two relay controller for real time trajectory generation and its...
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Tworelaycontrollerforrealtimetrajectorygenerationanditsapplicationtoinvertedorbitalstabilizationofinertia...
ArticleinAsianJournalofControl·January2012
DOI:10.1002/asjc.339
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Asian Journal of Control, Vol. 14, No. 2, pp. 1 9, March 2010Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/asjc.339
TWO RELAY CONTROLLER FOR REAL TIME TRAJECTORY
GENERATION AND ITS APPLICATION TO INVERTED ORBITAL
STABILIZATION OF INERTIA WHEEL PENDULUM VIA
QUASI-CONTINUOUS HOSM
Antonio Estrada, Luis T. Aguilar, Rafael Iriarte, and Leonid Fridman
ABSTRACT
A quasi-continuous high-order sliding mode (QC-HOSM) control isdeveloped to solve the tracking control problem for an inertia wheel pendulum.A first step towards the solution of the tracking control problem in underac-tuated systems is to find the set of reference trajectories. A reference modelbased on the two relay controller idea is then developed for generating a setof desired periodic trajectories for the pendulum centered at its upright posi-tion. The two relay controller produces oscillations at the scalar output of thereference underactuated system where the desired amplitude and frequencyare reached by choosing its gains. The HOSM will be capable of makingthe pendulum move, tracking the prescribed reference signals determined bythe trajectory generator. Performance issues of the controller constructed areillustrated in an experimental study.
Key Words: Variable structure control, orbital stabilization, inertia wheelpendulum, high-order sliding mode controller.
I. INTRODUCTION
The focus of this paper is to solve the trackingcontrol problem for the inertia wheel pendulum. Further3research applications in the control of underactuatedsystems have gone in many directions. For example,5fully-actuated robots where it is required that motioncontinues in spite of a failure of any of its actuators.7Other typical examples are systems where their desired
Manuscript received January 7, 2010; accepted September25, 2010.A. Estrada and L. Fridman are with Universidad Nacional
Autonoma de Mexico (UNAM), Department of Control, Engi-neering Faculty. C.P. 04510 Mexico D.F.L.T. Aguilar and R. Iriarte are with Instituto Politecnico
Nacional, Centro de Investigacion y Desarrollo de TecnologıaDigital, Avenida del parque 1310 Mesa de Otay Tijuana22510 Mexico. R. Iriarte is the corresponding author (e-mail:[email protected]).
operation mode is oscillatory; such as biped walking 9robots where a periodic trajectory is required to producea coordinated motion (see, e.g. [1]); and hopping robots 11where the thrust, decompression, flight, and compres-sion phase is also governed by a periodic motion (see, 13e.g. [2]) among others.
The present formulation is different from the 15typical formulation of the output tracking controlproblem for fully actuated mechanical systems [3, p. 17278], where the reference trajectory can be arbitrarilygiven, because underactuated systems are neither 19feedback nor input-state linearizables due to theircomplexity. Therefore, special attention is required in 21the selection of the desired trajectory for the systemunder study. Different approaches for the orbital stabi- 23lization have been proposed. In particular, Shiriaevet al. [4] introduces a constructive tool for the gener- 25ation and orbital stabilization of periodic solutionsfor underactuated nonlinear systems named virtual 27constraints approach. Grizzle et al. [5] demonstrate
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asymptotic tracking for an unactuated link by finding1conditions for the existence of a set of outputs thatyield a system with a one-dimensional exponentially3stable zero dynamics. Santiesteban et al. [6] introducedan asymptotic generator of the periodic motion through5a modified Van der Pol equation tested on a frictionpendulum to solve the swing-up problem for an inverted7pendulum with restricted travel (cart-pendulum). InAguilar et al. [7, 8], the idea of two relay controller9in order to generate oscillations at the output of anonlinear system was introduced. There are several11applications of the above mentioned procedures in theliterature (see, e.g. [9–12] and references therein).13
