Nonlinear state feedback controller design for underactuated mechanical system: a modified block backstepping approach.

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Nonlinear state feedback controller design for underactuated mechanical system: A modified block backstepping approach Shubhobrata Rudra n, Ranjit Kumar Barai, Madhubanti Maitra Department of Electrical Engineering, Jadavpur University, West Bengal, India

art ic l e i nf o

a b s t r a c t

Article history: Received 31 August 2013 Received in revised form 5 December 2013 Accepted 15 December 2013 This paper was recommended for publication by jeff piper

This paper presents the formulation of a novel block-backstepping based control algorithm to address the stabilization problem for a generalized nonlinear underactuated mechanical system. For the convenience of compact design, first, the state model of the underactuated system has been converted into the blockstrict feedback form. Next, we have incorporated backstepping control action to derive the expression of the control input for the generic nonlinear underactuated system. The proposed block backstepping technique has further been enriched by incorporating an integral action additionally for enhancing the steady state performance of the overall system. Asymptotic stability of the overall system has been analyzed using Lyapunov stability criteria. Subsequently, the stability of the zero dynamics has also been analyzed to ensure the global asymptotic stability of the entire nonlinear system at its desired equilibrium point. The proposed control algorithm has been applied for the stabilization of a benchmarked underactuated mechanical system to verify the effectiveness of the proposed control law in realtime environment. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Underactuated mechanical system Block backstepping Block-strict feedback form Zero dynamics Integral-action Real-time control Cart-pole system

1. Introduction Underactuated system is defined to be one for which the dimension of space spanned by the configuration vector is more than that of the space spanned by the control variables [1]. Simply stated, this refers to a particular class of a multi output mechanical system that has more degrees of freedom (DOF) than the number of control inputs applied. During the last few decades, there was a growing interest among the researchers pursuing studies regarding the control aspects of underactuated mechanical systems because of their wide application domain [1–11]. Nonetheless, the control task of a system with underactuation property is quite appealing, since use of fewer numbers of actuators allows reduction in cost and weight [9]. In addition, for such kind of mechanical systems the component failure rate would be sufficiently low [8,9]. However, the design of control algorithm for an underactuated system is more complicated than that of a fully actuated multipleinput multiple-output (MIMO) system. Most of the time, an underactuated system fails to satisfy Brockett0 s necessary condition for feedback linearization [12]. Consequently, different sophisticated control algorithms have been proposed in last few decades [13]. Design of a passivity based stabilizing controller has been n

Corresponding author. Tel.: þ 91 90 51 21 7543. E-mail addresses: [email protected] (S. Rudra), [email protected] (R. Kumar Barai), [email protected] (M. Maitra).

described in [14]. A constructive approach towards the design of a stabilizing controller for a special class of underactuated mechanical systems has been proposed in [15]. Based on the principle of averaging theorem, Tahmasian et al. have developed a state feedback control law for such systems [16]. In [17], the authors have proposed a ‘viability theory’ based control design approach to address the stabilization problem for this type of systems. Different sliding mode based stabilization controller design techniques have also been proposed in [3–5,8,10,11]. Design of an output feedback stabilization controller has been presented in [18]. The recent advancements in the field of soft computing have given rise to many intelligent methods for addressing the stabilization control problem of the underactuated mechanical systems [19,20]. The above-mentioned design schemes rigorously analyzed the stability of the system consisting of the plant and their proposed controllers (i.e. the controlled system). However, the mathematical formulations are often too complicated for real time implementation. In addition, the authors have often adhered to some simplifying assumptions constraining the scope of generality. Moreover, in most of the cases, the proposed control algorithms only ensure local stability of the control system around the desired equilibrium point. Hence, devising a generalized stabilization control algorithm for a truly generic underactuated mechanical system still remains an open problem [21–23]. The last few decades have witnessed significant advancements in the field of nonlinear control engineering [24] and backstepping technique has emerged as one of the most efficient feedback

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.021

Please cite this article as: Rudra S, et al. Nonlinear state feedback controller design for underactuated mechanical system: A modified block backstepping approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.021i

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S. Rudra et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

control algorithm for nonlinear systems [26–34]. Backstepping is a Lyapunov method based versatile control design approach for nonlinear systems that ensures the convergence of the regulated variables to zero [24–26]. The ordinary integral backstepping relies on the fact that the system under consideration is in strict feedback form [24–26]. Generally, single-input single-output (SISO) nonlinear systems satisfy this condition under some simplifying assumptions [24–26]. However, in case of multivariable control problem, quite often the system structure is not available in the strict feedback or semi strict feedback (i.e. lower triangular) form. Therefore, it is not possible to apply the integral backstepping technique in the usual manner to generate a control algorithm for MIMO systems. However, the state models of an underactuated system, which typically fall under MIMO systems, never possess the strict feedback form; therefore, we cannot simply apply the integral backstepping algorithm to address the stabilization problem for such systems. Although attempts have been made in the works of Reza Olfati Saber [35–44] to design a backstepping based control law for underactuated system after transforming the system state model to the normal form; the resulting control law is quite complicated and often seems to be inapt for real-time applications. Recently, several research endeavors have been directed towards arriving at more generalized backstepping algorithms that can efficiently address the stabilization problems of complex nonlinear systems [24–34]. Block backstepping technique has emerged as one of the most efficient backstepping based algorithm, which can address the control problem of different nonlinear multiple input multiple output (MIMO) systems [27–30]. At the first stage of block backstepping design, the state model of the system is transformed into a block strict feedback form by algebraic state transformation [25]. At the next stage, backstepping procedure is applied on each dynamic block to derive the expression of control input for overall nonlinear system [25]. Since, block backstepping extends the application of backstepping to the control problems of different nonlinear MIMO system [27–30]; therefore, block backstepping based design technique could be regarded as the most suitable alternative to a backstepping based feedback law for underactuated mechanical systems. Finally, we summarize our contributions as follows. In this paper, we have endeavored to devise a novel block-backstepping based algorithm to address the control problem of any underactuated system with nth degrees of freedom. Stability of the block-backstepping controller depends on the stability of its internal dynamics. Moreover, the stability of the internal dynamics can only be assessed by means of zero dynamics analysis [25]. Therefore, we have thoroughly analyzed the zero dynamics stability of the controlled system to ensure the global asymptotic stability of the underactuated system at a desired equilibrium point. However, mere stabilization capability of a controller may not be always adequate to ensure the reliable steady state performance of a dynamical system [45], and hence we have additionally incorporated the integral action in the proposed control law to ameliorate the steady state performance of the overall system. Finally, this paper successfully illustrates the application of the proposed control algorithm on a benchmark nonlinear underactuated system, and verifies the performance of the control algorithm in real time environment. Since, the inverted pendulum system is regarded as a very popular test bed for nonlinear underactuated system [46–53]; an inverted Pendulum [51] system has been selected as a real-time test bed to verify the effectiveness of the proposed control algorithm. The rest of the paper is organized as follows. In Section 2, we present the stepwise formulation of the control algorithm for a generic 2 DOF underactuated system. The subsections of Section 2 are organized as follows: Section 2.1 describes the systematic development of the control algorithm for a 2 DOF underactuated

