Orbital Stabilization of Inverted-Pendulum Systems ... - Semantic Scholar

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

Orbital Stabilization of Inverted-Pendulum Systems via Coupled Sliding-Mode Control Mun-Soo Park and Dongkyoung Chwa

Abstract—In this paper, we propose a coupled sliding-mode control (SMC) method for the periodic orbit generation and the robust exponential orbital stabilization of inverted-pendulum systems. We first design an SMC law to force a coupled sliding surface to be reached in finite time, such that the zero dynamics are generated in the form of a second-order undamped and forced nonlinear differential equation. Through the stability analysis, it is shown that there exist exponentially stable periodic solutions of the resulting zero dynamics (i.e., limit cycles around either the upright or downward equilibrium), even in the presence of the matched disturbance. Second, we design a target orbit stabilization control law by further introducing an auxiliary control law to the designed SMC law. This auxiliary control law utilizes the general integral of the autonomous zero dynamics, which preserves its zero value along the given target orbit, and thus, it can contribute to the exponential stabilization of the general integral. To demonstrate the validity of the proposed method, both the periodic orbit generation and target orbit stabilization control of the cart–pendulum, as an example among inverted-pendulum systems, are performed in numerical simulations. Index Terms—Coupled sliding-mode control (SMC), coupled sliding surface, inverted-pendulum systems, robust exponential orbital stabilization, second-order undamped and forced nonlinear differential equations.

I. I NTRODUCTION

I

N MANY real industrial applications, periodic motions have been considered as either goals to be achieved or obstructions to be removed. For example, whereas a limit cycle in the steady state of the precise positioning control system (e.g., x–y table) has to be eliminated since it severely degrades the positioning accuracy [1], the oscillatory motion of internal states (i.e., current and flux) in the torque control of rotating machines (e.g., induction motors) should be obtained to regulate the angular velocity or torque [2]. In particular, underactuated mechanical systems such as walking robots [3]– [8] and vertical takeoff and landing aircraft [9] are among the most crucial examples that require periodic motions and thus need to be orbitally stabilized.

Manuscript received February 1, 2008; revised April 10, 2009. First published April 28, 2009; current version published August 12, 2009. This work was supported by the Basic Research Program of the Korea Science and Engineering Foundation under Grant 2009-0075544. M.-S. Park was with the Department of Electrical and Computer Engineering, Ajou University, Suwon 443-749, Korea. He is now with the Korea Institute of Aerospace Technology, Daejeon 306-811, Korea. D. Chwa is with the Department of Electrical and Computer Engineering, Ajou University, Suwon 443-749, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2021178

A. Literature Review The control problem of underactuated mechanical systems has drawn much attention since the number of control inputs is less than that of variables to be controlled [10]–[13]. In the case that periodic motions are required [3], [4], [14], [15], the motion planning of such systems becomes more involved and more difficult due to the fact that zero dynamics should have periodic solutions (see [5] and [6] and the references therein). Recently, there has been a study on the constructive orbital stabilization via the virtual-constraint approach [16], which introduced the virtually constrained relations between configuration variables and synthesized a nonlinear feedback control law to make the constraints invariant. Then, an exponentially stable periodic solution is generated by forcing a general integral of the resulting zero dynamics to preserve its zero value. Other than the cart–pendulum [16], this method has been applied to various underactuated mechanical systems such as the Furuta pendulum [17], Pendubot [18], [19], and walking robots [7], [8]. This approach was further extended in [20], where the time-dependent virtual constraints are considered to have more involved motion, such that the pendulum oscillates around the vertical axis and the cart moves either forward or backward with the prescribed velocity. Even with its extensive applicability, the virtual-constraint approach has the following limitations: 1) much complexity occurs in solving the periodic differential Riccati equations required for the implementation of the control law; 2) only the local stability can be guaranteed in theory (in that initial conditions are assumed to belong to some vicinity of target orbits); and 3) robustness is not explicitly considered in the derivation of the feedback control law. In [21], a second-order sliding-mode control (SMC) has been presented for the orbital stabilization of the inverted pendulum. However, the designed controller is orbitally stabilizing only the actuated variables by forcing them to track the modified Van der Pol oscillator, and the stability analysis of the zero dynamics (the closed-loop unactuated dynamics) is not achieved. Thus, it is needed to study the orbital stabilization of underactuated mechanical systems, which involves rigorous stability analysis of periodic solutions of the zero dynamics, as well as the overall nonlinear system, and also considers practicality and robustness issues in the design and implementation of control law. In this paper, we introduce a novel robust and exponential orbital stabilization method for inverted-pendulum systems (as a benchmark example of underactuated mechanical systems) using the SMC method, which has been applied to many industrial applications such as electrical drives [22]–[25], servo system [26], crane system [27], robotics [28]–[30], and underactuated systems [31], [32].

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PARK AND CHWA: ORBITAL STABILIZATION OF INVERTED-PENDULUM SYSTEMS VIA COUPLED SMC

B. Problem Formulation Consider an inverted-pendulum system subject to a lumped disturbance D as follows: d dt



∂L ∂ q˙

 −

∂L = Bu + D ∂q

m11 (q) m12 (q) m21 (q) m22 (q)



S = ζ˙ + σζ

     n1 (q, q) q¨a u + d1 ˙ + = . (2) q¨u ˙ 0 n2 (q, q)

Note here that qu is the unactuated joint angle, and it satisfies 

qu ∈ D for the domain D = [−π, π], unless multiple equilibria 

are considered. For notational convenience, we introduce Cuh =  R × Suh (Clh = R × Slh , respectively) as the configuration space of the inverted-pendulum system in (2) over the upper half plane (over the lower half plane, respectively) by defin



ing Suh = {θ ∈ D| |θ| < π/2} (Slh = {θ ∈ D| |θ| > π/2}, respectively). Using the Legendre transformation in [33], the system (2) can be expressed as ˙ + ga (q)u + da q¨a = fa (q, q) q¨u = fu (q, q) ˙ + gu (q)u + du

(3)

where   fa = −n1 + m12 · m−1 22 · n2 · ga   −1 fu = m21 · m−1 22 · n1 − m11 · m22 · n2 · ga ga = m22 · det(M)−1

gu = −m21 · m−1 22 · ga

d a = ga · d 1

du = gu · d1

proposed scheme is outlined (see Fig. 4 for summary). First, we define a coupled variable in the form of ζ = λqa + qu as a linear combination of actuated and unactuated variables by using a positive coupling parameter λ, and then, we define a coupled sliding surface given by

(1)

where q = (qa , qu )T ∈ R2 and q˙ = (q˙a , q˙u )T ∈ R2 are the generalized coordinates and velocities, respectively; qa and qu are the actuated and unactuated variables, respectively; ˙ = B = (1, 0)T and u ∈ R is a control input variable; L(q, q) (1/2)q˙T M(q)q˙ − V(q) is a Lagrangian; M(q) is a symmetric positive-definite inertia matrix; V(q) is a potential energy of the systems; D = (d1 , 0)T is assumed to satisfy the classical matching condition; and d1 ∈ R includes parameter uncertainties, external disturbance, and unmodeled dynamics such as viscous and Coulomb friction forces exerted on the actuated joint, and it is bounded as |d1 | ≤ d¯1 for a known constant d¯1 . The inverted-pendulum system in (1) can be rewritten as 

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and da and du satisfy that |da | ≤ d¯a and |du | ≤ d¯u for known constants d¯a and d¯u , respectively. Our main goal is to control the inverted-pendulum system in (3) so as to result in a robust exponentially stable periodic motion via a so-called coupled SMC (i.e., an SMC based on the coupled sliding surface). In what follows, the procedure of the

(4)

where σ is a positive design parameter. Based on the invariance of sliding motion (i.e., S = 0) against the matched disturbance D, by virtue of SMC, we will show that the coupled sliding surface is reached within finite time t1 ≥ 0 (Theorem 1), in which case the zero dynamics of the inverted-pendulum system (3) can be obtained as the following second-order undamped and forced nonlinear differential equation (Theorem 2): qu + B(qu ) = C(t, qu ). A(qu )¨

