Open access peer-reviewed chapter

A New Control Strategy for Photovoltaic System Connected to the Grid via Three-Time-Scale Singular Perturbation Technique with Performance Analysis

Written By

Youssef Mchaouar, Abdelmajid Abouloifa, Ibtissam Lachkar and Mohammed Fettach

Submitted: 06 May 2019 Reviewed: 29 August 2019 Published: 01 April 2020

DOI: 10.5772/intechopen.89434

From the Edited Volume

Advanced Statistical Modeling, Forecasting, and Fault Detection in Renewable Energy Systems

Edited by Fouzi Harrou and Ying Sun

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This chapter addresses the problem of controlling single-phase grid-connected photovoltaic system through a full bridge inverter with L-filter. The control objectives are threefold: (i) forcing the voltage in the output of photovoltaic panel to track a reference. This reference has been obtained from the maximum power point tracking strategy; (ii) guaranteeing a tight regulation of the DC-link voltage; and (iii) ensuring a satisfactory power factor correction (PFC) at the grid such as the currents injected must be sinusoidal with the same frequency and the same phase as the grid voltage. The considered control problem entails several difficulties including: (i) the high dimension and strong nonlinearity of the system; (ii) the changes in atmospheric conditions. The problem is dealt with by designing a synthesized nonlinear multi-loop controller using singular perturbation technique, in which a three-time-scale dynamics is artificially induced in the closed-loop system. A formal analysis based on the three-time-scale singular perturbation technique and the averaging theory is developed to proved that all control objectives are asymptotically achieved up to small harmonic errors (ripples). The performance of the proposed approach and its strong robustness with respect to climate changes are evaluated based on the various simulations results carried out under Matlab/Simulink software.


  • single-phase grid-connected photovoltaic system
  • nonlinear control
  • three-time-scale singular perturbation technique
  • MPPT
  • power factor correction
  • averaging theory
  • stability analysis

1. Introduction

Due to dramatic increase in energy consumption and thrust to reduce carbon and greenhouse gas emissions from the traditional electric power generation systems, photovoltaic (PV) power generators have gained a great popularity in recent years. Indeed, photovoltaic systems produce electric power without harming the environment, transforming a free inexhaustible source of energy, solar radiation, into electricity. Furthermore, the major advantage of the photovoltaic systems is to meet the basic power requirement of non-electrified remote areas, where grid power has not yet reached. Also, there are other advantages such as the declining cost and prices of solar modules. On the other hand, the importance of PV systems in the solar industry makes these systems more efficient and reliable, especially for utility power in distributed generation (DG) at medium and low voltages power systems [1]. All these considerations assure a promising role for PV generation systems in the near future.

On the other hand, many technical problems, such as untimely failures, could be found on electronic systems related in particular to the transfer and conversion of this energy to the network. Today, most conversion systems often suffer from low yields in real production sites. To meet the requirements of the new international standards on expected performance on associated conversion systems, it is important to make a research effort to solve the many control problems associated with the static power converter and bring this area to a degree of sufficient maturity to make them industrial products in their own right. One of the difficulties caused by the use of a photovoltaic conversion chain is focused on the problem of non-perfect control of the chain between the photovoltaic generator itself and the continuous or alternative type of load.

The efficiency of a PV plant is affected mainly by three factors: the efficiency of the PV panel, the efficiency of the static power converter and its control, and the efficiency of the maximum power point tracking (MPPT) algorithm.

PV grid-connected systems represent the most important field applications of solar energy [2, 3, 4]. In general, the power converter interface, from PV module (the DC source) to the load or to the grid, consists of two-stage converters: The first-stage DC/DC converter is usually used to boost the PV voltage and to implement the maximum power point technique. While the second stage is used to convert this power into high-quality AC voltage, with power factor correction (PFC) respecting to the power supply grid (i.e. sinusoidal and in phase with the AC supply voltage).

Maximum power point tracking (MPPT) is required to match the PV array power to the environmental changes achieving to extract the maximum power output from a solar cell [5]. To this end, different MPPT techniques have been proposed such as incremental conductance [6], perturbation and observation (P&O) [7, 8], the hill-climbing, and some other special methods, such as neural networks, fuzzy logic technique [9]. Among all available techniques, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.

