A novel combined control based on an adaptive … · Advances in adaptive nonlinear control have...

SYSTEMS AND CONTROL ENGINEERING | RESEARCH ARTICLE

A novel combined control based on an adaptivegeneralized back-stepping method for a class ofnonlinear systemsMalek Ghanavati1, Karim Salahshoor2*, Mohammad Reza Jahed-Motlagh3, Amin Ramezani4

and Ali Moarefianpur1

Abstract: This paper was aimed at using an adaptive control to develop a back-stepping method for a special class of nonlinear systems, which requires no infor-mation on the upper bound of parametric uncertainty or disturbance. In thisresearch, a novel control approach, including a developed backstepping method andan adaptive controlling method, is introduced as an adaptive generalized back-stepping method (AGBM). Compared to the standard backstepping method, AGBM isfar more efficient because the standard method is applicable in strictly feedbacksystems, but AGBM expands this class of systems. Another advantage of AGBM isthat it can be applied to a special class of nonlinear systems with unmatcheduncertainty as well as unknown upper bound disturbance. In this method, anadaptive controlling method is used to compensate uncertainty and parametric

Malek Ghanavati

ABOUT THE AUTHORSMalek Ghanavati received his BSc in ElectricalPower, 2001, and his MSc degree in 2006, Iran.In 2017, he received his PhD in Control andSystem Engineering, Science and Researchbranch, Islamic Azad University, Iran. Hisresearch interests include nonlinear control andprocess control.

Karim Salahshoor is a professor at theDepartment of Automation andInstrumentation, Petroleum University ofTechnology, Iran. His research interests includeintelligent control, advanced process control,and intelligent monitoring.

Mohammad Reza Jahed Motlagh is a professorat the Department of Electrical Engineering, IranUniversity of Science and Technology, Iran. Hisresearch interests include control engineering,nonlinear system, and artificial intelligence.

Amin Ramezani is an assistant professor at theDepartment of Electrical Engineering, TarbiatModares University, Iran. His research interestsinclude process control and automation, discreteevent systems, and stochastic process control.

Ali Moarefianpour is an assistant professor ofElectrical Engineering Department, Science andResearch branch, Islamic Azad University, Iran. Hisresearch interests are convex optimization, indus-trial control, and artificial intelligence.

PUBLIC INTEREST STATEMENTThis paper focuses on using an adaptive controlto develop a backstepping method for a specialclass of nonlinear systems, which does notrequire any information about the upper boundof parametric uncertainty or disturbance. In thisresearch, a novel control approach (including thedeveloped backstepping method and the adap-tive controlling method) is introduced as theadaptive generalized backstepping method(AGBM). Compared to the standard backsteppingmethod, AGBM is far more efficient because thestandard method is applicable in strictly feed-back systems, but AGBM expands this class ofsystems. Another advantage of AGBM is that itcan be applied to a special class of nonlinearsystems with unmatched uncertainty as well aswith unknown upper bound disturbance. Thiscontrol method tends to stabilize the compressorsystem under surge instability.

Ghanavati et al., Cogent Engineering (2018), 5: 1471787https://doi.org/10.1080/23311916.2018.1471787

© 2018 The Author(s). This open access article is distributed under a Creative CommonsAttribution (CC-BY) 4.0 license.

Received: 11 February 2018Accepted: 28 April 2018First Published: 22 May 2018

*Corresponding author: KarimSalahshoor, Department ofAutomation and Instrumentation,Petroleum University of Technology(PUT), Tehran, IranE-mail: [email protected]

Reviewing editor:James Lam, University of HongKong, Hong Kong; Xinggang Yan,University of Kent, UK

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uncertainty besides stabilizing the controller against disturbances applied to thesystem. Finally, numerical simulation results demonstrate the advantages andfeasibility of the presented algorithm, showing that the AGBM controller guaranteesrobustness and system stability in the presence of disturbance and uncertainty.

Subjects: Systems and Control Engineering; Control Engineering; Systems and Controls

Keywords: nonlinear control; backstepping; strictly feedback; adaptive control; unmatcheduncertainty; adaptive generalized backstepping