(5)

In the case of the cart–pendulum as an illustrative example, the zero dynamics, with the coupling parameter λ chosen such that the zero dynamics are well defined over either the upper half plane or the lower half plane, will be shown to have the following properties. 1) If the zero dynamics of the cart–pendulum become exponentially autonomous (i.e., the forcing term C(t, qu ) in (5) vanishes exponentially), then there exists a general integral of the resulting zero dynamics, which is given in an explicit form for the given initial conditions and preserves its zero value along the solutions of the zero dynamics (Lemma 1). 2) As a consequence of 1), the trajectory of the autonomous zero dynamics for the given initial condition is bounded and periodic. That is, there exists a set of periodic orbits, parameterized by the given initial conditions (Lemma 2). In other words, with the properly chosen coupling parameter λ, if the zero dynamics in (5) are exponentially autonomous over the upper half plane (over the lower half plane, respectively), then they result in limit cycles around the upright equilibrium (around the downward equilibrium, respectively). Last, it will be proved that the solutions of the overall closed-loop dynamics of the cart–pendulum are periodic by showing that the zero dynamics in (5) become exponentially autonomous and by using properties 1) and 2) (Theorem 3). In addition to this periodic orbit generation, the stabilization toward a given target orbit will be established by introducing an auxiliary control law to the proposed coupled SMC law. By utilizing the general integral of the autonomous zero dynamics, this auxiliary control law is designed such that it contributes to the stabilization of the coupled sliding surface, as well as the general integral that preserves its zero value along the target orbit (Theorem 4). Unlike the previous method in [16] and [19], the proposed scheme guarantees robust stability of the target orbit against model uncertainty and external disturbance by virtue of SMC, and this property will be demonstrated via numerical simulations.

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choice of θ0 will be given later in Lemma 1. Also, we define a coupled sliding surface by modifying (4) as follows: ˙ Sc = ζ˜ + σ ζ˜

(9)

where ζ˜ = ζ − ζd and ζd is an exponentially vanishing reference for the coupled variable ζ, which is given by ζd (t) = ζ¯ · e−c(t−t0 )

Fig. 1. Cart–pendulum.

II. Z ERO D YNAMICS OF C ART –P ENDULUM W ITH C OUPLED SMC L AW The dynamics of the cart–pendulum shown in Fig. 1 can be described in the following as in [34]: 

γ β cos θ

β cos θ α

      x ¨ −β sin θθ˙2 u + d1 + = (6) θ¨ −η sin θ 0

where α = ml2 ; β = ml; γ = M + m; η = mgl (with M and m being the masses of the cart and pendulum, respectively; l being the length of the pendulum; and g being the gravitational acceleration); θ and x ¨ are the angle of the pendulum with respect to the vertical line and the moving distance of the cart from the zero position, respectively; and u is a control force applied to the cart. By using q = (x, θ)T , the state-space form of the cart–pendulum is given in the following as in (3): ˙ + gx (q)u + dx x ¨ = fx (q, q) ˙ + gθ (q)u + dθ θ¨ = fθ (q, q)

(7)

where |dx | ≤ d¯x and |dθ | ≤ d¯θ for known constants d¯x and d¯θ , respectively, and each function can readily be obtained as     β cos θ η sin θ · gx fx = β sin θθ˙2 − α     β γ cos θ β sin θθ˙2 + η sin θ · gx fθ = − α α   β α gθ = − cos θ · gx gx = γα − β 2 cos2 θ α d x = gx · d 1

dθ = gθ · d1 .

(8)

with constants ζ¯ and c > 0. Here, it should be noted that ζd ˙ θ) = is introduced just to avoid the zero solution [i.e., (x, ˙ x, θ, (0, 0, 0, 0)] and, thus, to generate periodic solutions, even when the system states start at the origin (more details of which will be explained in Section III-C and illustrated in Section V). Then, we can design a coupled SMC law such that the coupled sliding surface in (9) can be reached in finite time as in the following theorem. Theorem 1 (Coupled SMC Law): For the cart–pendulum in (7), consider the coupled sliding surface in (9) and a coupled SMC law given by uc = ueq + usw

(11)

where ueq = (−λfx − fθ − σ ζ˙ + (c2 − cσ)ζd )/(λgx + gθ ), usw = −(k · sgn(Sc ))/(λgx + gθ ), and k > λd¯x + d¯θ . If  ¯= (β/α) cos θu∗ the coupling parameter is chosen as λ = λ ∗ by using θu = (π/2) − for the arbitrarily small constant 0 <  π/2, then the coupled sliding surface can be reached within finite time t1 ≥ 0 over either the upper half plane ˙ θ) ∈ Clh ), and ˙ θ) ∈ Cuh ) or the lower half plane ((θ, ((θ, the resulting sliding mode, a so-called coupled sliding mode defined by Sc = 0, is guaranteed to be invariant, even in the presence of lumped disturbances dx and dθ . Proof: Consider a Lyapunov function given by Vc =

1 2 S . 2 c

(12)

By using (7), together with uc in (11) instead of u, and θ¨0 = θ˙0 = 0 and ζ¨d + σ ζ˙d = (c2 − cσ) · ζd [which follows from the definition of ζd in (10)], we can calculate the time derivative of Vc as follows:  V˙ c = Sc ζ¨ − ζ¨d + σ(ζ˙ − ζ˙d )  = Sc λfx + fθ + (λgx + gθ )uc + σ ζ˙ − ζ¨d − σ ζ˙d + λdx + dθ

Here, note that gx > 0 since (γα − β 2 cos2 θ) > 0 for all θ.

= Sc (−k · sgn(Sc ) + λdx + dθ )   ≤ − k − (λd¯x + d¯θ ) |Sc |

A. Coupled SMC Law

< 0.

Let us define a coupled variable for the cart–pendulum in ¯ where λ is a positive coupling parameter, (7) as ζ = λx + θ, θ¯ = θ − θ0 , and θ0 corresponds to either the upright (θ0 = 0) or downward (θ0 = ±π) equilibrium point. The details for the

(10)

(13)

This shows that if the coupled SMC law (11) is well defined (i.e., λgx + gθ = 0), then the coupled sliding surface Sc is reached within finite time t1 ≥ 0. In addition, according to the equivalent control method [35], the resulting sliding mode

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PARK AND CHWA: ORBITAL STABILIZATION OF INVERTED-PENDULUM SYSTEMS VIA COUPLED SMC

Sc = 0 is invariant for all t ≥ t1 against the coupled disturbance λdx + dθ since it satisfies the classical matching condition (i.e., λdx + dθ = (λgx + gθ )d1 ) for the coupled SMC law (11).  Now, let us consider two isolated regions given by Cˆuh = 