In order to provide a stable controller of DC/DC and DC/AC converters, many linear control methods have been proposed using many methods, such as a fuzzy-proportional integral controlled [10], a simple PR controller [11] where the performances have been illustrated by experimental result. However, in both proposed controllers, the problem of maximizing PV power transfer is not accounted for in the controller design. In contrast to linear control, nonlinear approaches can optimize the dynamic performance of system, such as sliding mode [12], fuzzy-sliding mode [13], feedback linearization [14], singular perturbation technique [15], and many others works [16, 17, 18]. In light of the previous descriptions, no theoretical analysis is made to formally prove that the closed-loop control performances are actually achieved.

In this chapter, a multi-loop nonlinear controller is designed and developed via singular perturbation technique (Chapter 11 in Refs. [19, 20]), as was shown in Refs. [21, 22], where three-time-scale dynamics is artificially induced in the closed-loop system. The control objectives are threefold: (i) achieving the MPPT for the PV array; (ii) ensuring a tight regulation of the DC-link voltage; and (iii) ensuring a grid connection with unity power factor (PF). These objectives must be met despite changes of the climatic variables (temperature and radiation). A theoretical stability analysis, for the closed-loop system, is provided using the three-time-scale singular perturbation technique [23, 24] and averaging technique (Chapter 10 in Refs. [19, 25]. The three-time-scale analysis allows to construct a suitable composite Lyapunov function candidate for the closed-loop photovoltaic system, and the stability properties of the resulting subsystems are analyzed providing mathematical expressions for the upper bounds of the singularly perturbed parameters.

Compared to previous works, the contribution of the new nonlinear controller enjoys several interesting features including the following:

  • Several control objectives are simultaneously taken into account such as: MPPT, DC regulation, and PFC, whereas only some of these objectives have been tackled in previous works [10, 11].

  • A theoretical analysis will prove, using three-time-scale singular perturbation and averaging technique, that the desired multiple objectives are achieved. Such a formal analysis was missing in the previous works [12, 13, 14, 15].

  • The nonlinearity of the controlled system was preserved in the controller design in order to keep all the properties of the studied system, whereas it is partly or totally ignored in previous controllers [14].

  • By including of three-time-scale dynamics in the full-order closed-loop system can ensure to achieve desired properties, such as robust zero steady-state error of the reference input realization, desired output performance specifications (overshoot, settling time), and insensitivity of the output transient behavior with respect to parameter variations and external disturbances.

The content of this chapter is outlined as follows: in Section 2, the grid-connected PV system is described and modeled. Section 3 is devoted to the cascade nonlinear controller design and its performances are formally analyzed in Section 4. The global performance of the closed-loop photovoltaic system will be illustrated by numerical simulation using MATLAB/SIMULINK tool in section 5. A conclusion and a reference list end the chapter.

2. System description

This section describes the modeling of photovoltaic system connected to the grid. The power circuit topology used in the proposed single phase grid connected to the photovoltaic array is shown in Figure 1. It consists of the following components: (i) a photovoltaic array which consists of an arrangement of Ns-series and Np-parallel strings; (ii) an input capacitor Cpv and a DC-DC boost converter used to increase the voltage level and achieve MPPT for photovoltaic array; (iii) a DC link capacitor Cdc; and (iv) a single-phase full-bridge inverter including four power semiconductors with L filter that is used to provide the energy to the grid and ensure power factor correction.

Figure 1.

Single phase grid-connected PV system.

Typical (Ip-Vp) characteristics of solar cells arranged in Np-parallel and Ns-series can be found in many places (see, e.g. [26]). The PV array module considered in this paper is of type KC200GT. In this chapter, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.

By analyzing the circuit and applying the well-known Kirchhoff laws, the system of Figure 1 can be described by the following set of differential equations:


where vpv and ipv are, respectively, the photovoltaic generator voltage and current. vdc and idc are, respectively, DC link voltage and current. ii designates the input current chopper, Cdc is DC link capacitor, ri and rg are, respectively, the equivalent series resistances (ESR) of input inductance Li and the filter inductance Lg. vg and ig are, respectively, the voltage and current of the grid. Here, the grid voltage is defined by vg=Egsinωgt, where Eg and ωg denote the constant amplitude and the constant angular frequency. The switching functions µ1 and µ2 are defined by:


The instantaneous model (1)(4) cannot be used directly for the development of continuous control laws since it involves, as input variables, the binary signal µ1 and µ2. To overcome this inconvenience, the average model is used [27]. Therefore, the state variables ii, vpv, ig, and vdc are replaced by their average values x1, x2, x3, and x4 over a cutting period. The control inputs u1 and u2 denote the average values of μ1 and μ2, respectively.


