous disturbances and uncertainties. A type of multivariable internal model for a class of nonlinear harmonic disturbances is constructed. Based on application of backstepping control law of the nominal system, a multivariable adaptive state feedback controller combined with multivariable internal model and adaptive control law is designed to guarantee the closed-loop system globally uniformly bounded, which is proved by a constructed Lyapunov function. The presented algorithm extends rejection of nonlinear single-input systems to multivariable globally defined normal form, the correctness and effectiveness of which are verified by the simulation results.


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## Body

## 1. Introduction

Along with the strong demand of electric power energy, conventional fossil fuels are gradually reduced. The exploiting of renewable energy is an inevitable route for development of sustainable society. According to the planning of EU Commission, the renewable energy goal is achieving 20% of energy consumption from sustainable sources by 2020 [1].Renewable energy sources include wind (onshore and offshore), solar PV, hydroenergy, biomass energy, geothermal energy, and tidal energy. Currently, two of main utilization approaches of renewable energy are off-grid power generation and connected to utility grid. In the past decade, the renewable energy sources connected to utility grid have achieved a considerable growth with the forceful needs of electric power energy [2]. Grid-connected inverter, which is one of the key technologies, supports renewable energy to be transmitted to the utility grid. In design, pulse width modulation (PWM) and the corresponding inverter control system require grid voltage as an ideal sine wave. Actually, it is difficult to keep a perfect sine wave of grid voltage under a variety of nonlinear loads and unexpected network failures. Furthermore, the existing researches have shown distorted grid voltage, such as dips [3], asymmetry [4], and harmonics [5], which will have a significantly adverse influence to grid power quality. In other words, rejection of current harmonics for grid-connected inverter under distorted grid voltage is a useful work for the normal operation of power networks [6].The rejection solutions of current harmonics for grid-connected inverter mainly consist of two categories: hardware implementation and software programming. A simple and practical scheme of hardware implementation is adding filters to inverter’s output. The aim of adding filters is to compensate for the fluctuations of grid voltage. In [7], LCL filters are designed for grid-connected converters. However, the application of LCL filters may encounter the resonance problem. Close by the resonant frequency, the impedance of the filters is small, and the harmonic current in corresponding frequency would be amplified, even beyond the harmonic standard [8]. In addition, new hardware will certainly increase the investment of the system. One of the rejection ways for software programming is to design a proportional resonant (PR) controller to improve the gain in background harmonic [9]. Nevertheless, with the increasing of the background harmonic frequency, the PR controller can reduce the phase margin of the system [10]. The other one is application of feedforward control of grid voltage to increase output impedance of the system. The feedforward control has no impact on phase margin of the system [11], but the scheme, which only depends on the proportion feedforward to regulation, cannot eliminate the influence of background harmonic [12].For the disadvantages of the above methods, some scholars attempt to apply modern control theory to improve the capability of harmonics disturbance rejection. The nonlinear control methods, such as repetitive control [13, 14], H∞ control [15, 16], prediction control [17–19], backstepping control [20–22], and fuzzy-adaptive control [23–26], can deal with the nonlinearity and uncertainty of nonlinear models better. However, these preexisting rejection control methods have the following problems more or less.(1)
The harmonics rejection algorithms are concerned primarily with the single-input field; the research of multiple-input field is rarely mentioned.(2)
For an operating grid-connected system, the parameters of resistance and inductance for the system are often time varying and cannot be measured accurately. Hence, these proportional-integral-based controllers, which take no account of the time-varying characteristics of the parameters, cannot work very efficiently.(3)
The parameter design of robust controller is relatively complicated and needs large amount of calculations.In the paper, a multivariable state feedback control algorithm based on nonlinear backstepping control with multivariable internal model principle (MIMP) and adaptive control law (ACL) for grid-connected inverter is proposed to reject the harmonic disturbance produced by a class of exosystem and uncertainties related with system states caused by modeling perturbations, parameter uncertainty or actuator end disturbance. The main contributions in the paper are the following.(1)
A new harmonic rejection algorithm based on nonlinear backstepping control with MIMP and ACL is proposed in the paper, which extends the disturbance rejection of the nonlinear single-input system to multivariable globally defined normal form in contrast with the previous literatures.(2)
A new type of nonlinear multivariable internal model for a class of nonlinear harmonic disturbances and uncertain items related with system states is constructed.(3)
The proposed algorithm is practically applied to reject current harmonic disturbances for grid-connected inverter under distorted grid voltage.The paper is organized as the following: it starts with an introduction to the research status of current harmonics rejection for grid-connected inverter in Section1. The modeling of state differential equation for grid-connected inverter is constructed and an analysis of current harmonics generation mechanism is given in Section 2. In addition, Section 2 describes the control and mathematical problem concerned in the paper. Nonlinear multivariable internal model is presented and designed in Section 3. In Section 4, nonlinear multivariable adaptive state feedback controller is proposed and the global robust stabilization is demonstrated. Section 5 gives the results of numerical simulation. Finally, the conclusions are summarized in Section 6.


## 2. Problem Formulation

### 2.1. Modeling of Grid-Connected Inverter

The model of grid-connected inverter can be shown in Figure1. In Figure 1, R and L represent the equivalent series resistance and inductor, respectively, C is the capacitance, vabc and iabc are three-phase output voltage and grid current of three-phase inverter, and eabc represents three-phase grid voltage.Figure 1
Modeling of grid-connected inverter.In terms of Figure1, the voltage equations of three-phase grid-connected inverter in stationary abc reference frame can be written as
(1)va=Ria+Ldiadt+ea,vb=Rib+Ldibdt+eb,vc=Ric+Ldicdt+ec.Application of the park transformation to (1) in rotating dq reference frame equation (1) is converted into
(2)vd=Rid+Ldiddt-ωLiq+ed,vq=Riq+Ldiqdt+ωLid+eq;
namely,
(3)[diddtdiqdt]=[-RLω-ω-RL][idiq]+[1L001L][vd-edvq-eq],
where vdq and idq represent the voltage and current of grid-connected inverter in rotating dq reference frame, edq is the grid voltage in dq reference frame, and ω is the synchronous angular velocity of the grid.Rearrange (2) in the form of state space as follows:
(4)x˙=Ax+B(v-e),y=Cx,
where x, v, and y are the state vector, the control input vector, and the output vector, respectively, e is the voltage vector of the grid,x=[idiq]T,v=[vdvq]T, y=x, e=[edeq]T,A=[-R/Lω-ω-R/L],B=[1/L001/L], and C=[1001].


