Arab J Sci EngDOI 10.1007/s13369-016-2144-0

RESEARCH ARTICLE - ELECTRICAL ENGINEERING

Output Feedback Control of a Class of Under-Actuated NonlinearSystems Using Extended High Gain Observer

Nasir Khalid1 · Attaullah Y. Memon1

Received: 19 May 2015 / Accepted: 14 April 2016© King Fahd University of Petroleum & Minerals 2016

Abstract The problem of output feedback stabilization of aclass of under-actuated benchmark nonlinear systems is dis-cussed. The proposed method utilizes an extended high gainobserver (EHGO)-based sliding mode control (SMC) tech-nique to control a class of nonlinear systems which may haveunstable zero dynamics. Starting with Lagrangian model ofthe system and using a suitable coordinate transformation,a generalized normal form representation is derived whichdecouples the system into an internal and external dynam-ics. The internal dynamics is utilized to derive an auxiliarysystem and the full-order EHGO thus obtained is used forestimation of derivative(s) of the system output that are fur-ther used in design of an output feedback control law. It isshown that the proposed output feedback controller stabilizesthe system and convergence of estimated states is demon-strated with suitable selection of observer parameters. Theproposed control scheme is applied to a benchmark nonlin-ear system, namely inertia wheel pendulum (IWP), in orderto demonstrate the efficacy of the technique by simulation.

Keywords Under-actuated benchmark systems · ControlLyapunov function · Sliding mode control · Extended highgain observer · Inertia wheel pendulum

1 Introduction

Under-actuated systems are electromechanical systems thatare characterized by lesser number of controls or actuators

B Nasir Khalidnasirkhalid@pnec.nust.edu.pk

1 Department of Electronics and Power Engineering, PNEngineering College, National University of Sciences andTechnology, Karachi, Pakistan

than number of degrees of freedom. Due to lesser num-ber of controls, some of configuration variables are unac-tuated and can only be driven by indirect coupling move-ment of actuated variables. These systems thus exhibit non-minimum phase behavior as some part of the system dynam-ics is unstable.Control of these systems is a challenging prob-lem, especially under output feedback (OFB) control, whencompared to fully actuated nonlinear systems. Examples ofunder-actuated systems are found in real-life problems suchas in robotics, automobiles, aircrafts, satellites, underwatervehicles,missiles etc, and in benchmark control systems suchas Cart Pole system, Translational Oscillator with RotatingActuator (TORA), Acrobot, Inertia Wheel Pendulum (IWP),Rotary Pendulum (ROTPEN) etc.

Control of such under-actuated systems has been an activefield of research for the last few decades. Work reported inthis area, however, mostly employ state feedback (SFB), andthis problem becomes more complex under output feedbackwhen only the system’s output is available for feedback. Theoutput feedback control problem of under-actuated systemsis therefore, an open problem [18]. Prominent work reportedfor control of under-actuated mechanical systems under statefeedback control includes [2,9,13–15,17], while for outputfeedback control the notable work done is reported in [3,4,12,18].

An extensive and comprehensive study on under-actuatednonlinear systems is reported in [13],where these systems areclassified into eight different classes based on kinetic symme-try properties of the system. These properties are utilized toreduce the complexity of the original system dynamics anda cascade nonlinear/linear mathematical form (called nor-mal form representation) is obtained using suitable changeof coordinates and control. This normal form is character-ized by a number of first order ODEs which can be con-veniently used to analyze controllability and observability

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properties of the system, as it bifurcates system dynamicsinto unstable internal dynamics and outer or external dynam-ics. According to the classification given in [13], a numberof benchmark under-actuated systems including the InertiaWheel Pendulum (IWP), Translational Oscillator with Rotat-ing Actuator (TORA) and Acrobot falls together in a similarsubclass, due to identical normal form representation called“strict-feedback form”. In [17], Spong et al. have proposeda passivity-based nonlinear controller to stabilize the IWPsystem. In [14], the formulation of [17] is extended to syn-thesize a controller structure that does not require velocitymeasurement and is non switching in nature. Here, prob-lem of IWP is addressed using interconnection and dampingassignment- a new formulation of passivity-based control-providing swing up and balance of pendulum. Lately, exten-sion inwork of [13] for development of stabilizing algorithmsof a class of under-actuated systems in strict-feedback formis presented in [15] where it is shown that disadvantagesrelatedwith previousmethods like energy shaping and damp-ing injection and back-stepping can be avoided by use ofMultiple Sliding Surfaces (MSS) Control and Dynamic Sur-face Control (DSC) techniques. These schemes are applied tounder-actuated benchmark systems of this subclass includingIWP, TORA and Acrobot.

While the above-mentioned work effectively addressesissues using dynamic state feedback, researchers investigatedstabilization and control of nonlinear under-actuated systemsunder observer-based output feedback. The importance ofoutput feedback increases when systems are not equippedwith velocity sensors due to economic reasons and also dueto the fact that it exhibits the ability of eliminating mea-surement noise in velocity signals. An important design tool,used in observer based OFB control, is known as High GainObserver (HGO) and has been successfully utilized for stateestimation and disturbance rejection [7,8]. HGO-based out-put feedback controller offers a very good solution for min-imum phase systems in terms of robustness and design sim-plicity. This technique has received widespread attention,especially after establishment of concept of separation prin-ciple as in [1,8], and presentation of a simpler but effectivesolution to peaking phenomenon [5]. The HGO-based con-troller design is, however, mostly done for minimum phasesystems [10] having stable zero dynamics while for under-actuated systems possibly having unstable zero dynamics,extended high gain observers (EHGO) have been success-fully implemented and have shown promising results in stateestimation and disturbance rejection. Nazrulla and Khalil in[12] utilize SMC and EHGO for stabilization of a nonmin-imum phase system, while Boker and Khalil in [3] estab-lish a full-order observer comprising of an EHGO and anextendedKalman filter (EKF) for the estimation of the outputand internal system states respectively. Both [3,12] demon-strated their results on a test-bench under-actuated Type-I

system namely translating oscillator with a rotating actuator(TORA). In [4], application of EHGO for disturbance rejec-tion is demonstrated on a bench mark inverted pendulumsystem.