3. Controller design

3.1 Input inductor current regulation and PFC objectives

3.1.1 Control law design

The first control objective is to enforce the photovoltaic voltage x2 to track, as closely as possible, the optimal point VM (called Regulator 2). However, it is well-known that the boost converter has a non-minimum phase feature. Such an issue is generally dealt with by resorting a cascaded loop design strategy that starts with the input current loop (Regulator 1), as it is shown in Figure 2. More specifically, the controller makes the input inductor current x1 to track a reference signal x1, the latter is determined from (Regulator 2).

Figure 2.

Schematic diagram of the proposed controller for single phase grid-connected PV system.

In parallel with the input current controller, the network current controller (Regulator 3) will be designed for power factor correction requirement that amounts to forcing the network current x3 to match the reference signal of the form x3=βsinωgt. It means that the grid current x3 should be sinusoidal and in phase with the AC grid voltage vg, with β is a signal. In fact, the latter is allowed to (and actually will) be time-varying but it must converge to a positive constant value. For both objectives (i.e. the input current x1 regulation and PFC objectives), let us consider Eqs. (7), (9) in the following form:


where X1,3=x1x3, M1=x4Lix4Lg, M2=riLix1+x2Lix4LirgLgx3vgLg, and U=u1u2.

Remark 1. Under condition that the DC-link voltage remains all the time positive, it can be showed that 0<M1minM1M1Max< and M2M2Max< are satisfied.

Let us introduce the following current tracking errors:


Then, the reference model can be constructed in the following form


where T1 and T3 denote the time constants and they are selected based on the desired settling time, respectively, for the currents x1 and x3. Based on (13), the realization errors of the desired behaviors of ẋ1 and ẋ3, namely Ω1 and Ω3, are given by


Therefore, the control problem limteit=0i=13 corresponds to the insensitivity condition defined by


Doing so, the behaviors of ẋ1 and ẋ3 with prescribed dynamics of expression (13) will be fulfilled. Replacing in Eq. (14)ẋ1 and ẋ3 by their expressions (7), (9), and since the requirement (15), one gets


Hence, there is an isolated root called the inverse dynamic solutions for U given by


The control variable, namely U, has emerged in Eq. (16). At this point, an appropriate control law with the first derivative in feedback has to be found, so that the (e1, e3)-systems are made asymptotically stable. As the objective is to drive the error to zero, it is natural to choose the Lyapunov functions candidate


From expressions (14) and (16), it can be easily checked that the following time-derivatives


are made negative-definite using the following control laws:


At this point, k1 and k3 are any design parameters, ε1 and ε2 are small positive parameters. In view of M1>0, it is easily seem from expressions (11) and (14) that


for a positive values of ki>0i=13. Therefore, from expressions (14) and (20), the discussed nonlinear control laws are formulated as follows


3.1.2 Singular perturbation system of the inner current loops

Consider the closed-loop system of inner loops composed of the Eqs. (7)(10) and the control laws (22), which can be rewritten in the following form


with Z=z1z2T=u1u2T, X=x1x2x3x4T, hXZt=k1e1T1x4Liz1+riLix1x2Li+x4Lik3e3T3x4Lgz2+vgLg+rgLgx3, fXZt=z1x4LiriLix1+x2Lix4Lix1Cpv+ipvCpvvgLgrgLgx3+x4Lgz2x1Cdcx1Cdcz1x3Cdcz2.

Now, we go to the fast time τ1=t/ε1ε2. Then for ε1=0, the ultra-fast dynamic subsystem (UFDS) is given by


After the rapid decay of transients in expression (25), the steady state (more precisely, quasi-steady state) tends toward an equilibrium Ze=h˜X. The manifold defined by Ze is called the slow manifold, which is given by


Remark 2.

  1. During the fast transient in expression (25), the variables X are treated as the frozen parameters.

  2. The equilibrium point given by expression (27) involves a division by the DC link voltage x4, from a practical point of view this division is not a problem because the DC link voltage remains all the time positive for the power converter to work correctly.