### 2.2. Current Harmonics Generation Mechanism for Grid-Connected Inverter

In terms of Figure1, the harmonic voltage equations of three-phase grid-connected inverter instationary abc reference frame can be written as
(5)vabck=Riabck+Ldiabckdt+eabck,
where k is the harmonic order, vabck is the control voltage vector, iabck is the current vector of harmonics, and eabck is the unbalanced harmonic voltage vector of the grid.Assume that the distorted voltage of the network contains the 5th and 7th harmonics. In stationaryabc reference frame, the rotating direction of the 5th harmonics is contrary to fundamental harmonics, and its rotating electrical angular velocity is -5ω.However, the rotating direction of the 7th harmonics is the same as the fundamental harmonics, and its rotating electrical angular velocity is 7ω.Indeed, the distorted voltage of the network may include more harmonics in addition to the 5th and 7th harmonics, such as the 3rd, 11th, and 13th. The rotating direction of the 11th harmonics is exactly identical to the 5th, except that the rotating electrical angular velocity is -11ω. A similar relationship exists in the 7th and 13th harmonics. In order to reveal the essence of the problem more simply, consider only the 5th and 7th harmonics and construct multiple synchronous rotating reference frames shown in Figure 2. For convenience, in what follows, +/− in superscript represents +/− rotating direction, respectively; 5 and 7 in superscript describe the rotating direction of the 5th and 7th harmonics, respectively; 1, 5, and 7 in subscript demonstrate the 1st, 5th, and 7th harmonics, respectively.Figure 2
Multiple synchronous rotating reference frames.In rotatingdq reference frame of the 5th harmonics, the voltage equation of the 5th harmonics can be written as
(6)vdq5=Ridq5+Ldidq5dt+edq5-j5ωLidq5,
where
(7)vdq5=vdq1+ej6ωt+vdq55-+vdq77+ej12ωt,idq7=idq1+ej6ωt+idq55-+idq77+ej12ωt,edq5=edq1+ej6ωt+edq55-+edq77+ej12ωt.
Equation (7) indicates that only voltage and current components of the 5th harmonics are DC signals in rotating dq reference frame of the 5th harmonics. Voltage and current components of the remaining harmonics are AC signals. One other thing to note is that all of the harmonic components in rotating dq reference frame of fundamental wave are AC signals.In a similar way, the voltage equation of the 7th harmonics can be written as(8)vdq7=Ridq7+Ldidq7dt+edq7+j7ωLidq7.
Equations (6) and (8) demonstrate that once the grid voltage becomes unbalance, the controller designed by the principle of ideal sine wave will not be able to compensate for harmonic voltage, and the current waveform of the grid will be distorted.


### 2.3. Control Problem Description

UseD(w) andδ(x,t)to describe the exogenous disturbance of voltage harmonics and the uncertain items related with system states caused by modeling perturbations, parameter uncertainties, or actuator end disturbances, respectively, then the state equation in (4) can be rewritten as a multivariable system as follows:
(9)x˙=Ax+B(v-e0-D(w)+δ(x,t))=f(x)+∑i=12gi(x)(vi-ei0-Di(w)+δi(x,t)),(vi-ei0-Di(w)+δi(x,t))δi(x,t)1≤i≤2,
wherex=[x1,x2]T=[id,iq]T, f(x)=[-(R/L)x1+ωx2-ωx1-(R/L)x2]=[-(R/L)id+ωiq-ωid-(R/L)iq], g1(x)=g2(x)=1/L, v=[v1,v2]T=[vd,vq]T, [e10,e20]=[ed0,eq0], and ed0 and eq0 are network voltages of fundamental wave in dq reference frame. For a specific network,ed0andeq0can be determined easily. The uncertain item δ(x,t) in the control input channel satisfies the matching condition. In the paper, the range of variable i is (1,2). For convenience and conciseness, in what follows, the range of variable i is no longer written.Assumption 1.
w
∈
R
q is an exogenous signal generated by the following exosystem:
(10)w˙=Wiw,
whereWiis a pending matrix depending on the exogenous signal.Assumption 2.
Considering disturbance-free and no modeling perturbations, the nominal system of (9) can be written as:
(11)x˙=f(x)+∑i=12gi(x)(vi-ei0)
there exists a control law of state feedback a follows:
(12)vi′=vi-ei0=αi(x),
let (11) be asymptotically stable. In addition, there exists a Lyapunov function V(x) which satisfies:
(13)d_(∥x∥)≤V(x)≤d-(∥x∥),(14)∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))≤-d0(∥x∥),(15)|∂V(x)∂x∑i=12gi(x)|2≤d0(∥x∥),
whered_,d-, andd0are all of class K∞ functions.Assumption 3.
There exist smooth functionsri(x):Rn→Rqwhich make
(16)∂ri(x)∂xgi(x)=Ki,
whereKiare nonzero constant vectors defined inRq.Assumption 4.
There exist a constantmi and a known smooth bounded function pi(x) making the uncertain item δ(x,t) satisfy
(17)|δi(x,t)|≤mipi(x).Remark 5.
Similar to the uncertain itemδ(x,t), the disturbance D(w) also satisfies the matching condition and can be injected in the input path. Nevertheless, a method of iterative design, such as adaptive backstepping, can extend the presented algorithm to more general instance of strict feedback. Therefore, the matching condition is not critical, and application of the assumption is reasonable.Remark 6.
One of the research points in the paper is stability problem for multivariable input system. Currently, some literatures focus on the point of stability problem, but the focused attention of these literatures is the field of single input and single output [27] or semiglobal stability [28]. For the multivariable system, the research of stability problem will be more challenging, and the core is to transform the stability problem from multivariable system to multiple single-input system [29].Remark 7.
For Assumption2, (13) and (14) automatically hold when the closed-loop system is asymptotically stable. Equation (15) holds when the closed-loop system is exponentially stable. Nevertheless, Assumption 2 is not a sufficient condition to make the closed-loop system exponentially stable [30].Remark 8.
For Assumption3, ifgi(x)is a nonzero constant vector, it will be easy to find a solution of(∂ri(x)/∂x)gi(x)=Kifor a non-zero constant vector Ki. For a nonconstant vectorgi(x), the solutions can be found more complex with the help of geometric tool [31].Remark 9.
Assumption4 is to guarantee the boundedness of the uncertain item δ(x,t) relative to the system state vector x. For instance, ifδ1(x,t)=x1sin(x1t),m1andp1(x)can be chosen as 1 andx1, respectively.The problem solved in the paper can be described by the following theorem.Definition 10.
For any given compact setDw∈Rq, state feedback controllervican always be found to ensure the solution of closed-loop system (9) existing under arbitrary initial conditions for allw(0)∈Dwand t≥0 and to reject exogenous disturbances and uncertain items.


