A linear differential flatness approach to controlling ... - Semantic Scholar

IMA Journal of Mathematical Control and Information (2007) 24, 31–45 doi:10.1093/imamci/dnl005 Advance Access publication on March 27, 2006

A linear differential flatness approach to controlling the Furuta pendulum ˜ † C ARLOS AGUILAR -I B A´ NEZ Centro de Investigación en Computación del Intituto Politecnico Nacional, Avenida Juan de Dios Bátiz s/n Esquina con Manuel Othon de Mendizabal, Colonia San Pedro Zacatenco, AP 75476, 07700 México, D.F., México AND

H EBERTT S IRA -R AMÍ REZ‡ Centro de Investigación y Estudios Avanzados del Intituto-Politecnico Nacional, Departamento de Ingenier´ıa Eléctrica, Sección de Mecatrónica, Avenida Intituto-Politecnico Nacional # 2508, Colonia San Pedro Zacatenco, AP 14740, 07300 México, D.F., México [Received on 18 February 2005; accepted on 10 October 2005] The aim of this work is to motivate the use of linear flatness, or controllability, to design a feedback control law for the rather challenging mechanism known as the ‘Furuta pendulum’. This is achieved by introducing three feedback controller design options for the stabilization and rest-to-rest trajectory tracking tasks: a direct pole placement approach, a hierarchical high-gain approach and a generalized proportional integral approach synthesized on the basis of measured inputs and outputs. Keywords: flatness; high-gain control; generalized proportional integral control; Furuta pendulum system.

1. Introduction The Furuta pendulum constitutes an interesting mechanical system which allows for challenging experiments in modern control systems theory courses. The Furuta pendulum consists of a direct drive motor acting as the actuator of a horizontal arm to which a free-rotating vertical pendulum is attached. The vertical pendulum is attached to the free end of the horizontal arm firmly linked to the rotating shaft of the motor. The study of an inverted pendulum on a rotating arm was first proposed by Prof. K. Furuta of Tokio Institute of Technology in Furuta et al. (1992). Ever since, the device has been popularly known as the ‘Furuta pendulum’. Since the angular acceleration of the vertical pendulum cannot be directly controlled, the Furuta pendulum constitutes an interesting example of an underactuated mechanical system, i.e. it has fewer actuators than degrees of freedom. As a result, the existing control techniques, developed for fully actuated mechanical robot manipulators, cannot be directly applied to the control of this interesting device. The Furuta pendulum model is also known to be non-feedback linearizable by means of static state feedback (see Jakubczyk & Respondek, 1980), and hence, it cannot be linearized by means of dynamic state feedback control either. This fact makes it especially difficult to carry out some controlled manoeuvres such as the ‘swinging up’ of the pendulum and other manoeuvres related † Email: [email protected] ‡ Email: [email protected] c The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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´ NEZ ˜ C. AGUILAR-IBA AND H. SIRA-RAMÍREZ