1. IntroductionThe nonlinear control community has paid much attention to the adaptive control of nonlinearsystems with parametric uncertainty (Isidori, 1995; Khalil, 2002; Krstic, Kanellakopoulos, &Kokotovic, 1995; Niu, Ahn, Li, & Liu, 2017; Niu & Li, 2017). Nonlinear systems with unknownparameters were exposed to the limitations of growth rate for uncertainties and matching condi-tions (Kanellakopoulos, Kokotovic, & Marino, 1991; Sastry & Isidori, 1989), which can be linearizedby adaptive controlled algorithms. The introduction of integrator backstepping design(Kanellakopoulos, Kokotovic, & Morse, 1991; Krstic et al., 1995; Seto, Annaswamy, & Baillieul,1994) has removed these limitations. Advances in adaptive nonlinear control have shifted thefocus toward the robust adaptive control problem for nonlinear systems with time-varying dis-turbances and un-modeled dynamics, which can be applied to many problems. Among these,adaptive backstepping is a widely used method (Ghanavati & Chakravarthy, 2015; Kanellakopouloset al., 1991; Krstic et al., 1995). Adaptive backstepping control is a successful method to gain globalstability and asymptotic tracking in a special class of nonlinear systems with parametric strictfeedback (Kanellakopoulos et al., 1991). Some studies (Krstic et al., 1995; Qian & Lin, 2002) havediscussed the adaptive controlling method in parametric strict feedback systems to generalize thebackstepping control to a greater set of nonlinear systems. Local stability was achieved in thesetwo research, and they proposed a number of adaptive methods for nonlinear systems with atriangular structure. The backstepping method has a major disadvantage – certain functionsshould be linear in unknown parameters (Zhang, Fidan, & Ioannou, 2003), and thus long andtiresome calculations are required to determine the regression matrices.

Recently, some studies have concentrated on robust adaptive control for a special class of nonlinearsystems with uncertainties, including nonlinearly appearing parametric uncertainty, uncertain nonli-nearity, and unmeasured input-to-state stable dynamics, to develop the scope of adaptive controldesign for nonlinear uncertain systems (Jiang & Hill, 1999; Jiang & Praly, 1998; Plycarpou & Ioannou,1995). A robust adaptive nonlinear control algorithm for nonlinear systems has been proposed inPlycarpou & Ioannou (1995), including both parametric uncertainty and unknown nonlinearity. It hasbeen assumed in this study that the unknown functions address the conditions of the triangularstructure. This study can be extended to the scope of universal adaptive stabilization methods for agreater group of uncertain systems. Jiang &Hill (1999) and Jiang & Praly (1998) have presented a robustcontrol algorithm for disturbed strict feedback nonlinear systems exposed to nonlinear parametricuncertainty, uncertain nonlinearity, and un-modeled dynamics. Some robust adaptive controllingmeth-ods have been discussed in Jiang&Hill (1999), Jiang&Praly (1998), and Plycarpou& Ioannou (1995) thatcan guarantee the uniform ultimate boundedness of closed-loop system signals. Ghanavati &Chakravarthy (2017) and Yao & Tomizukai (1997) combined the backstepping adaptive technique andformal deterministic robust control to produce an adaptive robust control method. The nonlinearsystems discussed in Jiang & Hill (1999), Jiang & Praly (1998), and Plycarpou & Ioannou (1995) sharesome characteristics. For example, systemuncertainties have a linearly parameterized form and systemvirtual control coefficients are considered equal to one. An adaptive robust controller using the back-stepping method was discussed in Sheng, Huang, Zhang, & Huang (2014), in which a fuzzy system isused to model unknown system characteristics. However, the main disadvantage of all the above-mentioned controllers is that the designer needs information on system characteristics and/or the


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disturbance bounds. In practical systems, information on system characteristics, uncertainty, andunknown upper bound is sometimes incomplete and there is no detailed information about them todesign a controller. Thus, robustness of the controlled system can cope with the above-mentionedproblems.

Inspired by the generalized backstepping method, Sahab & Zarif (2009) aimed at stabilizing andtracking a special class of autonomous nonlinear control systems. However, there is no constrainton the input and state variables so the present paper tries to address this issue. Therefore, thisstudy improves the findings of Sahab & Zarif (2009) in order to design a special class of nonlinearsystems where no information on the upper bound of uncertainties and disturbances is required.The method (AGBM) proposed in this paper has two major advantages over the adaptive robustbackstepping method with regulation functions (Yao, 2003; Zhang, Lu, & Xu, 2012). On the onehand, theoretically, this method guarantees the asymptotic stability of the state variables and, onthe other hand, AGBM has a higher efficiency over other methods (it can be employed in a specialgroup of nonlinear systems with unmatched uncertainty and unknown upper bound). A numericalsimulation was conducted to demonstrate the advantages and feasibility of the presented algo-rithm. Results of the simulation show that the AGBM controller guarantees robustness and, at thesame time, can control surge in serial compressors with disturbance and uncertainty. In theconclusion section, it can be seen that the proposed method has a higher performance incomparison with the improved backstepping method employed in Sahab & Zarif (2009) andSheng et al. (2014).