The methodology from Aguilar et al. [8] isused in the present paper, where a set of trajec-15tories, which involve inverted periodic motion arederived for the nonlinear equation of motion through17a relay feedback known as the two-relay controller.The proposed approach is based on the fact that all19second order sliding mode (SOSM) algorithms [13]produce chattering (periodic motions of relatively21small amplitude and high frequency) in the presence ofunmodelled dynamics. This method uses this property23of SOSM for the purpose of generating a relativelyslow motion with a significantly higher amplitude25and lower frequency than the respective frequency ofchattering. The frequency and amplitude of oscillations27consist in computing a two gains parameter only. In [7]three methods to compute these values were proposed:29describing function method, locus of the perturbedrelay system (LPRS), and Poincare maps. We refer the31reader to [11, 14] for experimental application of themethod.33
Originally in [8], a two relay controller wasdesigned for the inverted pendulum where the required35frequencies and amplitudes of periodic motions wereproduced without tracking of precomputed trajectories37(i.e. autonomous system), however the closed-loopsystem was sensitive to disturbances. In [14], the same39concept of [8] was introduced but identification ofdisturbances ensures robustness of the closed-loop41system. Now, the proposed framework for a real-timetrajectory generation via the two-relay controller and the43application of a high-order sliding mode state-feedbacktracking controller of an inertia wheel pendulum (non-45autonomous system) and its experimental verificationconstitute the main contribution of the paper.47
In the present paper, we use a new design algo-rithm for systems in strict-feedback form proposed49in Estrada and Fridman [15]. This algorithm achievesfinite-time exact tracking of the desired output in51the presence of unmatched perturbations and allowsreducing the gain of the discontinuous control as53
compared with the direct application of high-ordersliding modes. These features are accomplished via the 55usage of quasi-continuous high-order sliding modes(QC-HOSM) and a hierarchical design approach. At 57the first step, the desired dynamics for the first state isdefined by the desired tracking signal. Then the desired 59dynamic for each state is defined by the previous one.Each virtual control is divided into two parts, in the 61first one the part of the equivalent control that canbe constructed using the known part of the system is 63included while the second one is aimed at achievingthe desired dynamics in spite of perturbations. 65
Our paper is organized as follows: In Section IIthe dynamic model and problem formulation are 67described. In Section III the reference model with thetwo relay controller used to generate a set of desired 69trajectories for the inertia wheel pendulum around theopen-loop unstable equilibrium point is introduced. 71In Section IV, we present a quasi-continuous high-order sliding mode controller to achieve finite-time 73exact tracking controller of the desired output againstunmatched perturbation. Experimental results showing 75the effectiveness of the proposed method are givenin Section V. Finally, conclusions are provided in 77Section VI.
II. DYNAMIC MODEL AND PROBLEM 79STATEMENT
The dynamics of an inertia wheel pendulum can 81be described as follows [16]:[
J1 J2
J2 J2
][q1
q2
]+
[h sinq1
fs q2
]=
[0
1
]�+w (1)
83
where q1∈R is the absolute angle of the pendulum,counted clockwise from the vertical downward position; 85q2∈R is the absolute angle of the disk; fs q2 repre-sents the viscous friction force affecting the actuator 87where fs>0 is the viscous friction coefficient; J1, J2,and h are positive physical parameters that depend on 89the geometric dimensions and the inertia-mass distribu-tion; �∈R is the controlled torque applied to the disk 91(see Fig. 1); and w= (w1,w2) are the external distur-bances affecting the system. An upper bound Mi>0, i = 931,2 for the magnitude of the disturbances is normallyknown a priori 95
supt
|wi (t)|≤Mi , i =1,2. (2)
It should be noted that system (1) is nonlinear and under- 97actuated.
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Fig. 1. Inertia wheel pendulum.
Two goals are pursued in this paper: (i) Find the1set of oscillations qr that the inertia wheel pendulumcan follow around its upright position, and (ii) find �3such that
limt→∞‖q1(t)−qr1(t)‖= lim
t→∞‖�1(t)‖=0 (3)5
where �1∈R stands for the pendulum position error,t ∈R is the time, and qr1 is the desired trajectory of the7pendulum.
