On the Linear Control of Underactuated Nonlinear ...

On the Linear Control of Underactuated Nonlinear systems via tangent Flatness and Active Disturbance Rejection Control Mario Ram´ırez-Neria Departament of Automatic Control ´ CINVESTAV-IPN, Mexico Av. IPN 2580 Col. San Pedro Zacatenco ´ D.F. Mexico, C.P. 07360 Email: [email protected] Hebertt Sira-Ram´ırez Department of Electrical Engineering Mechatronics Section CINVESTAV - IPN Av. IPN 2580 Col. San Pedro Zacatenco ´ D.F. Mexico, C.P. 07360 Email: [email protected] ´ Garrido-Moctezuma Ruben Department of Automatic Control ´ CINVESTAV-IPN, Mexico Av. IPN 2580 Col. San Pedro Zacatenco ´ D.F. Mexico, C.P. 07360 Email: [email protected] ∗ ´ Alberto Luviano-Juarez ´ Instituto Politecnico Nacional UPIITA Av. IPN 2580 ´ Col. Barrio La Laguna Ticoman ´ D.F. Mexico, C.P. 07340 Email: [email protected]

In this article, a systematic procedure for controller design is proposed for a class of nonlinear under-actuated systems which are non feedback linearizable but exhibit a controllable (flat) tangent linearization around an equilibrium point. Linear Extended State Observer (LESO) based Active Disturbance Rejection Control (ADRC) is shown to allow for trajectory tracking tasks involving significantly far excursions from the equilibrium point. This is due to local approximate estimation and cancelation of the nonlinearities neglected by the linearization process. The approach is typically robust with respect to other endogenous and exogenous uncertainties and disturbances. The flatness of the tangent

∗ Corresponding

author

model provides a unique structural property that results in an advantageous low order cascade decomposition of the LESO design, vastly improving the attenuation of noisy and peaking components found in the traditional full order, high gain, observer design. The popular Ball and Beam System (BBS) is taken as an application example. Experimental results show the effectiveness of the proposed approach in stabilization, as well as in perturbed trajectory tracking tasks.

1 Introduction The analysis and control of Under-actuated Systems (UAS) has been an active topic of research in recent years.

It has been considered one of the most intricate problems in nonlinear control theory [1]. The fact that the cardinality of the control input set is lower than the degrees of freedom, has several adverse consequences. For instance, the fact that some under-actuated systems may not be input - output feedback linearizable via static or dynamic feedback [17] render most of the traditional control schemes inapplicable [30]. On the other hand, the need for less sensors and actuators in under-actuated systems leads to some design and operational advantages. These have allowed the practical use of this class of systems in some real world applications; such as damping systems [25], aerial and ground vehicles [3], [22], and [5], among others. The control problem of UAS has been addressed from many different perspectives: Sliding mode control [31], fuzzy control [4], neural networks [32], backstepping [21], and hybrid control [6]. In the work of Olfati-Saber [22], a comprehensive treatment of normal and cascade forms of under-actuated systems is provided. Generally speaking, UAS are not feedback linearizable and in multi-variable cases they may have an ill-defined vector relative degree [16]. This feature makes difficult the design of general controllers and/or observers [18]. Approximate feedback linearization, based on the tangent model, allows singularity avoidances at the expense of some weaknesses. Even if a well defined relative degree condition is achieved via dynamic extensions, the nonlinearities, as well as the un-modeled dynamics, degrade robust performance. To overcome this difficulty, robust options such as disturbance observers [12] have been shown to improve the behavior of the controlled system, while conserving some advantages. In this respect, exploitation of the flatness property [7] provides some remarkable advantages. The flatness of the linearized system is shown to naturally induce a low order cascade structure [24] which allows for a simpler (decoupled) and more efficient disturbance and state observer design. In this article, tangent flatness around an equilibrium point and ESO based ADRC are merged into a systematic procedure for robust feedback controller design in a class of under-actuated systems which are non feedback linearizable. The approach is shown to be suitable for trajectory tracking tasks involving significant excursions from the chosen, or natural, equilibrium point. A structural property, revealed by flatness, results in an advantageous low order cascade decomposition of the LESO design. This is shown to improve the attenuation of noise effects as well as reducing the peaking components commonly found in the full order, high gain, ESO design. The Ball and Beam System (BBS) is taken as a prototypical application example. Experimental results show the effectiveness of the proposed approach in stabilization, as well as in perturbed trajectory tracking tasks. This article is structured as follows: In section 2, some theoretical preliminaries are introduced. A control methodology for controlling a class of nonlinear under-actuated systems is proposed in section 3. The same section illustrates the procedure in a direct application to the BBS, where the flatness property of the tangent linearization mode is es-