Thework presented in this paper is an extension to our pre-vious work [6] on output feedback stabilization. It broadlyutilizes the earlier contributions [13,15] for the developmentof system mathematical model under state feedback designand [3,12] for the development of an EHGO to estimate vir-tual output to an auxiliary system under output feedback.While both this paper and [13,16] implement SFB designscheme, the controller design approach in our work is dif-ferent. It is based on SMC scheme that is derived usinga systematic, step by step generalized approach utilizingerror dynamics of external system and generating appro-priate output to stabilize internal system. Another differ-ence in this work, lies in generalized model transformationapplicable to class of under-actuated systems that utilizescollocated partial feedback linearization and a global coor-dinate transformation [13] to systematically generate normalform suitable for OFB control design. The resulting normalform representation bifurcates entire system into an inter-nal and external dynamics for utilization in EHGO-basedcontroller design. In the simulation section, it is shown thatthe closed loop system under the proposed OFB scheme,approach those of the state feedback control with the choiceof suitable EHGO gains. To reduce peaking effect due toinitial overshoot in the states some constraints are appliedon the system input which does not degrade controllerperformance.

The rest of the paper is organized as follows. Section 2presents the generalized dynamical model of the class ofunder-actuated systems, and formulates the problem. SMC-based controller design under output feedback is given inSect. 3 and corresponding stability analysis for generalizedsystem is provided in Sect. 4. Design is applied on an IWPand corresponding simulation results are discussed in Sect. 5.Finally, Sect. 6 draws the conclusions and points toward pos-sible avenues for future work.

2 General Mathematical Formulation

In this section, we formulate a generalized dynamical form ofa typical under-actuated nonlinear system. Startingwith clas-sical equations of motion for a nonlinear system, we identifykey factors to distinguish an under-actuated system with thatof a fully actuated system.UsingEuler-Lagrangian equationsof motion for this class of systems, firstly, the generalizedstate space representation of system is identified, and then,using suitable change of control/coordinates a typical normalform representation is derived.

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2.1 Nonlinear System Representation

A nonlinear system affine in control u ∈ �m , state vectorx ∈ �n and output y ∈ �n can be represented as

x = f (x) + g(x)u, x(0) = x0 (1)

y = h(x) (2)

where f : �n × �m → �n, g : �n × �m → �n andh : �n → � are nonlinear continuous smooth functions.x0 denotes initial states vector at t = 0 and x is defined asderivative of states x w.r.t. time (t). The differential Eqs. (1),(2) represent a class of nonlinear physical systems havingn degrees of freedom and requires n-dimensional position(configuration) vector θ � [θ1, θ2, . . . , θn] ∈ Q for localrepresentation.

We start with classical equation of motion for this type ofsystems, as given by

d

dt

[∂k

∂θi

]−

[∂k

∂θi

]= Ti , i = 1, 2, . . . , n (3)

where θ � [θ1, θ2, . . . , θn] are the generalized velocitiesand k is total kinetic energy of the system. The term Ti issum of forces acting on the system denoted by matrix T ext

iand forces obtained from potential energy function V of thesystem. This can be written as

Ti = −∂V

∂θi(θ) + T ext

i , i = 1, 2, . . . , n (4)

Lagrangian (L0) of this system can be represented as thedifference of energies of the system as follows

L0 = Kinetic Energy − Potential Energy (5)

or

L0 = k(θ , θ) − V (θ) (6)

Using (3), (4) and (6), the Euler Lagrange equations ofmotion for this system can be derived as

d

dt

[∂L0

∂θi

]−

[∂L0

∂θi

]= T ext

i , i = 1, 2, . . . , n (7)

and the total kinetic energy is given as

k(θ , θ) = 1

2θT M(θ)θ (8)

where M is inertia matrix of the system. Using (7), and con-sidering T ext

i as control input u, a simplified equation inmatrix form for a mechanical system is given as

M(θ)θ + C(θ, θ )θ + G(θ) = F(θ)u (9)

whereC(θ, θ ) is amatrix containing centrifugal and coriolis-effect terms, F(θ) is input or control matrix and G(θ) is thegravitymatrix. Using θ = p, Eq. (9) can be further simplifiedinto standard state space representation as

d

dt

[θ

p

]=

[p

M−1(θ)[C(θ, θ ) + G(θ)]]

+[

0M−1(θ)F

]u

(10)

Equation (10) resembles the compact mathematical form asgiven in (1).