III. GENERATING DESIREDTRAJECTORIES
Let us start by explaining how to find a set of11desired oscillations around the upright position. Tobegin with, let us rewrite (1) in terms of the reference13positions and velocities (qr , qr )[
J1 J2
J2 J2
][q1r
q2r
]+
[h sinq1r
0
]=
[0
1
]�r . (4)
15
We need to find �r to produce a set of desired periodicmotions of the underactuated link (y=q1r ) such that the17output has a periodic motion with desired frequency andamplitude. As will be shown later, viscous friction is not19required in the above equation since it acts as a dampingforce. Throughout, we confine our research interest in21desired oscillations around the upright position of thependulum which corresponds to the more difficult case,23
due to the fact that the open-loop system has an unstablezero dynamics. 25
In comparison to previous works [13, p. 12],where the describing function (DF) method was usedfor analysis of feedback relay systems, with linearplants, now we are dealing with a nonlinear plant.Therefore, linearization is required as a preliminary stepfor applying DF as a conventional method. Linearizing(4) around the equilibrium point (q∗
1r ,q∗2r , q
∗1r , q
∗2r )=
(�,0,0,0) we find that this equilibrium point isunstable, therefore the system becomes sensitive toinitial conditions in a neighborhood around the equi-librium point. However, the inertia wheel pendulumhas underactuation degree one and satisfies certainstructural properties noted in [5]. As a result, it ispossible to achieve exact linearization thus achievinglocal stability of the zero dynamics. Following [5], letus take
p1 = q1r −�+ J−11 J2q2r (5)
� = Kp1+ J1q1r + J2q2r (6)
where the output � yields an exponentially minimumphase system for all K>0, that is the zero dynamics 27have dimension one and are exponentially stable whilethe system is dynamically input–output decoupable. In 29our case, we are interested in that the trajectory of thezero dynamics vanishes around the equilibrium point �. 31It is easy to verify that
J1 p1=�−K p1 33
while
� = K J−11 J2q2r −h sin(q1r )+K q1r ,
� = −h cos(q1r )q1r −K J−11 h sin(q1r ),
¨� = R(q1r , q1r )+H(q1r )�r
where
H(q1r ) = h cos(q1r )
J1− J2,
R(q1r , q1r ) = h(q21r +H(q1r )) sin(q1r )
−hK
J1q1r cos(q1r ). (7)
Hence, we can take
�r =H−1(q1r )(u−a0�−a1�−a2�−R(q1r , q1r )), (8) 35
where H(qr ) is nonsingular around the equilibriumpoint (q�
1r , q�1r )= (�,0), a0,a1, and a2 are positive
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constants. Introducing the new state coordinates x =1(x1, x2, x3)= (�, �, �), we obtain⎡
⎢⎣x1
x2
x3
⎤⎥⎦ =
⎡⎢⎣
0 1 0
0 0 1
−a0 −a1 −a2
⎤⎥⎦
︸ ︷︷ ︸A
⎡⎢⎣x1
x2
x3
⎤⎥⎦+
⎡⎢⎣0
0
1
⎤⎥⎦
︸︷︷︸B
u,
p1 = − K
J1p1+ 1
J1y, y=[1 0 0]︸ ︷︷ ︸
C
x .
(9)
3
The following two-relay controller is used for excitinga periodic motion:5
u=−c1sign(y)−c2sign(y) (10)
where c1 and c2 are scalar parameters designed such7that the scalar-valued output function y(t) has a periodicmotion with the desired frequency� and amplitude Ap .9
Since we are interested in presenting the results inthe original coordinates q1r and q2r , let us show how to11compute an approximation for the amplitude of oscil-lations for the pendulum itself. Since p1+ J−1
1 Kp1=13J−11 �(t) we know that p1 exponentially converges to a
periodic steady-state function (the rate of convergence15can be regulated by K ).
Taking into account only the first harmonic17and letting the steady-state value for �(t) be �(t)≈Ap sin(�t) we can compute the approximate steady-19state value for p1(t) in the form
p1(t)≈ Ap√J 21 �2+K 2
sin
(�t+arg
{1
j J1�+K
}).