tablished along with a modified simplified system dynamics suitable for dealing with the stabilization and trajectory tracking problem in the context of ESO based ADRC. Experimental results are provided in Section 4. Finally, some conclusions and considerations for further research are found in Section 5.

2 Preliminary concepts Differentially Flat Systems It is said that a nonlinear system of the form x˙ = f (x, u), x ∈ Rn , u ∈ mathbbR, is differentially flat, i.e., linearizable by an endogenous (static) feedback [8], if there exists an endogenous variable, f , having the following properties: (i) Every system (state, input, or, output) variable may be expressed as a function of f and a finite number of its time derivatives. (ii) The variable f may be expressed as a function of the system state vector and a finite number of its time derivatives. (iii) the variable f does not satisfy any differential equation by itself. The flatness property allows an equivalence of the system with a linear controllable system, via state and input coordinate transformations. The desired flat output trajectory induces nominal system solutions which are directly obtained without integrating differential equations. A class of Under-actuated Systems Consider the following model of an under-actuated nonlinear mechanical system [31], [30]:

M11 (x)x¨1 + M12 (x)x¨2 + C1 (x, x) ˙ = b(x)u M21 (x)x¨1 + M22 (x)x¨2 + C2 (x, x) ˙ =0

(1)

where xT = (xT1 , xT2 ), x1 ∈ Rn−p, x2 ∈ R p are the states, u is the scalar control input. The functions C1 and C2 contain here Coriolis forces and gravity effects. The inertia matrix, defined by,



 M11 (x) M12 (x) M(x) = M21 (x) M22 (x) is positive definite and, hence, non-singular. Notice that the flat output does not, necessarily, constitute a measurable variable of the system. This may lead to some difficulties in certain cases. Generally speaking, however, the flat output represents a physically meaningful system variable. The advantages offered by its knowledge pays well in the trajectory planning tasks, controller design work and closed loop performance evaluation of the system. This is, precisely, the case of classical under-actuated systems such as: the Furuta, or rotational, pendulum, the ball

and beam system, the Vertical Take Off and Landing system, the TORA system, etc. (see [24], [16], [26] for further details).

2.1 Basic assumptions In this work, it is assumed that the SISO under-actuated system exhibits a controllable tangent linearization around a natural equilibrium point. The linearized system is therefore assumed to be flat. It will be desired to perform a meaningful trajectory tracking task that is translated into a corresponding flat output trajectory tracking task. The adopted trajectory is supposed to take the state of the system significantly away from the initial equilibrium point while, possibly, significantly exciting those nonlinearities neglected in the linearization process. The ESO based ADRC will take care of such neglected endogenous disturbances by direct estimation and canceling of their effects. In this estimation-canceling effort, are included as well those disturbances arising from un-modeled dynamics and exogenous perturbations. In many under-actuated systems cases, however, it is possible to establish the flatness property in the Tangent Linearized system around an certain equilibrium point. This allows for a systematic procedure in the robust flat output feedback controller design for these systems with the help of ESO based ADRC. In ADRC schemes, the control problem can be approached in terms of estimating the total disturbance affecting the system. This problem is naturally associated with the Linear Active Disturbance Rejection (LADR), which consists in using a Linear Extended State Observer in order to estimate the lumped disturbance input, reducing the control task to one of controlling a perturbed chain of integrators. An additional advantage of flatness is revealed in a structural property exhibited in the differential parametrization of the system variables via the flat output. This property places in simple terms of measurable (position) variables an intermediate high order time derivative of the flat output. The possibility arises of of building a natural cascade decomposition of the ESO design using two simple observers. This feature eliminates the need for the direct estimation, through a single observer, of all flat output time derivatives, thus reducing the natural measurement noise effects in the high order injected flat output phase variables estimation error dynamics. In the following section, a generalized procedure is applied to the popular Ball and Beam system.