## 2.1. Modeling of Grid-Connected Inverter

The model of grid-connected inverter can be shown in Figure1. In Figure 1, R and L represent the equivalent series resistance and inductor, respectively, C is the capacitance, vabc and iabc are three-phase output voltage and grid current of three-phase inverter, and eabc represents three-phase grid voltage.Figure 1
Modeling of grid-connected inverter.In terms of Figure1, the voltage equations of three-phase grid-connected inverter in stationary abc reference frame can be written as
(1)va=Ria+Ldiadt+ea,vb=Rib+Ldibdt+eb,vc=Ric+Ldicdt+ec.Application of the park transformation to (1) in rotating dq reference frame equation (1) is converted into
(2)vd=Rid+Ldiddt-ωLiq+ed,vq=Riq+Ldiqdt+ωLid+eq;
namely,
(3)[diddtdiqdt]=[-RLω-ω-RL][idiq]+[1L001L][vd-edvq-eq],
where vdq and idq represent the voltage and current of grid-connected inverter in rotating dq reference frame, edq is the grid voltage in dq reference frame, and ω is the synchronous angular velocity of the grid.Rearrange (2) in the form of state space as follows:
(4)x˙=Ax+B(v-e),y=Cx,
where x, v, and y are the state vector, the control input vector, and the output vector, respectively, e is the voltage vector of the grid,x=[idiq]T,v=[vdvq]T, y=x, e=[edeq]T,A=[-R/Lω-ω-R/L],B=[1/L001/L], and C=[1001].


## 2.2. Current Harmonics Generation Mechanism for Grid-Connected Inverter

In terms of Figure1, the harmonic voltage equations of three-phase grid-connected inverter instationary abc reference frame can be written as
(5)vabck=Riabck+Ldiabckdt+eabck,
where k is the harmonic order, vabck is the control voltage vector, iabck is the current vector of harmonics, and eabck is the unbalanced harmonic voltage vector of the grid.Assume that the distorted voltage of the network contains the 5th and 7th harmonics. In stationaryabc reference frame, the rotating direction of the 5th harmonics is contrary to fundamental harmonics, and its rotating electrical angular velocity is -5ω.However, the rotating direction of the 7th harmonics is the same as the fundamental harmonics, and its rotating electrical angular velocity is 7ω.Indeed, the distorted voltage of the network may include more harmonics in addition to the 5th and 7th harmonics, such as the 3rd, 11th, and 13th. The rotating direction of the 11th harmonics is exactly identical to the 5th, except that the rotating electrical angular velocity is -11ω. A similar relationship exists in the 7th and 13th harmonics. In order to reveal the essence of the problem more simply, consider only the 5th and 7th harmonics and construct multiple synchronous rotating reference frames shown in Figure 2. For convenience, in what follows, +/− in superscript represents +/− rotating direction, respectively; 5 and 7 in superscript describe the rotating direction of the 5th and 7th harmonics, respectively; 1, 5, and 7 in subscript demonstrate the 1st, 5th, and 7th harmonics, respectively.Figure 2
Multiple synchronous rotating reference frames.In rotatingdq reference frame of the 5th harmonics, the voltage equation of the 5th harmonics can be written as
(6)vdq5=Ridq5+Ldidq5dt+edq5-j5ωLidq5,
where
(7)vdq5=vdq1+ej6ωt+vdq55-+vdq77+ej12ωt,idq7=idq1+ej6ωt+idq55-+idq77+ej12ωt,edq5=edq1+ej6ωt+edq55-+edq77+ej12ωt.
Equation (7) indicates that only voltage and current components of the 5th harmonics are DC signals in rotating dq reference frame of the 5th harmonics. Voltage and current components of the remaining harmonics are AC signals. One other thing to note is that all of the harmonic components in rotating dq reference frame of fundamental wave are AC signals.In a similar way, the voltage equation of the 7th harmonics can be written as(8)vdq7=Ridq7+Ldidq7dt+edq7+j7ωLidq7.
Equations (6) and (8) demonstrate that once the grid voltage becomes unbalance, the controller designed by the principle of ideal sine wave will not be able to compensate for harmonic voltage, and the current waveform of the grid will be distorted.