21
Now, using the equations �(t)≈�Ap cos(�t) and−h sin(q1r )= �−K p1 it can be concluded that q123converges exponentially to a periodic steady-stateoscillation, provided it is small.25
Finally, for q1r close enough to � we havesin(q1r )≈�−q1r , and so
q1r (t) ≈ �− �Ap
hcos(�t)
− �Ap
h√J 21 �2+K 2
sin
(�t+arg
{1
j J1�+K
}).
This expression gives us an estimation on the amplitudeof achieved oscillations of the pendulum around � to be27
Ar ≈ �Ap
h
√1+ 1
J 21 �2+K 2.
Since q2r = J1(p1−(�−q1r ))/J2, the steady-state29amplitude of q2r can be estimated as well.
3.1 Gain computation procedure 31
Before giving the gain computation formulas, wepresent a brief description of how the formulas were 33obtained [7, 8].
Let firstly the linearized plant be given by: 35
x= Ax+Bu
y=Cx, x ∈Rn, y∈R, n=2m (11)
which can be represented in the transfer function form 37as follows:
W (s)=C(s I−A)−1B. 39
Let us assume that matrix A has no eigenvalues at theimaginary axis and that the relative degree of (11) is 41greater than 1.
The Describing Function (DF), N , of the vari-able structure controller (10) is the first harmonic ofthe periodic control signal divided by the amplitude ofy(t) [17]:
N = �
�Ap
∫ 2�/�
0u(t) sin(�t)dt
+ j�
�Ap
∫ 2�/�
0u(t)cos(�t)dt (12)
where Ap is the amplitude of the input to the nonlin- 43earity (of y(t) in our case) and � is the frequency ofy(t). However, algorithm (10) can be analyzed as the 45parallel connection of two ideal relays where the inputto the first relay is the output variable and the input to 47the second relay is the derivative of the output variable.For the first relay the DF is 49
N1= 4c1�Ap
,
and for the second relay is [17] 51
N2= 4c2�A2
,
where A2 is the amplitude of dy/dt . Also, take into 53account the relationship between y and dy/dt in theLaplace domain, which gives the relationship between 55the amplitudes Ap and A2: A2= Ap�, where � is thefrequency of oscillation. Using the notation of algorithm 57(10) we can rewrite this equation as follows:
N =N1+sN2= 4c1�Ap
+ j�4c2�A2
= 4
�Ap(c1+ jc2), (13)
59
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where s= j�. Let us note that the DF of algorithm (10)1depends on the amplitude value only. This suggests thetechnique for finding the parameters of the limit cycle -3via the solution of the harmonic balance equation [17]
W ( j�)N(a)=−1, (14)5
where a is the generic amplitude of the oscillation atthe input to the nonlinearity, and W ( j�) is the complex7frequency response characteristic (Nyquist plot) of theplant. Using the notation of algorithm (10) and replacing9the generic amplitude with the amplitude of the oscil-lation of the input to the first relay, this equation can be11rewritten as follows:
W ( j�)=− 1
N(Ap), (15)
13
where the function at the right-hand side is given by
− 1
N(Ap)=�Ap
−c1+ jc24(c21+c22)
.15
Equation (14) is equivalent to the condition of thecomplex frequency response characteristic of the open-17loop system intersecting the real axis in the point(−1, j0). The graphical illustration of the technique of19solving (14) is given in Fig. 2. The function −1/N isa straight line, the slope of which depends on c2/c121ratio. The point of intersection of this function and ofthe Nyquist plot W ( j�) provides the solution of the23periodic problem.
Nyquist quadrant identification. Identify thequadrant in the Nyquist plot where the desiredfrequency is located, that is, � can belong to any of thefollowing sets (see Fig. 2):
Q1 = {�∈R :Re{W ( j�)}>0, Im{W ( j�)}≥0}Q2 = {�∈R :Re{W ( j�)}≤0, Im{W ( j�)}≥0}
Fig. 2. Example of a Nyquist plot of the open-loop systemW ( j�) with the two-relays controller.