R × Sˆuh and Cˆlh = R × Sˆlh for Sˆuh = {θ ∈ D| |θ| < θu∗ } and  Sˆlh = {θ ∈ D| |θ| > θl∗ }, with θl∗ = (π/2) + , as subsets of Cuh and Clh [which are defined below (2)], respectively. By using gθ = −((β/α) cos θ) · gx from (8) and λ = (β/α) cos θu∗ , we have λgx + gθ = (β/α)(cos θu∗ − cos θ) · gx . Then, it is obvious that λgx + gθ = 0 for all θ ∈ Sˆuh (θ ∈ Sˆlh , respec˙ θ) ∈ Cˆlh , respectively) ˙ θ) ∈ Cˆuh ((θ, tively) and, thus, for all (θ, since gx > 0, as shown before [below (8)]. Therefore, we can conclude that the coupled SMC law in (11) is well defined over the upper half plane (over the lower half plane, respectively) as → 0. This completes the proof of the theorem.  Remark 1: In fact, the proposed control law in (11) is well defined for any constant value of λ such that λgx + gθ = 0 for all |θ| < π/2 over the upper-half plane (or for all π/2 < |θ| ≤ π over the lower half plane). For example, one can choose λ = 2(β/α) in the case of the upper half plane, satisfying λ > β/α so as to prevent the denominator of the control law from becoming zero. However, this choice cannot be admissible in our problem since it renders the zero dynamics unstable, as will be explained later in Section III. To this end, λ should be chosen such that λ < (β/α) cos θ for all θ over the upper half plane or the lower half plane, as stated in Theorem 1. In other words, the coupled SMC law in (11) guarantees the boundedness and asymptotic stability of ζ˜ and, thus, those of the coupled variable ζ over each isolated region (i.e., Cuh or Clh ). However, it is notable that this implies neither the boundedness nor the (orbital) stability of actuated and unactuated variables (i.e., θ and x, respectively) of the closed-loop system in ζ. In what follows, we will show that the coupled SMC law results in a second-order undamped and forced nonlinear differential equation, which is the zero dynamics1 of the closedloop system (7) and (11). Based on this property, we will rigorously investigate the orbital stability of the overall closedloop dynamics in the next section. B. Zero Dynamics The next theorem states that the coupled SMC law in (11) generates the zero dynamics of the cart–pendulum in (7) in the form of a second-order undamped and forced nonlinear differential equation. Theorem 2 (Zero Dynamics): Consider the coupled sliding surface in (9), and suppose that there exists the coupled SMC law uc in (11) such that the coupled sliding mode (i.e., Sc = 0) is reached within finite time t1 ≥ 0 and that it is invariant along the solution of the closed-loop dynamics for all t ≥ t1 . Then, the zero dynamics of the closed-loop cart–pendulum become A(θ, λ)θ¨ + B(θ, λ) = C(t, θ, p, ζd ),

t ≥ t1

(14)

1 In this paper, the closed-loop dynamics in the coupled sliding mode are referred to as the zero dynamics of the cart-pendulum in the sense that they are constrained by the coupled sliding mode (Sc = 0) rather than the zero output (ζ = 0).

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where p = (λ, σ, c) is a design parameter vector, and each function is given by A(θ, λ) = β cos θ − λα B(θ, λ) = λη sin θ C(t, θ, p, ζd ) = σ 2 β cos θ · χ(t, p) χ(t, p) = (ζ1 − ζd1 ) · e−σ(t−t1 ) +

c2 ζd . σ2

(15)

Proof: By following Theorem 1, we have Sc = 0 for all t ≥ t1 by the coupled SMC law in (11). Then, using θ¨0 = θ˙0 = 0, we obtain from (9) ¯ = ζ˙d + σζd λx˙ + θ˙ + σ(λx + θ)

(16)

which can be integrated into λx + θ¯ = (ζ1 − ζd1 ) · e−σ(t−t1 ) + ζd

(17)

where ζ1 and ζd1 denote ζ(t1 ) and ζd (t1 ), respectively. Also, using ζ¨d + σ ζ˙d = (c2 − cσ) · ζd from the definition of ζd in (10), (16) can be differentiated as 2 1 σ ˙ + c − cσ ζd . x ¨ = − θ¨ − (λx˙ + θ) λ λ λ

(18)

Then, using (16), (17), and ζ˙d = −cζd from (10), (18) readily becomes   c2 1 σ2 x ¨ = − θ¨ + (ζ1 − ζd1 ) · e−σ(t−t1 ) + 2 ζd . (19) λ λ σ Finally, substituting x ¨ in the second row of dynamics (6) into (19) yields (β cos θ − λα)θ¨ + λη sin θ − σ 2 β cos θ · χ(t, p) = 0

(20)

which gives (14). This completes the proof of the theorem.  Remark 2: The zero dynamics in (14) differ from the dynamics induced by the virtual constraint in [7], [8], [16]–[20], in that (14) is in the form of undamped and forced differential equation, whereas the virtually constrained dynamics include a quadratic term in velocity. Since the forcing term C(t, θ, p, ζd ) in (14) exponentially vanishes (as will be shown later in Theorem 3), the zero dynamics in (14) can simplify the proposed method in control design and stability analysis of the overall system dynamics. On the other hand, the quadratic force term in the virtually constrained dynamics makes it much more difficult and complicated to analyze the existence and stability of the periodic solution of the controlled system [20]. III. E XISTENCE OF P ERIODIC O RBITS In this section, we investigate properties of the zero dynamics in (14), together with the existence of its periodic solutions. First, in the case that the zero dynamics are autonomous (i.e., C(t, θ, p, ζd ) = 0), it is shown that there exists a first integral for the zero dynamics, and also, every solution of the

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autonomous zero dynamics is bounded and periodic. Then, by showing that the zero dynamics in (14) exponentially become autonomous, it is proved that there exist bounded and periodic solutions of the overall closed-loop system (7) and (11). A. Autonomous Zero Dynamics Suppose that the zero dynamics in (14) become autonomous as θ¨ + w(θ, λ) = 0

t ≥ t2

(21)

where t2 > t1 and w(θ, λ) = B(θ, λ)/A(θ, λ). Here, note that w(θ, λ) is well defined over either the upper half plane or the lower half plane, which, along with the property of the autonomous zero dynamics in (21), will be explained in the following lemma. Lemma 1 (General Integral): Consider the zero dynamics in ˙ θ) of the zero (21), and suppose that there exists a solution (θ, ˙ ˙ dynamics with the initial conditions θ2 = θ(t2 ) and θ2 = θ(t2 ) over either the upper half plane or the lower half plane. Then, a general integral of the zero dynamics in (21), which preserves its zero value along this solution, is given by ¯ λ) + I2 ˙ θ, θ˙2 , θ2 , θ0 , λ) = θ˙2 + 2W ¯ (θ, (22) I(θ,

¯ λ) = ¯ w(s, λ)ds, and I2 = −θ˙2 − ¯ (θ, where θ¯ = θ − θ0 , W 2 θ ¯ ¯ 2 · W (θ2 , λ). Proof: Based on the fact that 2θ¨ = dθ˙2 /dθ holds along ˙ θ) of the autonomous zero dynamics in (21), the solution (θ, one can define a new variable Y(θ) = (dθ/dt)2 satisfying 2θ¨ = (dY/dθ). Then, (21) can be rewritten in the equivalent form dY = −2w(θ, λ) dθ

(23)

which is a linear first-order differential equation of Y with respect to θ. Also, its general solution has the following form: θ Y(θ) = Y(θ2 ) − 2

w(s, λ) ds.

(24)

θ2

From (15) and (21), and using η = βg from (6), we have ¯ λ) = λg w(θ,

β sin θ¯ β cos θ¯ − λα

(25)

where θ¯ = θ − θ0 , as defined in Section II-A, and θ0 = 0 (θ0 = ±π, respectively) corresponds to the upright equilibrium point (downward equilibrium points, respectively). Here, note that one can choose θ0 = π (θ0 = −π, respectively) for π/2 < θ ≤ 3π/2 (−3π/2 ≤ θ < −π/2, respectively) such that −π/2 < θ¯ < π/2 for all θ ∈ Slh [where Slh is defined in the paragraph below (2)].2 Now, recall from Theorem 1 that 2 Since θ ∈ [−π, π] can be considered as the whole space of the pendulum angle, we have two downward equilibrium points (i.e., θ0 = π and θ0 = −π), which coincide with each other, and thus, they make no difference in physical sense.