Compared to the above-mentioned results, content obtained from this paper indicates thatAGBM has several advantages, such as guaranteeing asymptotic stability of the state variablesand capability of a special class of nonlinear systems with unmatched uncertainty and unknownupper bound. A numerical simulation was conducted to demonstrate the advantages and achiev-ability of the presented algorithm. Simulation results show that the AGBM controller guaranteesrobustness and, at the same time, it is able to control surge in serial compressors with disturbanceand uncertainty.

The rest of the paper is organized as follows. Section 2 presents the process of generalizing therobust backstepping method for a special group of nonlinear systems with unmatched uncertainty,along with the related assumptions. In Section 3, AGBM is employed in designing a surge controllerfor serial compressors with flow and pressure disturbance. Section 4 presents the simulationresults and further discussions. Finally, Section 5 presents our conclusions. A summary of thenotations is presented in Table 1.

1.1. Nomenclature

2. Problem formulation and proving AGBMSahab & Zarif (2009) used the extended backstepping method for a special class of systems, whichis bigger than the normal backstepping method. This class of nonlinear systems is

_X ¼ F Xð Þ þ G Xð Þη

_η ¼ f X; ηð Þ þ g X; ηð Þu (1)

where f, g, F, and G denote known functions and u 2 R is the control input.Moreover, X ¼ x1; x2; � � � ; xn½ � 2 Rn; η 2 R.

To develop a new method using the approaches employed in Sahab & Zarif (2009) and adaptivecontrol, which can be applied to a special class of nonlinear systems with unmatched uncertaintyand unknown upper bound, this paper has considered the nonlinear systems as follows:

_xi ¼ fi Xð Þ þ gi Xð Þηi þ δi t; Xð Þ; 1 � i � n


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_ηi ¼ fn þ i X; ηið Þ þ gnþi X; ηið Þui þ δnþi t;X; ηið Þ (2)

where fi;gi; i ¼ 1;2; � � � ;2n are known functions, δi; i ¼ 1;2; � � � ;2n are unknown functions, andDi; i ¼ 1;2; � � � ;2n are upper unknown bounded. ui 2 R; i ¼ 1;2; � � � ;n indicates the control inputs.Moreover, we have: X ¼ x1; x2; � � � ; xn½ � 2 Rn and ηi 2 R; i ¼ 1;2; � � � ;n. In addition, it is consideredthat fi;gi;hi; δi; i ¼ 1;2; � � � ;2n for all t;X; ηið Þ 2 0;1½ Þ � Rn � R; i ¼ 1;2; � � � ;n, in relation to t theyare piecewise continuous and in relation to X; ηi; i ¼ 1;2; � � � ;n they are locally Lipschitz. Furthermore,δi; i ¼ 1;2; � � � ;n is uniformly bounded for all t; Xð Þ, and δnþi; i ¼ 1;2; � � � ;n is uniformly bonded forall t; X; ηið Þ; i ¼ 1;2; � � � ;n.

Assuming the positive definite function V xð Þ with a derivative negative definite as follows:

V xð Þ ¼ 12∑n

i¼1xi2 þ ci~Di

2� �

(3)

Table 1. Dimensional and nondimensional variables (Uddin & Gravdahl, 2012)

Variable Dim Nondim

Pressure rise coefficient p ψ ¼ p12ρU

2

Compressor pressure rise pc ψc

Mass flow coefficient m φ ¼ mρUAc

Throttle Mass flow coefficient mT ΦT

Valve gain K

Compressor and duct length Lc

Effective flow-passage lc lc ¼ lI þ 1a þ lE

Compressor and duct length lc

Exit duct length lE

Compressor duct area Ac

Greitzer’s parameter B B ¼ U2as

ffiffiffiffiffiVpAc lc

qFluid density ρ

Speed of sound as

Mean rotor speed U

Plenum volume VP

Time t � ¼ UR t

Reciprocal time lag of compressorpassage

a

Compressor characteristic semi-width W

Compressor characteristic semi-height H

Rotor mean radius R

Subscripts

Compressor C

Throttle T

Plenum P

Inlet I

Close-coupled valve V

Exit E

Throttle control valve TCV

Inlet Guide Vanes IGV


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where ~Di ¼ Di � D̂i and D̂i are the estimation parameters of Di and ci is a regulating parameter, wehave

_V xð Þ ¼ ∑n

i¼1xi fi þ gi φi þ ψ ið Þ þ δið Þ � ciD

,iD̂i

� �(4)