to the stabilization around the unstable vertical position equilibrium point while achieving an angular trajectory tracking on the part of the horizontal arm. Furuta et al. (1992) proposed a robust swing-up control strategy using a subspace projection method. Their controller uses a bang–bang pseudostate feedback control method and it is not suitable for rest-to-rest manoeuvres. Yamakita et al. (1995) proposed several different methods to swing up a single and a double rotational pendulum. The methods are based either on an energy approach or on an ad hoc robust feedback control method. Iwashiro et al. (1996) considered a ‘golf swinger’ robot based on a rotational Furuta pendulum. The system uses an algorithm based on an efficient mechanical energy management approach. Olfati-Saber (1999) derived a control law for the stabilization of the pendulum in its upright unstable position from any arbitrary initial condition. The controller described by OlfatiSaber is based on a back-stepping procedure. A similar proposal, using a sliding mode control approach, ¨ is presented in a recent book by Utkin et al. (1999). Astrom & Furuta (2000) proposed, based on a simplified non-linear model of the Furuta pendulum, an interesting feedback controller based on energy considerations. Recently, Fantoni & Lozano (2002) proposed a controller for swinging the pendulum and raising it to its uppermost unstable equilibrium point through an hybrid controller based on a homoclinic orbit stabilization technique. Their control method is based on the passivity properties of the system. It is well known that a linearized tangent model of the Furuta pendulum system is locally controllable around the unstable equilibrium point (Akesson, 2001). This means that the system is locally ‘flat’ and, hence, the stabilization problem can be solved by a direct pole placement procedure (Fliess, 2000). Flatness allows for the complete local controllability of all system variables, including the nonminimum phase outputs (Fliess et al., 1998), without any undesirable internal dynamics. Moreover, since the tangent linearization model of the system around its unstable equilibrium point is independent of the nominal angular position of the horizontal arm, the trajectory tracking problem for large angular displacements of the arm should offer no particular difficulty. Since these last two aspects are not widely recognized, we treat, in some detail, the stabilization and trajectory tracking problem for the Furuta pendulum from a linearized viewpoint. In this paper, we present a differential flatness approach for stabilization and trajectory tracking in a Furuta pendulum system. The adopted model assumes that the inverted pendulum travels within a sufficiently small vicinity of its unstable equilibrium point, while the horizontal arm covers a large angle manoeuvre in a prespecified time interval. Differential flatness of the tangent approximation is then exploited in three different proposed controller design approaches: a pole placement-based controller; a hierarchical high-gain controller, related to the celebrated back-stepping design procedure; and, finally, a generalized proportional integral (GPI) approach, recently developed by Fliess et al. (2002). The last technique exploits traditional linear state feedback, but uses no state measurements and no asymptotic state observers. Rather, the state vector is replaced by a linear combination of iterated integrals of the input and output, which purposely neglects initial conditions. The effect of the initial conditions is later compensated, thanks to the superposition principle, via a sufficient number of iterated integrations of the output tracking error. The dynamic feedback control scheme is found to be internally exponentially asymptotically stable via a suitable and natural nesting of the open-loop integrations. In all the controller design cases presented, the flatness property of the linearized Furuta pendulum system considerably simplifies the controller design tasks and trivializes the trajectory specification, or trajectory planning, aspects of the problem. The proposed approaches provide systematic controller design options for many other interesting mechanical systems which are locally controllable around their unstable operating equilibria. In Section 2, we introduce the normalized non-linear model of the Furuta Pendulum; we next obtain the linearization of the system around the unstable equilibrium; finally, we show that the obtained

A LINEAR DIFFERENTIAL FLATNESS APPROACH TO CONTROLLING THE FURUTA PENDULUM

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incremental linear model is flat. Section 3 presents a linear pole placement-based controller for the stabilization of the underactuacted Furuta pendulum system. The equilibrium-to-equilibrium transfer of the system, within a finite time interval, is also accomplished by a pole placement-based trajectory tracking controller exploiting the flatness property. In order to obtain a more robust controller design, a hierarchical high-gain feedback controller scheme is also derived in Section 4. In Section 5, we present a GPI feedback controller for the stabilization and trajectory tracking problems for the Furuta pendulum. The proposed GPI controller requires only precise input and output measurements which, in this case, are represented by the applied input torque and the obtained angular position of the arm. In the three controller design cases, we present computer simulation results depicting the performance of the system to the particular feedback controller option. In Section 6, we estimate the domain of attraction (DOA) when the system is controlled by a flatness-based pole placement controller designed for the stabilization problem. Section 7 is finally devoted to present the conclusions of the article. 2. The Furuta pendulum The Furuta pendulum is a system consisting of an inverted pendulum connected to a horizontal rotating arm (see Fig. 1) acted upon by a direct current motor drive. The non-linear model of the mechanical part of the system, which can be derived from either the Newton or Euler–Lagrange formalism (see Lozano & Fantoni, 2002), is given by f = [I0 + m 1 (L 20 + l12 sin2 θ1 )]θ¨0 + m 1l1 L 0 cos θ1 θ¨1 + m 1l12 sin(2θ1 )θ˙0 θ˙1 − m 1l1 L 0 sin θ1 θ˙12 , 0 = m 1l1 L 0 cos θ1 θ¨0 + [J1 + m 1l12 ]θ¨1 − m 1l12 sin θ1 cos θ1 θ˙02 − m 1 gl1 sin θ1 ,

(2.1)

where θ0 is the rotational angle of the arm, θ1 is the rotational angle of the pendulum and f is the input torque applied to the joint of the horizontal arm. The torque f acts as the only control input. The parameters m 1 and J1 stand for the mass of the pendulum and the inertia of the pendulum around its centre of gravity, respectively. J0 is the inertia of the horizontal arm. L 0 and l1 are the total lengths of the arm and the vertical distance of the centre of gravity of the pendulum to its pivot point on the extreme of the horizontal arm.

FIG. 1. The Furuta pendulum system (FP).


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We first proceed to normalize the equations of the above system. Define the scaled variables g f t, u = τ= l1 gm 1l1

(2.2)

and the positive constants ε=

I0 , m 1l12

µ=

L0 , l1

η=

J1 + 1, m 1l12

κ = ε + µ2 .