λ = (β/α) cos θu∗ , together with θu∗ = π/2 − for  π/2, and from the proof of Theorem 1 that two isolated regions Cˆuh and Cˆlh , together with both Sˆuh and Sˆlh , are defined. Since we ¯ < θ∗ for all θ ∈ Sˆlh and appropriately chosen θ0 , as exhave |θ| u ¯ plained earlier, ˆthe denominator of w(θ, λ) in (25)¯is positive for ˆ it is obvious that w( all θ ∈ Suh Slh . Therefore,

θ, λ) in (25) is ¯ λ) = ¯ w(s, λ)ds = ¯ (θ, well defined in Cˆuh Cˆlh , and also, W θ −λg ln(β cos θ¯ − λα) is well defined in the same region. That is, they are well defined over either the upper half plane Cuh or the lower half plane Clh as → 0. Then, (24) can be rewritten by ¯ λ) + 2W ¯ (θ, ¯ (θ¯2 , λ). Y(θ) = θ˙22 − 2W

(26)

Therefore, we have (22) from (26), which is a general integral of the autonomous zero dynamics in (21) for a constant I2 that is determined by the initial conditions.  Lemma 1, in other words, states that if there exists a solution of (21) with the coupling parameter chosen such that (21) is well defined in Cuh Clh as in Theorem 1, then the solution is a curve that completely lies on the level set I = 0, which is well defined in the same region. However, for the given initial conditions, the existence of a general integral does not suffice to ensure the periodicity of the solution, and thus, it is required to show that the solution specified by the general integral (22) represents a periodic orbit. In [36], for example, it was shown via phase-plane analysis that a second-order nonlinear differential equation having quadratic force term in velocity has nontrivial periodic solutions, provided that the continuous spring force term is odd with respect to an equilibrium point in its neighborhood. On the other hand, it is shown in [37] that all solutions of the undamped nonlinear differential equation are bounded and periodic under similar conditions. Now, based on these results, it will be shown in the following lemma that the solution of (21), satisfying (22) for the given initial conditions in Lemma 1, is bounded and periodic. Lemma 2 (Bounded and Periodic Solutions): For the given initial condition (θ˙2 , θ2 ) = (0, 0) ∈ Cuh Clh , suppose that ˙ θ) of the zero dynamics in (21), such there exists a solution (θ, that I = 0 as in Lemma 1. Then, the solution, which is well defined over either the upper half plane Cuh or the lower half plane Clh , is bounded and necessarily periodic. Proof: Let us first recall λ = β/α cos θu∗ , together with ∗ θu = π/2− for  π/2, and two isolated regions Cˆuh and Cˆlh , together with both Sˆuh and Sˆlh , as in the proof of Theorem 1. With the assumption that I = 0, we have from (22) ¯ λ) = −I2 ¯ (θ, θ˙2 + 2W

(27)

¯ λ) = −λg · ln(β cos θ¯ − β cos θ∗ ) is obtained by ¯ (θ, where W u using the proof of Lemma 1 and λ = β/α cos θu∗ , and I2 is as defined below (22). Note here that, with appropriately chosen ¯ λ) is positive for all ¯ (θ, θ0 as in the proof of Lemma 1, W ¯ λ) → ∞ as θ¯ → θ∗ since ¯ (θ, θ ∈ Sˆuh Sˆlh , and we have W u − ln(x) → ∞ as x → 0. Let us suppose that θ2 ∈ Sˆuh Sˆlh 

(i.e., |θ¯2 | = |θ2 − θ0 | < θu∗ for the appropriately chosen θ0 as in the previous paragraph). Then, we have −I2 < ∞, and thus, (27) implies θ˙ ∈ L∞ and θ ∈ Sˆuh Sˆlh from the fact that

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˙ < ∞ and W ¯ λ) < ∞. Otherwise, the left-hand side of ¯ (θ, |θ| (27) would become infinite, which leads to the contradiction. This property holds for all initial conditions (θ˙2 , θ2 ) ∈ Cuh Clh as → 0. Therefore, it is true that the solution satisfying (22) for the given initial condition is bounded in the domain Cuh Clh as → 0. ˙ θ) does not oscillate. Then, Now, suppose that the solution (θ, θ must be monotonic (i.e., we must have θ˙ ≥ 0 or θ˙ ≤ 0 for ¯ all t > t2 ). Assume that θ is positive for the appropriately chosen θ0 in Cuh Clh (a similar argument works for the case when θ¯ is negative). First, when θ˙ > 0, θ keeps increasing, which is contradictory to the boundedness of θ. Second, when θ˙ < 0, θ decreases below −π/2, which is again contradictory ˙ to the boundedness of θ. Finally,

t when limt→∞ θ = 0, the ¯ ˙ ˙ integration of (21) as θ = θ2 − t2 w(θ(s), λ)ds and the fact ¯ λ) = w(¯ c, λ), where w(¯ c, λ) is a positive that slims→∞ w(θ(s), ∞ constant (due to θ¯ > 0 as assumed) for c¯ = c + θ0 and t2 ds = ∞, lead to the contradiction to the boundedness of θ. With ˙ θ) must these arguments, we can conclude that the solution (θ, oscillate. In addition, by following the arguments in [36], it is true that the curve (27) in the phase plane is symmetric and ˙ θ) satisfying (27) is periodic. closed, and thus, the solution (θ, This can also be seen by using the Poincare–Bendixson theorem ˙ θ) of the [38], which can be employed as the solution (θ, autonomous zero dynamics remains in a finite region.  Consequently, it can be stated from Lemmas 1 and 2 that the autonomous zero dynamics in (21) generate a closed  ˙ θ) ∈ Cˆuh Cˆlh | I(θ, ˙ θ, θ˙2 , curve, which is defined as O = {(θ, θ2 , θ0 , λ) = 0, t ≥ t2 }3 for the given coupling parameter λ, appropriately chosen θ0 , and the time instant t2 ≥ t1 .4 It should be noted here that the periodic orbit O is parameterized by only two constants λ and t2 .5 Therefore, the shaping of the target orbit (which is one of the most crucial parts in the orbital stabilization) is simplified (more details will be illustrated in the next section). On the other hand, from the general integral in (22) defining a specific orbit for the given initial conditions, one can find a so-called first integral6 as in [19], which specifies a family of orbits corresponding to its level sets. As is well known, a first integral (which is an energy function) can be used as a classical Lyapunov function. In what follows, we will show that there exist exponentially stable periodic orbits of the original zero dynamics in (14) by constructing a Lyapunov function based on the first integral of the autonomous zero dynamics in (21).7 Consequently, it will be shown that this result implies the

domain Cˆuh Cˆlh is specified by an arbitrarily small constant . Thus, in theory, this domain can be enlarged almost up to Cuh Clh (the whole plane except the horizon) by letting  → 0. 4 Here, t is the reaching time of the coupled sliding surface S , as stated in c 1 Theorem 1, and t2 denotes the time instant when the original zero dynamics (14) become the autonomous one (21). 5 θ determines only the region between both isolated regions Cˆ ˆ 0 uh and Cuh , where an orbit is generated. Thus, the target orbit can be shaped by λ and t2 . 6 Following the definition in [38], a first integral is different from a general one in the sense that it preserves its value along any solution of a system, independently of the initial conditions. 7 In other words, any periodic orbits of the autonomous zero dynamics in (21) (i.e., O with specific λ and t2 ) are exponentially stable. 3 The

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exponential orbital stabilization of the cart–pendulum by the coupled SMC law in (11). B. Overall Closed-Loop Dynamics By following [19, Lemma 9], one can obtain the first integral of the autonomous zero dynamics in (21) from the general integral (22) derived in Lemma 1 as follows: ˙ θ, θ0 , λ) = θ˙2 + 2W (θ, ¯ λ) V(θ,

(28)

where λ is the same as in Theorem 1 and θ0 is appropriately chosen as in Lemma 1. Note here that the first integral in (28) satisfies I = V − Va for Va = −I2 , where Va = ˙ a ) and θa = θ(ta ) V(θ˙a , θa , θ0 , λ) is a constant, and θ˙a = θ(t for arbitrary time instant ta ≥ t1 . Also, it is obvious from (22) that V in (28) preserves its constant value along the solution of the autonomous zero dynamics in (21). Using Lemma 2, the constant level set V = Va can be used to define a family of periodic orbits of (21) parameterized by λ and ta , which is given by 