3 A control design procedure through a Case Study Consider the well known Ball and Beam System, shown in the Figure 1, consisting of a ball placed on a beam, which undergoes an angular displacement around a certain pivot, actuated by a dc motor directly coupled to the beam by means of a pulley. The dynamical model is represented as follows:

Fig. 1. A schematic of the Ball and Beam system

I2 )¨r − m2 rθ˙ 2 + m2 g sin θ = 0 R2 ˙ (m2 r2 + m1 l12 + I1 + I p)θ¨ + 2m2 rr˙θ+ (m2 +

+m2 gr cos θ + m1 gl1 sin θ = Nτ

(2)

(3)

where r denotes the position of the ball from the center of mass of the beam, θ is the angular position of the beam, m2 and R denote, respectively, the ball mass and its radius. I2 = 25 m2 R2 is the ball inertia, I1 denotes the beam inertia, m1 is the beam mass and I p represents the inertia of the pulley system, and N is a ratio of distances concerning the pulley/motor actuator. The control input is the motor torque, denoted by τ, which can be expressed as a function of the voltage input through the approximate relation: τ(t) = kτ /RaV (t), where kτ is the motor torque constant, and Ra is the motor’s armature electric resistance. Problem formulation It is desired to transfer the ball position from an initial value, r(0) , to a given final value r(t f inal ) in a finite, prescribed, interval of time [0,t f inal ]. The desired reference position is specified as a rest to rest trajectory, by means of interpolation polynomials of the B´ezier type (see [11]). Step 1:Obtain the linearized system around an equilibrium point The system desired equilibrium point is described by:

r¯ = 0, r¯˙ = 0, θ¯ = 0, θ¯˙ = 0, V¯ = 0

(4)

The approximate linearization of the system (2) - (3) around the equilibrium point (4) is given by

I2 )¨r + m2 gθδ = 0 R2 δ kτ N V (m1 l12 + I1 + I p)θ¨ δ + m2grδ + m1 gl1 θδ = Ra δ (m2 +

(5) (6)

where rδ = r − r¯ = r, r˙δ = r˙ − r¯˙ = r˙, θδ = θ − θ¯ = θ, θ˙ δ = θ˙ − θ¯˙ = θ˙ and Vδ = V − V¯ = V . Step 2: Verify the existence of the flatness property, and obtain a flat output as well as the input-to-flat output relationship The linearized model (5) - (6) is controllable and, hence, flat. The incremental flat output is given by the incremental position of the ball, rδ = r − r¯ = r, coincident here with the actual ball position r. All the variables in the BBS can be differentially parameterized in terms of the flat output rδ (i.e., each variable is a function of r and a finite number of its time derivatives). From equation (5), we have the following relations:

θδ = −



   (m2 + I2 /R2 ) (m2 + I2/R2 ) (3) r¨δ , θ˙ δ = − rδ m2 g m2 g (7)

Step 3:Find the cascade form of the underactuated system In this case, a simple relation exists between the second order time derivative of the flat output and the angular position of the beam. This step can reduce the complexity of the extended state observer associated with the ADRC control scheme. For this system, define, α = m2 g/(m2 + I2/R2 ). Then, the linearized acceleration of the ball can be expressed in terms of the angular position of the beam through the following relation: r¨δ = −α θδ Using the linearized dynamics of the beam (6) and equation (7), we obtained the control input Vδ (t) parameterized in terms of the flat output rδ = r and its time derivatives as: (m1 l12 + I1 + I p) (4) m1 gl1 kτ N Vδ (t) = − rδ − r¨ + m2 grδ (8) Ra α α δ The incremental input-to-flat output relationship for the BBS (8), is represented by means of a fourth order linear, time-invariant, system (i.e., n = 4):

(4)

first one controlled by the input voltage Vδ (t), with the corresponding output given by the flat output incremental acceleration r¨δ = r¨. This output coincides, modulo a constant factor, α, with the beam incremental angular position, θδ = θ, i.e., r¨δ = −αθδ . The signal −αθδ , acts then as an auxiliary, measurable, input to the second block, which consists of an elementary chain of two integrators rendering, simple enough, the estimation of the phase variables r˙δ and rδ (see Figure 2). This cascade property simplifies and decouples the ESO design task in the ADRC scheme. The net result is that on-line observer based estimations are limited to first order time derivatives estimations instead of consecutive time derivatives up to a third order.