By selecting

ψ i ¼�1gi

D̂i tanhxiμi

� �(5)

we have

_V xð Þ ¼ ∑n

i¼1xi fi þ giφið Þ þ ∑

n

i¼1xi δi � D̂i tanh

xiμi

� �� ciD

,iD̂�i

� �

¼ ∑n

i¼1xi fi þ giφið Þ þ ∑

n

i¼1xi δi � D̂i tanh

xiμi

� �� D

,i xi tanh

xiμi

� �� ciD̂

�i

� �(6)

To negate the derivative of Lyapunov function for stability, the adaptive law is defined as follows:

D̂�

i¼ 1

cixi tanh

xiμi

� �(7)

According to Sheng et al. (2014)

0 � xj j � x tanhxμ

� �� 0:2785μ (8)

We have

_V xð Þ � ∑n

i¼1xi fi þ giφið Þ þ 0:2785 ∑

n

i¼1μiDi

� �w xð Þ þ 0:2785 ∑n

i¼1μiDi (9)

where w xð Þ denotes a positive definite function and φi;ψ i are states feedback control law. Equation(2) can be rewritten as follows:

_xi ¼ fi þ gi φi þ ψ ið Þ þ gi ηi � φi � ψ ið Þ þ δi; 1 � i � n

_ηi ¼ fnþi þ gnþiui þ δnþi (10)

Then, if we use the following inverse variation, we obtain:

zi ¼ ηi � φi � ψ i; 1 � i � n (11)

_zi ¼ fnþi þ gnþiui þ δnþi � ∑n

j¼1

@φi

@xjþ @ψ i

@xj

� �fj þ gjηj þ δj� �

(12)

Having rewritten (10), we obtain

_xi ¼ fi þ gi φi þ ψ ið Þ þ gizi þ δi; 1 � i � n

_zi ¼ fnþi þ gnþiui þ δnþi � ∑n

j¼1

@φi

@xjþ @ψ i

@xj

� �fj þ gjηj þ δj� �

(13)

Next, the Lyapunov function mentioned in (2) is chosen as follows:


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VT X; ηð Þ ¼ 12∑n

i¼1xi

2 þ ci~Di2

� �þ 12∑n

i¼1zi2 þ cnþi

~Dnþi2

� �

¼ V xð Þ þ 12∑n

i¼1zi2 þ cnþi

~Dnþi2

� �(14)

This function is a positive definite function. Therefore, the only thing to do is to specify whether itsderivative is negative definite. We differentiated Equation (14) in relation to time to reach thefollowing function:

_VT X; ηð Þ ¼ ∑n

i¼1

@V@xi

fi þ gi φi þ ψ ið Þ þ δið Þ þ ∑n

i¼1

@V@xi

gizi þ ∑n

i¼1zi _zi � ∑

n

i¼1cnþi D

,

nþiD̂�

nþi(15)

Having (9) in mind, we can rewrite (15) in a simpler way:

_VT X; ηð Þ � �w xð Þ þ 0:2785 ∑n

i¼1μiDi þ ∑

n

i¼1

@V@xi

gizi

þ ∑n

i¼1zi fnþi þ gnþiui þ δnþi � ∑

n

j¼1

@φi

@xjþ @ψ i

@xj

� �fj þ gjηj� �

� ∑n

j¼1

@φi

@xjþ @ψ i

@xj

� �δj

!

� ∑n

i¼1cnþi

~Dnþi D̂�

nþi(16)

Then, the control inputs, ui, can be regarded as

ui ¼1

gnþi�fnþi þ ∑

n

j¼1

@φi

@xjþ @ψ i

@xj

� �fj þ gjηj� �

� xigi � ki ηi � φi � ψ ið Þ � D̂nþi tanhzi

μnþi

� �þ vi

!; ki > 0

(17)

where vi denotes the virtual control input and gnþi Xð Þ � 0. Having applied the control inputs in(17), we are able to rewrite (16) as

_VT X; ηð Þ � �w xð Þ þ 0:2785 ∑n

i¼1μiDi � ∑

n

i¼1kiz2i

þ ∑n

i¼1zi vi � ∑

n

j¼1

@φi

@xjþ @ψ i

@xj

� �δj þ ∑

n

i¼1zi δnþi � D̂nþi tanh

ziμnþi

� �� cnþi

~Dnþi D̂�nþ1

!