system, and then Section 2.2 presents the detailed zero dynamic analysis of the controlled system. The next section of the paper, Section 3, presents the derivation of the control algorithm for a generic n DOF underactuated system (where n is any arbitrary integer). Similar to Section 2, Section 3, is also divided in several subsections as follows: Section 3.1, describes the stepwise derivation of the control law for nth DOF underactuated mechanical system, finally, Section 3.2, presents the zero dynamic stability analysis of the controlled system. In Section 4, the proof of stability of the proposed control algorithm has been given. Section 5 presents the real-time verification of the proposed control algorithm on Feedback Instruments Ltd, UK, make Digital Pendulum setup. Finally, Section 6 states the conclusions of the paper.

2. Development of the control algorithm for a generic 2-DOF underactuated system In order to ensure better comprehensibility, in this section we present the systematic derivation of the control law for a generic 2-DOF system. Then at the next section, we will present the more generalized version of the control algorithm, which would be able to address the stabilization control problem of a generic nth DOF underactuated system. Assumption 1. In this article, we have considered a class of underactuated mechanical system that is described by the continuous differential equation [54]. Let us consider the state model of a 2 DOF underactuated system [36–38] as shown in Eq. (1) below: q_ 1 ¼ p1 q_ 2 ¼ p2 p_ 1 ¼ f 1 ðq; pÞ þg 1 ðqÞu p_ 2 ¼ f 2 ðq; pÞ þg 2 ðqÞu

ð1Þ

where q ¼ colðq1 ; q2 Þ A ℝ is the configuration state vector of a generic 2-DOF underactuated mechanical system, and the time derivative of the configuration state vector q, is expressed as p ¼ col ðp1 ; p2 Þ A ℝ2 , and u A ℝ is the control input (where “col” denotes column vector). Actually, f1(q,p) and f2(q,p)are the nonlinear functions of q and p defined as f 1 ðq; pÞ : ℝ4 -ℝ and f 2 ðq; pÞ : ℝ4 -ℝ. In addition g1(q) and g2(q) are the functions of configuration vector q defined as g 1 ðqÞ : ℝ2 -ℝ and g 2 ðqÞ : ℝ2 -ℝ. Also f1(q,p), f2(q,p), g1(q) and g2(q) are all C1 functions. In addition, f 1 ð0; 0Þ ¼ 0 andf 2 ð0; 0Þ ¼ 0. Moreover, we can express the state vector X as X ¼ ½ q1 q2 p1 p2 T A ℝ4 : Now, the objective is to design a nonlinear state feedback control input u for the underactuated system shown in (1) employing block backstepping technique such that it would ensure the asymptotic stability of the system. In other words, if the state error E is defined as E ¼ XðtÞ X 0 , where X(t) denotes the system states at time t and X0 denotes the desired equilibrium point in the state space; then the goal of the control law u is to ensure that |E|-0 as t-1. 2

2.1. Systematic derivation of the control algorithm for 2 DOF underactuated system In this subsection, we present the systematic development of a block backstepping based control algorithm to address the stabilization problem of 2-DOF system. Since the state model of the underactuated system shown in Eq. (1) is not in strict feedback form, hence we cannot directly apply the integral backstepping technique to design a state feedback controller for the underactuated system [9,25]. Therefore, at the outset of the design, we transform the state model into a block-strict feedback form. The interested readers may



find the detailed mathematical formulation of block-strict feedback form in Ref. [25]. At the first stage of design, we transform the system into a reduced order state model and find the expression of control input u to stabilize the reduced order system at its origin. The derivation of the control algorithm has been shown in the following steps 1 to 4 as given below:

ð2Þ

∂g 2 ∂g ðf þ g 1 uÞ þ 2 ðf 2 þ g 2 uÞ ∂q1 1 ∂q2

þ p1

∂2 g 2 2 ∂2 g 2 ∂2 g 2 2 p þ2 p p þ p ∂q1 ∂q2 1 2 ∂q22 2 ∂q21 1

(

∂g 1 ∂g ðf þ g 1 uÞ þ 1 ðf 2 þ g 2 uÞ ∂q1 1 ∂q2

þ p2

∂2 g 1 2 ∂2 g 1 ∂2 g 1 2 p þ2 p p þ p ∂q1 ∂q2 1 2 ∂q22 2 ∂q21 1

z 2 ¼ p2 α 1

f 1 þg 1 u þf 1 dg2 p þ ðf 1 þ g 1 uÞdg2 p

∂f 1 ∂f ∂f ∂f p þ 1 ðf þ g 1 uÞ þ 1 p2 þ 1 ðf 2 þ g 2 uÞ ∂q1 1 ∂p1 1 ∂q2 ∂p2 ∂g 2 ∂g ðf 1 þ g 1 uÞ þ 2 ðf 2 þ g 2 uÞ þ p1 ∂q1 ∂q2 !) ∂2 g 2 2 ∂2 g 2 ∂2 g 2 2 p þ 2 p p þ p þ p1 ∂q1 ∂q2 1 2 ∂q22 2 ∂q21 1 (

h ∂g ∂g i In the above representation, dg ¼ ∂q1 ∂q2 and for ease of representationf 1 ðq; pÞ, f 2 ðq; pÞ, g 1 ðqÞ, g 2 ðqÞare denoted by f1,f2, g1, g2, respectively. Step 2: In this step, a suitable stabilizing function has been selected to find out the desired value of the virtual input for the first subsystem. A conventional choice for stabilizing function is given below in (4):

f 2 dg1 p þ ðf 2 þg 2 uÞdg1 p

∂f 2 ∂f p þ 2 ðf þ g 1 uÞ ∂q1 1 ∂p1 1 ∂f ∂f þ 2 p2 þ 2 ðf 2 þ g 2 uÞ ∂q2 ∂p2 ∂g 1 ∂g ðf 1 þ g 1 uÞ þ 1 ðf 2 þ g 2 uÞ þ p2 ∂q1 ∂q2 þ g1