(2.3)

We obtain, after some algebraic manipulations, the following normalized system: u = [κ + sin2 θ1 ]θ¨0 + µ cos θ1 θ¨1 + sin(2θ1 )θ˙0 θ˙1 − µ sin θ1 θ˙12 , 0 = µ cos θ1 θ¨0 + ηθ¨1 − sin θ1 cos θ1 θ˙02 − sin θ1 ,

(2.4)

where, with an abuse of notation, the ‘dot’ stands for differentiation with respect to the normalized (dimensionless) time τ. Linearization of the normalized system around the unstable equilibrium point θ0 = X = constant, θ1 = 0, θ˙0 = 0 and θ˙1 = 0 is achieved by defining the incremental errors as follows: θ0δ = θ0 − X, θ1δ = θ1 and u δ = u; we obtain κ θ¨0δ + µθ¨1δ = u δ ,

µθ¨0δ + ηθ¨1δ = θ1δ .

(2.5)

The incremental flat output, denoted by Fδ , is, in this case, given by the following relation: Fδ = µθ0δ + ηθ1δ . Indeed, all the variables can be parameterized in terms of the flat output Fδ and a finite number of its time derivatives as follows: θ1δ = F¨δ ,

θ0δ =

1 [Fδ − η F¨δ ], µ

uδ =

κ ¨ σ (4) Fδ + Fδ , µ µ

(2.6)

where σ = µ2 −ηκ. Note that σ is a negative constant as it follows from the definition of the parameters µ, η and κ (see (2.3)). Finally, note that the differential parameterization (2.6) allows for some analysis of the behaviour of the various system incremental variables. For instance, the incremental angular position θ0δ is a nonminimum phase output of the system since its zero dynamics, corresponding to θ0δ = 0, is given by the unstable dynamics Fδ . F¨δ = η The zero dynamics associated with the incremental angular position θ1δ satisfies the differential equation F¨δ = 0, which has the origin as a critically stable equilibrium.


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3. A flatness-based pole placement approach for stabilization and tracking 3.1

Stabilization around the unstable equilibrium point

The state-dependent input-coordinate transformation, uδ =

κ ¨ σ Fδ + v δ , µ µ

(3.1)

shows that the linearized system (2.6) is equivalent to the following chain of integrations: (4)

Fδ

= vδ .

(3.2)

A stabilizing feedback controller may be readily obtained by setting (3) v δ = −(k0 Fδ + k1 F˙δ + k2 F¨δ + k3 Fδ )

= −(k0 [µθ0δ + ηθ1δ ] + k1 [µθ˙0δ + ηθ˙1δ ] + k2 θ1δ + k3 θ˙1δ ),

(3.3)

where the set of coefficients {k0 , k1 , k2 , k3 } is chosen such that the closed-loop characteristic polynomial of the linearized system, defined as p(s) = s 4 + k3 s 3 + k2 s 2 + k1 s + k0 , is a Hurwitz polynomial. 3.2

Simulation results

Figure 2 shows the closed-loop responses, to the derived feedback controller, when applied to the original non-linear system (2.1). A significant initial deviation of the angular position of the pendulum and the angular position of the arm were hypothesized to be θ0δ = −0.02 rad and θ1δ = 0.15 rad. In the simulation, we used the following numerical values of the physical parameters: I0 = 0.175 m,

L 0 = 0.25 m,

m 1 = 0.15 kg,

l1 = 0.25 m,

J1 = 0.19 N m s2 .

The closed-loop characteristic polynomial was chosen to be p(s) = (s 2 +2ζ wn s +wn2 )2 with ζ = 0.707 and wn = 0.9.

FIG. 2. Closed-loop responses of flatness-based controlled Furuta pendulum system to stabilizing controller.


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3.3 Trajectory tracking for a rest-to-rest manoeuvre Based on the linearized system model (2.5), we can attempt a trajectory tracking approach to the problem of changing the angular position of the horizontal arm between an initial and final value while keeping the pendulum around its unstable equilibrium position. The rest-to-rest manoeuvre is to be accomplished within a finite time interval, following a smooth trajectory with no restriction on the magnitude of the horizontal arm angular displacement. We first assume that the horizontal arm is located, at time τ = τinitial , at a resting point θ0 (τinitial ) = X initial , with the pendulum standing in its unstable equilibrium position θ1 (τinitial ) = 0. The corresponding incremental state equilibrium point is characterized by θ0δ (τinitial ) = 0,

θ˙0δ (τinitial ) = 0,

θ1δ (τinitial ) = 0,

θ˙1δ (τinitial ) = 0.