2.2 Under-Actuated System Dynamics

An electromechanical nonlinear system with configurationvector θ ∈ Q and having dynamical model (9) is calledfully actuated or under-actuated system depending on rankof F(θ). If rank of F(θ) = n, where n is dim (Q), then sys-tem is called fully actuated system and if rank of F(θ) < nthen, it is called under-actuated system. Unlike fully actuatedsystems, under-actuated systems do not allow exact feed-back linearization, due to lesser number of actuators thandimensions of the configurationmanifold. In such cases, onlypartial feedback linearization is possible. For this purpose,the configuration space θ ∈ Q can be bifurcated into an outerlinear systemand an inner nonlinear system.The equations ofmotion for an under-actuated nonlinear system with controlinput v can be expressed as

M(θ)θ + C(θ, θ )θ + G(θ) = F(θ)v (11)

where v ∈ �m is the control, F(θ) ∈ �n×m is an exter-nal non-square control matrix with m < n and having fullcolumn rank taking the form as F(θ) = [0, Im]T . The n-dimensional configuration vector space can now be dividedup as θ = (θx , θs) ∈ �n−m × �m where θx , θs representunforced and forced components of configuration vectorspace, respectively. This presence and absence of actuationin some part of configuration manifold determines the shapeof the system such that the inertia matrix of these systemsis independent of a subset of configuration variables. Thissubset of variables is termed as external variables and com-pliment of these are called as shape variables. In other words,shapevariables are thosevariables that appear in inertiamatrixM(θ) of a simple under-actuated system. This further impliesthat for external variables where ∂M(θ)/∂(θx ) = 0, the fol-lowing identity holds

∂k(θ, θ )

∂(θx )= 0 (12)

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which shows kinetic symmetry w.r.t external variables. ThusLagrangian (6) for such systems can now be represented as

L0 = 1

2θT M(θs)θ − V (θ) (13)

and corresponding equation of motion can be decomposedas:

d

dt

[∂L0

∂θx

]−

[∂L0

∂θ

]= Fx (θ)v, v ∈ �m

d

dt

[∂L0

∂θs

]−

[∂L0

∂θ

]= Fs(θ)v, v ∈ �m

⎫⎪⎪⎬⎪⎪⎭

(14)

The Input matrix F in under-actuated systems (14) is repre-sented by F = (Fx , Fs) and the rank of F is always less thann. To analyze these type of systems, a number of cases can bedefined depending on whether the shape configuration vari-ables are fully actuated, partially actuated or unactuated, andthe presence or absence of input coupling due to the inputmatrix F . Reza in [13], using certain structural properties ofnonlinear under-actuated systems, classified these systemsinto eight different classes, described normal form repre-sentations and pointed toward the various controller designprocedures.

We consider the class of under-actuated nonlinear sys-tems, termed as Type-I systems in [13], represented in gen-eralized form as

[mxx (θs) mxs(θs)

msx (θs) mss(θs)

] [θxθs

]+

[hx (θ, θ )

hs(θ, θ )

]=

[0v

](15)

where mi j ’s are block matrices obtained after partitioning ofinertia matrix M(θ) and hx , hs are position/ velocity com-ponents of unforced and forced system dynamics. A partic-ular feature of this class of systems is that it shows lack ofcontrol in its first part and hence exact feedback lineariza-tion is not possible using any suitable change of control.However, as shown in [15] and the references therein, it ispossible to partially linearize the system, so that dynam-ics of the actuated part is transformed into a linear sys-tem using a method called Collocated Partial Feedback Lin-earization. The generalized mathematical representation ofthis class of systems is termed as strict-feedback normalform and is obtained after a suitable change of controlas

v = α(θx )τ + β(θ, θ ) (16)

where the m × m positive definite and symmetric matrixα(θx ), and β(θ, θ) are defined as

Linear System Nonlinear Systemτ ξ1 y

Fig. 1 Cascade interconnection of Type-I under-actuated system

α(θx ) = mss(θx ) − msx (θx )m−1xx (θx )mxs(θx )

β(θ, θ ) = hs(θ, θ ) − msx (θ)m−1xx hx (θ, θ )

Using a global change of variables or diffeomorphism as

η1 = θx + γ (θs)

η2 = mxx (θs)θx + mxs(θs)θs

}(17)

The resulting generalized form for this class of (Type-I) sys-tems is given as

η1 = m−1xx (ξ1)η2 + γ (ξ1)

η2 = κ(η1, ξ1)

}(18)

ξ1 = ξ2

ξ2 = τ

}(19)

where

γ (ξ1) =∫ ξ1

0

mxs(ξ1)

mxxds, κ(η1, ξ1) = −∂Vr (η1, ξ1)

∂η1

and Vr (η1, ξ1) = V (η1 − γ (ξ1), ξ1)

The dynamical model of such under-actuated system can berepresented as a cascaded interconnection of linear and non-linear subsystems (18), (19) as shown in Fig. 1.

The systems (18) and (19) can be further represented incompact notation with original control v as

η = φ0(η, ξ)

ξ = Aξ + B[β(η, ξ) + α(ξ, v)]y = ξ1

⎫⎪⎬⎪⎭ (20)

With η = [η1 η2]T , φ0 = [m−1xx (ξ1)η2 + γ (ξ1) κ(η1, ξ1)]T ,

and the m × m matrix A,m × 1 input matrix B representchain of m integrators. ξ1 is taken as the system output y.

3 Output Feedback Control Design

In this section, we proceed with the design of an SMC-basedOFB control law for the system (20). The required states areestimated using full-order EHGO. Toward that end, we fol-low the approach of [3] and begin by considering an auxiliary

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system consisting of internal system η and a virtual output σas follows

η = φ0(η, ξ) (21)

σ = β(η, ξ) (22)

With above in view, the full-order EHGO can be written as

˙ξ = Aξ + B[σ + α(ξ , v)] + H1[y − ξ1] (23)˙σ = β(η, ξ ) + H2[y − ξ1] (24)˙η = φ0(η, ξ ) + K [σ − β(η, ξ )] (25)

y = ξ1 (26)

where H1 = [α1/ε α2/ε2 . . . αm/εm]T , H2 = [αm+1/ε

m+1]and K are the observer gains chosen such that the observerdynamics remain faster than those of the overall closed loopsystemunderSFB.Thegainparametersα1, α2, . . . , αm, αm+1

are chosen so that roots of the polynomial pm+1 + α1 pm +· · · + αm p + αm+1 are in the open left-half plane, and theparameter ε is a small positive constant.