ð4Þ

Rt where χ 1 ¼ 0 z1 dtand λ is a design constant. Step 3: The second error variable ðz2 A ℝÞ has been defined as the difference between stabilizing function and the virtual control in such a manner that asymptotic stabilization of second variables automatically implies the stabilization of the first subsystem. Here, it has been defined as following:

!)#

¼ ðf 2 þ g 2 uÞ þ c1 ðz2 c1 z1 λχ 1 Þ þ λz1 "(

þ g2

ð5Þ

þ p2

z_ 1 ¼ p2 kðp1 þ g 2 p_ 1 þ g_ 2 p1 g 1 p_ 2 g_ 1 p2 Þ _ ¼ p2 kðp1 þ g 2 ðf 1 þ g 1 uÞ þ p1 dg2 U q_ g 1 ðf 2 þ g 2 uÞ p2 dg1 qÞ ¼ p2 kðp1 þ g 2 f 1 þ p1 dg2 p g 1 f 2 p2 dg1 pÞ ð3Þ

α1 ¼ c1 z1 λχ 1 þ kðp1 þ g2 f 1 þ p1 dg2 p g 1 f 2 p2 dg1 pÞ

!)

_ þ ðf þ g uÞdg p f 2 dg1 p þ g 1 df 2 X 2 2 1

k

In order to enhance the steady-state performance of the system an integral action has been incorporated with the state feedback law. Consequently, stabilizing function of Eq. (4) has been modified as follows:

þ p1

where, k is a design constant. Now, the dynamics of z1 is expressed in Eq. (3) shown below:

α1 ¼ c1 z1 þ kðp1 þ g2 f 1 þ p1 dg2 p g1 f 2 p2 dg1 pÞ

_ þðf þ g uÞdg p f 1 þg 1 u þf 1 dg2 p þ g 2 df 1 X 1 1 2

k

Step 1: The first step of block backstepping design is to define a new control variable z1 A ℝ in terms of the original states of the given model in such a manner that the asymptotic stabilization of z1 dynamics ensures the stabilization of the systems at the desired equilibrium point. Quite often, the variable z1 is also called as first error variable. z1 ¼ q2 kðq1 þ g 2 p1 g 1 p2 Þ

3

"(

þ p2

∂2 g 1 2 ∂2 g 1 ∂2 g p1 þ 2 p1 p2 þ 21 p22 2 ∂q1 ∂q2 ∂q1 ∂q2

!)#

¼ ψ u þ λz1 þ c1 ðz2 c1 z1 λχ 1 Þ þ ϕ where,

ð8Þ

∂f 1 ∂f þ g 22 1 þ g 1 dg2 p ∂p1 ∂p2 ∂g ∂g ∂f ∂f þ p1 2 g 1 þp1 2 g 2 g 21 2 g 1 g 2 2 ∂q1 ∂q2 ∂p1 ∂p2 ∂g ∂g g 2 dg1 p p2 1 g 1 p2 1 g 2 ∂q1 ∂q2

ψ ¼ g 2 k g1 þ g1 g2

ð6Þ

ð9Þ

and Now, with the above definition of second control variable, the dynamics of first error variable becomes: z_ 1 ¼ z2 c1 z1 λχ 1

ð7Þ

Dynamics of second control variable z2 can be expressed as: _ 1 ¼ f 2 þ g 2 u þ c1 z_ 1 þ λz1 z_ 2 ¼ p_ 2 α "( k p_ 1 þ g_ 2 f 1 þ g 2 f_ 1 þ p_ 1 dg2 p þ p1 dg2 p_ þ p1 (

∂2 g 2 2 ∂2 g 2 ∂2 g p1 þ 2 p1 p2 þ 22 p22 2 ∂q1 ∂q2 ∂q1 ∂q2

!)

g_ 1 f 2 þ g 1 f_ 2 þ p_ 2 dg1 p þ p2 dg1 p_ þ p2

∂2 g 1 2 ∂2 g 1 ∂2 g p1 þ 2 p1 p2 þ 21 p22 2 ∂q1 ∂q2 ∂q1 ∂q2

¼ ðf 2 þ g 2 uÞ þ c1 ðz2 c1 z1 λχ 1 Þ þ λz1

"(

∂g 2 ∂g f 1 þ 2 f 2 :: ∂q1 ∂q2 ∂f 1 ∂f ∂f ∂f p þ 1p þ 1f þ 1f þ g2 ∂q1 1 ∂q2 2 ∂p1 1 ∂p2 2 !) ∂2 g 2 2 ∂2 g 2 ∂2 g 2 2 p þ 2 p p þ p þ p1 ∂q1 ∂q2 1 2 ∂q22 2 ∂q21 1 ∂g 1 ∂g 2f 2 dg1 p þp2 f 1 þ 1f 2 ∂q1 ∂q2 ∂f 2 ∂f 2 ∂f 2 ∂f þ g1 p1 þ p2 þ f 1 þ 2f 2 ∂q1 ∂q2 ∂p1 ∂p2 !)# 2 2 ∂ g2 2 ∂ g2 ∂2 g 2 2 p þ 2 p p þ p þ p2 1 2 ∂q1 ∂q2 ∂q21 1 ∂q22 2

ϕ ¼ f 2 k

f 1 þ 2f 1 dg2 p þ p1

ð10Þ

!)# In the design Steps 1 to 3 stated above, the choice of the two new state variables z1, z2 and their derivatives actually transform the original state model of the underactuated system described in



4

Eq. (1) into a block-strict feedback form. This approach results into a compact method of backstepping, which is novel and distinguishes it from the ordinary integral backstepping [25]. Step 4: It is the final step and gives the control law u so as to attain the desired dynamics of the z1 and z2 i.e. z_ 1 ¼ λ1 χ 1 c1 z1 þ z2 and z_ 2 ¼ z1 c2 z2 . The desired dynamics of z2 is expressed in Eq. (8) as following: z_ 2 ¼ z1 c2 z2 ð11Þ Consequently, from Eqs. (8) and (11) the desired control input can be derived in the following manner:

ψ u þ λz1 þ c1 ðz2 c1 z1 λχ 1 Þ þ ϕ ¼ z1 c2 z2 or; ψ u ¼ λz1 c1 ðz2 c1 z1 λχ 1 Þ ϕ z1 c2 z2 or; u ¼ ψ 1 ½ ð1 c21 þ λÞz1 ðc1 þ c2 Þz2 þ λc1 χ 1 ϕ

ð12Þ

The proposed control law of Eq. (12) transforms the closed loop system in following forms: z_ 1 ¼ z2 c1 z1 λχ 1 z_ 2 ¼ z1 c2 z2

ð13Þ

2.2. Analysis of zero dynamics of the controlled system Section 2.1, presents the systematic derivation of the control law for the two degrees of freedom underactuated mechanical systems. The expression of control input has been shown in Eq. (12). However, the control input u (described in Eq. (12)) only ensures the stability of the transformed system in z1, z2 coordinate system. Nevertheless, the algebraic state transformation that has been described in Eqs. (2) and (6) transforms the original fourth order underactuated state model of Eq. (1) into a reduced order state model (described by z1 and z2). The first order differentiation of z1 yields the dynamics of Eq. (3). Thus, the second order differentiation of z1 will result in the following dynamics. (On the assumption that, z1 is the output of the system as described by Eq. (1)): z€ 1 ¼ z_ 2 c1 z_ 1 λz1 ¼ ψ u þ λz1 þ c1 z_ 1 þ ϕ c1 z_ 1 λz1 ¼ ψuþϕ

ð14Þ

So it is quite evident from the above expression (14) that the two successive differentiations of z1 establishes an explicit relationship between z1 (output) and u (input). Therefore, if z1 is treated as an output of the system then the relative degree [55–60] of the overall underactuated system becomes two. Hence, from the concept of output feedback linearization [55–59] one can conclude that the state transformation described in Eq. (2) yields an unobservable internal dynamics of order two. However, the stability of the total control system depends on the stability of the internal dynamics [25,55–60]. Moreover, the stability of the internal dynamics can only be assessed by means of zero dynamics analysis [59,60]. In order to ensure the global asymptotic stability of the proposed controller, zero dynamics of the controlled underactuated system has been analyzed in the following manner. At first, a suitable control input, which renders the variable z1 identically equal to zero, has been found. Next, a configuration variable and its derivative have been chosen to describe the zero dynamics of the transformed system. In this paper, state variables q1 and p1 have been selected to describe the zero dynamics of the system. Since, two successive differentiation of z1 is required to establish an explicit relationship between input and output, the variable z1, the first derivative of z1 ðz_ 1 Þ and second derivative of z1 ðz€ 1 Þ must be equal to zero [59] for the derivation of zero dynamics: Now, z1 ¼ q2 kðq1 þ g 2 p1 g 1 p2 Þ ¼ 0 ) q2 ¼ kg 1 p2 þ kðq1 þ g 2 p1 Þ

ð15aÞ

z_ 1 ¼ p2 þkðg 1 f 2 þ p2 dg1 pÞ kðp1 þg 2 f 1 þ p1 dg2 pÞ ¼ 0 ) p2 ¼ kðg 1 f 2 þ p2 dg1 pÞ þ kðp1 þ g 2 f 1 þ p1 dg2 pÞ

ð15bÞ

z€ 1 ¼ ψ u þ ϕ ¼ 0 ) u ¼ ψ 1 ϕ

ð15cÞ

Consequently, one can represent the dynamics of q1, p1 subsystem together with the input u of Eq. (15c) as: q_ 1 ¼ p1 p_ 1 ¼ f 1 þ g 1 u ¼ f 1 g 1 ψ 1 ϑ

ð16Þ

Furthermore, after a few algebraic manipulations, it is always possible to replace all the terms containing q2 and p2 by the expressions of Eqs. (15a) and (15b), respectively. Hence, it is possible to describe the dynamics of the transformed system with the reduced order model as shown in Eq. (13) and subsequently the zero dynamics model as shown in Eq. (17), q_ 1 ¼ p1 p_ 1 ¼ Fðq1 ; p1 Þjz1 ¼ 0

ð17Þ

Now, if the parameter k is selected in such a manner to ensure the zero dynamics (vide Eq. (17)) stability of the transformed system, then it will ensure the asymptotic convergences of q1 and p1 to their desired equilibriums. Now, convergences of z1, q1 and p1, automatically ensure the asymptotic convergence of each of q2 and p2 to their desired equilibrium points. It is evident from the expressions of ψ,ϑ, and Eq. (17) that the zero dynamics stability of the system depends on the choice of the parameter k. Therefore, k is to be selected in a manner to assure the stability of the zero dynamic system, which in turn would guarantee the global asymptotic stability of the entire system. Remark 1. The above formulation is not rigid or specific for a particular underactuated system. Similar type of control algorithms can be obtained by defining a new set of state variables like: z1 ¼ q1 kðq2 þ g 1 p2 g 2 p1 Þ. Remark 2. The control law relies on the fact that ψ is invertible. In case of control law design for real-time implementations, it is always possible to select a value of k that will ensure the invertibility of ψ and the stability of the zero dynamics, simultaneously. Remark 3. The proposed control law asymptotically globally stabilizes the equilibrium of the original fourth order state model of the 2-DOF underactuated system. Remark 4. The state model of the underactuated system shown in Eq. (1) is not in strict feedback form. Therefore, we have used the transformations of Eqs. (2), (5), (6), and (11) to convert the system into a reduced order strict feedback form. The aforementioned transformation together with the control input u of Eq. (12) results in the reduced order system of Eq. (13) and the zero dynamics Equation of (17). Remark 5. Another significant feature of the proposed block backstepping design is that the transformation of the system into block-strict feedback form has been carried out during the design of control algorithm, which makes it more compact than its predecessor backstepping based approaches [35–44]. Remark 6. The integral action has been incorporated into the proposed backstepping technique by modifying the conventional stabilizing function. Thus, the ultimate control input derived by the proposed modified block-backstepping technique would enhance the steady state performance of the system. In the next Section 3, we will present the formulation of a generalized block-backstepping control algorithm, which will address the control problem of nth DOF underactuated system.