αkτ N m1 gl1 Vδ (t) − r¨δ (m1 l12 + I1 + I p)Ra (m1 l12 + I1 + I p) αm2 g + rδ 2 (m1 l1 + I1 + I p)

rδ = −

Notice that the linearized system naturally decomposes into a cascade connection of two independent blocks; the

Fig. 2.

Cascade structure of ball and beam system

Step 4: Implement the LADR control for the linear approximation system To apply the Active Disturbance Rejection Controller [12], [15], [33], [2], and to exploit the underlying flatness property, it is necessary to express the system as an additively disturbed chain of integrators. The incremental (linearized) system can be expressed as follows:

(4)

αkτ N αm2 g Vδ (t) + rδ (m1 l12 + I1 + I p)Ra (m1 l12 + I1 + I p) m1 gl1 − r¨δ + H.O.T. (m1 l12 + I1 + I p)

rδ = −

where H.O.T. stands for high order nonlinear terms. Let us define the flat output trajectory tracking error as,

erδ := rδ − r∗ (t) = r − r¯ − (r∗ (t) − r¯) Clearly, due to the nature of the desired equilibrium state and input, this term precisely coincides with the tracking error r − r∗ (t). On the other hand, consider the feedforward control input as Vδ∗ (t).

Using the above relations, the following dynamics is obtained for the flat output trajectory tracking error, (4)

αkτ N αm2 g eV (t) + erδ (m1 l12 + I1 + I p)Ra (m1 l12 + I1 + I p) m1 gl1 e¨rδ + H.O.T. − (m1 l12 + I1 + I p)

while the rest of the error system is given by,

e˙2 = e3

erδ = −

e˙3 = −

αkτ N V (t) + ξ(t) 2 (m1 l1 + I1 + I p )Ra

(10)

The key step in flatness-based ADRC schemes [28] is to treat the tracking error dynamics as the following simplified perturbed model:

(4)

e rδ = −

αkτ N eV (t) + ξ(t) (m1 l12 + I1 + I p)Ra

where ξ(t) is the total disturbance, representing the effects of the neglected incremental state dependent linearities and, specially, the nonlinearities collected in the higher order terms, eliminated by the linearization process. The possibly un-modeled dynamics, and all the external unknown disturbances affecting the system are also included in this term. In other words:

ξ(t) =

αkτ N V ∗ (t) + 2 (m1 l1 + I1 + I p)Ra −

αm2 erδ (t) 2 (m1 l1 + I1 + I p)

m1 gl1 e¨rδ (t) + H.O.T. 2 (m1 l1 + I1 + I p )

Linear Extended State Observer Design (i) Denote, erδ simply as ei , i = 0, 1, . . . , n − 1. The flat output trajectory tracking error perturbed state space model is given by,

State and disturbance observers schematics

Thanks to the flatness property, a set of two decoupled, high gain, second order Linear Extended State Observers (LESO), or GPI observers, can be proposed for the simultaneous estimation of the flat output tracking error phase variables associated with erδ and of the perturbation signal ξ(t) = z1 , The LESO observers are considerably simplified. They are also noise rejection efficient. Using the cascade property it follows: For the phase variables e0 and e1 in equation (9), one sets,

e˙ˆ0 = eˆ1 + ρ1(e0 − eˆ0 ) e˙ˆ1 = eˆ2 + ρ0(e0 − eˆ0 ) while for for the rest of the dynamics, associated with the phase variables, e2 and e3 , in (10), including the disturbance z1 , one synthesizes:

e˙0 = e1 e˙1 = e2 e˙2 = e3 e˙3 = −

Fig. 3.

e˙ˆ2 = eˆ3 + λ p+1(e2 − eˆ2 ) αkτ N e˙ˆ3 = − V (t) + zˆ1 + λ p(e2 − eˆ2 ) (m1 l12 + I1 + I p)Ra zˆ˙ j = zˆ j+1 + λ p− j (e2 − eˆ2 ), j = 1, . . . , p − 1