It is desired to move the angular position of the horizontal arm to a new position θ0 (τfinal ) = X final within a finite time interval T = τfinal − τinitial , in such a way that when the arm reaches the desired final position, the pendulum exhibits no oscillations around the unstable equilibrium point. This means that the horizontal arm angular position θ0δ and the pendulum angular position θ1δ adopt the following equilibrium point, at time τ = τfinal : θ0δ (τfinal ) = X final − X initial ,

θ˙0δ (τfinal ) = 0,

θ1δ (τfinal ) = 0,

θ˙1δ (τfinal ) = 0.

We next assume that we can accomplish the manoeuvre with a small angular deviation of the pendulum from the vertical line, so that we can still use the linearized system (2.5) for deriving the feedback controller. Naturally, we test the performance of the resulting linear feedback controller on the non-linear system model (2.1). Let us note that under equilibrium conditions, the value of the flat output Fδ is related to the incremental angular position of the arm θ0δ by a static relation given by Fδ = µθ0δ . Therefore, the original control problem is transformed into one of controlling the incremental flat output Fδ between two different equilibrium points by following a prescribed trajectory, which we define by Fδ∗ (t) = µθ0∗ (t), where θ0∗ (t) may be specified by a smoothly interpolating polynomial between the two given equilibria θ0∗ (τinitial ) = 0 and θ0∗ (τfinal ) = X final − X initial . Let us recall, from (2.6), the control input differential parameterization of the linearized system κ σ (4) u δ = F¨δ + Fδ . µ µ A controller which achieves the equilibrium-to-equilibrium transfer, via trajectory tracking, is given by the following expression: κ σ u δ = F¨δ − − [Fδ∗ (t)](4) + k0 (Fδ − Fδ∗ (t)) + k1 ( F˙δ − F˙δ∗ (t)) µ µ + k2 ( F¨δ − F¨δ∗ (t)) + k3 (F (3) − [Fδ∗ (t)](3) ) , δ

where the set of coefficients {k0 , k1 , k2 , k3 } is chosen so that the closed-loop characteristic polynomial of the linearized system, given by p(s) = s 4 + k3 s 3 + k2 s 2 + k1 s + k0 , is Hurwitz. In terms of the normalized incremental state variables, the tracking feedback controller is given by κ σ u δ = θ1δ − k0 (Fδ − Fδ∗ (t)) + k1 ( F˙δ − F˙δ∗ (t)) µ µ + k2 (θ1δ − F¨δ∗ (t)) + k3 (θ˙1δ − [Fδ∗ (t)](3) ) . (3.4)


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FIG. 3. Closed-loop system responses for a large horizontal arm angular position rest-to-rest excursion using two transition time intervals T .

3.4

Simulation results

Figure 3 shows the closed-loop responses of the non-linear system (2.1) to the derived tracking feedback controller given in (3.4). The figure shows several runs for two different desired transfer times, T = τfinal − τinitial , T = 1.6 s and T = 3.4 s. The desired arm angular displacement, in actual coordinates, was set to be given by θ0∗ (τinitial ) = 0 rad and θ0∗ (τfinal ) = 1 rad. In the presented simulations, the equilibrium-to-equilibrium trajectory Fδ∗ (t), for the flat output Fδ , was implemented by the following function: Fδ∗ (τ ) = Fδ (τinitial ) + [Fδ (τfinal ) − Fδ (τinitial )]ψ(τ, τinitial , τfinal ),

(3.5)

where ψ(τ, τinitial , τfinal ) is a polynomial in the normalized time variable τ satisfying ψ(τinitial , τinitial , τfinal ) = 0,

ψ(τfinal , τinitial , τfinal ) = 1.

We have prescribed such a polynomial, smoothly interpolating between 0 and 1, as follows: ψ(τ, τinitial , τfinal ) =

τ − τinitial T

5 6 τ − τinitial i−1 i−1 (−1) ri , T i=1

with r1 = 252, r2 = 1050, r3 = 1800, r4 = 1575, r5 = 700 and r6 = 126. As seen from Fig. 3, for large and also for reasonably short transfer time intervals, the linearizationbased tracking controller performs rather well. 4. A hierarchical high-gain approach for stabilization and tracking The previously derived feedback controller (3.4) is not quite robust with respect to unmodelled perturbations affecting the arm motions. Due to this fact, the control design should be split into a two-level