� �w xð Þ þ 0:2785 ∑n

i¼1μiDi � ∑

n

i¼1kizi2 þ ∑

n

i¼1zi vi � ∑

n

j¼1ð@φi@xj

þ @ψ i@xj

!δj

þ ∑n

i¼1zi δnþi � Dnþitanh

ziμnþi

� �� D

,nþi zi tanh

ziμnþi

� �� cnþiD̂

�nþi

� �� (18)

To negate the derivative of the Lyapunov function for stability, the parameter of the adaptivelaw is presented as follows:

D̂�nþ1 ¼ 1

cnþizi tanh

ziμnþi

� �(19)

Considering (8), we have

_V x; ηð Þ � �w xð Þ þ 0:2785 ∑n

i¼1μiDi þ 0:2785 ∑

n

i¼1μnþiDnþi � ∑

n

i¼1kizi2

þ ∑n

i¼1zi vi � ∑

n

j¼1

@φi

@xjþ @ψ i

@xj

� �δj

!(20)

At this moment, assuming (21), one can search for the virtual control variables, vi

∑n

i¼1zi vi � ∑

n

j¼1

@φi

@xjþ @ψ i

@xj

� �δj

!(21)


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By choosing

vi ¼ �γizihi22; γi > 0 (22)

Then we have

∑n

i¼1�γizi

22hi

22 þ zi2hi2Di

� �(23)

where Di denotes the upper unknown bound on δi. In this equation,

zi2hi2 ¼ Di

2γi(24)

Equation (23) reaches the following maximum:

∑n�1

i¼1

Di2

4γi(25)

Thus, it is obtained that

_VT X; ηð Þ ¼ �w xð Þ � ∑n

i¼1kizi2 þ ∑

n

i¼1

Di2

4γiþ 0:2785 ∑

n

i¼1μiDi þ μnþiDnþi� �� ∑

n

i¼1γi zihi � Di

2γi

� �2

(26)

Because w xð Þ belongs to class K1, it is always true that _V is negative outside some ball, andaccording to Lemma 14.1 from Khalil (2002), which addresses the principle of designing a nominalcontroller at first and then the corrective term controller (using the same nominal Lyapunovfunction) to a matched uncertainty term with desirable stability properties, the solution of theclosed-loop system is uniformly bounded for any initial state x t0ð Þ.

3. An illustrated exampleIn this section, a developed backstepping controller is designed, which requires no information ondisturbance upper bound or parametric uncertainty and throttle gain and throttle valve feature.Thus, we showed the adaptive of the controlled system to cope with the time-varying flow andpressure disturbances. Moor and Greitzer (Gravdahl & Egeland, 1997; Moor & Greitzer, 1986)provided a dynamic model for a centrifugal compressor. As it can be seen in Figure 1, this dynamicmodel (MG-Model) consists of a gas compressor, a pressure tank, a close coupled valve (CCV), athrottle valve, and the inlet and exit ducts.

The equations of the modes of three serial compressors description, shown in Figure 1, arepresented as follows:

Figure 1. The generalized com-pressor system of the MG-model (Gravdahl & Egeland,1997).


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_p1 ¼ 1

4B12lc1

m1 � φT1 p1ð Þ þ δ1� �

_m1 ¼ 1lc1

ψc1 m1ð Þ � p1 � u1 þ δ4� �

_p2 ¼ 1

4B22lc2

m2 � φT2 p2ð Þ þ δ2� �

_m2 ¼ 1lc2

ψc2 m2ð Þ � p2 � u2 þ δ5� �

_p3 ¼ 1

4B32lc3

m3 � φT3 p3ð Þ þ δ3� �

_m3 ¼ 1lc3

ψc3 m3ð Þ � p3 � u3 þ δ6� �

(27)

By choosing

x1 ¼ p1; x2 ¼ p2 ; x3 ¼ p3; η1 ¼ m1; η2 ¼ m2; η3 ¼ m3 (28)

Given Gravdahl & Egeland (1999), the following is obtained:

δ1 ¼ 0:15 e�0:015t: cos 0:2 t

δ2 ¼ 0:02 sin 0:1tþ 0:02 cos 0:4t

δ3 ¼ 0:02 sin 0:1tþ 0:02 cos 0:4t

δ4 ¼ 0:1 e�0:005t: sin 0:3 t

δ5 ¼ 0:02 sin 0:1tþ 0:02 cos 0:4t

δ6 ¼ 0:02 sin 0:1tþ 0:02 cos 0:4t

8>>>>>>>>><>>>>>>>>>:

(29)

Equation (27) can be rewritten as

_x1 ¼ 1

4B12lc1

η1 � φT1 x1ð Þ þ δ1� �

_x2 ¼ 1

4B22lc2

η2 � φT2 x2ð Þ þ δ2� �

_x3 ¼ 1

4B32lc3

η3 � φT3 x3ð Þ þ δ3� �

_η1 ¼ 1lc1

ψc1 η1ð Þ � x1 � u1 þ δ4� �

_η2 ¼ 1lc2

ψc2 η2ð Þ � x2 � u2 þ δ5� �

_η3 ¼ 1lc3

ψc3 η3ð Þ � x3 � u3 þ δ6� �

(30)