3. Development of the control algorithm for a generic n-DOF underactuated system

5

The dynamics of z1 is expressed in Eq. (23): z_ 1 ¼ p2 Kðp1 þ f þ gu gu DðgÞp2 Þ ¼ p2 Kðp1 þ f DðgÞp2 Þ

ð23Þ

In this section, we present the systematic derivation of the control law for a generic n-DOF underactuated system. At first, we precisely describe the dynamics of a generalized n- DOF underactuated mechanical system and then devise a generalized blockbackstepping based control algorithm to address the control problem of the n DOF underactuated mechanical system. Let us consider the generic Lagrangian model of a n- DOF underactuated system [35] as shown in Eq. (18)

expressed as Dðg ij Þ ¼ ∑nk ¼ 1 ∂qij pk , where i and j indicates the

m11 ðqÞq€ 1 þ m12 ðqÞq€ 2 þ h1 ðq; pÞ ¼ 0 m21 ðqÞq€ 1 þ m22 ðqÞq€ 2 þ h2 ðq; pÞ ¼ BðqÞτ

α1 ¼ c1 z1 λχ 1 þ Kðp1 þ f DðgÞp2 Þ

ð18Þ

where, q1 A ℝn1 ,q2 A ℝn2 and q ¼ colðq1 ; q2 Þ A ℝn is the configuration vector of nth degree of freedom underactuated mechanical system. The dimension of the overall configuration manifold is n1 þ n2 ¼ n [35]. The time derivative of the configuration vector q is expressed as p ¼ colðp1 ; p2 Þ A ℝn , and τ A ℝn2 is the control input. In the above representation, h1 ðq; pÞ : ℝ2n -ℝn1 andh2 ðq; pÞ : ℝ2n -ℝn2 contain the coriolis, centrifugal and gravity terms [35]. Whereas, mij ðqÞ q¼ 1, 2, represents the components of the n n inertia matrix which is symmetric and positive definite for all q. BðqÞ A ℝn2 n2 represents a full rank matrix. Due to Spong [61], there exists an invertible change of the control input τ as shown in Eq. (19) 1 τ ¼ B 1 ðqÞðh2 ðq; pÞ m21 ðqÞm11 ðqÞh1 ðq; pÞÞ 1 ðqÞm12 ðqÞÞu þ B 1 ðqÞðm22 ðqÞ m21 ðqÞm11

ð19Þ

which transform the Lagrangian model of Eq. (20) into

In above representationDðgÞ A ℝn1 n2 represents the time differentiation of matrix g(q), where any element of D(g) can be ∂g

k

position of the elements of D(g) matrix. Step 2: Choice of stabilizing function for z1 subsystem would be done after incorporating the integral action for enhancing the steady-state performance of the system as shown below: ð24Þ

Rt

where χ1 ¼ 0 z1 dtand λ and c1 is arbitrary positive design constant. Step 3: The second error variable z2 A ℝn2 is defined as: z 2 ¼ p2 α 1

ð25Þ

with the above definition of second control variable, the dynamics of first error variable becomes: z_ 1 ¼ z2 c1 z1 λχ 1

ð26Þ

Dynamics of second error variable z2 can be expressed as: z_ 2 ¼ p_ 2 a_ 1 ¼ u þc1 z_ 1 þ λz1 Kðp_ 1 þ f_ DðgÞp_ 2 D2 ðgÞp2 Þ ¼ u þc1 ðz2 c1 z1 λχ 1 Þ þ λz1 K½f þ gu þDf q1 p1 þ Df q2 p2 þ Df p1 ðf þ guÞ

q_ 1 ¼ p1 q_ 2 ¼ p2

þDf p2 u DðgÞu D2 ðgÞp2

p_ 1 ¼ f ðq; pÞ þ gðqÞu p_ 2 ¼ u

ð20Þ n1 n2

where, f ðq;pÞ : ℝ -ℝ , gðqÞ A ℝ 2n

n1

are given by

1 f ðq; pÞ ¼ m11 ðqÞh1 ðq; pÞ 1 gðqÞ ¼ m11 ðqÞm12 ðqÞ

ð21Þ

The state vector can be expressed as X ¼ ½ q1 p1 q2 p2 A ℝ2n . Now, the objective is to design a nonlinear state feedback control input u for the underactuated system shown in (20) employing block backstepping technique such that it would ensure the asymptotic stability of the system. In other words, if the state error E is defined as E ¼ XðtÞ X0 , where X(t) denotes the system states at time t and X0 denotes the desired equilibrium point in the state space; then the goal of the control law u is to ensure that |E|-0 as t-1.

In Eq. (27), Df q1 A ℝn1 n1 , Df q2 A ℝn1 n2 , Df p1 A ℝn1 n1 and Df p2 A ℝn1 n2 represent the matrices of partial derivatives of f vector with respect to different sub-component of state vector such as q1, q2, p1 and p2. D2 ðgÞ A ℝn1 n2 is given by D2 ðgÞ ¼ ½D21 ðgÞ þ D22 ðgÞ. The definition of D21 ðgÞand D22 ðgÞ are as follows: ∂2 gij pk pl and k ∂ql

D21 ðg ij Þ ¼ ∑nl¼ 1 ∑nk ¼ 1 ∂q

can be easily inferred that D22 ðgÞp2 can be further partitioned into three parts as shown in the following equation: D22 ðgÞp2 ¼ D22p1 ðgÞf þ D22p1 ðgÞgu þ D22p2 ðgÞu

ð22Þ

where, K A ℝn2 n1 is a constant matrix such that K ij ¼ k only when i¼j or K ij ¼ 0 otherwise. In the above equation, we have represented gðqÞ by g. Henceforth, for the ease of representation, we will denote f ðq; pÞ and gðqÞ by f and g, respectively.

ð28Þ

where the elements of each fragment are: n2

z1 ¼ q2 Kðq1 þ p1 gp2 Þ

k

elements of p vector. Hence, from the structure of D22 ðgÞmatrix it

i¼1

Similar to the previous case of 2-DOF underactuated mechanical system, at first, we transform the state model of a n-DOF system (shown in (20)) into a reduced order state model in blockstrict form and then find the expression of control input u to stabilize the reduced order system at its desired equilibrium. The derivation of the control algorithm has been shown in the following steps 1 to 4 as given below: Step 1: Define the new control variable z1 A ℝn2 as follows:

∂g

D22 ðg ij Þ ¼ ∑nk ¼ 1 ∂qij p_ k

In the above expression, pk and pl represents the individual

D22p1 ðg mr Þ ¼ ∑

3.1. Systematic derivation of the control algorithm for n DOF underactuated system

ð27Þ

∂g mi p ; where D22p1 ðgÞ A ℝn1 n1 ∂q1r 2i

m ¼1,…,n1 indicates the row index of g matrix, q1r is the elements of the vector q1, where r denotes the index of the particular configuration variables and p2i denotes element of p2 vector. ∂g mi 2 n1 n2 2 D22p2 ðg mr Þ ¼ ∑ni ¼ m ¼1,…,n1 indi1 ∂q p2i , where D2p2 ðgÞ A ℝ 2r

cates the row number of g matrix, q2r is the elements of the vector q2, where r denotes the index of the particular configuration variables. Hence, z_ 2 ¼ u þ c1 ðz2 c1 z1 λχ 1 Þ þ λz1 K½ f þ gu þ Df q1 p1 þ Df q2 p2 þDf p1 ðf þ guÞ þ Df p2 u i DðgÞu D21 ðgÞp2 D22p1 ðgÞf D22p1 ðgÞgu D22p2 ðgÞu ¼ ψ u þ λz1 þ c1 ðz2 c1 z1 λχ 1 Þ þ Φ

ð29Þ



6

Consequently, one can represent the dynamics of q1, p1 subsystem together with the input u of Eq. (36c) as: q_ 1 ¼ p1 p_ 1 ¼ f 1 þ g 1 u ¼ f 1 g 1 ψ 1 Φ

ð37Þ

Furthermore, after a few algebraic manipulations, it is always possible to replace all the terms containing q2 and p2 by the expressions of Eqs. (36a) and (36b), respectively. Hence, it is possible to describe the dynamics of the transformed system with the reduced order model as shown in Eq. (34) and subsequently the zero dynamics model as shown in Eq. (38), q_ 1 ¼ p1 p_ 1 ¼ Fðq1 ; p1 Þjz1 ¼ 0

Fig. 1. Schematic diagram of the control system.

where the expressions of ψ A ℝn2 n2 and Φ A ℝn2 are shown in following equations:

ψ ¼ ½I Kðg Df p1 g Df p2 þ DðgÞ þ D22p1 ðgÞg þ D22p2 ðgÞÞ

ð30Þ

and

Φ ¼ K½ f þ Df q1 p1 þ Df q2 p2 þ Df p1 f D21 ðgÞp2 D22p1 f

ð31Þ

In Eq. (30), I denotes an identity matrix of order n2. Similar to the case of 2-DOF underactuated system (described in Section 2.1), in the above three steps we transformed the state model of n-DOF underactuated system of (20) into the block-strict feedback form. Step 4: The control law u has been designed to ensure the desired dynamics for z2. The desired dynamics of z2 is expressed in Eq. (32) as following: z_ 2 ¼ z1 c2 z2

ð32Þ

where c2 is an arbitrary positive design constant. Consequently, from Eqs. (28) and (32) the desired control input can be derived in the following manner: or, u¼ψ

1

½ ð1 c21 þ

λÞz1 ðc1 þ c2 Þz2 þ λc1 χ 1 Φ

ð33Þ

Above choice of control input u results in the following dynamics z_ 1 ¼ z2 c1 z1 λχ 1 z_ 2 ¼ z1 c2 z2

ð34Þ

3.2. Analysis of the zero dynamics of the controlled system Similar to the analysis of zero dynamics stability presented in Section 2.2, it is also possible to analyze the stability of the zero dynamics for the n DOF underactuated system (20). Akin to the previous case of 2 DOF underactuated mechanical system, two successive differentiation of z1 yields an explicit relationship between the input vector u and the output vector z1. From Eq. (26), we can write: z€ 1 ¼ z_ 2 c1 z_ 1 λz1 ¼ ψ u þ λz1 þ c1 z_ 1 þ Φ c1 z_ 1 λz1 ¼ ψuþΦ

ð35Þ

Hence, we can proceed in a similar manner to derive the expressions of the zero dynamics system (we exactly follow Eqs. (15a), (15b), and (15c) to derive the expression of zero dynamics). Consequently, z1 ¼ 0 ) q2 ¼ Kðq1 þ p1 gp2 Þ

ð36aÞ

z_ 1 ¼ p2 Kðp1 þ f DðgÞp2 Þ ¼ 0 ) p2 ¼ Kðp1 þ f DðgÞp2 Þ

ð36bÞ

z€ 1 ¼ ψ u þ Φ ¼ 0 ) u ¼ ψ 1 Φ

ð36cÞ

ð38Þ

The schematic diagram of the overall control system is shown in Fig. 1. Now, if the elements of matrix K is selected in such a manner to ensure the zero dynamics (vide Eq. (38)) stability of the transformed system, then it will ensure the asymptotic convergences of q1 and p1 to their desired equilibriums. It is evident from the expressions of ψ, and Φ of Eqs. (36c) and (38) that the zero dynamics stability of the system depends on the choice of the elements (parameters) of K matrix. Therefore, K should be selected in a judicial manner to assure the stability of the zero dynamic system, which in turn would guarantee the global asymptotic stability of the entire system (shown in Eq. (20)). Remark 7. The dimension of the reduced order system described by Eq. (34) is 2n2 and the dimension of the zero dynamics described in (38) is 2n1. The dimension of the overall system state model (20) is 2n. Hence, the above results agree with the fact that the order of overall system (2n) ¼the order of reduced order system in z1, z2 (2n2)þthe order of zero dynamics(2n1). Remark 8. The stability of the zero dynamics system (38) ensures the asymptotic convergence of q1 and p1 to zero (i.e. q1 -0 and p1 -0; as t-1). 4. Proof of stability

Lemma 1. If the underactuated system is represented by the state model of Eq. (20), then the system can be transformed into a block strict feedback form by a global state transformation as described by Eqs. (22)–(27). Moreover, there exists a state feedback control law as given in Eq. (33), which ensures the global stabilization of the equilibrium for the underactuated system. Proof. The first part of the Lemma (i.e. the transformation of the underactuated system in block strict feedback form) can be proved by direct calculation the stability property of the control law, and the detailed mathematical formulation is available in [35]. Now, the proof of the stabilizing property of the control law is given below. The proposed control law of Eq. (33) transforms the closed loop system in the closed loop forms of Eq. (34). Now, the following Lyapunov function has been defined for the transformed system described by Eq. (39) as following: 1 1 1 V ¼ λχ T1 χ 1 þ zT1 z1 þ zT2 z2 2 2 2

ð39Þ

The derivative of the Lyapunov Function can be computed as: _ ¼ λχ T z1 þ zT ðz2 c1 z1 λχ Þ þ zT ðz1 c2 z2 Þ V 1 1 1 2 ¼ c1 zT1 z1 c2 zT2 z2