38

strategy as proposed by Fliess et al. (1998) for the crane control problem, which, incidentally, belongs to the same family of underactuated systems as our treated example. The high-gain hierarchical approach is as follows: We first use a lower-level high-gain feedback controller to regulate the position of the arm towards the trajectory of an auxiliary variable reference signal, qδ (τ ). We now consider the influence of the variable set point qδ (τ ) on the ‘slow dynamics’ of the system and proceed to determine the law of variation of this reference signal so that the flat output converges towards a desired stabilizing reference trajectory. From the differential parameterization (2.6), in terms of the flat output Fδ , we may obtain the following two relations: 1 F¨δ = (Fδ − µθ0δ ), η

θ¨0δ =

µ η θ1δ − u δ . σ σ

(4.1)

Consider then the following high-gain controller: uδ =

σ [k1 θ˙0δ + k2 (θ0δ − qδ (t))], η

(4.2)

where qδ (t) is an auxiliary variable reference trajectory for the normalized incremental arm angular position θ0δ . Then the closed-loop horizontal arm dynamics is governed by θ¨0δ + k1 θ˙0δ + k2 [θ0δ − qδ (t)] =

µ θ1δ . σ

(4.3)

If the gain k1 and k2 are sufficiently large, the motion of the closed-loop system (4.3) asymptotically converges towards the following reduced-order dynamics, valid on the controlled system’s slow manifold: k2 (4.4) θ˙0δ = − [θ0δ − qδ (t)] k1 and, thus, θ0δ exponentially converges to qδ (t). Now the pendulum dynamics, represented by the first equation of (4.1), is governed by the reference signal qδ (t). We then have 1 µ F¨δ = Fδ − qδ (t). η η

(4.5)

We may now regard the auxiliary reference signal qδ (t) as a virtual control input, which is to be chosen to guarantee that the flat output Fδ follows the desired reference trajectory Fδ∗ (t). We then have qδ (t) =

1 η (µθ0δ + ηθ1δ ) + − F¨δ∗ (t) + 2ζ wn (µθ˙0δ + ηθ˙1δ − F˙δ∗ (t)) µ µ + wn2 (µθ0δ + ηθ1δ − Fδ∗ (t)) .

(4.6)

4.1 Simulations results Figure 4 shows the performance of the designed hierarchical feedback control system acting on the nonlinear system (2.1). The values of k1 and k2 were chosen to be k1 = 25 and k2 = 10. The reference trajectory Fδ∗ (τ ) for the flat output Fδ (τ ) was set to be identically zero, as required by the stabilization task. The rest of the controller parameters were set to be ζ = 0.70707 and wn = 0.9. In this case, we obtained a control force which was nearly five times larger than in the previous design.


39

FIG. 4. High-gain hierarchical feedback controller performance in a stabilization task.

FIG. 5. High-gain hierarchical feedback trajectory tracking controller performance for several transition time intervals T .

Figure 5 shows the performance of the non-linear system (2.1) to the proposed hierarchical feedback controller (4.6) designed for tracking purposes. The reference trajectory Fδ∗ (τ ) for the flat output Fδ (τ ) was set to be the same interpolating polynomial (3.5) used in Section 3. This trajectory causes the angular position of the arm to go from the initial value 0 rad to that of 1 rad for two equilibriumto-equilibrium transfer time intervals, T = 1.5 s and T = 3.2 s, respectively. The designed highgain hierarchical controller (4.3)–(4.6) allows for smaller transfer times than those obtained with the pole placement-based controller. These are accomplished without pendulum destabilization between the two desired equilibria. The disadvantage is that of significantly higher feedback control input signal amplitudes. 5. A GPI approach for stabilization and tracking Let us consider the linearized system and define the incremental output yδ = θ0δ . The linearized system is observable with respect to this output and hence it is constructible. Therefore, we can obtain an integral input–output parameterization of the incremental flat output Fδ and its time derivatives, modulo

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the unknown initial conditions. Indeed, from (2.5) we obtain the following structural estimates of the incremental state variables: τ λ

1 = u ()d dλ − κ y θ δ δ , 1δ µ 0 0 σ τ η τ λ ˙θ1δ = yδ (λ)dλ + u δ (υ)dυ d dλ, µ 0 µ 0 0 0 θ0δ = yδ , τ 1 ˙ u δ (λ)dλ − µθ1δ . y˙δ = κ 0 τ Φ), are also used without We denote by ( Φ) the quantity 0 Φ(λ)dλ. Iterated integrals, such as ( the integration variables and integration limits. An integral input–output parameterization of the incremental flat output and its time derivatives, modulo the unknown initial conditions, is obtained from the flat output definition and the previous state variables’ input–output parameterization. We use a ‘hat’ notation to denote the obtained structural estimates of the incremental flat output and its time derivatives: 1 Fδ = uδ σ yδ + η , µ 1 ˙ Fδ = u δ − κ yδ , µ (5.1) 1 ¨ Fδ = u δ − κ yδ , µ µ κ ˙