Then, by choosing

z1 ¼ η1 � φ1 � ψ1

z2 ¼ η2 � φ2 � ψ2

z3 ¼ η3 � φ3 � ψ3 (31)

Given Equation (12), the overall Lyapunov function is chosen as

V x; zð Þ ¼ 2B12lc1x1

2 þ 2B22lc2x2

2 þ 2B32lc3x3

2 þ lc12z21 þ lc2

2z22 þ lc3

2z23 þ c1

2~D1

2 þ c22~D2

2

þ c32~D3

2 þ c42~D4

2 þ c52~D5

2 þ c62~D6

2

(32)


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Thus, it can be observed that this equation is a positive definite function. To stabilize the overallsystem, the only thing to do is to have a negative Lyapunov equation δð Þ derivative.

_V x; zð Þ ¼ x1 z1 þ φ1 þ ψ1 � φT1 x1ð Þ þ δ1� �þ x2 z2 þ φ2 þ ψ2 � φT2 x2ð Þ þ δ2

� �þ x3 z3 þ φ3 þ ψ3 � φT3 x3ð Þ þ δ3

� �þ z1 ψc1 η1ð Þ � x1 � u1 þ δ4 � lc1 _φ1 þ _ψ1ð Þ� �þ z2 ψc2 η2ð Þ � x2 � u2 þ δ5 � lc2 _φ2 þ _ψ2ð Þ� �þ z3 ψc3 η3ð Þ � x3 � u3 þ δ6 � lc3 _φ3 þ _ψ3ð Þ� �� c1~D1D̂

�1 � c2~D2D̂

�2 � c3~D3D̂

�3 � c4~D4D̂

�4 � c5~D5D̂

�5 � c6~D6D̂

�6

(33)

The virtual control vectors φi; ψ i and i = 1, 2, 3 are chosen as

ψ1 ¼ �D̂1 tanhx1μ1

� �; ψ2 ¼ �D̂2 tanh

x2μ2

� �; ψ3 ¼ �D̂3 tanh

x3μ3

� �; μ1; μ2; μ3 > 0 (34)

D̂�

1¼ 1

c1x1 tanh

x1μ1

� �; D̂�

2¼ 1

c2x2 tanh

x2μ2

� �; D̂�

3¼ 1

c3x3 tanh

x3μ3

� �; μ1; μ2; μ3 > 0 (35)

φ1 ¼ φ2 ¼ φ3 ¼ 0 (36)

We obtain:

�V�x; zð Þ ¼ �x1φT1 x1ð Þ � x2φT2 x2ð Þ � x3φT3 x3ð Þ þ 0:2785μ1D1 þ 0:2785μ2D2 þ 0:2785μ3D3

þ z1 ψc1 η1ð Þ � u1 � lC1 :1

4B12lC1

@ψ1

@x1η1 � φT1 x1ð Þ þ δ1� �þ δ4

!

þ z2 ψc2 η2ð Þ � u2 � lC2 :1

4B22lC2

@ψ2

@x2η2 � φT2 x2ð Þ þ δ2� �þ δ5

!

þ z3 ψc3 η3ð Þ � u3 � lC3 :1

4B32lC3

@ψ1

@x3η3 � φT3 x3ð Þ þ δ3� �þ δ6

!� c4~D4 4D̂

�4�c5~D5D̂

�5 � c6~D6D̂

�6

(37)

By choosing

u1 ¼ ψc1 η1ð Þ � 1

4B12

@ψ1

@x1η1 � φT1 x1ð Þ� �þ k1z1 þ D̂4 tanh

z1μ4

� �� v1; k1; μ4 > 0 (38)

D̂�

4¼ 1

C4z1 tanh

Z1μ4

� �; c4 > 0 (39)

u2 ¼ ψc2 η2ð Þ � 1

4B22

@ψ2


z2μ5

� �� v2; k2; μ5 > 0 (40)

D̂�

5¼ 1

C5z2 tanh

Z2μ5

� �; c5 > 0 (41)

u3 ¼ ψc3 η3ð Þ � 1

4B32

@ψ3


z3μ6

� �� v3; k3; μ6 > 0 (42)

D̂�

6¼ 1

C6z3 tanh

Z3μ6

� �; c6 > 0 (43)