αkτ N V (t) + ξ(t) (m1 l12 + I1 + I p )Ra

Notice that the variable e2 may be expressed as:

z˙p = λ0 (e2 − eˆ2)

e2 = r¨δ − r¨δ∗ (t) = −αθδ − r¨δ∗ (t) which represents a known input to the second order pure integration system (see Figure 3),

e˙0 = e1 e˙1 = e2 = −αθδ − r¨δ∗ (t)

(9)

(11)

where p ∈ Z+ is the order of the dynamic state extension in the observer, which can be chosen of rather arbitrary order (customarily, p = 1, has been shown to produce good results). In the present application example, the order p = 6 was systematically chosen (see the Experimental Results section). The observation error, e˜0 = e0 − eˆ0 , of the incremental flat output tracking error satisfies the following perturbed dynamics:

with e¨˜0 + ρ1e˙˜0 + ρ0 e˜0 = η(e˙˜0 , e˜0 , e2 ) where η(e˙˜0 , e˜0 , e2 ) is a perturbation term (which is assumed to be uniformly absolutely bounded) depending on the estimation and tracking errors, considering e2 as a perturbation term, depending on e0 , e1 (see [29] for a comprehensive analysis of this estimation error behavior and [14] for rigorous proofs of the convergence properties). An appropriate choice of the design coefficients: {ρ1 , ρ0 }, placing the roots of the corresponding characteristic polynomial into the left half of the complex plane, renders an asymptotically, exponentially, decreasing estimation error phase variables, e˜0 , e˙˜0 and e¨˜0 towards previously chosen arbitrarily small neighborhoods of the origin. In this case, the parameters ρ1 , ρ0 are selected such that a stable second order dynamics with characteristic polynomial s2 + 2ζo ωo s + ω2o , being ζo , ωo ∈ R+ , is matched. Then, the parameters are chosen as follows: ρ1 = 2ζo ωo , and ρ0 = ω2o , with ω0 and ζ0 positive constants. The tracking error velocity for the flat output e˙rδ = e1 is, thus, accurately estimated for feedback purposes by eˆ1 . In the same manner, consider the observation error, e˜2 = e2 − eˆ2 , of the flat output acceleration tracking error. It generates the following linear reconstruction error dynamics:

(p+2)

e˜2

(p+1)

+ λ p+1e˜2

+ . . . + λ1e˙˜2 + λ0 e˜2 = [ξ(t)](p)

A necessary and sufficient condition for having the incremental flat output acceleration estimation error e˜2 , and (p+1) its associated phase variables, e˜2 , e˙˜2 , . . . , e˜2 , ultimately, uniformly, convergent towards a sufficiently small neighborhood of the acceleration estimation error phase space, consists in choosing the observer gains: λk , {k = 0, ..., p + 1}, sufficiently large and such that the linear injected error dynamics becomes Hurwitz. The efficient pole placement procedure, developed in [19], is hereby adopted. For an arbitrary p + 2-th order characteristic polynomial, such a pole placement procedure is described as follows: Let T be a desired settling time adopted as the generalized time constant. Let α1 be the desired damping which will be an adjustable parameter taken to be a real positive constant larger than 2, i.e., α1 > 2. Define: 

αi = 

sin



iπ p+2



2 sin

  π + sin p+2  α1 ,   iπ p+2

λ p+2 =

T p+2 a0 p+1

α p+1 α2p α3p−1 · · · α1

The rest of the λ parameters in po (s) are computed via: λi =

T i a0 , 3 2 αi−1 αi−2 αi−3 · · · αi−1 1 λ p+2

i = 1, 2, 3, ..., p + 1

Thus, the pole location produces reasonable estimation error responses, with minimal “peaking effects”, but sufficiently fast rise times (as shown in [20]). The controller may be readily synthesized as a disturbance canceling ADRC, properly removing the total input disturbance, ξ(t), while imposing a desired exponentially dominated tracking error dynamics. This canceling action is achieved by using the total disturbance estimated value zˆ1 . Simultaneously, the tracking error phase variables estimates eˆ1 and eˆ3 are obtained from the proposed observers. The controller is thus given by:

V=

  (m1 l12 + I1 + Ip )Ra zˆ1 + k3 eˆ3 + k2 e2 + k1 eˆ1 + k0 e0 αkτ N

where, naturally, the tracking errors, e0 and e2 , themselves are used instead of their estimates. This is due to the fact that these variables are measurable through the variables r and θ as revealed by flatness. The coefficients of the feedback controller are chosen using considering the differential equation of the closed loop flat output tracking error: (4)

(3)

e0 + k3 e0 + k2e¨0 + k1e˙0 + k0e0 = ξ(t) − zˆ1 The set of coefficients {k3 , k2 , k1 , k0 } are chosen so that the associated characteristic polynomial of the dominantly linear closed loop dynamics, pc (s) = s4 + k3 s3 + k2 s2 + k1 s + k0 becomes a Hurwitz polynomial. In this case, one sets: k3 = 4ζc ωc , k2 = 2ω2c + 4ζ2c ω2c , k1 = 4ζc ω3c , k0 = ω4c , with 0 < ζc < 1, ωc ∈ R+ . One may force the closed loop tracking error dynamics to exhibit the desired stable fourth order dynamics with characteristic polynomial coincident with (s2 + 2ζc ωc s + ω2c )2 .

i = 2, 3, ..., p + 1

Let a0 be an arbitrary strictly positive constant. Choose, λ0 =

Ta0 λ p+2

4 Experimental results A schematic of the experimental platform is shown in Figure 4. The experimental device consists of a DC motor from NISCA Motors (model NC5475B) which drives an aluminium beam via a synchronous belt and a pulley with a

Block diagram of the ball and beam control system

ratio N = 6 : 1. The angular position of the beam is measured using an incremental optical encoder of 2500 pulses per revolution. A linear sensor, fixed on the beam, consists on an etched wire made of a nickel-chromium wire on which a stainless steel ball can roll with negligible friction. The ball acts as a wiper, similar to that of a potentiometer. The position along the beam can be obtained by measuring the output voltage of the linear sensor with a resolution of 25 [mm/V].

rδ∗ (t)

rδ (t)

0.1

0.05

[m]

Fig. 4.

ζo = 1.3, ωo = 13. The observer gain parameters for observation error e˜2 were set: p = 6, T = 6, a0 = 4, α1 = 4.1. The controller design parameters were taken as: ζc = 1 , ωc = 15. Figure 6 shows the performance of the closed loop trajectory tracking, from the initial position: rδ (0) = 0, towards the equilibrium position: rδ (9) = 0.12. Once stabilized at the previous position, the ball is further moved to the final resting position: rδ (18) = −0.12. The tracking error evolution is shown in Figure 7. The position error is restricted to a an interval of [−0.004, 0.004] [m]. The produced control voltage V is depicted in Figure 9. Notice that the lumped disturbance estimation (Figure 10) shapes the control input due to its disturbance canceling efforts.

0

−0.05

−0.1

0

5

10

15

20

25

30

time [s]

Fig. 6.

Performance of closed loop reference trajectory tracking

−3

x 10

erδ = rδ (t) − rδ∗ (t)

6

4

Fig. 5.

Ball and beam system experimental prototype

A power amplifier from STMicroelectronics model TDA7293 drives the motor. The data acquisition is carried out through a data card, model QPIDe, from Quanser. This card reads signals from the optical incremental encoder, from the linear sensor voltage and, in turn, it supplies the computed control voltage to the power amplifiers. The control strategy described before was implemented in the Matlab-Simulink Quarc platform. Finally, the sampling time was set to be 0.001[s]. The parameters for the beam are: Inertia of beam, I1 = 0.0045[Kg-m2]. Mass of the beam m1 = 0.065[Kg]. Distance of the center of mass of beam to the axis of rotation l1 = 0.015[m]. Inertia of the pulley system, I p = 0.001[Kg-m2]. Mass of the ball, m2 = 0.065[Kg], and inertia, I2 = 0.0045[Kg-m2]. The radius of the ball is, R = 0.0127[m]. The parameters of the motor are: Ra = 12.1[Ω], kτ = 0.0724[Nm/A]. The initial conditions for the position variables in the system were: [r = 0, θ = 0]. The observer gain parameters for the observation error e˜0 , were set to be :

[m]

2

0

−2

−4

−6

0

5

10

15

20

25

30

time [s]

Fig. 7.