## 2.3. Control Problem Description

UseD(w) andδ(x,t)to describe the exogenous disturbance of voltage harmonics and the uncertain items related with system states caused by modeling perturbations, parameter uncertainties, or actuator end disturbances, respectively, then the state equation in (4) can be rewritten as a multivariable system as follows:
(9)x˙=Ax+B(v-e0-D(w)+δ(x,t))=f(x)+∑i=12gi(x)(vi-ei0-Di(w)+δi(x,t)),(vi-ei0-Di(w)+δi(x,t))δi(x,t)1≤i≤2,
wherex=[x1,x2]T=[id,iq]T, f(x)=[-(R/L)x1+ωx2-ωx1-(R/L)x2]=[-(R/L)id+ωiq-ωid-(R/L)iq], g1(x)=g2(x)=1/L, v=[v1,v2]T=[vd,vq]T, [e10,e20]=[ed0,eq0], and ed0 and eq0 are network voltages of fundamental wave in dq reference frame. For a specific network,ed0andeq0can be determined easily. The uncertain item δ(x,t) in the control input channel satisfies the matching condition. In the paper, the range of variable i is (1,2). For convenience and conciseness, in what follows, the range of variable i is no longer written.Assumption 1.
w
∈
R
q is an exogenous signal generated by the following exosystem:
(10)w˙=Wiw,
whereWiis a pending matrix depending on the exogenous signal.Assumption 2.
Considering disturbance-free and no modeling perturbations, the nominal system of (9) can be written as:
(11)x˙=f(x)+∑i=12gi(x)(vi-ei0)
there exists a control law of state feedback a follows:
(12)vi′=vi-ei0=αi(x),
let (11) be asymptotically stable. In addition, there exists a Lyapunov function V(x) which satisfies:
(13)d_(∥x∥)≤V(x)≤d-(∥x∥),(14)∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))≤-d0(∥x∥),(15)|∂V(x)∂x∑i=12gi(x)|2≤d0(∥x∥),
whered_,d-, andd0are all of class K∞ functions.Assumption 3.
There exist smooth functionsri(x):Rn→Rqwhich make
(16)∂ri(x)∂xgi(x)=Ki,
whereKiare nonzero constant vectors defined inRq.Assumption 4.
There exist a constantmi and a known smooth bounded function pi(x) making the uncertain item δ(x,t) satisfy
(17)|δi(x,t)|≤mipi(x).Remark 5.
Similar to the uncertain itemδ(x,t), the disturbance D(w) also satisfies the matching condition and can be injected in the input path. Nevertheless, a method of iterative design, such as adaptive backstepping, can extend the presented algorithm to more general instance of strict feedback. Therefore, the matching condition is not critical, and application of the assumption is reasonable.Remark 6.
One of the research points in the paper is stability problem for multivariable input system. Currently, some literatures focus on the point of stability problem, but the focused attention of these literatures is the field of single input and single output [27] or semiglobal stability [28]. For the multivariable system, the research of stability problem will be more challenging, and the core is to transform the stability problem from multivariable system to multiple single-input system [29].Remark 7.
For Assumption2, (13) and (14) automatically hold when the closed-loop system is asymptotically stable. Equation (15) holds when the closed-loop system is exponentially stable. Nevertheless, Assumption 2 is not a sufficient condition to make the closed-loop system exponentially stable [30].Remark 8.
For Assumption3, ifgi(x)is a nonzero constant vector, it will be easy to find a solution of(∂ri(x)/∂x)gi(x)=Kifor a non-zero constant vector Ki. For a nonconstant vectorgi(x), the solutions can be found more complex with the help of geometric tool [31].Remark 9.
Assumption4 is to guarantee the boundedness of the uncertain item δ(x,t) relative to the system state vector x. For instance, ifδ1(x,t)=x1sin(x1t),m1andp1(x)can be chosen as 1 andx1, respectively.The problem solved in the paper can be described by the following theorem.Definition 10.
For any given compact setDw∈Rq, state feedback controllervican always be found to ensure the solution of closed-loop system (9) existing under arbitrary initial conditions for allw(0)∈Dwand t≥0 and to reject exogenous disturbances and uncertain items.


## 3. Design of Nonlinear Multivariable Internal Model

Application of internal model principle (IMP) to reject the exogenous disturbanceD(w) and uncertain item δ(x,t) in the paper is chosen as an indirect method. In other words, an appropriate equation of internal model should be constructed to estimate the nonlinear disturbances and uncertain items. The nonlinear equation of exogenous disturbance internal model is designed as
(18)D^i(w)=Viw^i(t),(19)w^i(t)=η^i(t)+ri(x),(20)η^˙i(t)=(Wi-KiVi)(η^i(t)+ri(x))-Kivi′(x)-∂ri(x)∂xf(x),
wherew^(t)is the estimation value of exogenous signalw(t),η^iis the introduced auxiliary vector,η^i=[η^i1η^i2⋯η^ip],ri(x)is a vector constructed by the smooth functionsri(x), ri(x)=[ri(x)0⋯0]1×p, and the matrix Ki∈Rq is selected satisfying Assumption 3 and makes the matrix(Wi-KiVi)be Hurwitz; that is, positive definite matrices Pi and Qi are always existing and satisfy
(21)Pi(Wi-KiVi)+(Wi-KiVi)TPi=-Qi.In terms of Assumption4, the adaptive internal model of uncertain item is devised as
(22)δ^i(x,t)=miKipi(x)tanh(η^iTPiTKipi(x)λi),
wheremiandpi(x)indicate a constant and a known smooth function defined in Assumption 4, respectively, andλiis a designed constant.Define an auxiliary error vector as follows:(23)eri(t)=wi(t)-w^i(t).With the derivative of (23) along with (9), (10) and (19), we can obtain
(24)eri·(t)=w˙i(t)-w^˙i(t)=Wiwi-η^˙i(t)-∂ri(x)∂x∂x∂t=Wiwi-(Wi-KiVi)(η^i(t)+ri(x))+Kivi′(x)+∂ri(x)∂xf(x)-∂ri(x)∂x×(+δi(x,t))f(x)+gi(x)(vi-ei0-Di(w)+δi(x,t)))=Wiwi-(Wi-KiVi)(η^i(t)+ri(x))+Kivi′(x)-∂ri(x)∂xgi(x)(vi-ei0-Di(w)+δi(x,t))=Wiwi-(Wi-KiVi)(η^i(t)+ri(x))+Kivi′(x)-Ki(vi-ei0-Di(w)+δi(x,t))=Wiwi-(Wi-KiVi)w^i(t)+Kivi′(x)-Ki(vi-ei0-Viw(t)+δi(x,t))=Wiwi-(Wi-KiVi)w^(t)+KiViw(t)-Kiδi(x,t)=(Wi-KiVi)eri(t)-Kiδi(x,t).