(3) Fδ = uδ − Fδ. σ σ Based on the previously derived controller (3.1)–(3.3), the above structural estimates of the incremental flat output and its time derivatives can be used in the synthesis of a GPI incremental feedback controller (See Fliess et al., 1998, for more details). One simply replaces the flat output Fδ and its time derivatives by the obtained integral input–output parameterization presented in (5.1). Taking into account the neglected initial conditions, we proceed to compensate their destabilizing effect via a sufficient number of integrals of the output stabilization (or tracking) error. The required stabilizing GPI controller is obtained as follows: κ ¨ σ

(3) Fδ − k3 Fδ + k4 F˙ δ + k5 F¨ δ + k6 Fδ − ρ1 , u= µ µ ρ˙1 = −β2 yδ + ρ2 ,

ρ1 (0) = 0,

ρ˙2 = −β1 yδ + ρ3 ,

ρ2 (0) = 0,

ρ˙3 = −β0 yδ ,

ρ3 (0) = 0.

Then the closed-loop dynamic system is governed by the following expression:

κ ¨ σ (4) σ κ ¨ (3) 2 ˙ Fδ + Fδ = − k3 Fδ + k4 Fδ + k5 − Fδ + k6 Fδ + (k 0 + k 1 t + k 2 t ) − ρ1 , µ µ µ µ

(5.2)


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where k 0 , k 1 and k 3 are unknown constants which depend on the initial conditions. The closed-loop system is thus given by (7)

Fδ

(6)

+ k6 Fδ

(5)

+ k5 Fδ

(4)

+ k4 Fδ

(3)

+ k3 Fδ

+ β2 y¨δ + β1 y˙δ + β0 yδ = 0

(5.3)

from the second equation of (2.6); recalling that yδ = θ0δ , we have that yδ = (Fδ −η F¨δ )/µ. Substituting this last expression into (5.3), we obtain the following closed-loop dynamics for the flat output:

ηβ2 ηβ1 (7) (6) (5) (4) (3) Fδ + k6 Fδ + k5 Fδ + k4 − Fδ + k3 − Fδ µ µ

β0 η ¨ β2 β1 ˙ β0 − Fδ + Fδ = 0. + (5.4) Fδ + µ µ µ µ It is clear that by appropriate choice of the design parameters {k6 , k5 , k4 , k3 , β2 ,β1 , β0 }, we guarantee that the roots of the closed-loop characteristic polynomial

ηβ2 4 ηβ1 3 β2 β0 η 2 β1 β0 s + s + s+ s + k3 − − =0 s 7 + k6 s 6 + k5 s 5 + k4 − µ µ µ µ µ µ may be conveniently located in the left half of the complex plane. Thus, we guarantee that the closedloop system is locally exponentially asymptotically stable with respect to the origin. Finally, the rest-to-rest manoeuvre, by means of suitable trajectory tracking, can also be accomplished using only the angular position measurements of the arm and the applied input force, by considering the following GPI feedback tracking controller: uδ =

κ σ δ − Fδ∗ ) F¨δ − − [Fδ∗ ](4) + k3 ( F µ µ

(3) ∗ (3) ˙δ − [ F˙δ∗ ]) + k5 ( F ¨δ − [ F¨δ∗ ]) + k6 ( F + k4 ( F δ − [Fδ ] ) − ρ1 ,

ρ˙1 = −β2 (yδ − yδ∗ ) + ρ2 ,

= 0,

ρ˙2 = −β1 (yδ −

= 0,

ρ˙3 = −β0 (yδ −

ρ1 (0) ∗ yδ ) + ρ3 , ρ2 (0) yδ∗ ), ρ3 (0) = 0,

where yδ∗ is the desired horizontal arm, rest-to-rest, trajectory. 5.1 Simulation results Figure 6 shows the closed-loop responses to the GPI feedback controller for the stabilization problem. The initial conditions and the physical parameters were taken as indicated in Section 3. The closed-loop characteristic polynomial was selected as follows: p(s) = (s 2 + 1.27262s + 0.81)3 . 6. Computing the DOA Consider a finite-dimensional non-linear system x˙ = f (x), with the state x ∈ Rn , under the assumption that it is locally stable. Then, the DOA is defined as the subset of Rn formed by all the initial states

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FIG. 6. GPI controlled stabilization responses of the Furuta pendulum system.