Radial position trajectory tracking error

5 Conclusions and suggestions for future research The problem of controlling a class of under-actuated systems, in a trajectory tracking task, was solved by means of an exploitation of the flatness associated with the linearized model and via an Extended State Observer based linear Active Disturbance Rejection Control. The scheme uses the tangent linearization system model of the Ball and Beam system

0.25

r ¨δ∗ (t)

r ¨δ (t) = −αθ(t)

0.2 0.15

[ m/s2 ]

0.1 0.05 0

−0.05 −0.1 −0.15 −0.2 −0.25

0

5

10

15

20

25

30

time [s]

Fig. 8.

Radial acceleration for decoupled observer design

4

a simplified, bold, reformulation of the system dynamics to an either first, or second order, paradigm, while on-line implementing algebraic estimation methods [9], [27] to overcome the lack of information of the high order plant. Model free control can be used as an interesting alternative to our proposal in those cases where the order of the system is unknown, or purposefully chosen to be ignored. The use of a lower order disturbance observer (LESO) is also possible, until the observer extension becomes one dimensional (see [23], [13], and references therein). The predominantly linear control scheme exhibits interesting advantages associated with low computational costs. The scheme may be applied in autonomous systems, or in embedded hardware implementations. This area may benefit from further studies and developments in the realm of sampled data control, or exact discretization, techniques in linear approximations of nonlinear discrete systems of flat nature.

V (t)

3 2

[V]

1 0 −1 −2 −3 −4

0

5

10

15

20

25

30

25

30

time [s]

Fig. 9.

Voltage input control signal

150

ˆ ξ(t) =z ˆ1 (t)

100

4

[ m/s ]

50

0

−50

−100

−150

0

5

10

15

20

time [s]

Fig. 10.

On-line estimation of total disturbance

around an arbitrary equilibrium point. The traditional drawbacks present in the customary linear control scheme were overcome by using a particular structure of the Linear Extended State Observer, of the Luenbeger type, induced by the flatness of the system. The scheme achieves highly competitive experimental results when compared with other linear and nonlinear controllers. A new alternative technique, denoted as Model Free Control [10], has been attracting attention in many control engineering research areas. This control scheme is based on

References [1] Aguilar-Ibanez, C., and Guti´errez-Frias, O., 2008. “Controlling the inverted pendulum by means of a nested saturation function”. Nonlinear Dynamics, 53(4), pp. 273–280. [2] M.A. Arteaga-P´erez, A. G.-G., and Weist, J., 2015. “On the observability and the observer design of differential pneumatic pistons”. ASME, Journal of Dynamic Systems, Measurement, and Control, 137(8), p. 081006. [3] Bayezit, I., and Fidan, B., 2013. “Distributed cohesive motion control of flight vehicle formations”. IEEE Transactions on Industrial Electronics, 60(12), pp. 5763–5772. [4] Begovich, O., Sanchez, E., and Maldonado, M., 2002. “Takagi-sugeno fuzzy scheme for real-time trajectory tracking of an underactuated robot”. IEEE Transactions on Control Systems Technology, 10(1), pp. 14–20. [5] Fantoni, I., and Lozano, R., 2002. Non-linear control for underactuated mechanical systems. Springer. [6] Fierro, R., Lewis, F., and Lowe, A., 1999. “Hybrid control for a class of underactuated mechanical systems”. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 29(6), pp. 649–654. [7] Fliess, M., L´evine, J., Martin, P., and Rouchon, P., 1995. “Flatness and defect of non-linear systems: Introductory theory and applications”. International Journal of Control, 61, pp. 1327–1361. [8] Fliess, M., L´evine, J., Martin, P., Ollivier, F., and Rouchon, P., 1997. “Controlling nonlinear systems by flatness”. In Systems and Control in the Twenty-first Century. Springer, pp. 137–154. [9] Fliess, M., Sira-Ram´ırez, H., 2003 “An algebraic framework for linear identification” ESAIM , Control Optimization and Calculus of Variations 9, pp. 151– 168. [10] Fliess, M., and Join, C., 2013. “Model-free control”. International Journal of Control, 86(12), pp. 2228– 2252. [11] Forrest, A., 1972. “Interactive interpolation and ap-