By substituting of this equilibrium Ze into Eq. (24), the slow dynamic subsystem (SDS) of inner loops takes place on the slow manifold, according to the equation


Proposition 1. Consider the closed-loop system composed of Eqs. (23) and (24). For ε10, the system takes the singular perturbation form where the UFDS is defined by equation (25), while the SDS of inner loop is defined by equation (28). Under the considerations given by Remark 2, one has the following properties

  1. If the design parameters ki (i=1,3) are positives, the UFDS (25) will be exponentially stable, and Z converge exponentially fast to Ze.

  2. The behaviors of xi (i=1,3) are prescribed by the stable reference equations of the form dxi/dt=xixi/Ti. Then, the requirements limtei=limtxixi=0 are maintained.

3.2 MPPT and DC bus voltage regulation objective

The second step consists in completing the inner control loops by outer control loops for PV voltage (Regulator 2) and DC-link voltage (Regulator 4). The aim is now to enforce the photovoltaic voltage x2 and the DC-link voltage x4 to track, respectively, the optimal point x2=VM and a given reference voltage x4=Vdc, such that tuning laws for the ratio β and x1 must be designed. According to the three-time-scale design methodology that is employed in this work, the general formulation of the three-time-scale singular perturbed systems requires the system to possess three different time scales. To this end, the voltage loops will be slow compared to the transients of the current loops. Therefore, the design parameters for voltage loops, in particular (ε2, T2, and T4) must satisfy: 0<ε1ε2ε21, and T1T3<T2T4. In addition, the steady states for the current x1 and x3 yield, respectively, x1 and x3. Therefore, the SDS of inner loop given by equation (28) will be reduced to


Now, the first step is to establish the relation between the ratio β (which acts as the control input of the outer loop) and the DC-link voltage x4 (representing the output of the outer loop).

3.2.1 Relation between β and x4, and the control law

The relation between the ratio β and the DC-link voltage x4 is the subject of the following proposition.

Proposition 2. We consider the second equation of expression (29) and the power factor correction requirement defined by x3=βsinωgt.

  1. The relation between x4 and β is described in the following results


  2. Therefore, the squared-voltage xr,4=x42 varies, in response to the tuning ratio β, according to the following first-order time-varying nonlinear equation:


The second step is to establish control laws for the outer loops, in which Y=y1y2T=x1βT represent the new control inputs, while Xr=xr,2xr,4T=x2x42T represent the new output variables. To this end, introduce the following tracking errors


Then, let the desired behavior of Xr be assigned by


The error of the desired dynamic realization it follows


Then, the insensitivity condition is given by


Similar to the previous subsection and bearing in mind the fact that β and their first derivative must be available, and in order to meet the requirement (35), we should apply the control law given by the following structure


3.2.2 Singular perturbation system of the outer voltage loops

Combining expressions (31) and (36), and the first equation of expression (29), one obtains


with f2=2xr,2y1riy12CdcEgvg2y2+rgvg2y22CdcEg2. The fast dynamic subsystem (FDS) is obtained by transforming the slow time-scale t to the fast time-scale τ2=t/ε2, then, by setting ε2=0


Notice that expression (39) has an isolated equilibrium at Ye=y1ey2e, which will be determined (in the mean) in Appendix. As the FDS (39) is nonlinear, the stability properties of its equilibrium can be checked through the analysis of the Jacobian matrix of the linearized version defined as follows


with AF=4riy1eCdc2xr,2Cdc, AF=2vg2CdcEg+4rgvg2y2eCdcEg2. Taking into account that y2e is positive, we conclude that all eigenvalues of AF satisfy Reλ1,2<0 for the negative values of k2 and k4. Therefore, AF is Hurwitz matrix. By substituting of the equilibrium Ye into expression (38), the reduced SDS of outer loops takes place on the slow manifold, according to the equation


Proposition 3. Consider the system closed loop composed of expressions (37) and (38). For ε20, this system takes the singular perturbation form, where the FDS is given by expression (39) and the reduced SDS of outer loop is given by expression (42). One has the following properties

  1. If the design parameters k2 and k4 are negative, the FDS (39), will be exponentially stable and Y converge exponentially fast to Ye.

  2. The behaviors of xr,j (j=2,4) are prescribed by the stable reference equations of the form dxr,j/dt=xjxr,j/Tj. Then, the requirements limtej=limtxjxr,j=0 (j=2,4) are maintained.

4. Control system analysis

The objective of the global stability of closed-loop system can be analyzed in the following theorem. It is shown that the control objectives are achieved (in the mean) with an accuracy that depends on the network frequency ωg and the small parameters εii=12.