## 4. Design of Nonlinear State Feedback Controller

In comparison with the nominal system (11), the original system (9) adds two items: exogenous disturbanceD(w)and uncertain itemsδ(x,t). On the basis of control law of the nominal system, nonlinear state feedback controller of the original system (9) should add two items to compensate forD(w)andδ(x,t). In terms of the idea, nonlinear state feedback controller of the original system (9) is designed as
(25)vi′=αi(x)-D^i(w)+θi(·),
whereθ(·)is a pending design function to compensate for an uncertain item such as modeling perturbation.Apparently, designing of nonlinear internal modelD^i(w)is accomplished in Section 3. In what follows, we complete the design ofvi′in two steps. The first step is application of backstepping theory to deviseαi(x), and the second is to complete an adaptive solution ofθ(·)to reject uncertain item δ(x,t).(1) Designing of
α
i
(
x
)
. It is assumed that the parameters of grid-connected inverter are known and invariant. The control target of designingαi(x)is to regulateid(x1)andiq(x2)to track the respective reference valuex1refandx2ref.Firstly, we define the errors as follows:(26)ez1=x1-x1ref,ez2=x2-x2ref;
the dynamics derived from (26), we can get with
(27)ez·1=x˙1-x˙1ref,ez·2=x˙2-x˙2ref.
Substituting (9) into (27), we obtain
(28)ez·1=x˙1-x˙1ref=-RLx1+ωx2+1Lα1(x)-x˙1ref,(29)ez·2=x˙2-x˙2ref=-RLx2-ωx1+1Lα2(x)-x˙2ref.
Define a quadratic function as the following:
(30)Q1=12c1ez12,
wherec1is a positive real number.Derivative ofQ1along (28) can be written as
(31)Q˙1=c1ez1ez·1=c1ez1(-RLx1+ωx2+1Lα1(x)-x˙1ref).
Assume that
(32)-RLx1+ωx2+1Lα1(x)-x˙1ref=-ez1c1.
That is,
(33)α1(x)=-Lez1c1+Rx1-Lωx2+Lx˙1ref.
Replacing (33) into (31), the derivative ofQ1is given by
(34)Q˙1=c1ez1ez·1=-ez12≤0.
Define another quadratic function as the following:
(35)Q2=12c2ez22,
wherec2is also a positive real number.Derivative ofQ2along (29) is given by
(36)Q˙2=c2ez2ez·2=c2z2(-RLx2-ωx1+1Lα2(x)-x˙2ref).
Suppose that
(37)-RLx2-ωx1+1Lα2(x)-x˙2ref=-ez2c2;
that is,
(38)α2(x)=-Lez2c2+Rx2+Lωx1+Lx˙2ref.
Replacing (38) into (36), the derivative ofQ2can be written as
(39)Q˙2=c2ez2ez·2=-ez22≤0.Now, from (30) and (35), the Lyapunov function V(x) defined in Assumption 2 can be written as
(40)V(x)=Q1+Q2=12c1ez12+12c2ez22.Hence, from (34) and (39), we can see that the controllersα1(x)shown in (33) andα2(x)shown in (38) can stabilize the disturbance-free closed-loop system.(2)  Designing of
θ
(
·
)
. Under the actual operating environments, the parameters of grid-connected inverter are not always known and invariant. For instance, the inductance L varies with the environment temperature, and the resistance R changes nonlinearly with heating. Hence, the parametric uncertainties in the process of system modeling should be considered to reflect the real condition of the operating system. Due to the uncertainty of modeling perturbations,θ(·)should be an adaptive controller to reject the perturbations. In terms of the internal model of uncertain items shown in (22), the controller is devised as
(41)θi(x,t)=mipi(x)tanh((∂V(x)/∂x)gi(x)pi(x)ξi),
whereξiis the designed constant.Convergence Proof of the Proposed Controller. For a start, we give a lemma as the following.Lemma 11.
For any positive numberε>0, there exists a smooth function h making the inequality |x|≤xh(x)+ε, for all x∈R hold, and f(0)=0.Remark 12.
If we chooseh(x)=(1/4ε)x, for all x∈R, the above inequality holds obviously. In [32], another function meeting the above requirements is given as h(x)=tanh(βx/ε), for all x∈R, whereβ=e-(β+1)andβ>0, henceβ<1/2.In terms of (23) and (40), construct a new Lyapunov function as