Q3 = {�∈R :Re{W ( j�)}≤0, Im{W ( j�)}<0}Q4 = {�∈R :Re{W ( j�)}>0, Im{W ( j�)}<0}.Gain parameters computation.The frequency of 25
the oscillations depends only on the c2/c1 ratio, and itis possible to obtain the desired frequency � by tuning 27the �=c2/c1 ratio:
�= c2c1
=− Im{W ( j�)}Re{W ( j�)} . (16) 29
Since the amplitude of oscillations is given by
Ap = 4
�|W ( j�)|
√c21+c22, (17) 31
then the c1 and c2 values can be computed as follows:
c1 =
⎧⎪⎪⎨⎪⎪⎩
�
4
Ap
|W ( j�)|(√1+�2)−1 if �∈Q2∪Q3
−�
4
Ap
|W ( j�)|(√1+�2)−1 elsewhere
(18)
c2 = � ·c1. (19)
Figure 3 shows a control block diagram applied to theinertia wheel pendulum with the real time trajectory 33generator. We refer the reader to [7] for two additionalmethods for computing c1 and c2: the LPRS method 35and the Poincare map design.
Fig. 3. Block diagram of the two-relay controller forreal-time trajectory generation for orbital stabilizationof inertia wheel pendulum.
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IV. TRACKING OF THE PREDICTEDPERIODIC OSCILLATIONS
As mentioned in the introductory section, our goal3is to design a controller that ensures exact tracking of q1rin spite of the uncertainty and disturbances present in5the real plant with respect to the reference model. Due tosome structural properties noted in [5], the inertia wheel7pendulum (IWP) model can be transformed to the strict-feedback form. Thus the design algorithm reported in9[15] can be applied. Following [5] the strict-feedbackform of the IWP model is:11
z1 = −h sin(q1)
q1 = J−11 z1− J−1
1 J2z2
z2 = h sin(q1)
J1− J2+ J1
J2(J1− J2)�
(20)
where q2= z2.13The hierarchical design procedure in [15] is based
on the so-called QC-HOSM algorithms [18]. Its main15advantage is that it allows reduction of the gain ofthe discontinuous control as compared with the direct17application of them. The gain reduction is achieved byconstructing virtual controls in which part of the equiv-19alent control is included. This is done making use of theknown nominal part of the system. Due to uncertainties21and disturbances, the exact construction of the equiv-alent control is impossible, nevertheless a QC-HOSM23is also introduced in each virtual control in orderto reject those unknown terms. Each virtual control25requires some degree of smoothness, determined by itsrelative degree with respect to the control input, which27is achieved introducing the discontinuous term througha proper number of integrators which in turn define the29order of the QC-HOSM used. For the IWP, the designstarts by taking the state z2 in (20) as a virtual controller,31�1(q1), for the state q1, which has relative degreetwo, of the system (20). Since the desired tracking33signal is q1r , it has a smooth second derivative andthus fulfills the smoothness condition for the hierarchic35design then:
Step 1. The first sliding surface is chosen as �1�q1−37q1r (t). The 2-sliding homogeneous quasi-continuouscontroller is included in �1(q1)39
�1(q1) = J1 J−12 {J−1
1 z1+u1,1}
u1,1 = −1�1+|�1|1/2sign(�1)
|�1|+|�1|1/2 .(21)
The derivative �1 is calculated by means of the 41following robust differentiator [19]
s0 = −2L1/2|s0−�1|1/2sign(s0−�1)+s1
s1 = −1Lsign(s1− s0).(22)
43
Step 2. Now for state z2, �2�z2−�1(q1)
� = J2 J−11 {h sin(q1)+(J1− J2)u2,1}
u2,1 = −2sign(�2).(23)
45
Remark 1. Notice that in the sliding mode
q2= z2= J1 J−12
{J−11 z1+
∫u1,1
},
47
and due to chattering analysis [20], it can be provedthat the term inside the integral is bounded due to the 49absolute continuity of the desired trajectory. Thus it canbe proved that q2 remains bounded, nevertheless, the 51bound depends on the initial conditions q1(t0), q1(t0).