Theorem. Consider the overall control system composed of the Pv panel, boost DC-DC converter and DC-AC inverter, described by the model (7)(10), in closed loop with the multi-cascade multi-loop composed of:

  • The inner regulators (23), where (ε1ε2, k1, k3, T1, T3) are the design parameters;

  • The outer regulators (37), where (ε2, k2, k4, T2, T4) are the design parameters.

Then, one has the following property

  1. The augmented state vector Z=u1u2T=z1z2T, Y=x1βT=y1y2T, and X=x1x2x3x4T=x1x2x3x42T undergoes the following state equations



    h=k1e1T1z1x4Li+rix1x2+x4Lik3e3T3x4Lgz2+vg+rgx3Lg, f=riLix1+x2Lix4Li+z1x4Liy1Cpv+ipvCpvx4Loz2vgLgroLgx32x2y1riy12Cdc2vg2y2CdcEg2rgvg2y22CdcEg2, g=k2e2T2+y1CpvipvCpvk4e4T4+2vg2y2CdcEg+2rgvg2y22CdcEg22x2y1riy12Cdc.

  2. Let the control design parameters be selected, such that the following inequalities hold T1<T2, T3<T4, k1>0, k2<0, k3>0, k4<0, and 0<ε1ε2ε21.

Then, there exist positive constants ρ, εnn=12, such that for and 0<ρ<ρ0<εn<εn1n=12 the system (43)(45) has an asymptotically stable 2π/ωg periodic solution Xtεnρ, Ytεnρ, Ztεnρ that continuously depends on εnn=12 and ρ=1/ωg. Furthermore, when (εn=0 and ρ=0), one has


where X0=x1,0x2,0x3,0x4,0T, Y0=y1,0y2,0T, and Z0=z1,0z2,0T with x2,0=VM, x4,0=Vdc, z1,0=1x2,0x4,0+rix1,0x4,0, z2,0=rox3,0x4,0, y1,0=ipv,0, y2,0=Eg+2Eg/22+2rox2,0ipv,0riipv,022ro.

See Appendix for the proof.

5. Simulation and discussion of results

The experimental setup is described by Figure 2 and the nonlinear controller, developed in Section 3, including the control laws (23) and (37), will now be evaluated by simulation in MATLAB/SIMULINK platform using the electromechanical characteristics of Table 1.

PWM switching frequencyfPWM14kHz
DC capacitanceCdc7mF

Table 1.

PV system and single-phase grid characteristics.

The numerical values used for the design parameters are chosen as follows: ε1=5.81×102, ε2=7.21×105, T1=2.1×104, T2=9.3×103, T3=9.51×105, T4=1.43×102, k1=7.1×107, k2=2.56×106, k3=2.3×106, K4=8.1×108. These values have proved to be suitable based on several trials respecting the singular perturbation technique. In this simulation, we consider the KC200GT type of PV array module with Ns=54 and Np=6.

The performances of the proposed controller are illustrated by Figures 35. Figure 3 shows that the DC-link voltage x4 is well regulated and quickly settles down after each change in the signal reference (stepping from 600V to 750V at t=0.4s). The wave frame of the output current x3 is showed in Figure 4. The current is sinusoidal and in phase with the network voltage complying with the PFC requirement. This is further demonstrated by Figure 5, which shows that the ratio β takes a constant value after transient periods following the changes in reference signals.

Figure 3.

DC-link voltage.

Figure 4.

Power factor correction checking.

Figure 5.

Tuning parameter β.

5.1 Radiation variation effect

Figure 6ac illustrates the resulting closed-loop control performances in presence of radiation changes. Specifically, the radiation takes a low, medium, and high value (equal to 800, 1000 and step to 600 W/m2 at times 0, 0.4, and 0.7 s, respectively), meanwhile the temperature is kept constant, equal to 298.17K (i.e. 25C). Figure 6a shows that the captured PV voltage varies between 182, 212, and 138 V. These values correspond to the maximum points. Figure 6b shows that the DC-link voltage regulation is recovered after a short transient period following each change of the irradiation. Figure 6c shows that the current amplitude changes whenever the radiation varies. It is seen that the output current x3 and the grid voltage vg are actually sinusoidal and in phase. Hence, the converter connection to the supply network is done with a unitary power factor.

Figure 6.