For the control law synthesis, we first consider the internalsub-system (18) which is unforced and unperturbed complexnonlinear system and can be stabilized using control Lya-punov function (CLF) method as in [13]. Treating ξ1 ≡ ω asvirtual control input to the system, there exist a state feedbacklaw as

ω(η1, η2) = αr (η1) − a�(c1η1 + c2η2) (27)

It can be seen that (27) can be effectively utilized for stabi-lization of the internal system (21). Here, αr (η1) is a smoothfunction such thatκ(η1, αr (η1)) = 0, αr (0) = 0 and�(c1η1+c2η2) in (27) is any scalar sigmoidal function. The externalsubsystem control law is designed in such a way that itsoutput, acting as virtual input for internal subsystem, gener-ates the desired stabilization function. An equivalent slidingmode control scheme is utilized for generation of requisitestabilization function. The generalized system (20) and inter-nal system feedback control law (27) satisfies the necessaryassumptions needed to design sliding mode controller forexternal system.

we startwith error dynamics of external configuration vec-tor as

e1 = ξ1 − ξ1d (28)

e2 = ξ2 − ξ2d (29)

where e1 is the error in the generation of stabilization functionξ1d = ω(η1, η2) by ξ1 and e2 is the error in the generationof ξ2d and ξ2. The time-derivatives of the error dynamics

equations are

˙e1 � ˙ξ1 − ˙

ξ1d = f1(η, ξ , ξ1d ,˙ξ1d) (30)

˙e2 � ˙ξ2 − ˙

ξ2d = f2(ξ2d ,˙ξ2d , τeq) (31)

where τeq is equivalent control law obtained from sliding sur-face s and is used to obtain final control law. The derivatives˙ξ1d ,

˙ξ2d can be easily calculated using numerical differen-

tiation with sufficient accuracy. This will, however, requirecontroller to be implemented in discrete form. To avoid thiscomplexity a low-pass filter with small positive time con-stant can be used. Function f1 is calculated such that theerror dynamics approach to zero and a time varying slidingsurface s is given as

s = λe1 + e2 (32)

The time derivative of sliding surface is used to calculate anequivalent control law τeq. A slidingmode control law is thenevaluated in terms of system states and error dynamics suchthat the time derivative of sliding surface tends to zero thatrenders the system asymptotically stable and is given by

˙s = f3(η, ξ ,

˙ξ1,

˙ξ2, τeq

)(33)

τ = τeq − k(sign

(s))

(34)

where k > 0 is a constant. The control law τ obtained in suchway shows high frequency oscillations known as chatteringin the sliding phase which are very much undesirable. Thechattering effect can beminimized by replacing sign functionwith a suitable saturation function in (34) and the resultingcontrol law then takes the form

τ = τeq − ksat

(s

μ

)

v = ατ + β

⎫⎪⎬⎪⎭ (35)

where μ is a small positive constant defining boundary layerfor sliding surface.

4 Stability Analysis

Stability analysis is performed in two steps. First, under statefeedback and then under output feedback that uses a full-order EHGO. In the case of OFB design, observer dynamicsbecome significant and need to be analyzed to ensure closedloop system stability. In first step, we analyze internal non-linear system and cascaded system stability, using suitableanalytical tools under the stated assumptions. We use Lya-punov method to analyze the subsystem represented by (25)with virtual control law (27) and follow the steps as in [13].

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For the state feedback stability analysis, we consider the sys-tem (23)–(26) and suppose (η, ξ ) = (η, ξ). As ξ act as virtualcontrol input for (21), we choose a positive definite functionV◦(η) = 1

2 η2 as Lyapunov function candidate and examine

V◦(η) = (η)[φ0(η, ξ )] for stability analysis of the subsystem.Assuming ξ = ω, the virtual control input; using a sigmoidalfunction as �(η1, η2) and a smooth function αr (η1) in (27),it can be easily verified that

V◦(0) = 0; V◦(η) > 0; V◦(0) = 0; and V◦(η) ≤ 0

Using the above conditions and with the help of LaSalle’sinvariance principle, it can be shown that the origin of thenonlinear subsystem is globally asymptotically stable.The composite system (18), (19) under control law (35) canbe represented in terms of internal states and sliding surfaceas

˙η1 = m−1xx (ω(η1, η2) + s)η2 + γ (ω(η1, η2) + s)

˙η2 = κ(η1, ω(η1, η2) + s)

}(36)

s = λe1 + e2

˙s = f3(η, ξ ,˙ξ1,

˙ξ2, τeq)

}(37)

The stability of the composite system (36), (37) can be con-sidered to conclude stability of overall closed loop system.To proceed in this manner, we first consider few assumptionsregarding the class of under-actuated systems under consid-eration.

The inner, nonlinear subsystem (36) with zero input fromouter system (37) is globally Lipschitz. This assumptionmayappear as stringent, but it holds for all the benchmark systemsbelonging to the class of under-actuated systems (see for ref.[15]) and further meets the assumption for global Lipschitzproperty as system partial derivatives up to second order arebounded [16]. A suitable sigmoid function, when chosen asinput to internal system, will not alter said property of system(36).A zero input fromouter system, further renders the innersystem as globally asymptotically stable [13].