ð40Þ



That is, with t-1

The expression of Eq. (40) reveals the negative-definiteness of _ Vand also implies the fact that VðtÞ r Vð0Þ. Therefore, it ensures the boundedness of z1 and z2. Now, define the following function:

q1 -0andp1 -0

NðtÞ ¼ c1 zT1 z1 þ c2 zT2 z2

q2 þ Kgp2 -0

ð41Þ

Integration of Eq. (41) results the following expression: Z t _ τÞdτ Vð Vðt Þ ¼ Vðχ 1 ð0Þ; z1 ð0Þ; z2 ð0ÞÞ þ 0

¼ Vðχ 1 ð0Þ; z1 ð0Þ; z2 ð0ÞÞ

Z

0

t

NðτÞdτ

ð42Þ

Thus, Z

t 0

NðτÞdτ ¼ Vðχ 1 ð0Þ; z1 ð0Þ; z2 ð0ÞÞ VðtÞ

ð43Þ

_ r 0and VðtÞ Z 0, the following results can be Considering VðtÞ obtained easily: Z t lim NðτÞdτ r1 ð44Þ t-1

7

ð47Þ

Therefore, with t-1we have left with, ð48Þ

The above Eq. (48) reveals the fact that q2 þ Kgp2 must converge to zero when z1 converges to the zero. Now according to the definition of the state model of Eq. (20) q_ 2 ¼ p2 . Consequently, we can conclude that the state q2 and p2 are orthogonal to each other. In addition, K is a constant matrix. Furthermore, by definition, g(q) is not a null matrix (vide Eq. (21)). Therefore, asymptotic convergence of q2 þ Kgp2 to zero clearly implies that the individual element q2 and p2 must converge to zero when t-1. Hence, the proposed control law ensures the global stabilization of the original n-DOF underactuated system (shown in Eq. (20)).

5. Real time validation

0

_ The derivative of Vcan be expressed as € ¼ ½2c1 zT z_ 1 þ 2c2 zT z_ 2 V 1 2

ð45Þ

Since z1 ; z2 ; z_ 1 ; z_ 2 are bounded, Eq. (45) implies the fact that _ VðtÞis a continuous function of time. Hence, with the application of Barbalat0 s Lemma [62] it can be proved that z1 and z2 converge to zero as t-1. Lemma 2. The global asymptotic stability of the reduced order system of (34) together with the global asymptotic stability of the zero dynamics system (38) ensures the global asymptotic stability of the original underactuated system of (20). Proof. The global asymptotic stability of the reduced order system of Eq. (34) ensures that the state vectors of the reduced order system (i.e. z1, z2) will asymptotically converge to the desired equilibrium. That is z1 will asymptotically converge to zero as t-1. z1 ¼ q2 Kðq1 þ p1 gp2 Þ-0

ð46Þ

In addition, the global asymptotic stability of the zero dynamic system (vide Eq. (38)) ensures that the q1 and p1 will asymptotically converge to zero as t-1(Vide Remark 8).

To verify the effectiveness of the proposed control algorithm, it has been applied on a cart-pole system. The Digital Pendulum System manufactured by Feedback Instruments Ltd, UK, bearing model number 33-005-PCI [38] has been used as a laboratory prototype of cart-pole system. The total experimental setup is shown in Fig. 2. The task is to stabilize the pendulum in vertical upright position while regula0ting the motion of the cart along a desired trajectory. The state model of the cart-pole system [39] is shown below (where x1 ¼x and x2 ¼ θ) x_ 1 ¼ x3 x_ 2 ¼ x4 x_ 3 ¼ x_ 4 ¼

μgl sin 2x2 μx24 sin x2 bx3 au 2d

d

μg sin 2x2 μ 2d

x24

d

þ

d

sin x2 bx3 lu þ d d d

ð49Þ

In the above state model, μ ¼ lðM þ mÞd ¼ J þ μl sin 2 x2 and 2 a ¼ l þ J=M þ m, where M is the mass of the cart, m is the mass of the pendulum bob, l represents the length of the rod, J represents the moment of inertia, and b represents the coefficient of viscous friction between cart0 s wheels and rails.

Fig. 2. Experimental setup of the Digital Inverted Pendulum.



8

The drift vector field for x3 and x4 are as follows: f1 ¼ f2 ¼

μgl sin 2x2 μ

2d

x24

sin x2 bx3 d d

μg sin 2x2 μx24 sin x2 bx3 2d

d

d

Table 1 Parameters of the proposed block backstepping controller.

ð50aÞ ð50bÞ

c1

c2

λ

k

10

10

0.001

1

The control fields are given by, g 1 ¼ a=d and g 2 ¼ l=d. Moreover, d_ ¼ μlx4 sin 2x2

ð51Þ

d€ ¼ 2μlx24 cos 2x2 þ μlx_ 4 sin 2x2

ð52Þ

μx3 cos x2 b μgl cos 2x2 2μx4 sin x2 lu au 4 x4 f2þ f1þ f_ 1 ¼ d d d d d d ð53Þ and μx3 cos x2 b μg cos 2x2 2μx4 sin x2 lu au f_ 2 ¼ 4 x4 f2þ f1þ d d d d d d ð54Þ Now, z1 is defines as following: z1 ¼ x2 kðx1 þldx3 adx4 Þ

ð55Þ

Hence, the dynamics of z1 becomes: _ z_ 1 ¼ x4 kðx3 þ dðlx 3 ax4 Þ þ dðlf 1 af 2 ÞÞ The stabilization function

ð56Þ

α1 is defined as:

_ α1 ¼ λχ 1 c1 z1 þ kðx3 þ dðlx 3 ax4 Þ þ dðlf 1 af 2 ÞÞ

ð57Þ

The second control variable z2 is defined as z2 ¼ x4 α1 and the dynamics of z2 is defined as following: lu z_ 2 ¼ f 2 þ þ λz1 þ c1 ðz2 c1 z1 λχ 1 Þ h d au 2 k f 1 þ þ 2μlx24 cos 2x2 ðlx3 ax4 Þ þ μgl x4 cos 2x2 d lu þ 2μlx4 sin 2x2 ðlf 1 af 2 Þ þ μl sin 2x2 f 2 þ d lu 2μlx4 sin x2 f 2 þ μlx34 cos x2 blf 1 d blau lu μagx4 cos x2 2μax4 sin x2 f 2 þ d d ba2 u 3 μax4 cos x2 baf 1 d

Fig. 3. (a) Angular variation of the pendulum with time; (b) Angular velocity of the pendulum.