Radiation variation effect: (a) Photovoltaic voltage. (b) DC-link voltage. (c) PFC checking.

5.2 Temperature variation effect

The perfect MPPT is illustrated by Figure 7a. Here, the temperature steps from 298.15 to 318.15 K, then to 308.15 K while the radiation λ is kept constant equal to 1000W/m2. Figure 7b shows that the DC-link voltage x4 is tightly regulated: it quickly settles down after each change in the temperature. Figure 7c illustrates the current amplitude changes whenever the temperature varies. The current remains (almost) sinusoidal and in phase with the network voltage complying with the PFC requirement.

Figure 7.

Temperature variation effect: (a) Photovoltaic voltage. (b) DC-link voltage. (c) PFC checking.

6. Conclusion

In this chapter, an advanced controller is developed for PV grid-connected system. The latter is described by fourth-order nonlinear averaged model. The multi-loops nonlinear controller has been designed and developed using three-time singular perturbation technique and averaging theory.

Using the theoretical analysis (via three-time-scale singular perturbation technique and averaging theory) and simulation, it is proved that the controller does meet the performances for which it was designed, namely: (i) Maximum power point tracking of PV array; (ii) tight regulation of the DC bus voltage; (iii) perfect power factor in the grid; and (iv) global asymptotic stability of the all system.

Several simulation results have been made that illustrate the high performances of the proposed controller in ideal operating conditions (in the presence of meteorological constant) and its robustness against radiation and temperature change.

Appendix (Proof)

Part 1: Eqs. (43)(45) are immediately obtained from expressions (37), (38), (23), and the first and the third equation of expression (24).

Part 2: The stability of the time-varying system (43)(45) will now be performed in two steps using the averaging theory (e.g. Chapter 10 in Refs. [19, 25]) and the singular perturbation theory (e.g., Chapter 11 in Refs. [19, 23, 24]). The next step consists in using the averaging theory. To this end, let us introduce the time-scale change t¯=ωgt. Using this time, one gets ε1ε2Z¯̇t¯=ρh¯X¯Z¯t¯, ε2Y¯̇t¯=ρg¯X¯Y¯t¯, and X¯̇t¯=ρf¯X¯Y¯Z¯t¯ with ρ=1/ωg. In view of expressions (43)(45), it is seem that the functions h¯X¯Z¯t¯, g¯X¯Y¯t¯, and f¯X¯Y¯Z¯t¯ as functions of t¯ are periodic with period–2π, let us introduce the averaged functions:


Since the systems here studied present equilibrium different from zero and in order to satisfy this requirement, a change of variables is introduced such that defines the new system in terms of its error dynamics. Therefore, the error dynamics are defined by introducing: Z˜0=z˜1,0z˜2,0T=z1,0z1,0z2,0z2,0T, Y˜0=y˜1,0y˜2,0T=y1,0y1,0y2,0y2,0T, and X˜0=x˜1,0x˜2,0x˜3,0x˜4,0=x1,0x1,0x2,0x2,0x3,0x3,0x4,0x4,0, where the constant x1,0, x2,0, x3,0, x4,0, y1,0, y2,0, z1,0, and z2,0 represent the desired average values of the state variables. Then, since expressions (43)(45) and according to expressions (47)(49), the system can be rewritten into its error-dynamics formulation thus defining the closed-loop error dynamics as:




In order to get stability results regarding the system of interest (43)(45), it is sufficient (thanks to averaging theory) to analyze the averaged system (50)(52). Now, the asymptotic stability of the resulting three-time-scale photovoltaic single phase grid system (50)(52) is discussed, which is based on the sequential (double) time-scale analysis similar to the one presented in Refs. [23, 24], it is an extension of the two-time-scale analysis presented [19].

The use of theory of the three-time-scale singular perturbations for the stability analysis is based on the idea that, for 0<ε1ε2ε21, the trajectories in X˜0, Y˜0, and Z˜0 of the system (50)(52) can be approximated by three models: the slow dynamic subsystem (SDS) of full system, the fast dynamic subsystem (FDS), and the ultra-fast dynamic subsystem (UFDS). We can thus find Lyapunov functions for each one of the singularly perturbed subsystems.