The stability of external, driving system (37) canbe inferredby considering a quadratic Lyapunov function candidate interms of s as follows

V = 1

2s2

Taking time derivative of V and substituting the values of uand ueq, we can obtain V as

V = s ˙s = −sksat

(s

μ

)

Outside the boundary layer, i.e., When ‖s‖ ≥ μ

V = −ks

(s

‖s‖)

= −k‖s‖ < 0

Inside the boundary layer, i.e., when ‖s‖ ≤ μ, we have

V = −sk

(s

μ

)= −

(k

μ

)‖s‖2 < 0

which shows that by choosing k > 0, the trajectories of theclosed loop system starting with arbitrary initial conditionswill reach the boundary layer s = μ in finite time and remaininside it thereafter.

Next, we consider the system under output feedback.Toward this end, we closely follow the procedure in [10],[7, theorem14.6] to show trajectories convergence andbound-edness of system states. For the observer based OFB system,scaled estimation error dynamics is formulated and trajectoryconvergence /boundedness of states is analyzed by appropri-ate choice of Lyapunov functions.

We define the the scaled estimation error χ (χη, χξ,σ ) forstates (η, ξ , σ ) as

χη = η − η (38)

χξi = (ξi − ξi )/εm+1−i , 1 ≤ i ≤ m (39)

χσ = β(η, ξ) − σ (40)

Let D(ε) = diag[εm, εm−1, . . . , ε], χξ,σ = [χTξ χσ ] and

D1(ε) = diag[D, 1], then (38)–(40) can be written as

D(ε)χξ = ξ − ξ (41)

D1(ε)χξ,σ =[(ξ − ξ )T β(η, ξ) − σ

]T(42)

Consider the EHGO-based OFB system (23)–(26). The esti-mation error dynamics for this system can be written as

˙χη = φ0(χη + η, Dχξ + ξ ) − φ0(η, ξ )

−H [(β(χη + η, Dχξ + ξ ) − χσ ) − β(η, ξ )] (43)

� gr (η, ξ , χη, D1χξ,σ , t) (44)

ε ˙χξ,σ = Λχξ,σ + ε[B1�β + B2�α

](45)

where �β = β(χη + η, ξ + Dχξ , v) − β(η, ξ , v),

�α = �α/ε,�α = α(ξ + Dχξ , v) − α(ξ , v) and

Λ =

⎡⎢⎢⎢⎢⎢⎣

−h1 1 0 · · · 0−h2 0 1 · · · 0

......

.... . . 0

−hm 0 0 · · · 1−hm+1 0 0 · · · 0

⎤⎥⎥⎥⎥⎥⎦

, B1 =[0B

], B2 =

[B0

]

It is evident from Eqs. (43)–(45) that system exhibit two-time scale properties. Using the results [3,12] and [10], we

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can show boundedness and convergence of the trajectories ofthe closed loop system. Toward that end, we define compactsets Ω,Σ such that η0 ∈ Ω, (ξ0, σ0) ∈ Σ and show thattrajectories starting in Ω ×Σ enter an appropriately definedpositively invariant set S in finite time. We define Lyapunovfunction candidate for χη as V1(t, χη) = (χη)

T P−1χη satis-fying the identity 0 < a1 In−m ≤ P−1 ≤ a2 In−m where P−1

is a positive definite matrix and a1, a2 are arbitrary constants.Then it can be shown that V1 satisfies

a1‖χη‖2 ≤ V1(t, χη) ≤ a2‖χη‖2 (46)∂V1∂χη

gr (.) ≤ −a3‖χη‖2, ∀‖χη‖ ≤ c1, ∀t ≥ t0 (47)

where a3, c1 are positive constants independent of ε. Theinequalities (46), (47) will ensure that the compact set sat-isfies Ω ⊂ �n−m . The boundary layer system is obtained

from τb = t/ε and setting ε = 0 for scaled error as dχξ,σ

dτb=

Λχξ,σ and the Lyapunov function candidate W (χξ,σ ) forthis subsystem is given as W (χξ,σ ) = χT

ξ,σ P0χξ,σ whereP0 is positive definite solution of Lyapunov equation P0Λ+ΛT P0 = −I . It can be seen that W satisfies

λmin(P0)‖χξ,σ ‖2 ≤ W (χξ,σ ) ≤ λmax (P0)‖χξ,σ ‖2 (48)∂W

∂χξ,σ

Λχξ,σ ≤ −‖χξ,σ ‖2 (49)

and the time derivative of V1,W satisfy

V1 ≤ − a1‖χη‖2 + k2‖χξ,σ ‖ (50)

W ≤ − 1

ε‖χξ,σ ‖2 + k3‖χξ,σ ‖‖P0‖ (51)

Here k1, k2 are positive constants independent of ε. Eqs. (50),(51) show that trajectories of estimation error dynamics arebounded.

Furthermore, considering the closed loop system com-prising of (20), (33), (43), (45) and compactly representingit as χ = fr (χ, χξ,σ , χη), where χ = [ηT , ξ T , v]T andχ0 = [ηT0 , ξ T0 , v0], it can be shown as in [10] and the refer-ences therein that the trajectories starting in Ω × Σ remainbounded and converge asymptotically.

The above-mentioned conclusions can be summarized asthe following theorem.

Theorem 1 Under the stated assumptions and suitable val-ues of controller parameters a, k,C = c1, c2, . . . ∈ �m−n,

observer gains H1, H2, K, the closed loop system compris-ing of composite system (20), the full-order EHGO-basedobserver (23)–(25) and the output feedback control law (35),there exists μ∗ > 0 and ε∗ > 0 such that for every 0 < μ <

μ∗ and 0 < ε < ε∗, the trajectories of closed loop systemare bounded and approach the origin as t −→ ∞.