x0 = x(0) which tend asymptotically to x(∞) = 0. Thus, we compute the region of attraction or DOA based on the quadratic Lyapunov function (more details are found in Barreiro et al., 2002). This technique consists of proposing an energy function V (x) = x P x/2, with P > 0, and then considering ellipsoids of bounded energy in the form: E P (v 2 ) = {x ∈ Rn : x P x v 2 }. Thus, when the system is locally stable, an estimation of the attraction basin, obtained with P > 0 fixed, is given by E 0P = E P (v 02 (P)), where the ‘largest-stable’ energy level v 02 (P) is obtained as ∂ V (x) f (x) 0 for all x ∈ E P (v 2 ) . (6.1) v 02 (P) = max v 2 : ∂x Therefore, E 0P is the largest P-ellipsoid compatible with the non-increasing energy condition V˙ < 0. This means that all trajectories starting at x ∈ E 0P are bounded, and whenever V˙ < 0, they tend asymptotically to the origin. Based on the definition of the DOA, we compute the region of attraction when the Furuta pendulum is fed back by means of a ‘flatness-based linear pole placement-based controller’ for the stabilization. To solve the problem, we proceed as follows: We obtain a simple state-space realization of the closed-loop system and we propose an energy function. Finally, we estimate a bound for the largest-stable energy label. 6.1

The state-space realization and energy function

Let us consider the normalized non-linear system (2.4) when it is fed back by (3.1)–(3.3), and let us assume that |θ1 | is very small. In this particular case, the equation of the closed-loop system can be rewritten as Y˙δ = AYδ + R(Yδ ), (6.2) where Yδ is the error of the incremental vector, defined as Yδ = [θ0δ , θ1δ , θ˙0δ , θ˙1δ ] and A is the matrix given by ⎡ ⎤ 0 0 1 0 ⎢ 0 ⎥ 0 0 1 ⎥ A=⎢ ⎣ k0 η (1 + k2 η + k0 η2 )/µ k1 η (k3 η + k1 η2 )/µ ⎦ . −k2 − k0 η −k1 µ −k3 − k1 η −k0 µ Note that the characteristic polynomial of the matrix A is given by p(s) = s 4 + k3 s 3 + k2 s 2 + k1 s + k0 . It is chosen such that A is Hurwitz. The vector R(·) is defined as R (Yδ ) = [0

0 (M0−1 (Yδ )) ],


where

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2 −2θ1δ θ˙0δ θ˙1δ + µθ1δ θ˙1δ κ µ . , (Yδ ) = M0 = µ η θ1δ θ˙ 2

0δ

Clearly, the closed-loop equation (6.2) is locally asymptotically stable around Yδ = 0, since A is a stability matrix. Hence, there always exist two positive matrices P and Q, such that A P + P A = −Q. A simple energy function can be established as V (Yδ ) = 12 Yδ PYδ . Finally, the derivative of V (·) along the trajectories of the system is given by V˙ (Yδ ) = −Yδ QYδ + 2Yδ P R(Yδ ). 6.2

Estimation of the largest-stable energy level

Given V˙ (·), we have to find the sign of V˙ (Yδ ) checked for all Yδ ∈ ∂ E P (2v i2 ) where ∂ E P (v i2 ) = {x ∈ 2 − v2 = ε > 0 Rn : x P x = v i2 }, where the given discrete energy levels v i2 are chosen such that v i+1 i and i = {1, 2, . . . , n}. A bound v ∗2 = v i2 is found at the ith step, if there are two discrete energy levels 2 , such that V˙ (Y ) 0 for all Y ∈ ∂ E (2v 2 ) and V˙ (Y ) > 0 for some Y ∈ ∂ E (2v 2 ). v i2 and v i+1 δ δ P δ δ P i i+1 2 2 Then, we can say that v i is a bound of the largest-stable energy level denoted by v 0 (P). The smaller the ε, the better the bound estimation. To verify whether V˙ (Yδ ) 0 or V˙ (Yδ ) > 0 for Yδ ∈ ∂ E P (2v i2 ), a mesh M is built, which in 4D problems can be expressed as M = {Y (x jkl ) ∈ ∂ E P (2v i2 ) : 0 j, k, l N } where x jkl denotes the three independent coordinates necessary so that Y (x jkl ) lies on the hipper-surface of the ith P-ellipsoid. Roughly speaking, the set of discrete vectors Y (x jkl ) is generated, provided that V (Y (x jkl )) = 2v i2 for all {0 j, k, l N }. 6.3 Numerical results Let us take the same physical and control parameters as described in Section 3 {κ = 5.66, µ = 1, η = 1.0002, k0 = 0.656, k1 = 2.06, k2 = 3.24, k3 = 2.54}. The matrix A is computed as ⎡ ⎤ 0 0 1 0 ⎢ ⎥ 0 0 0 1 ⎥ A=⎢ ⎣ 0.6562 4.8971 2.0642 4.6091 ⎦ −0.6561 −3.8962 −2.0619 −4.6081 and selecting Q = Diag{1, 1, 1, 1}. The Lyapunov equation is solved for P and the unique solution is given by ⎤ ⎡ 45.58 −9.81 −0.50 0.5 ⎢−9.81 3.43 −0.50 −0.5 ⎥ ⎥ ⎢ P=⎢ ⎥. ⎣−0.50 −0.50 16.87 −7.05⎦ 0.50 −0.50 −7.05 3.61 The numerical implementation for computing the largest-stable energy level v ∗ was carried out by using the set of energy levels {v 1 = 0.15, v 2 = v 1 + ε, v 2 = v 1 + 2ε, . . ., v 30 = 0.35} with ε = 0.01. The parametrization of the mesh M was generated by using N = 30. The result was the hipper-ellipsoid of energy level given by v ∗ = 0.31. The algorithm to find bounds was written in Matlab 5.3 for Windows.