  Δ ˙ θ, θ0 , λ) = Va , t ≥ ta ˙ θ) ∈ Cˆuh Cˆlh  V(θ, W = (θ, (29) for λ = β/α cos θu∗ as in Theorem 1, appropriately chosen θ0 , and the time instant ta ≥ t1 . Then, based on the first integral in (28), one of our main results [i.e., the (uniform) exponential orbital stability of the original zero dynamics in (14) and that of the overall closed-loop system (6) and (11)] can be stated as in the following theorem. Theorem 3: Suppose that there exists the coupled SMC law given by (11) for (6), which results in the zero dynamics in (14) within finite time t1 ≥ 0. Then, every solution of the zero dynamics (14) converges uniformly and exponentially to the periodic orbits in W in (29), which is determined by λ and the arbitrary time instant ta . In addition, as the coupled output (ζ) and its reference (ζd ) vanish exponentially, the actuated and unactuated variables of the overall closed-loop system (11) for (6) are uniformly exponentially orbitally stable over either the upper half plane Cuh or the lower half plane Clh . Proof: In the first step, the uniform exponential stability of the zero dynamics in (14) will be proved. To this end, we will show the boundedness of solutions of the zero dynamics (14) in the domain Cˆuh Cˆlh , together with its exponential convergence to the autonomous one in (21). First, we rewrite the zero dynamics in (14) as follows: ¯ λ) + h(t, θ, ¯ p, ζd ), θ¨ = −w(θ,

t ≥ t1

(30)

¯ λ) is as in (25) and h(t, θ, ¯ p, ζd ) = C(t, θ, ¯ p, ζd )/ where w(θ, ¯ λ) from (15), with θ¯ = θ − θ0 . Let us consider a conA(θ, ¯ p, ζd )| for an tinuous function ψ, such that ψ ≥ ρ · |h(t, θ, arbitrary positive constant ρ, and ψ → 0 as h → 0, and thus,  t limt→∞ Ψ(t) = Ψ∞ < ∞ for Ψ(t) = t1 ψ(τ )dτ . Note that the existence of ψ and Ψ for all t ≥ t1 follows from the exponential decrease of χ in (15). Now, let us consider a continuous

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 ˙ θ) ¯ = and positive function defined as Vˆ (θ, V + Vˆ0 for an arˆ ˆ ∞ for all ˆ bitrary positive ˆconstant V0 , such that V0 < V ≤ ˙ ˆ (θ, θ) ∈ Cuh Clh from the fact that 0 < V ≤ ∞.8 Also, con ˙ θ, ¯ t) = sider a continuous increasing function given by Vz (θ, ¯ ˙ θ), ¯ such that ν1 (Θ) ≤ Vz (θ, ˙ ¯ e−Ψ(t) Vˆ (θ, ˆ θ, t) ≤ ν2 (Θ) for T ˙ ¯ ˆ all Θ = (θ, θ) in the domain Cuh Clh and for all t ≥ t1 . ¯ Here, Ψ(t) = Ψ(t) − Ψ∞ ≤ 0, and ν1 (·) and ν2 (·) are continuous increasing functions. Using (28), (30), and Vˆ = V + Vˆ0 from definition, we then obtain the time derivative of Vz as follows: ¯ ¯ V˙ z = − ψe−Ψ Vˆ + e−Ψ V˙

 ¯ ˙ ¯ p, ζd ) . = − e−Ψ ψ Vˆ − 2θh(t, θ,

(31)

¯ p, ζd )| ≥ 0 from definiUsing the properties ρ−1 ψ ≥ |h(t, θ, ¯ λ) > 0 for all θ ∈ Sˆuh Sˆlh and appropriately tion, and W (θ, chosen θ0 , and also, letting Vˆ0 = ρ−2 , it follows from (28) and (31) that  ¯ ˙ V˙ z ≤ − ψe−Ψ θ˙2 + ρ−2 − 2ρ−1 |θ|  2 ¯ ˙ − ρ−1 ≤ − ψe−Ψ |θ| ≤0

¯ λ) + h(t, ¯ θ, ¯ p, ζd ), θ¨ = −w(θ,

t ≥ t0

(33)

¯ θ, ¯ p, ζd ) = −β cos θ¯ / (β cos θ¯ − λα)(σ ζ˙ + k · where h(t, sgn(Sc ) + (c2 − cσ)ζd ). Then, one can choose a continuous ¯ such that it possesses similar properties function ψ¯ for h, ˙ ζd , and Sc are of ψ chosen for h, from the fact that ζ, bounded and vanish exponentially by following the results ˙ θ) [solutions of in [39]. Therefore, the boundedness of (θ, the closed-loop unactuated dynamics in (33)] in the reaching mode follows from the aforementioned result, and thus, ˆuh Sˆlh for any initial conditions in S θ˙ ∈ L∞ , and θ remains ˙ 0 ), θ(t0 )) ∈ Cˆuh Cˆlh , even in the reaching mode (i.e., (θ(t Sc = 0). Consequently, the solutions of the closed-loop unactuated dynamics converge uniformly and exponentially to the periodic orbits in W for the given initial conditions. In addition, since we have x = −(1/λ)θ¯ in the coupled ˙ x) sliding mode (i.e., Sc = 0) as ζ, ζd → 0, the solutions (x, of the closed-loop actuated dynamics are also periodic. This completes the proof.  C. Properties of Generated Periodic Orbits

(32)

which implies the uniform boundedness of Vz , and thus, that ˙ of the solutions ˆ (θ, θ) of the zero ˙dynamics in (14) in the ˆ domain Cuh Clh . That is, we have θ ∈ L∞ , and θ remains in the domain Sˆuh Sˆlh ; thus, we can conclude that there exists ˙ θ) of the zero dynamics for any initial a bounded solution (θ, ˙ value (θ(t1 ), θ(t1 )) ∈ Cˆuh Cˆlh . From the fact that h → 0 as t → tb for sufficiently large tb (due to the exponential convergence of χ(t, p) in (15) to zero), it is also clear that the zero dynamics in (30) are exponentially autonomous to ˙ θ, ¯ t) → Vˆ (θ, ˙ θ) ¯ since Ψ ¯ →0 (21). Accordingly, we have Vz (θ, ˙ ¯ ˆ (i.e., Ψ → Ψ∞ ), and thus, the resulting V (θ, θ) is a constant  (i.e., Vˆb (θ˙b , θb , θ0 , λ) = Vb + Vˆ0 ) since V˙ z ≤ 0 and Vz is lower bounded (i.e., Vz converges to a constant). Thus, we have V = Vb . Following Lemmas 1 and 2 with this result, we can ˙ therefore ˆ conclude that, for the initial condition (θ(t1 ), θ(t1 )) ∈ ˆ Cuh Clh , any solution of the zero dynamics in (14) or (30) converges uniformly exponentially to a periodic orbit in W in (29) with Va = Vb and ta = tb , which is parameterized by λ and tb . Second, we show that the aforementioned result [i.e., the uniform exponential orbital stability of the zero dynamics in (14) or (30)] implies the uniform exponential orbital stability of the overall closed-loop system (6) and (11). To do this, it necessarily requires to show the boundedness of the closed-loop unactuated dynamics in the reaching phase (i.e., for all t such that t0 ≤ t ≤ t1 ). To this end, we first con∗ ) ≤ W (θ, ¯ λ) ≤ follows from the fact that 0 < W (0, (β/α) cos θu ∗ , together with θ ∗ = (π/2) −  and arbitrary small ∞ for λ = (β/α) cos θu u constant   π/2, in the domain Sˆuh Sˆlh as in Lemma 2. 8 This

sider the following closed-loop unactuated dynamics9 instead of (30):