Fig. 2 Inertia wheel pendulum (IWP)

5 Example: Inertia Wheel Pendulum (IWP)

Inertia wheel pendulum is a constructionally simple test-bench nonlinear mechanical system as shown in Fig. 2.The system consists of a simple pendulum with an actuatedwheel attached to its independent and floating end, whereasthe other pendulum end is connected to an unactuated rotat-ing joint. When the wheel is rotated at a fairly high speed,angular acceleration is produced and resultant torque is usedas an input for controlling the pendulum angle. The controlproblem is to stabilize the pendulum in an upright equilib-rium position, while the wheel stops rotating at an arbitraryangle. IWP, thus, falls into the category of under-actuatedsystems, since one end of the pendulum remains unactuated,and IWP is controlled by a single actuator.

5.1 Model Transformation of an IWP

To obtain a mathematical model of the IWP, we start withthe Lagrangian of the system L(θ, θ ) = [K.E − P.E] withθ = [θ1, θ2] as the configuration variables

L(θ, θ ) = 1

2θT M θ − V (θ1) (52)

[m11 m12

m21 m22

] [θ1θ2

]+

[−m0 sin(θ1)0

]=

[0v

](53)

where V (θ1) = m0 cos(θ1), m0 = mg, m = m1lc1 + m2l1,m11 = m1l2c1 + m2l21 + I1 + I2, m12 = m21 = m22 = I2, m1 = Pendulum mass (kg), m2 = Wheel mass (kg),θ1 = Pendulum Angle (rad), θ2 = Wheel Angle (rad), v =

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Control Input or Torque (Nm), l1 = Pendulum length (m),lc1 = Distance to the center of the mass (m), I1 =Pendulum moment of inertia (kg m2), I2 = Wheel momentof inertia (kg m2), g = Acceleration due to gravity (m/sec2)

and , M =[m11 m12

m21 m22

].

We start with transformation of IWP system model (53)in terms of θ1 and θ2 as given by set of Eq. (54)

θ1 = − m12

m11m22 − m12m21v

+ m22m0

m11m22 − m12m21sin(θ1)

θ2 = m11

m11m22 − m12m21v

− m21m0

m11m22 − m12m21sin(θ1)

y = θ1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(54)

where pendulum angle θ1 is taken as system output y. Toderive a state space representation of plant model, we assumesystem states as follows

x1 = θ1,

x1 = θ1 = x2

x3 = θ2,

x3 = θ2 = x4

Then state space representation of IWP System model andits output can be represented as following set of equations

x1 = x2

x2 = − m12

m11m22 − m12m21v

+ m0m22

m11m22 − m12m21sin(x1)

x3 = x4

x4 = − m11

m12m21 − m22m11v

+ m0m21

m12m21 − m11m22sin(x1)

y = x1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(55)

For the output y, it can be easily seen that it is an under-actuated system and its relative degree m is given as m =2 < n. The degree of under-actuation further depicts thatoverall system can be further bifurcated into internal andexternal dynamics in terms of [η(η1, η2), ξ(ξ1, ξ2)]. To deter-mine (η, ξ) dynamics, we follow procedure as in [7] and

apply change of coordinates as follows

η1 = −m11x1 − m12x3

η2 = −m11x2 − m12x4

ξ1 = x1

ξ2 = x2

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(56)

The change of coordinates mentioned in (56) gives normalform representation of IWP model. The internal dynamics isobtained as

η1 = η2

η2 = −m0 sin(ξ1)

}(57)

Corresponding external dynamics and output are given as

ξ1 = ξ2

ξ2 = m0m22

m11m22 − m12m21sin(ξ1)

− m12

m11m22 − m12m21v

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(58)

y = ξ1 (59)

5.2 OFB Controller Design

In this section, an output feedback stabilizing controller usingextended high gain observer is implemented. Toward thatend, we assume that the pendulum angle θ1 can be measuredand is available for feedback. The normal form representation(57), (58) renders system configuration as an interconnectionof internal and external system dynamics. An auxiliary sys-tem that utilizes internal dynamics of system is derived and afull-order extended high gain observer is then implemented.The estimated state variables are provided to an SMC-basedOFB nonlinear controller.

5.2.1 EHGO-Based Control Law

Normal form of an IWP as in Eqs. (57), (58) and (59) can begeneralized similar to (20) as follows

η = A1η + φ0(ξ1)

ξ = Aξ + B[β(η, ξ) + α(ξ, v)]y = ξ1

⎫⎪⎬⎪⎭ (60)

Here for IWP system, A1 =[0 10 0

], φ0 =

[0

−m0sin(ξ1)

],

A =[0 10 0

], B =

[01

], β =

[m0m22

m11m22−m12m21sin(ξ1)

],

α =[

− m12m11m22−m12m21

v].

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Auxiliary system having virtual output σ is given as

η = A1η + φ0(ξ) (61)

σ = β(η, ξ) (62)

Using virtual output σ and IWP internal and external dynam-ics full-order EHGO similar to (23), (25) is given as

˙ξ = Aξ + B[σ + α(ξ , v)] + h1[y − ξ1] (63)˙σ = β(η, ξ ) + h2[y − ξ1]˙η = A1η + φ0(ξ ) + h3[σ − β] (64)

y = ξ1

where h1 = [α1/ε α2/ε2]T , h2 = [α3/ε

3]. h3 is chosenhere as small positive constant. The estimated system statesobtained from EHGO block (63), (64) are utilized in pro-posed SMC-based controller design, and an output feedbackcontrol law is derived to achieve stabilization of the IWP sys-tem. Toward that end, we begin with transforming the system(53) into the normal form by using collocated partial feed-back linearization and global coordinate transformation forchange of control and input decoupling [13].