(42)W=V(x)+∑i=12eriTPieri.With the derivative of (42) along the system (9), (24), and (25), we obtain
(43)W˙=∂V(x)∂x(f(x)+∑i=12gi(x)(vi-ei0-Di(w)+δi(x,t)))+∑i=12(eriT(Pi(Wi-KiVi)+(Wi-KiVi)TPi)eri)+2∑i=12(eriTPi(-Kiδi(x,t)))=∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))+∂V(x)∂x∑i=12gi(x)(D^i(w)-Di(w))-∂V(x)∂x∑i=12gi(x)(θi(x,t)-δi(x,t))-∑i=1m(eriTQieri)+2∑i=12(eriTPi(-Kiδi(x,t)))=∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))+∂V(x)∂x∑i=12gi(x)(Viw^(t)-Viw(t))-∂V(x)∂x∑i=12gi(x)((∂V(x)∂xgi(x)pi(x)ξi)mipi(x)tanh(mipi(x)tanh)mipi(x)×((∂V(x)/∂x)gi(x)pi(x)ξi)-∑i=12((∂V(x)∂xgi(x)pi(x)ξi)mipi(x)tanh-δi(x,t)(∂V(x)∂xgi(x)pi(x)ξi))-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t))).In terms of Assumption4, (43) can be rewritten as(44)W˙≤∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))+∂V(x)∂x∑i=12gi(x)(Vieri(t))-∂V(x)∂x∑i=12gi(x)(mipi(x)tanh((∂V(x)/∂x)gi(x)pi(x)ξi)-|mipi(x)|)-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t)))=∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))+∂V(x)∂x∑i=12gi(x)(Vieri(t))+mi∑i=12(∂V(x)∂xgi(x)|pi(x)|-∂V(x)∂xgi(x)pi(x)tanh((∂V(x)/∂x)gi(x)pi(x)ξi))-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t))).According to Lemma11, the following inequality holds:
(45)0≤|x|-xtanh(xε)≤12ε.
That is,
(46)∂V(x)∂xgi(x)|pi(x)|-∂V(x)∂xgi(x)pi(x)tanh((∂V(x)/∂x)gi(x)pi(x)ξi)≤12ξi.
Substituting (46) into (44), we can get
(47)W˙≤∂V(x)∂x(f(x)+∑i=12gi(x)αi(x))+∂V(x)∂x∑i=12gi(x)(Vieri(t))+mi12ξi-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t))).
With the application of (14) in Assumption 2 into (47), we can obtain
(48)W˙≤-d0(∥x∥)+∂V(x)∂x∑i=12gi(x)(Vieri(t))+mi12ξi-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t)))≤-d0(∥x∥)+|∂V(x)∂x∑i=12gi(x)|2|Vieri(t)|+mi12ξi-∑i=1m(eriTQieri)-2∑i=12(eriTPi(Kiδi(x,t))).
Assuming that λmin(Qi) represent the minimal eigenvalue of the matrixQi, (48) can be given by
(49)W˙≤-d0(∥x∥)+|∂V(x)∂x∑i=12gi(x)|2|Vieri(t)|+mi12ξi-∑i=1mλmin(Qi)∥eri∥2+∑i=12(2μi∥PiKi∥2∥eriT∥2+12μi|mipi(x)|2),
whereμiis a positive number.In terms of permanent establishment inequality2ab≤ca2+c-1b2, (49) is rewritten as
(50)W˙≤-d0(∥x∥)+(|∂V(x)∂x∑i=12gi(x)|2+14∑i=12∥Vi∥2∥eri∥2)+mi12ξi-∑i=12λmin(Qi)∥eri∥2+∑i=12(2μi∥PiKi∥2∥eriT∥2+12μi|mipi(x)|2).
With the application of (15) in Assumption 2 into (50), we can obtain
(51)W˙≤-d0(∥x∥)+(d0(∥x∥)+14∑i=12∥Vi∥2∥eri∥2)+mi12ξi-∑i=12λmin(Qi)∥eri∥2+∑i=12(2μi∥PiKi∥2∥eriT∥2+12μi|mipi(x)|2)=14∑i=12∥Vi∥2∥eri∥2+mi12ξi-∑i=12λmin(Qi)∥eri∥2+∑i=12(2μi∥PiKi∥2∥eriT∥2+12μi|mipi(x)|2)≤14∑i=12∥Vi∥2∥eri∥2-∑i=12λmin(Qi)∥eri∥2+mi12ξi+∑i=12(2μi∥PiKi∥2∥eriT∥2+mi2μi|pmax|2),
where|pmax|indicates the maximum value of|pi(x)|.Suppose thatdi=λmin(Qi)-(1/4)∥Vi∥2-2μi∥PiKi∥2>0 and choose properξito satisfy
(52)W˙≤-∑i=1mdi∥ei∥2+mi12ξi+∑i=12(mi2μi|pmax|2)≤0.
As can be seen above, the proposed controller can ensure all the signals of the closed-loop system uniformly bounded. Consequently, the paper comes to the following conclusion.Theorem 13.
There exist positive definite matricesPi and Qi satisfying formula (21) and nonzero constant vectorKi∈Rqsatisfying Assumption 3, such that the matrixWi-KiViis Hurwitz. Furthermore, the formula (52) holds. For the multivariable nonlinear system (9) and exosystem (10) satisfying the Assumptions (1) to (5), the nonlinear multivariable internal models (18) and (22) and the input feedback control (25) can make the closed-loop system globally uniformly bounded.