" ð58Þ

Hence, comparison of Eq. (58) with Eq. (8), yields the following expression for ψ andϕ: l a μl sin 2x2 2μl sin x2 x4 ψ ¼ k þ d d d d ba 2μalx4 sin x2 ðl aÞ þ d d 2

2

ϕ ¼ f 2 k½f 1 þ2μlx24 cos 2x2 ðlx3 ax4 Þ 2 þ 2μlx4 sin 2x2 ðlf 1 af 2 Þ þ μgl x4 cos 2x2 3 2μlx4 sin x2 f 2 μlx4 cos x2 blf 1 μagx4 cos x2 þ 2μax4 sin x2 f 2 þ μax34 cos x2 þ baf 1 þ μlf 2 sin 2x2

ð59Þ

þ k ld

1 þ kda þ ak dμgðl aÞ 2

2

# x3

k μgðl aÞx1 þ k½1 þ dl þ bða lÞ þlkμgdx3 2

x4 ¼

1 þ dl þ bða lÞ þ lkμgd

2

1 þ kda þ ak dμgðl aÞ 2

2

As we have stated in Section 2.2, Eqs. (15a)–(15c) depict the zero dynamics structure for a 2 degree of freedom system. Consequently, in the special case of the inverted pendulum (ðz1 ¼ 0andz_ 1 ¼ 0Þ) yield the following expression of x2 and x4. " # 2 2 adk μgðl aÞ x1 x2 ¼ k 1 2 2 1 þkda þ ak dμgðl aÞ

ð61bÞ

Now, if we replace the x2 and x4 terms of f1, ψ, and ϕ by the expressions of (61a) and (61b), respectively, then the equations of zero dynamics for the inverted pendulum will take the following form: x_ 1 ¼ x3 x_ 3 ¼ f 1 ðx1 ; x3 Þ g 1 ðx1 ; x3 Þψ 1 ðx1 ; x3 Þϕðx1 ; x3 Þ

ð60Þ

ð61aÞ

ð61cÞ

In Eq. (61c), f 1 ðx1 ; x3 Þ; g 1 ðx1 ; x3 Þ; ψ 1 ðx1 ; x3 Þ and ϕðx1 ; x3 Þ represent the same functions f1, g1, ψ-1, and ϕ, with the only difference being that the last equation, all the x2 and x4 terms have been replaced by the expression of (61a) and (61b), respectively. The gain k is selected in such a manner that it will ensure the stability of the zero dynamic system of (61c). The control input u is defined according to Eq. (10). The design parameters c1, c2, λ can be chosen arbitrarily, provided the condition c1 4 0, c2 40 and λ 40 is satisfied. Constant k is selected in such a manner that it ensures the stability of the zero dynamic



9

Fig. 5. Control input to the system.

be observed that the cart is able to follow its desired trajectory within few seconds. Moreover, the control input to the system (the voltage applied on the dc motor armature) is varying within a safe limit, which ensures that the control input will never generate high stress on the associated mechanical accessories. Thus, the block backstepping controller with integral action manifests a very fast response during the motion control of the inverted pendulum on cart system and maintains the stability of the pendulum in its vertical upright position even in the face of impulse disturbance. [One video file (video 1) of real-time experimentation has also been attached as a supplementary video file.] Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.isatra.2013.12.021. Fig. 4. (a) Cart displacement; (b) Cart Velocity.

6. Conclusions system of (61c). The control parameters values have been shown in Table 1. The length and weight of the pendulum are 0.402 m and 0.095 kg. The cart mass is 1.12 kg. The moment of inertia of the inverted pendulum is 0.014 kg-m2. The dynamic friction coefficient between cart and rail is 0.05 kg/s. These values of the physical parameters of the digital pendulum setup are obtained from the Users’ Manual of the Digital Pendulum System [38]. Although the maximum force that can be applied on the Cart is 16.5 N, the maximum control force actually applied on the cart has been limited to 6 N for smooth operation of actuator. The resolution of the encoder is 2048 increments per one revolution. The experiment has been performed for 100 s. As already mentioned, Matlabs Simulink environment and Real-time Windows Target has been used to implement and test the proposed control algorithm in real-time. The control algorithm has been developed as a Simulink model and then it has been passed through the “Build” operation to create all the executable files that are necessary for the real-time implementation. In order to verify the effectiveness of this proposed controller in real situation, an external disturbance has been introduced by applying an impulse force on the pendulum after 47 s in an experimental run while it was in the vertical upright position. The dynamic responses of the pendulum for the angular position in space and angular velocity have been shown in Fig. 3(a) and (b) respectively during the application of this impulse disturbance. The position of the cart on the rail and the velocity of the cart during the application of the impact disturbance are shown in Fig. 4(a) and (b), respectively. The variation of the input voltage on the driving motor terminal has been shown in Fig. 5. From these results, it can be observed that the cart is moving quickly to reduce the impact of the disturbance on the pendulum. In addition, it can

This paper presents a theoretical framework for designing a block backstepping controller for nonlinear underactuated mechanical system. The proposed control algorithm ensures the global asymptotic stability of the origin of the underactuated system. At the outset of the design, a global change of coordinates has been introduced to transform the state model of the underactuated system into a block strict feedback form, which is convenient for backstepping design for MIMO systems. Thereafter, a nonlinear block-backstepping control law has been designed for the generic underactuated system. Integral action has been incorporated in the control law to enhance the steady state performance of the controller. In addition, the zero dynamic stability of the controller has thoroughly been analyzed to ensure the global asymptotic stability of the overall nonlinear system. The effectiveness of the proposed control algorithm has been established by experimental studies in hard real-time. The experimental results establish the fact that the theoretical design of the control law is apt for real-time applications. Finally, to conclude, the authors take liberty to claim that the proposed control law is not only able to ensure theoretical asymptotic global stability for a large class of underactuated systems, but it also exhibits an excellent performance in the case of real-time applications.

Acknowledgment This work was supported in part by the Ministry of Science and Technology under Fellowship scheme A.20020/11/97-IFD. [Inspire Fellowship scheme]. Authors are very much thankful to Mr. Bedadipta Bain for his constant support and valuable suggestions during the preparation of this article. Last but not the least, the first author would like to express his deepest gratitude to all the


10


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