For the UFDS, it is necessary to ensure that the dynamic of expression (50) does not to shift from the quasi-steady-state equilibrium Z˜0=h˜0X˜0. Then, the associated Lyapunov function candidate is obtained by introducing a change of variables Ẑ0=Z˜0h˜0X˜0, so that its equilibrium is centered at zero. By letting τ1=t¯/ε1ε2, the UFSD in function of Ẑ0 is defined as follows


in which X˜0 is treated as a fixed parameter. Thus, the associated Lyapunov function can be defined by


In view of ki>0i=13 and x˜4,0+x4,0>0, it is clear that the solutions PU1 and PU2 can be chosen as


with qU1 and qU2 are positive. Similar to the ultra-fast subsystem, it is necessary to ensure that the dynamic of expression (51) does not to shift from the equilibrium Y˜0=g˜0X˜0. Then, by introducing a change of variables Ŷ0=Y˜0g˜0X˜0, so that its equilibrium is centered at zero and by letting τ2=t¯/ε2, the ultra-fast subsystem can be rewritten in the following form


with ψ=2k4riŷ1,02Cdc+rgŷ2,022Cdc+ψŷ2,0Cdc+2riCpvCdcx˜2,0T2+ipv,0Cpvx˜2,0+x2,0Cdcŷ1,0.

Since expression (61), it is seem that the fast subsystem is nonlinear. According to the proposition 4 (part i), it is shown that this subsystem can be made asymptotically stable by letting kj>0 (for j=2,4). Moreover, we can find a Lyapunov function, which takes the following quadratic form


with the solutions to the associated Lyapunov given as


in which X˜0 is treated as a fixed parameter. Finally, the new slow dynamic subsystem is obtained by substituting of the ultra-fast subsystem equilibrium Z˜0=h˜0X˜0 and the fast subsystem equilibrium Y˜0=g˜0X˜0 into expression (52)


It is easy to define the associated Lyapunov function for SDS as follows


where pS1=T1qS1/2, pS2=T2qS2/2, pS3=T3qS3/2, pS4=T4qS4/2, and qS1, qS2, qS3, qS4 are the positive constants.

Based on these Lyapunov function candidates, the double application of the standard two-time-scale stability analysis is divided in two stages: in the first stage, the stability analysis focusses on proving the stability properties of the degenerated SF-subsystem (slow-fast subsystem). The results obtained will be used in the second stage in order to prove the stability properties for the full SFU-system (slow-fast-ultra-fast subsystem).

SFStability analysis

In the first stage, the standard method for two-time-scale systems is applied in which the previously derived Lyapunov functions for the slow and fast subsystems, that is, VS and VF, respectively, must satisfy certain inequalities.

  • Isolated equilibrium of the origin for theSF-subsystem

The origin X˜0=0Y˜0=0 is an isolated equilibrium of the SF-subsystem


moreover, Y˜0=g˜0X˜0 is the root of 0=ĝ0X˜0Y˜0 which vanishes at X˜0=0, and g˜0X˜0<ϑ1X˜0 where ϑ1 is a κ function.

  • Reduced system condition for theSF-subsystem

Using the Lyapunov function (67) and substituting ρf˜0X˜0Y˜0h˜0X˜0 yields


where α11 and ϕ1X˜0=qS12x˜1,02+qS22x˜2,02+qS32x˜3,02+qS42x˜4,02.

  • Boundary-layer system condition for theSF-subsystem

Since expression (62)–(65), it is seem that


where QF1 and QF2 are defined as follows


For physical point of view and domain of working principle, it is supposed that all physical state variables are bounded in domain of interest, where X˜0minX˜0X˜0Max, Y˜0minY˜0Y˜0Max, and Z˜0minZ˜0Z˜0Max. Therefore, QF1 and QF2 can take positives minimum possible values QF1,min>0 and QF2,min>0. This can be done by ensuring the appropriate selection of qF1, qF2, k2, and k4 as follows


Therefore, the boundary-layer system condition for the SF-subsystem is defined as follows


where α21 and ϕ2Ŷ0=QF2minŷ2,02+QF1minŷ1,02.

Now, we consider the composite Lyapunov function candidate of the SF -subsystem given as follow


with 0<η1<1. The derivative of VSF presents new terms which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions


where the constants θ1, θ2, and γ1 are non-negative.