## 5. Numerical Simulations and Analysis

To illustrate the performance of the present control algorithm, some numerical simulations are performed in the section. The whole simulation time is 5 seconds with the sampling interval 0.001 s. The simulation parameters are chosen as follows.For the grid, the rms value of network voltagee0=380 V the synchronous angular velocityω=100π rad/s. For the three-phase grid-connected inverters, the filter inductance L=1.0 mH and equivalent series resistance R=0.02Ω.Substituting the relevant parameters into the original state equation (9), the nonlinear system (9) can be described by
(53)x˙=f(x)+∑i=12gi(x)(vi-ei0-Di(w)+δi(x,t)),1≤i≤2,
where x=[x1,x2]T, f(x)=[-20x1+100πx2-100πx1-20x2], g1(x)=g2(x)=1000, and[e10,e20]=[380,0]. The control inputv=[v1,v2]T. It is thus clear that the system (53) has two-variable input, and the conventional single-input algorithm cannot solve the problem.Example 14.
The network is immersed 3% 3rd harmonics to fundamental wave.
The exosystem matrices for exogenous disturbanceD(w) represented in (10) and (18) are given asW1=W2=[02ω-2ω0], V1=V2=[11.40], and the uncertain items satisfy
(54)δ1(x,t)=x1sin(x1t),δ2(x,t)=x2sin(x2t).
It is worth noticing that the rotating electrical angular velocity of the 3rd harmonics in stationaryabc reference frame is in accordance with that of the 2nd harmonics in rotating dq reference frame of fundamental wave. Hence,W1=W2=[02ω-2ω0]in dq coordinated system represents the 3rd harmonics in stationary abc coordinated system. Furthermore, the values of V1and V2 represent that the network is immersed 3% 3rd harmonics to fundamental wave.In terms of (33) and (38), the control law of the nominal system is given by
(55)α1(x)=-Lz1+Rx1-Lωx2+Lx˙1ref=-0.001(x1-x1ref)+0.02x1-0.1πx2+0.001x˙1ref,α2(x)=-Lez2+Rx2+Lωx1+Lx˙2ref=-0.001(x2-x2ref)+0.02x1+0.1πx2+0.001x˙2ref.Supposing that(56)V(x)=12c1(x1-x1ref)2+12c2(x2-x2ref)2,we can obtain(57)∂V(x)∂x(f(x)+∑i=12gi(x)αi)=-(x1-x1ref)2-(x2-x2ref)2,(58)∂V(x)∂x∑i=12gi(x)=c1(x1-x1ref)L+c2(x2-x2ref)L.
Supposing x′=[(x1-x1ref)(x2-x2ref)T], in terms of (56), (57) and (58), and choosing c1=L and c2=L make the following inequalities hold
(59)12L∥x′∥2≤V(x)≤L∥x′∥2,∂V(x)∂x(f(x)+∑i=12gi(x)αi)≤-∥x′∥2,|∂V(x)∂x∑i=1mgi(x)|2≤∥x′∥2.
That is, Assumption 2 is satisfied.Choosing(60)r1(x)=9Lx1,r2(x)=9Lx1,
we can obtain
(61)K1=∂r1(x)∂xg1(x)=[90]T,K2=∂r2(x)∂xg2(x)=[90]T.
Therefore, Assumption 3 holds.According to (61), we can obtain
(62)W1-K1V1=[-102.6000628.3185-628.31850],(63)W2-K2V2=[-102.6000628.3185-628.31850].
Furthermore, letP1=P2=[4-224]; substituting (62) and (63) into (21), respectively, we obtain
(64)Q1=[820.8000-205.2000205.20000],Q2=[820.8000-205.2000205.20000].
Letμ1=μ2=0.005; substitutingQi, Vi, Pi, Ki, andμiinto (52), through some arithmetical operations, we can obtain
(65)d1=λmin(Q1)-14∥V1∥2-2μ1∥P1K1∥2=54.9832-14×129.9600-2μ1×1620>0,d2=λmin(Q2)-14∥V2∥2-2μ2∥P2K2∥2=54.9832-14×129.9600-2μ2×1620>0.Hence, through the presented algorithm, the final internal models of disturbance and uncertain items are given by(66)D^1(w)=V1w^1(t),D^2(w)=V2w^2(t),w^1(t)=η^1(t)+r1(x),w^2(t)=η^2(t)+r2(x),η^˙11(t)=-102.6η^11(t)-0.7434x1+628.3185η^12(t)-9u1-2.8274x2,η^˙12(t)=-628.3185η^11-5.6549x1,η^˙21(t)=-102.6η^21(t)+628.3185η^22(t)-9u1+1.9036x1+0.18x2,η^˙22(t)=-628.3185η^21-5.6549x1,δ^1(x,t)=m1K1p1(x)tanh([η^11η^12]P1TK1p1(x)λ1)=[9tanh((36η^11-18η^12)x1λ1)0],(67)δ^2(x,t)=m2K2p2(x)tanh([η^3η^4]P2TK2p2(x)λ2)=[9tanh((36η^21-18η^22)x2λ2)0].Choosingm1=m2=1, ξ1=ξ2=0.1, the controller is designed as
(68)α1(x)=-Lez1c1+Rx1-Lωx2+Lx˙1ref=-(x1-x1ref)+0.02x1-0.3142x2,α2(x)=-Lez2c2+Rx2+Lωx1+Lx˙2ref=-(x2-x2ref)+0.02x2+0.3142x1,v1′=α1(x)-D^1(w)+m1p1(x)tanh((∂V(x)/∂x)g1(x)p1(x)ξ1)=-(x1-x1ref)+0.02x1-0.3142x2-11.4(η^11(t)+0.009x1)+x1tanh((x1-x1ref)x1ξ1)-380,v2′=α2(x)-D^2(w)+m2p2(x)tanh((∂V(x)/∂x)g2(x)p2(x)ξ2)=-(x2-x2ref)+0.02x2+0.3142x1-11.4(η^21(t)+0.009x1)+x2tanh((x2-x2ref)x2ξ2).Let the initial condition of the simulation bex(0)=[00]T, η^(0)=[0000]T, w^(0)=[10]T. The reference values of the output current are chosen asxref=[4,0]T. Figures 3, 4, 5, and 6 show the simulation results. Figure 3 demonstrates the system states (output currents in dq reference frame). Figure 4 indicates the control inputs in dq axis, respectively. The disturbances produced by the exosystem and their estimates are shown in Figure 5. Figure 6 demonstrates the estimating errors under the existence of exosystem disturbances and uncertain items. As shown in Figure 3, the system states are asymptotically convergence to the expected references under the existence of exogenous disturbances and uncertain modeling perturbations. Figures 5 and 6 also indicate that the designed internal models can produce the exogenous disturbance successfully and the estimating errors converge to zero.Figure 3
System states (output currents indq reference frame).Figure 4
Control inputs indq axis, respectively.Exogenous disturbances ofw1 and w2 and their estimates.
(a)