We have


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�V�x; ηð Þ ¼ �x1φT1 x1ð Þ � x2φT2 x2ð Þ � x3φT3 x3ð Þ � k1z12 � k2z22 � k3z32 þ 0:2785μ1D1

þ 0:2785μ2D2 þ 0:2785μ3D3 þ 0:2785μ4D4 þ 0:2785μ5D5 þ 0:2785μ6D6

þ z1 v1 � 1

4B12

@ψ1

@x1:δ1

� �þ z2 v2 � 1

4B22

@ψ2

@x2:δ2

� �þ z3 v3 � 1

4B32

@ψ3

@x3:δ3

� � (44)

By selecting

v1 ¼ �γ1z1 � h12; h1 ¼ � 1

4B12

@ψ1

@x1; γ1>0 (45)

v2 ¼ �γ2z2 � h22; h2 ¼ � 1

4B22

@ψ2

@x2; γ2>0 (46)

v3 ¼ �γ3z3 � h32; h3 ¼ � 1

4B32

@ψ3

@x3; γ3>0 (47)

Finally, we have

_V X; ηð Þ ¼ �x1φT1 x1ð Þ � x2φT2 x2ð Þ � x3φT3 x3ð Þ � k1z12 � k2z22 � k3z32 þ 0:2785μ1D1

þ 0:2785μ2D2 þ 0:2785μ3D3 þ 0:2785μ4D4 þ 0:2785μ5D5 þ 0:2785μ6D6 � γ1 z1h1 � D1

2γ1

� �2

� γ2 z2h2 � D2

2γ2

� �2

� γ3 z3h3 � D3

2γ3

� �2

þ D12

4γ1þ D2

2

4γ2þ D3

2

4γ3(48)

Given by Eq. (2.45) from Gravdahl & Egeland (1999), it is supposed that the throttle is passive, i.e.

ψ̂ Φ̂T ψ̂ð Þ � 0"ψ̂ . We can obtain

ψ̂ Φ̂T ψ̂ð Þ � 0 )�xiφTi xið Þ � 0; i ¼ 1;2;3 (49)

And given in Lemma 2.26 from Krstic et al. (1995), which expresses the closed-loop system ISSstability with respect to disturbance input and global uniform boundedness condition, it can beconcluded that _V is negative whenever w xð Þ > Δk k21

4k .

4. Simulation results and discussionsIn this paper, we applied the above-mentioned AGBM controller in three serial compressors, whichare shown in Figure 2. It was conducted using MATLAB Software. The MG mfodel was employed tomodel the compressors and they all were equipped with a CCV (Gravdahl & Egeland, 1997).

Compressors have similar parameters, and based on Greitzer (1976), they are assumed to be

B1 ¼ 1:8; lc1 ¼ 3; H1 ¼ 0:18; W1 ¼ 0:25; Ψc01 ¼ 0:3B2 ¼ 1:8; lc2 ¼ 3; H2 ¼ 0:18; W2 ¼ 0:25; Ψc02 ¼ 0:3B3 ¼ 1:8; lc3 ¼ 3; H3 ¼ 0:18; W3 ¼ 0:25; Ψc03 ¼ 0:3

8<: (50)

Based on Gravdahl & Egeland (1999), flow disturbance can be a result of the compressorupstream processes or their serial pattern. Therefore, in the simulations presented in thissection, the impact resulting from the serial pattern of compressors is denoted as flow

Figure 2. Arrangement of threeserial compressors (Menon,2005).


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disturbance in the equations. A comparison was made to make our investigation moreaccurate on the above-mentioned controller and to reach a deeper understanding of thecontroller performance in different states. This comparison is between the output pressurediagram and the compressor output flow diagram before and after the use of controllingforce. Despite the fact that there is no information on the parametric uncertainty upperbound in this research, it can be seen that the designed controller performance with AGBMis robust and more effective in all investigated states compared with the extended back-stepping method presented in Sahab & Zarif (2009). The approach employed by them, unlikeour model, needs information about nonlinear uncertain systems under matching conditionsand it can be easily seen that this model is not robust to bounded uncertainties.

There are several applications (Ghanavati, Salahshoor, Jahed Motlagh, Ramezani, &Moarefianpour, 2017; Sheng et al., 2014; Yao, Deng, & Jiao, 2015; Yao, Jiao, & Ma, 2015) thatcan be used for the controller to demonstrate the effectiveness of our method. In this section, weconsider the system in Sheng et al. (2014), and our goal is to compare the method presented inthis paper with the existing backstepping controllers. In the first step, the system is simulated withno control input. Then, in compressor 1, at t = 100 s, in compressor 2, at t = 120 s, and incompressor 3, at t = 150 s, the throttle valve gain coefficient, γT, was reduced from 0.65 to 0.6.The flow and pressure outputs for compressors 1–3 are shown in Figure 3.