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It was tested in a personal computer Dell Dimension 4300, with an Intel Pentium processor at 1.6 GHz, 128 Mb in RAM. The algorithm took around 3 min to finish and provide the expected results. Finally, the related results with the simulation are shown in Figure 2, where the initial condition is taken as Yδ (0) = (−0.02 rad, 0.15 rad, 0, 0). Clearly, Yδ (0) belongs to the attraction region because V (Yδ (0)) = 0.077 < (0.31)2 = v ∗2 . 7. Conclusions Differential flatness allows to systematically solve a number of interesting non-linear control problems. In this instance, we have exploited flatness for the efficient regulation and tracking of the Furuta pendulum, which happens to be only locally controllable (i.e. flat) around its unstable equilibrium. Since the system is also observable and, hence, constructible, an integral input–output parametrization of the incremental state variables and of the incremental flat output and its various time derivatives is possible. This allows us to obtain a trajectory tracking controller for the incremental flat output behaviour guaranteeing the desired rest-to-rest manoeuvre. The input–output integral parametrization allows one to locally control the system, by simply resorting to a standard pole placement procedure on a higherorder system devoid of internal instabilities. This feedback regulation scheme is carried out without use of full state measurements or without devising dynamic systems devoted to solve the state estimation problem, as traditionally provided by asymptotic state observers. The three types of controllers outlined in this paper can be used for stabilization and trajectory tracking problems in some other interesting underactuated mechanical systems, such as the inverted pendulum on a cart, some swinging cranes and other such systems. Finally, when the Furuta pendulum is controlled via a flatness-based pole placement controller for stabilization, the DOA was estimated by means of digital computer calculations, with highly satisfactory results. Acknowledgements This research was supported by the Centro de Investigación en Computación of the IPN, by the Coordinación de Posgrado e Investigación of the IPN, under Research Grant 20020247, by the Centro de Investigación y Estudios Avanzados del IPN and by the Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT-México), under Research Grant 32681-A. R EFERENCES A KESSON , J. (2001) Safe manual control of unstable system. Masters Thesis, Luna Institute of Tecnology, Luna, Sweden. ¨ STROM , K. J. & F URUTA , K. (2000) Swinging up a pendulum by energy control. Automatica, 36, 287–295. A BARREIRO , A., A RACIL , J. & PAGANO , D. (2002) Detection of attraction domains of non-linear systems using bifurcation analysis and Lyapunov functions. Int. J. Control, 75, 314–327. FANTONI , I. & L OZANO , R. (2002) Nonlinear Control of the Underactuacted Mechanical System. London: Springer. F LIESS , M. (2000) Sur des Pensers Nouveaux Faisons des Vers Anciens. Actes Conférence Internationale Francophone d’Automatique. Lille, pp. 26–36. ´ F LIESS , M., M ARQUEZ , R., D ELALEAU , E. & S IRA -R AMÍ REZ , H. (2002) Correcteurs Proportionnels-Intégraux Généralisés. ESAIM: Control Optim. Calc. Var., 7, 23–41 Available at http://www.emath.fr/cocv/. ´ , R. (1998) Regulation of non-minimun phase outputs: a flatness F LIESS , M., S IRA -R AMÍ REZ , H. & M ARQUEZ based approach. Perspectives in Control: Theory and Applications (D. Normand-Cyrot ed.). London: Springer, pp. 143–163.


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