(b)Estimating errors ofw1 and w2.
(a)
(b)Example 15.
The network is immersed 1.5% 3rd harmonics and 1% 5th harmonics to fundamental wave.
The exosystem matrices for exogenous disturbanceD(w) represented in (10) and (18) are given asW=[02ω00-2ω0000004ω00-4ω0],V1=V2=[5.703.80], and the uncertain items also satisfy
(69)δ1(x,t)=x1sin(x1t),δ2(x,t)=x2sin(x2t).The design of the nominal system and the verification of Assumption2 are the same as in Example 14, their derivation processes are no longer described in detail.Choosing(70)r1(x)=r2(x)=[9Lx19Lx1]T,
we can obtain:
(71)K1=K2=∂r1(x)∂xg1(x)=[9090]T.
Therefore, Assumption 3 holds.In terms of (71), we can obtain
(72)W1-K1V1=W2-K2V2=[-51.3000628.3185-34.20000-628.3185000-51.30000-34.20001256.63700-1256.6370].
Let
(73)P1=P2=[3.970777-0.159154-0.072142-0.300626-0.1591544.1371510.0707350.045912-0.0721420.0707355.920095-0.079577-0.3006260.045912-0.079576.010020];
substituting (73) into (21), we obtain
(74)Q1=Q2=[200.0011100.000099.999699.9993100.0000199.998899.9989100.000799.999699.9989200.0100100.000199.9996100.0007100.0004199.9900].
Letμ1=μ2=0.01; substituting Qi, Vi, Pi, Ki, andμiinto (52), through some arithmetical operations, we can obtain
(75)d1=λmin(Q1)-14∥V1∥2-2μ1∥P1K1∥2=99.9924-14×46.93-2μ1×4013.57>0,d2=λmin(Q2)-14∥V2∥2-2μ2∥P2K2∥2=99.9924-14×46.93-2μ1×4013.57>0.Hence, through the presented algorithm, the final internal models of disturbance and uncertain items are given by(76)D^1(w)=V1w^1(t),D^2(w)=V2w^2(t),w^1(t)=η^1(t)+r1(x),w^2(t)=η^2(t)+r2(x),η^˙11(t)=-51.3η^11+628.3185η^12-34.2η^13-9u1+5.3731x1-2.8274x2,η^˙12(t)=-628.3185η^11-5.6549x1,η^˙13(t)=-51.3η^11-34.2η^13+1256.637η^14-9u1-0.2817x1-2.8274x2,η^˙14(t)=-1256.637η^13,η^˙21(t)=-51.3η^21+628.3185η^22-34.2η^23-9u2+2.3657x1+0.18x2,η^˙22(t)=-628.3185η^21-5.6549x1,η^˙23(t)=-51.3η^21-34.2η^23+1256.637η^24-9u2+2.3657x1+0.18x2,η^˙24(t)=-1256.637η^23,δ^1(x,t)=m1K1p1(x)tanh×([η^11η^12η^13η^14]P1TK1p1(x)λ1)=[9tanh((35.0877η^11-0.7958η^12+52.6316η^13-3.4218η^14)x1λ1)09tanh((35.0877η^11-0.7958η^12+52.6316η^13-3.4218η^14)x1λ1)0],δ^2(x,t)=m2K2p2(x)tanh×([η^21η^22η^23η^24]P2TK2p2(x)λ2)=[9tanh((35.0877η^21-0.7958η^22+52.6316η^23-3.4218η^24)x2λ2)09tanh((35.0877η^21-0.7958η^22+52.6316η^23-3.4218η^24)x2λ2)0].Choosingm1=m2=1, ξ1=ξ2=0.1, the controller is designed as
(77)v1′=α1(x)-D^1(w)+m1p1(x)tanh((∂V(x)/∂x)g1(x)p1(x)ξ1)=-(x1-x1ref)+0.02x1-0.31415x2-5.7(η^11(t)+0.009x1)-3.8η^13(t)+x1tanh((x1-x1ref)x1ξ1)-380,v2′=α2(x)-D^2(w)+m2p2(x)tanh((∂V(x)/∂x)g2(x)p2(x)ξ2)=-(x2-x2ref)+0.02x2+0.31415x1-5.7(η^21(t)+0.009x1)-3.8η^23(t)+x2tanh((x2-x2ref)x2ξ2).Let the initial condition of the simulation bex(0)=[00]T, η^(0)=[00000000]T, and w^(0)=[1010]T. The reference values of the output current are also chosen asxref=[4,0]T. Figures 7, 8, 9, and 10 show the simulation results. Figure 7 display that the system states are asymptotically convergence to the expected references under the existence of exogenous disturbances and uncertain modeling perturbations. Figures 9 and 10 also indicate that the designed internal models can produce the exogenous disturbances successfully.Figure 7
System states (output currents indq reference frame).Figure 8
Control inputs indq axis, respectively.Exogenous disturbances ofw1 and w2 and their estimates.
(a)
(b)Estimating errors ofD1 and D2.
(a)
(b)


## 6. Conclusions

In this paper, a nonlinear backstepping control with multivariable adaptive internal model principle for grid-connected inverter is proposed to reject the harmonic disturbance produced by a class of exosystems under the existence of uncertain items related with system states. Due to the nonlinearity and multiple variables for the original system, a nonlinear and multivariable internal model is constructed. In addition, in order to compensate for the effect of bounded uncertain items, an adaptive control law is designed to realize the real-time estimation of the perturbation. Based on the backstepping control law of the nominal system, a state feedback controller combined with the multivariable internal model and the adaptive control law is designed. A Lyapunov function is constructed and theoretically proves that all the signals of the multivariable closed-loop system are global boundedness. The simulation results show that the proposed control algorithm can guarantee the closed-loop system asymptotically converge to expected references quickly and the designed internal model can produce the exogenous disturbances successfully.Restricted by the actual problem of grid-connected inverter control under distorted grid voltage, the proposed algorithm in the research aims at rejection of harmonic disturbances, and without considering the nonharmonic disturbances. However, the nonharmonic disturbances may induce adverse impacts, for example, noise and precision reduction. The future research should extend the algorithm to reject the nonharmonic disturbances.



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*Source: 749798-2013-09-03.xml*