The results of simulation on the uncontrolled system are presented in Figure 3. It can beobserved in the figures that when the throttle gain is reduced from 0.65 to 0.6, the compressionsystem becomes unstable. The simulation results clearly show the transformation of the compres-sor’s operating range and its entrance into the surge instability area, which creates a limit cycle.Figures 4–6 show the control signals. As it can be seen, this signal is positive and follows theactuator restriction (valve capacity requires the constraint 0 ≤ u(t) ≤ 1 to hold), whereas the controlsignal derived from Sheng et al. (2014) is negative. Figures 7–9 show the status of output flowstudied by applying the control signal through the AGBM controller. The pressure outputs forcompressors 1–3 using the AGBM controller are shown in Figures 10–12, respectively.

As it can be seen in Figures 7–12, the flow and pressure for compressors 1, 2, and 3 were stabilizedby using the designed AGBM controller and the system was controlled in the presence of disturbancewhile the equilibrium point is located in the unstable area. The AGBM controller, as derived in Section2, can damp the unstable modes as the operation of the compression system is pushed into the surgearea by restricting the flow through the throttle valve. Given that the control signal is applied to theCCV actuator, it cannot take a negative value and this is an advantage of this research and the

0 50 100 150 200 250 300 350 400 450 500-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (Sec)

1

1

ψ

φ

ψ ,

φ

Figure 3. The output flow andpressure for compressors 1, 2,and 3.


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designed controller, whereas the control signal from Sheng et al. (2014) is negative and physicallyimpractical. In addition, the controller presented in Sahab & Zarif (2009) cannot control any of thesecompressors by applying the control signal using the method of Sahab & Zarif (2009) and all three

Figure 4. Control signal forcompressor 1.




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Figure 8. Output flow for com-pressor 2.




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Figure 10. Output pressure forcompressor 1.




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compressors studied in this research enter the surge area and become unstable. Because Sahab’swork is for systems without disturbance, disturbed nonlinear systems are considered in this article. Itis clear that Sahab’s method does not respond to this class of systems.

5. ConclusionThe simulation results indicate that AGBM can present the outputs with a high precision. It can be easilyseen in the simulation figuresof this paper that theAGBMcontroller can stabilize three serial compressors(with flow and pressure disturbances) with a high precision. The AGBM controller can retain its highperformance inparametricuncertaintyandunmatcheddisturbanceconditions. Becauseof theminimumlevel of error and better tracking in the presented method, when there is unmatched uncertainty andunknown upper bound disturbance, AGBM is more appropriate for a special class of nonlinear systemsthan the extended backsteppingmethod. Additionally, unlike the standard backstepping method, therewasno instability in the results obtainedby simulation. Thus, it canbe said that theAGBMproposed in thisstudy ismore robust to bounded uncertainties, which can be considered as another advantage of AGBM.Finally, it should be noted that AGBM provides accurate outputs for nonlinear systems with a strictlyfeedback form, no matter if there is a bounded uncertainty or disturbance.

AcknowledgmentsThe authors would like to thank the referees for theirconstructive comments.

FundingThis research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

Author detailsMalek Ghanavati1

E-mail: [email protected] ID: http://orcid.org/0000-0001-9794-6377Karim Salahshoor2

E-mail: [email protected] ID: http://orcid.org/0000-0002-9751-9540Mohammad Reza Jahed-Motlagh3

E-mail: [email protected] ID: http://orcid.org/0000-0001-8662-1155Amin Ramezani4

E-mail: [email protected] ID: http://orcid.org/0000-0001-7901-7945Ali Moarefianpur1

E-mail: [email protected] ID: http://orcid.org/0000-0003-4778-89421 Department of Electrical Engineering, Science andResearch Branch, Islamic Azad University, Tehran, Iran.

2 Department of Automation and Instrumentation,Petroleum University of Technology (PUT), Tehran, Iran.

3 Computer Engineering School, Iran University of Scienceand Technology, Tehran, Iran.

4 Electrical and Computer Engineering Department,Tarbiat Modares University, Tehran, Iran.

Competing interestsThe authors declare that there is no conflict of interest.

Citation informationCite this article as: A novel combined control based on anadaptive generalized back-stepping method for a class ofnonlinear systems, Malek Ghanavati, Karim Salahshoor,Mohammad Reza Jahed-Motlagh, Amin Ramezani & AliMoarefianpur, Cogent Engineering (2018), 5: 1471787.

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