We have shown that the coupled SMC law in (11) can be used to generate a family of periodic orbits W in (29), which is parameterized by the coupling parameter λ and initial conditions (θ˙a , θa ). For the given initial condition (θ˙a , θa ) (coupling parameter λ, respectively), a periodic orbit can be specified by the coupling parameter λ (initial condition (θ˙a , θa ), respectively), as shown in Figs. 2 and 3. These figures show that the proposed coupled SMC method can shape the generated periodic orbits of the zero dynamics in (21) in the following two different ways: 1) For a fixed initial condition of (θ˙a , θa ), each value of λ determines the frequency of each limit cycle (see the left columns in Figs. 2 and 3 for θ ∈ Suh and θ ∈ Slh , respectively), and 2) for a fixed value of λ, each initial condition (θ˙a , θa ) determines both amplitude and frequency of each limit cycle (see the right columns in Figs. 2 and 3 for θ ∈ Suh and θ ∈ Slh , respectively). On the other hand, when the states of the cart–pendulum ˙ θ) = (0, 0, 0, 0) at t = 0) in (6) start at the origin (i.e., (x, ˙ x, θ, and ζ¯ in (10) is set to zero, the states remain at the origin all the time without any oscillation. Therefore, ζd can be used with appropriately chosen ζ¯ to have (θ˙a , θa ) = (0, 0) such that the orbital stabilization of the cart–pendulum is guaranteed for any initial condition in Cuh Clh , as stated in Lemma 2 (see Fig. 5). D. Further Discussions From a practical point of view, the zero dynamics may not be exactly the same as in (14) since there may exist unmodeled (possibly discontinuous and high-frequency) dynamics due to 9 The closed-loop unactuated dynamics in (33) are obtained by the substitution of the coupled SMC law (11) in the second equation of (6). Here, the disturbance dθ is neglected since it can effectively be rejected by virtue of sliding-mode control.

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Fig. 2. Family of orbits (around the upright equilibrium point: θ0 = 0) of the autonomous zero dynamics (21) for fixed initial conditions [(left) (θa , θa ) = (0, 30)] and fixed coupling parameter [(right) λ = 0.1228], respectively. In each case, periodic orbits are generated for various coupling parameters [(left) λ = [0.1, 0.5, 1, 2, 3, 4, 5, 6]] and initial conditions [(right) (θa , θa ) = [(0, 5), (0, 10), (0, 20), (0, 40), (0, 60), (0, 80), (0, 87)]], respectively. (a) Phase portraits for θa = 160◦ . (b) Phase portraits for λ = 0.1228. (c) Pendulum angle for θa = 160◦ . (d) Pendulum angle for λ = 0.1228. (e) Angular velocity for θa = 160◦ . (f) Angular velocity for λ = 0.1228.

the following reasons: 1) The disturbance rejection by the coupled SMC law in (11) may be restricted in real mechanical systems due to the magnitude limitation of built-in actuators, and 2) friction forces, such as dry friction in actuator, may cause actuator nonlinearities such as dead-zone characteristics. Therefore, due to such unmodeled dynamics10 in the zero dynamics (14), the generation of periodic orbits of real plants by using the proposed coupled SMC law may not be exactly the same, as explained in this section. Moreover, there may exist mismatched disturbance d2 (including unmodeled friction in the unactuated joint) in the second row of the cart–pendulum (6), which is the case where the sliding-mode dynamics are disturbed as follows: ˙ + σ(x + θ) ¯ = ∇ · d2 λ(x˙ + θ)

(34)

where ∇(t) and d2 (t) are assumed to be differentiable and bounded functions. Considering the uncertainty factor ∇ · d2 10 It should be noted that these uncertainties can be sufficiently compensated by using disturbance observers as in [40]–[42].

and following the derivation of (14) [or, equivalently, (33)], we can obtain the practical zero dynamics as follows: ¯ λ) = h(t, θ, ¯ p, ζd ) + hd (∇ · d2 ) θ¨ + w(θ,

(35)

  ¯ hd (∇ · d2 ) = β cos θ/(β cos θ¯ − λα)d¯2 , ∇ = (γ −  ¯ ¯ and d¯2 = ˙ d˙2 −σ∇d2 + σ 2 L(∇· λβ cos θ)/(αγ −β 2 cos2 θ), ∇· d2 ). Here, σL(∇ · d2 ) denotes the low-pass-filtered signal of ∇ · d2 . Then, the general integral of (35) can be obtained for large t as follows:

where

˙ θ, θ˙2 , θ2 , θ0 , λ) = θ˙2 + 2W ¯ λ) + I2 + I¯2 ¯ (θ, I(θ,

(36)

where I¯2 = θ hd (∇ · d2 )ds. This implies that the solutions of (35) deviates from the periodic orbit, passing through the point (θ˙2 , θ2 ), by the amount of |I¯2 |. In this case, it can be stated that the solutions of (35) are almost periodic [43]. In other words, there exists a time interval T = T (φ), such that |θ(t) − θ(t + T )| < φ, where φ = φ(I¯2 ) and I¯2 are assumed to be bounded. This argument can also be validated via several numerical simulations.

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Fig. 3. Family orbits (around the downward equilibrium point: θ0 = ±π) of the autonomous zero dynamics (21) for fixed initial conditions [(left) (θa , θa ) = (0, 160)] and fixed coupling parameter [(right) λ = 0.1228], respectively. In each case, periodic orbits are generated for various coupling parameters [(left) λ = [0.1, 0.5, 1, 2, 3, 4, 5, 6]] and initial conditions [(right) (θa , θa ) = [(0, 185), (0, 190), (0, 200), (0, 220), (0, 240), (0, 260), (0, 267)]], respectively. (a) Phase portraits for θa = 160◦ . (b) Phase portraits for λ = 0.1228. (c) Pendulum angle for θa = 160◦ . (d) Pendulum angle for λ = 0.1228. (e) Angular velocity for θa = 160◦ . (f) Angular velocity for λ = 0.1228.

IV. S TABILIZATION T OWARD T ARGET O RBIT In general, orbital stabilization is achieved by forcing the trajectory of a system to converge to a given target orbit. In this regard, it is required to modify the proposed coupled SMC law in (11) for the target orbit stabilization, such that it stabilizes a given target orbit without destabilizing the coupled sliding mode (Sc = 0). As stated in Lemmas 1 and 2, the generated periodic orbits are closely related to the general integral of the autonomous zero dynamics in (21), in that the general integral preserves its zero value (i.e., I = 0) along the generated periodic orbits. Based on this property, we now propose a target orbit stabilization control law by introducing an auxiliary control law to the coupled SMC law in (11). Let us first consider a T -periodic target orbit (θ˙ω , θω ) ∈ W such that θω (t) = θω (t + T ), along which the general integral given from (22) by  ¯ λ) + Iωa (37) ˙ θ, θ˙ω (ta ), θω (ta ), θ0 , λ = θ˙2 + 2W ¯ (θ, I θ, ¯ (θ¯ω (ta ), preserves its zero value. Here, Iωa = −θ˙ω2 (ta ) − 2W ¯ λ) and θω (ta ) = θω (ta ) + θ0 ; λ and θ0 are the same as in

Theorem 1 and Lemma 1, respectively; and (θ˙ω (ta ), θω (ta )) can be selected from any point along the target orbit (θ˙ω , θω ). Then, we propose a target orbit stabilization control law as follows: ¯sw + υ u ¯c = ueq + u

(38)

¯sw = −k · ρ · sat(Sc /ρ)/(λgx + where ueq is given as in (11), u gθ ), and υ is an auxiliary control to be designed. Note here that we replaced sgn(·) with sat(·) in the switching control law, where sat(·) is a saturation function defined for sufficiently small constant ρ such that sat(Sc /ρ) = Sc /ρ for |Sc | ≤ ρ and sat(Sc /ρ) = sgn(Sc /ρ) for |Sc | > ρ. In what follows, the term (c2 − cσ)ζd is omitted in ueq for simplicity in analysis. Note, however, that it can also be used to prevent the trivial zero solution as in (11). Now, the auxiliary control law υ is designed such that (38) stabilizes the target orbit (θ˙ω , θω ) ∈ W, even in the presence of the matched disturbance as in the following theorem. Theorem 4: Consider the cart–pendulum (6) and the control law in (38). Suppose that the auxiliary control law is given by ˙ υ = −kI θI

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PARK AND CHWA: ORBITAL STABILIZATION OF INVERTED-PENDULUM SYSTEMS VIA COUPLED SMC

Fig. 4.