In particular, using Eq. (53), the applied torque can berepresented as

v = ατ + β (65)

where α = m22 − m12m21m11

, β = m21m11

m0 sin(θ1), τ = θ2 Tofurther decouple the system’s input, we use a global changeof variables from [15], as given below

z1 = m11θ1 + m12θ2, z2 = θ1, z3 = θ2 (66)

Using observed states fromEHGOblock (63), (64) and coor-dinate transformation in (56), the estimated system states(η, ξ ) in terms of (z1, z2, z3) are given as

z1 = −η2, z2 = ξ1, z3 = − 1

m22η2 − m11

m12ξ2 (67)

The advantage of the above change of coordinates is that ityields the system equations in the following strict-feedbackform

˙z1 = m0 sin(z2

)(68)

˙z2 = (z1 − m12 z3)

m11(69)

˙z3 = τ (70)

The resulting system (68)–(70) can now be represented as acascade connection of two subsystems; namely, a linear sub-system (68)–(69) and a nonlinear subsystem (70), as shownin Fig. 3.

˙z2 = (z1−m12z3)m11

˙z3 = τ

Linear Block

˙z1 = m0 sin(z2)

Nonlinear Block

z2τ y

Fig. 3 Cascade interconnection of inertia wheel pendulum (IWP)

5.2.2 Internal System Control Law

We first apply Control Lyapunov Function method to makethe subsystem represented by (68) globally asymptoticallystable. We follow the steps given in [13], by choosing z2 asan input for the system and any positive definite functionV◦(z1) as Lyapunov function candidate and then taking itstime derivative to find a control law as shown below:

V◦(z1) = 1

2z21 (71)

V◦(z1) = m0 z1 sin(z2)

V◦(z1) = z1m0 sin(α(z1)) (72)

where α(z1) = z2. From (71) and (72), it can be seen that asigmoidal function α(z1) will make V◦(z1) as negative defi-nite and hence choosing

α(z1) = −a arctan(c1 z1) (73)

with 0 < a ≤ π/2 and c1 > 0 will render origin of thenonlinear system (68) as globally asymptotically stable.

5.2.3 Composite System Control Law

By using sliding mode control for subsystem (69)–(70), thecontrol law for entire system is derived. We take error sur-faces for z2 and z3 as:

e1 = z2 − z2d (74)

e2 = z3 − z3d (75)

where e1 is the estimated error in the generation of stabiliza-tion function z2d = α(z1) by z2 and e2 is the estimated errorin the generation of z3d by z3.

The time-derivatives of the Eqs. (74)–(75) are

˙e1 = ˙z2 − ˙z2d (76)˙e2 = ˙z3 − ˙z3d (77)

Substituting ˙z2 and ˙z3 from (69), (70) in (76), (77) gives

˙e1 = 1

m11z1 − m12

m11z3 − ˙z2d (78)

˙e2 = τ − ˙z3d (79)

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In order to make e1 zero, using (78), we define

z3 = m11

m12

(k1e1 + 1

m11z1 − ˙z2d

)

where k1 > 0. The outputs z3d and ˙z3d obtained from thelow-pass filter with small positive time constant v3d can beused [15] to generate the desired control signal as

v3d ˙z3d + z3d = z3

with initial conditions z3d(0) = z3(0) = 0. Next, we definea time varying surface s(t) in space �2 by a scalar equations(e; t) = 0

s = λe1 + e2

˙s = λ

(1

m11z1 − m12

m11z3 − ˙z2d

)+ τeq − ˙z3d

τeq = ˙z3d − λ

(1

m11z1 − m12

m11z3 − ˙z2d

)

u = τeq − ksign(s) (80)

where k is a strictly positive constant. The control law τ

obtained shows high frequency oscillations known as chat-tering in sliding phasewhich can be removed using saturationfunction as in [11]. The resulting control law τ under outputfeedback is given as follows

τ = τeq − ksat

(s

μ

)(81)

and finally control input to IWP system is given as

v = ατ + β (82)

is the desired control law.

5.3 Simulation Results

In this section, we present simulation-based results for theproposed OFB control scheme to control the IWP system.For performance comparison with previously demonstratedresults as in [13,15,17] and [6], we have used the samemodelparameters. We present a comparison of simulation resultsbetween the state feedback controller as in [6] and the outputfeedback controller (81), (82). The IWP system parameterschosen for this purpose are given inTable 1, and the controllerparameters used in the SFB/OFBdesign are shown inTable 2.

Figures 4–7 show the simulation results for various states/control input of the IWP system under state feedback control,output feedback control and the corresponding convergenceerrors, respectively. These results show convergence to theequilibrium of the pendulum angle, pendulum velocity and

Table 1 IWP system parameters

Parameter Value Unit

l1 0.124 m

lc1 0.063 m

m1 0.020 kg

m2 0.300 kg

I1 47e−6 kgm2

I2 32e−6 kgm2

m11 4.83e−3 kgm2

m12,m21,m22 32e−6 kgm2

g 9.81 m/sec2

Table 2 Controller parameters

Parameter Value Parameter Value

a π/2 c1 9

k1 4 λ 10

μ 0.01 k 100

α1 2 α2 5

α3 20 ε 0.001

v3d 0.035

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time[second]

Pen

dulu

m A

ngle

[rad]

(a) Pendulum Position: SFB vs OFB

Actual AngleEstimated Angle

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time[second]

Err

or in

Pen

dulu

m A

ngle

[rad]

(b) Error in Position

Fig. 4 Pendulum angle (θ1) and convergence error plot

0 1 2 3 4 5−12

−10

−8

−6

−4

−2

0

2

Time[second]