Therefore, from the singular perturbation theory (e.g. Theorem 11.3 in Ref. [19]), it follows that the derivative of VSF is negative-definite for


SFU Stability Analysis:

The stability of full system is analyzed now. In this step, the results obtained in above section will be used in order to prove the asymptotic stability properties of the full SFU-system, which, for convenience, is first rewritten as


where χ˜0=x˜1,0x˜2,0x˜3,0x˜4,0y˜1,0y˜2,0T, similarly to the SF general asymptotic stability analysis presented in the above section, the SFU-system is treated like a two-time-scale singularly perturbed system, where the SF-subsystem is treated as the new slow augmented reduced order. The stability of the full system implicates that the previously derived Lyapunov functions for the new slow and new fast subsystems, that is, VSFχ˜0 and VUẐ0, respectively, must satisfy certain inequalities.

  • Isolated equilibrium of the origin for theSFU-subsystem

The origin χ˜0=0Z˜0=0 is an isolated equilibrium of the SFU-subsystem


Moreover, Z˜0=h˜0χ˜0 is the unique root of 0=ĥ0χ˜0Z˜0, which vanishes at χ˜0=0, and h˜0χ˜0<ϑ2χ˜0 where ϑ2 is a κ function.

  • Reduced system condition for theSFU-subsystem

Using Lyapunov function candidate VSFχ˜0 given by expression (76), it is easily shown that


Recalling that Ŷ0=Y˜0g˜0X˜0, and since (61)(65), one gets



pF3ŷ1,0x˜2,0=CpvCdc2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,04T2k4ψψ+Cdck2/2k4CpvqF2=A9qF2, pF3ŷ1,0x˜4,0=0


pF1ŷ1,0x˜4,0=0, pF3ŷ2,0x˜4,0=Cdc22riCpvx˜2,0T2+ipvCpvx˜2,0+x2,04T4k4ψ2ψ+Cdck2/2k4CpvqF2=A12qF2, pF2ŷ2,0x˜4,0=Cdc24k4T4ψ2qF2=A11qF2.

Substituting the expressions of ρF˜0χ˜0h˜0χ˜0 and by using absolute value version of Young’s inequality aba2+b2/2, it shows that


where Μ1=1η1qS12, Μ2=1η1qS221η12T22qS22η121T22, Μ3=1η1qS32, Μ4=1η1qS421η12T42qS42η121T42, M5=qF12M51η12Cpv2, and M6=qF22M6η122ψCdcrgCdcŷ2,0Max2


Now, it needs to ensure that Μ1x˜1,02Μ2x˜2,02Μ3x˜3,02Μ4x˜4,02Μ5ŷ1,02Μ6ŷ2,02<0. This can be done by ensuring the appropriate selection of qS1, qS2, qS3, qS4, qF1, and qF2, such that qS1 and qS3 have sufficient large positive values, qF1 and qF2 have sufficient small positive values, and qS2 and qS4 are limited as follows 11η11η1T222<qS2<1+1η11η1T222, 11η11η1T422<qS4<1+1η11η1T422 (for η1<13). Therefore, the reduced system condition for the SFU system is defined as follows


where α31 and ϕ3χ˜0=Μ1x˜1,02+Μ2x˜2,02+Μ3x˜3,02+Μ4x˜4,02+Μ5ŷ1,02+Μ6ŷ2,02.

  • Boundary-layer system condition for the SFU-system

Using the Lyapunov function given by expression (58)–(60), and substituting ρĥ0X˜0Ẑ0 of expression (57), one obtains


where α41 and ϕ4Ẑ0=qU12ẑ1,02+qU22ẑ2,02.

Now, we consider the composite Lyapunov function candidate of the full system given as follow


with 0<η2<1. The derivative of VSFUχ˜0ẑ0 presents new terms, which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions:


the constants θ3, θ4, and γ2 are non-negative. Therefore, from the singular perturbation theory (e.g. Theorem 11.3 in Refs. [19, 23, 24]), it follows that the derivative of VSFU is negative-definite when


with ε2=α1α2α1γ1+θ1θ2 and η1=θ1θ1+θ2.

Therefore, it can be inferred that the equilibrium X0=x1,0x2,0x3,0x4,0T, Y0=y1,0y2,0T, and Z0=z1,0z2,0T of full system, is asymptotically stable for all εi<εii=12. Therefore, Part 2 immediately follows from the averaging theory (e.g. Theorem 10.4 in Ref. [19]). Theorem is established.


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Written By

Youssef Mchaouar, Abdelmajid Abouloifa, Ibtissam Lachkar and Mohammed Fettach

Submitted: 06 May 2019 Reviewed: 29 August 2019 Published: 01 April 2020