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Procedure of the proposed scheme.

where I is the general integral given in (37). If the switching control gain k in (38) is chosen such that  1 ¯ k + (λd¯x + d¯θ ) k> (40) ρ

Once |Sc | is bounded as |Sc | ≤ ρ in sufficiently short time [by virtue of sufficiently large k in (38)], we can use sat(Sc /ρ) = Sc /ρ by definition and therefore obtain the overall closed-loop ˙ as follows: dynamics from (41), (43), and using υ = −kI θI

˙ |I|, then there exists where k¯ > kI · (β − αλ)/(αγ − β 2 )|θ| a positive constant kI for the given sufficiently large k such that the modified coupled SMC law u ¯c in (38) renders both the coupled sliding surface in (10) and the general integral in (37) asymptotic stable, even in the presence of the matched disturbance. Therefore, the given target orbit is robustly asymptotically stabilized. Proof: Using (7)–(9) and (38), the closed-loop dynamics of the coupled sliding surface can be obtained as follows:   Sc ˙ Sc = −k · ρ · sat (41) − Λ1 υ + λdx + dθ ρ

X˙ = ΞX + Δ



where Λ1 = −(λgx + gθ ) = β cos θ¯ − αλ/αγ − β cos θ¯ is positive for all θ ∈ Sˆuh Sˆlh , as explained in Lemma 1. In addition, we obtain the closed-loop unactuated dynamics from the second equation in (7), together with (38), as follows:    Sc ˙ ¨ ¯ θ = −w(θ, λ) − Λ2 σ ζ + k · ρ · sat − Λ2 Λ1 υ ρ (42) 2

2

 ¯ where Λ2 = β cos θ/(β cos θ¯ − αλ) > 0. Here, dθ in the second equation in (7) is neglected since it satisfies the matching condition and thus can be rejected by the proposed control law (as explained in the following). Now, let us consider the Lyapunov function in (12) again. Then, its time derivative can be obtained from (41) and using (40) as V˙ c ≤ −(k¯ − |Λ1 υ|)|Sc | for |Sc | > ρ. This implies that |Sc | is upper bounded by a sufficiently small bound ρ since ˙ |I|. k¯ is larger than |Λ1 υ| as k¯ > kI · (β − αλ/αγ − β 2 )|θ| ˙ θ) of (42) is also bounded, which Accordingly, the solution (θ, follows from the argument [below (33)] in Theorem 3. On the other hand, one can obtain the time derivative of the general integral in (37) by using (42) as follows:  ¯ λ) θ˙ I˙ = θ¨ + w(θ,    Sc ˙ ˙ (43) θ˙ − Λ2 Λ1 θυ. = − Λ2 σ ζ + k · ρ · sat ρ



(44)



where X = (Sc , I)T , Λ3 = Λ1 Λ2 , and     −k Λ1 kI θ˙ 0 Ξ= Δ = . −σΛ2 ζ˙ −Λ2 k θ˙ Λ3 kI θ˙2 Following the results in [14], the overall closed-loop dynamics in (44) can be rewritten as ´ = ΞX ¯ + Δ(τ, ¯ X) X

(45)

where´denotes the derivative with respect to the time evolution τ along the target orbit (i.e., ∂/∂τ )   −k Λ1 (θ˙ω , θω )kI θ˙ω ¯ Ξ= −Λ2 (θ˙ω , θω )k θ˙ω Λ3 (θ˙ω , θω )kI θ˙ω2 ¯ X) satisfies Δ(τ, ¯ 0) = 0. Here, note that the homoand Δ(τ, ´ ¯ in (45) is the so-called linearized geneous equation X = ΞX transverse dynamics to the given target orbit (θ˙ω , θω ). Since ˙ ζ → 0 as Sc → 0, it follows from the result ¯ → 0 and ζ, Δ in [14] that the given target orbit (θ˙ω , θω ) is exponentially ´ = ΞX ¯ in (45) is asymptotically stable. stable if and only if X ´ = ΞX ¯ The asymptotic stability of the homogeneous system X can be obtained by showing that the eigenvalues of the monodromy matrix ΦΞ (0, t) (or, equivalently, Z(T ) when Z(0) = I)11 belong to the open unit disk (i.e., they are less than one). Through the numerical calculation of ΦΞ (0, t) over the given target trajectory as usual (for example, [18], [44], and [45]), one can find a positive gain kI for the given sufficiently large k such that the overall closed-loop dynamics in (44) are asymptotically stable, and thus, the target orbit is exponentially stabilized. Consequently, the modified coupled SMC law in (38) exponentially stabilizes the given target orbit (θ˙ω , θω ) for the cart–pendulum in (7), even in the presence of the matched 11 Here, Φ (t, τ ) and Z(t) are the state transition and fundamental matrices, Ξ ´ = ΞX. ¯ respectively, for the homogeneous system X

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Fig. 5. Periodic orbit generation (around the upright equilibrium point: θ0 = 0) of the cart–pendulum when the initial condition is given as (x, ˙ x, θ˙a , θa ) = (0, 0, 0, 0). (a) Cart position and pendulum angle [(solid) θ and (dotted) x]. (b) Coupled sliding surface and coupled variable [(solid) Sc and (dotted) ζ]. ˙ (f) Phase portrait (x versus x). (c) Control input. (d) Function V in (28). (e) Phase portrait (θ versus θ). ˙

disturbance, provided that the switching control gain in (38) ˙ |I|12 to is chosen such that k¯ kI · ((β − αλ)/(αγ − β 2 ))|θ| satisfy k¯ |Λ1 υ|, and thus, V˙ c ≤ 0 for |Sc | > ρ. This completes the proof.  Remark 3: The modified coupled SMC law (i.e., target orbit stabilization control law) in (38) utilizes the general integral in (37), which preserves its zero value along the given target orbit (θ˙ω , θω ). In the case of cart–pendulum, this general integral can be obtained in a closed form as in (22), which is due to the fact that the equation of motion of the cart–pendulum system is relatively simple (i.e., it does not possess gyroscopic force term unlike the Furuta pendulum, Pendubot, Acrobot, etc.). Even in cases when the general integral cannot be given in a closed form as in the Furuta pendulum, we could see that the modified coupled SMC law in (38) is still feasible since it can be obtained by numerical integration of the term w(θ, λ) in the autonomous zero dynamics (21) as in [7], [8], and [16]–[20]. The details are omitted due to space constraints. Remark 4: In the case of the cart–pendulum in (6) or (7), the cart position is also orbitally stabilized by the proposed method 12 Since |I| and |θ| ˙ are upper bounded by virtue of sliding-mode control, the ¯ sufficiently switching control gain in (38) is feasible in general by choosing k large.

in (38) since the coupled sliding mode Sc = 0 results in the relation x = −1/λθ, as explained in the proof of Theorem 3. In other words, if the target orbit (θ˙ω , θω ) for the pendulum angle is stabilized, then the cart position evolves on the periodic orbit defined by xω = −1/λθω . On the other hand, all periodic orbits in (29) are not feasible for the real cart–pendulums since they have limited length of cart traveling distance and limited velocity due to the current limit in the servo systems. In this case, the aforementioned property, together with the properties illustrated in Section III-C, can be used to guide the successful choice of the feasible target orbit to be stabilized. The overall procedure of the proposed scheme is shown in Fig. 4. V. N UMERICAL S IMULATIONS A. Generation of Periodic Orbits Numerical simulations were performed to evaluate the performance of the periodic orbit generation for the cart–pendulum (6) using the proposed coupled SMC law uc in (11). The physical parameters in (6) are M = 1.12 kg, m = 0.11 kg, l = 0.1407 m, and g = 9.81 m/s2 . Simulation results for two different initial states are shown in Figs. 5 and 6. In the first case, an