Pen

dulu

m V

eloc

ity[ra

d/s]

(a) Pendulum Velocity: SFB vs OFB

Actual VelocityEstimated Velocity

0 1 2 3 4 5−12

−10

−8

−6

−4

−2

0

2

Time[second]

Err

or in

Pen

dulu

m V

eloc

ity[ra

d/s]

(b) Error in Velocity

Fig. 5 Pendulum velocity (θ1) and convergence error plot

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Arab J Sci Eng

0 1 2 3 4 5−500

0

500

1000

1500

2000

2500

3000

3500

Time[second]

Whe

el V

eloc

ity[ra

d](a) Wheel Velocity: SFB vs OFB

Actual VelocityEstimated Velocity

0 1 2 3 4 5

−500

0

500

1000

1500

2000

2500

3000

3500

Time[second]

Err

or in

Whe

el V

eloc

ity[ra

d/s]

(b) Error in Velocity

Fig. 6 Wheel velocity (θ2) and convergence error plot

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

Time[second]

Inpu

t Tor

que[

N−m

]

(a) Control Input: SFB vs OFB

SFB ControllerOFB Controller

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

Time[second]

Err

or in

Inpu

t Tor

que[

N−m

]

(b) Error in Control Input

Fig. 7 Control input (v) and error plot

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−12

−10

−8

−6

−4

−2

0

2

Pendulum Angle[rad]

Pen

dulu

m V

eloc

ity[ra

d/s]

Phase Portrait Plot: SFB vs OFBSFB ControllerOFB Controller

Fig. 8 Phase portrait plot (θ1 vs θ1)

wheel velocity. These states converge to equilibrium pointin fairly small interval of time. As wheel angle θ2 does notplay important role in IWP dynamics, it is not shown as statevariable here.

Figures 4a–7a show simulation results for the SFB con-troller as in [6] versusOFB controllerwhich uses an extendedhigh gain observer (81), (82). Figures 4b–7b show the cor-responding convergence error between actual and estimatedstates/control input. Figure 8 shows phase portrait plot underSFB andOFBcontrol.We investigate the convergence of sys-tem estimated states using EHGO with state feedback-basedcontroller. The results show that estimated states converge

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time[second]

Pen

dulu

m A

ngle

[rad]

(a) Pendulum PositionNominal ModelPerturbed Model

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time[second]

Err

or in

Pen

dulu

m A

ngle

[rad]

(b) Error in Position

0 1 2 3 4 5−12

−10

−8

−6

−4

−2

0

2

Time[second]P

endu

lum

Vel

ocity

[rad/

s]

(a) Pendulum Velocity

Nominal ModelPerturbed Model

0 1 2 3 4 5−14

−12

−10

−8

−6

−4

−2

0

2

Time[second]

Err

or in

Pen

dulu

m V

eloc

ity[ra

d/s]

(b) Error in Velocity

0 1 2 3 4 5−1000

0

1000

2000

3000

4000

5000

Time[second]

Whe

el V

eloc

ity[ra

d/s]

(a) Wheel VelocityNominal ModelPerturbed Model

0 1 2 3 4 5−2000

−1000

0

1000

2000

3000

4000

5000

6000

Time[second]

Err

or in

Whe

el V

eloc

ity[ra

d/s]

(b) Error in Velocity

Fig. 9 System states plots (θ1, θ1, θ2): Nominal versus Perturbedmodel and corresponding convergence error

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

Time[second]

Inpu

t Tor

que[

N−m

]

(a) Control Input

Nominal ModelPerturbed Model

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

Time[second]

Err

or in

Inpu

t Tor

que[

N−m

]

(b) Difference in Control Input

Fig. 10 Control input (v) plot: nominal versus perturbed model anderror plot

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Arab J Sci Eng

to actual states arbitrarily fast. However, small delay is seenin the initial phase. As mentioned earlier, in output feed-back design only pendulum angle is used for estimation ofall the system states. Initial values of these system states arenot precisely known. Furthermore, in order to avoid peak-ing phenomenon, we use a saturation function in our controllaw. All these factors result in a slight degradation in the sys-tem transient response. Therefore resulting behavior of statesunder EHGO-based output feedback controller is expectedto be slightly delayed as compared with actual states

The control law is designed without taking into accountthe parametric uncertainties and modeling errors. However,inherent properties of SMC-based control scheme and HGO-based observer design in general ensures a certain level ofrobustness in system response under such conditions. To ver-ify these aspects simulations are performed with modelingerrors and as shown in Figures 9–10, it is observed that con-troller can safely stabilize the system up to 30% variation inparameters around nominal values and only small degrada-tion in transient response is observed.

6 Conclusion

In this paper, we presented an EHGO-based output feedbackstabilizing controller for a class of under-actuated benchmarknonlinear systems under slidingmode control framework. Byusing Lagrangian based nonlinear model, a suitable coordi-nate transformation and a change of control is applied to getnormal form representation of the system. For an EHGO-based state estimation, the system is decoupled into a linearand nonlinear subsystem, and the estimated states obtainedare further utilized for development of SMC-based nonlinearcontrol law.

The design is validated by simulation on a benchmark non-linear system model, namely, IWP. Using simulation results,it is shown that the system response under output feedbackconverges to that of the state feedback controller. It is alsoshown that the estimated states converge to actual states arbi-trarily fast by appropriate selection of the observer designparameters. The analysis and simulation results indicate thatthe proposed output feedback control technique can be imple-mented on other under-actuated systems of similar class,which includes systems like TORA, Acrobot etc, and thisconstitutes a direction of the future work.

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