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26] [27]

proximation by b´ezier polynomials”. The Computer Journal, 15(1), pp. 71–79. Gao, Z., 2006. “Active disturbance rejection control: A paradigm shift in feedback control system design”. In American Control Conference, pp. 2399–2405. Gao, Z., 2014. “On the centrality of disturbance rejection in automatic control”. ISA transactions, 53(4), pp. 850 – 857. Guo, B., and Zhao, Z., 2011. “On the convergence of extended state observer for nonlinear systems with uncertainty”. Systems and Control Letters, 60, pp. 420– 430. Han, J., 2009. “From PID to active disturbance rejection control”. IEEE Transactions on Industrial Electronics, 56(3), pp. 900–906. Hauser, J., Sastry, S., and Kokotovic, P., 1992. “Nonlinear control via approximate input-output linearization: the ball and beam example”. IEEE Transactions on Automatic Control, 37(3), pp. 392–398. B. Jakubczyk, W. R., 1980. “On linearization of control systems”. Bull. Acad. Polonaise Sci. Ser. Sci. Math., 28, pp. 517–522. Jo, N., and Seo, J., 2000. “A state observer for nonlinear systems and its application to ball and beam system”. IEEE Transactions on Automatic Control, 45(5), pp. 968–973. Keel, Y. K. L., and Bhattacharyya, S., 2003. “Transient response control via characteristic ratio assignment”. IEEE Transactios on Automatic Control, 48(1), pp. 2238–2244. Ma, C., Cao, J., and Qiao, Y., 2013. “Polynomialmethod-based design of low-order controllers for twomass systems”. IEEE Transactions on Industrial Electronics, 60(3), pp. 969–978. Mazenc, F., Pettersen, K., and Nijmeijer, H., 2002. “Global uniform asymptotic stabilization of an underactuated surface vessel”. IEEE Transactions on Automatic Control, 47(10), pp. 1759–1762. Olfati-Saber, R., 2002. “Normal forms for underactuated mechanical systems with symmetry”. IEEE Transactions on Automatic Control, 47(2), pp. 305–308. Radke, A., and Gao, Z., 2006. “A survey of state and disturbance observers for practitioners”. In American Control Conference, 2006, IEEE. Ramirez-Neria, M., Sira-Ramirez, H., GarridoMoctezuma, R., and Luviano-Juarez, A., 2014. “Linear active disturbance rejection control of underactuated systems: The case of the furuta pendulum”. ISA Transactions, 53(4), pp. 920 – 928. Scheller, J., and Starossek, U., 2011. “Design and experimental verification of a new active mass damper for control of pedestrian-induced bridge vibrations”. In Fourth International Conference Footbridge. Sira-Ram´ırez, H., Agrawal, S., 2004. Differentially Flat Systems. Marcel Dekker, New York. Sira-Ram´ırez, H., Agrawal, S., 2014. Algebraic Identification and Estimation Methods in Feedback Control Systems. Wiley, Sussex, United Kingdom.

[28] Sira-Ram´ırez, H., Luviano-Ju´arez, A., and Cort´esRomero, J., 2012. “Flatness-based linear output feedback control for disturbance rejection and tracking tasks on a chua’s circuit”. International Journal of Control, 85(5), pp. 594–602. [29] Sira-Ramirez, H., Linares-Flores, J., GarciaRodriguez, C., and Contreras-Ordaz, M., 2014. “On the control of the permanent magnet synchronous motor: An active disturbance rejection control approach”. IEEE Transactions on Control Systems Technology, 22(5), September, pp. 2056–2063. [30] Spong, M., 1998. “Underactuated mechanical systems”. In Control Problems in Robotics and Automation. Springer, pp. 135–150. [31] Xu, R., and Ozguner, U., 2008. “Sliding mode control of a class of underactuated systems”. Automatica, 44(1), pp. 233–241. [32] Yan, Z., and Wang, J., 2012. “Model predictive control for tracking of underactuated vessels based on recurrent neural networks”. IEEE Journal of Oceanic Engineering, 37(4), pp. 717–726. [33] Zheng, Q., Chen, Z., and Gao, Z., 2009. “A practical approach to disturbance decoupling control”. Control Engineering Practice, 17(9), pp. 1016–1025.