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Fig. 6. Periodic orbit generation (around the upright equilibrium point: θ0 = 0) of the cart–pendulum when the initial condition is given as (x, ˙ x, θ˙a , θa ) = (0, 1, 0, 30). (a) Cart position and pendulum angle [(solid) θ and (dotted) x]. (b) Coupled sliding surface and coupled variable [(solid) Sc and (dotted) ζ]. ˙ (f) Phase portrait (x versus x). (c) Control input. (d) Function V in (28). (e) Phase portrait (θ versus θ). ˙

initial condition is given by the origin [i.e., (x˙ 0 , x0 , θ˙0 , θ0 ) = (0, 0, 0, 0)], and the results are shown in Fig. 5. In this case, the design parameters in (9) are given as λ = 0.1228 (which corresponds to the choice of θu∗ = 0.989 · π/2 as in Theorem 1), σ = 5, ζ¯ = π/6, and c = 1. Also, the switching control gain in (11) is given by k = 20. As stated in Section III-C, we used a nonzero exponentially vanishing reference ζd for the generation of a nontrivial periodic orbit, even in the case that the initial states are at the origin. Second, the other initial condition is given by (x˙ 0 , x0 , θ˙0 , θ0 ) = (0, 1, 0, 30), and its simulation results are shown in Fig. 6. In this case, the design and control parameters are the same as in the first case, except for ζ¯ = 0. In both cases, we used the sigmoid function, given by Sc /(|Sc | + ε) together with ε = 10−4 , instead of the sign function in (11). From both figures, one can see that the coupled sliding surface is reached within a very short time, and the sliding mode is invariant thereafter, as expected [see Figs. 5(b) and 6(b)]. In addition, it can be seen that the generated periodic orbits in steady state correspond to each level set of the function V in (28) [see Figs. 5(d) and 6(d)]. This corroborates the arguments in Theorem 3. Note that there exist some biases in each control input [see Figs. 5(c) and 6(c)] due to the use of the sigmoid function, which, however, does

not affect the shape of the generated periodic orbits (i.e., they are almost exactly symmetric with respect to the equilibrium point). The periodic orbits are generated as in Figs. 5(e) and (f) and 6(e) and (f). B. Orbital Stabilization To evaluate the robust orbital stabilization performance of the proposed target orbit stabilization control law in (38), we also performed numerical simulations for the cart–pendulum in comparison with the previous method (i.e., virtual constrained approach) reported in [7], [8], and [16]–[20]. In the simulations, the target orbit (θ˙ω , θω ) for the proposed method was chosen such that it passes two points (0, 50) and (60, 0). This target orbit is stabilized from the initial condition (x˙ 0 , x0 , θ˙0 , θ0 ) = (0, 1, 0, 10). With other design parameters as in Section V-A, the proposed control law in (38) is implemented by using ρ = 0.5, kI = 30, and k = kI · ((β − αλ)/(αγ − ˙ |I| + 450. Here, the choice of k and kI follows from β 2 ))|θ| Theorem 4. To implement the previous control law in [16], we chose L = 6.1 that defines the output variable as y = x + L sin θ. In this method, since the general integral [which differs from (37)] used by the previous control law is not given in a

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Fig. 7. Target orbit stabilization of the cart–pendulum (without model uncertainty and external disturbance) when the initial condition is given as (x, ˙ x, θ˙a , θa ) = (0, 10, 0, 0). (a) Pendulum angle [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (b) (Dash–dot) Coupled sliding surface Sc and (solid) coupled variable ζ of the proposed method and (dotted) the output variable y of the previous method. (c) Control input [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (d) General integral I [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (e) Phase portrait θ versus θ˙ using the previous method [(solid) target orbit and (dashed) trajectory of pendulum]. (f) Phase portrait θ versus θ˙ using the proposed method [(solid) target orbit and (dashed) trajectory of pendulum].

closed-loop form (due to a quadratic force term, as explained in Remark 1), we obtained it via numerical integration as in [16]. On the other hand, the auxiliary control law, which was designed by the modified linear-quadratic-regulator-based method in [16], is replaced by the alternative method developed in [19] for implementation convenience. The target orbit for the previous method was also chosen such that it passes two points (0, 50) and (60, 0) for the comparison with the proposed method. Note that both of the target orbits are not the same since they correspond to different general integrals (see [16] for details). Simulations were carried out in two cases to evaluate the robustness of the proposed method. In the first case, simulations were performed without system uncertainty and external disturbance, and the results are shown in Fig. 7. On the other hand, Fig. 8 shows the simulation results performed in the presence of both of them. In this case, 10% deviation in each model parameter is considered as the system uncertainty, and external disturbance d1 is applied to the cart–pendulum in (6), as shown in Fig. 9. In Fig. 7, it is shown that the coupled sliding surface is bounded in transient response and finally converges to zero [see Fig. 7(b)]. In addition, the trajectory

of the cart–pendulum converges to the given target orbit [see Fig. 7(f)] as the general integral is stabilized [see Fig. 7(d)]. On the other hand, the previous method results in steady-state error in the output variable and pendulum angle, i.e., the given target orbit is not sufficiently stabilized [see Fig. 7(e)]. By comparing the results in Figs. 7 and 8, it can be seen that the proposed method stabilize the given target orbit, even in the presence of system uncertainty and external disturbance as desired, while the previous method shows the worse performance of the target orbit stabilization and even blows up as in Fig. 8(b), (d), and (e). VI. C ONCLUSION A novel SMC method was proposed for the periodic orbit generation and target orbit stabilization of inverted-pendulum systems. In the proposed method, we first designed a coupled SMC law to render the coupled sliding surface reached in finite time, such that orbitally stable zero dynamics can be generated. Accordingly, it was shown that the orbitally stable zero dynamics has its general integral, which preserves zero value along

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Fig. 8. Target orbit stabilization of the cart–pendulum (with model uncertainty and external disturbance) when the initial condition is given as (x, ˙ x, θ˙a , θa ) = (0, 10, 0, 0). (a) Pendulum angle [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (b) (Dash–dot) Coupled sliding surface Sc and (solid) coupled variable ζ of the proposed method and (dotted) the output variable y of the previous method. (c) Control input [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (d) General integral I [(solid) proposed method (38) and (dotted) previous method [16], [19]]. (e) Phase portrait θ versus θ˙ using the previous method [(solid) target orbit and (dashed) trajectory of pendulum]. (f) Phase portrait θ versus θ˙ using the proposed method [(solid) target orbit and (dashed) trajectory of pendulum].

control. The validity and feasibility of the proposed method for practical applications were demonstrated via several numerical results. R EFERENCES

Fig. 9.

Time history of external disturbance.

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Mun-Soo Park received the B.S. degree from the School of Electrical and Electronics Engineering and the M.S. and Ph.D. degrees from the Department of Electronics Engineering, Ajou University, Suwon, Korea, in 1998, 2000, and 2007, respectively. From 2007 to 2009, he was a Postdoctoral Research Associate with Ajou University. In 2009, he was a Senior Research Scientist with Korea Institute of Industrial Technology, Ansan, Korea. Since 2009, he has been a Senior Research Scientist with Korea Institute of Aerospace Technology, Daejeon, Korea. His current research interests include adaptive and robust nonlinear control theories with their applications to robotics and underactuated mechanical systems, fuzzy and neural network systems, control-oriented system identification, and inertial navigation systems for autonomous unmanned vehicles.

Dongkyoung Chwa received the B.S. and M.S. degrees from the Department of Control and Instrumentation Engineering and the Ph.D. degree from the School of Electrical and Computer Engineering, Seoul National University, Seoul, Korea, in 1995, 1997, and 2001, respectively. From 2001 to 2003, he was a Postdoctoral Researcher with Seoul National University. In 2003, he was a Visiting Research Fellow with The University of New South Wales, Australian Defence Force Academy, Canberra, Australia, and the Honorary Visiting Academic with the University of Melbourne, Melbourne, Australia. In 2004, he was a BK21 Assistant Professor with Seoul National University. Since 2005, he has been with the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea, where he is currently an Associate Professor. His research interests include nonlinear, robust, and adaptive control theories and their applications to robotics, underactuated systems including wheeled mobile robots, underactuated ships, cranes, and guidance and control of flight systems.

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