V. EXPERIMENTAL RESULTS 53
In this section, we present experimental resultsusing the laboratory inertial wheel pendulum (Mecha- 55tronic Kit) manufactured by Quanser Inc., depicted inFig. 1 where J1=4.572×10−3, J2=2.495×10−5, and 57h=0.3544 (see [16]). The viscous friction coefficientfs =8.80×10−5 was identified by applying the proce- 59dure from Kelly et al. [21]. Experiments were carriedout to achieve the orbital stabilization of the unactu- 61ated link (pendulum) q1 around the equilibrium pointq� = (�,0). 63
For the experiments, we select �=2� [rad/s] andAp =0.05 as desired frequency and amplitude, respec- 65tively. Following the procedure given in Subsection 3.2to compute c1 and c2 we first find, by plotting the 67Nyquist plot of (9) under K =1×10−4, a0=350,a1=155, and a2=22, that the desired frequency 69belongs to the third quadrant, i.e. �∈Q3. SinceRe{ j2�}=−6.73×10−4 and Im{ j2�}=−6.85×10−4 71and by using (18), (19) we find that c1=5.7177 andc2=−5.82. Figure 4 shows the periodic reference 73signal around �. The constants for the controller (21),(23) are taken as 1=−8 and 2=200 and the constants 75for the differentiator (22) are taken as 1=1.1,2=1.5,and L=10. 77
The initial conditions for the inertia wheelpendulum, selected for the experiments, were q1(0)=3 79[rad] and q2(0)=0 [rad], whereas all the initial velocityconditions were set to q1(0)= q2(0)=0 [rad/s]. 81
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0 2 4 6 8 10 12 14 16 18 203.1
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
time [s]
q 1r [
rad]
Fig. 4. Periodic reference signal at �=2� [rad/s] andAp =0.07 generated by the two-relay controllerreference model under the parameters c1=2,c2=−0.1, K =1×10−4, a0=350, a1=155, anda2=22.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
100
200
time [s]
Fig. 5. Tracking error of the underactuated link � under nodisturbances and velocity error of the disk ( ˙q2).
Experimental results for the inertia wheel 1pendulum, driven by the sliding mode trackingcontroller (21), (23) are depicted in Figure 5 for the3disturbance-free case. In order to test the robust-ness of the orbitally stabilizing controller (21), (23)5external disturbances were randomly added by lightlyhitting the pendulum at time instants t1≈137 [s] and7t2≈147 [s]. Figure 6 shows experimental results forthe perturbed case.
9
VI. CONCLUSIONS
State feedback sliding mode tracking control is11used in an underactuacted mechanical system. The
105 110 115 120 125 130
0
0.1
0.2
105 110 115 120 125 130
0
500
time [s]
Fig. 6. Tracking error of the underactuated link � underdisturbances and perturbed velocity of the disk (q2).
desired periodic orbit is centered at the upright position 13where the open-loop plant becomes a non-minimum-phase system. The designed controller takes the 15trajectories of the inertia wheel pendulum into a set ofinverted desired trajectories which have been generated 17by an IWP reference model governed by a two-relaycontroller. The experimental verification, made for a 19laboratory prototype, demonstrates the effectiveness ofthe developed approach. 21
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application to swing up/balancing control,” Int.1J. Robust Nonlin. Control, Vol. 18, No. 4–5,pp. 544–556 (2008).3
7. Aguilar, L., I. Boiko, L. Fridman, and R.Iriarte, “Generating self-excited oscillations for5underactuated mechanical systems via two relaycontroller,” Int. J. Control, Vol. 82, No. 9,7pp. 1678–1691 (2009).
8. Aguilar, L., I. Boiko, L. Fridman, and R. Iriarte,9“Generating self-excited oscillations via two-relaycontroller,” IEEE Trans. Autom. Control, Vol. 54,11No. 2, pp. 416–420 (2009).
9. Canudas de Wit, C., B. Espiau, and C. Urrea,13“Orbital stabilization of underactuated mechanicalsystems,” Proc. 15th World Cong., Barcelona, Spain15(2002).
10. Shiriaev, A. S., L. B. Freidovich, and S. V. Gusev,17“Transverse linearization for controlled mechanicalsystems with several pasive degrees of freedom,”19IEEE Trans. Autom. Control, Vol. 55, No. 4, pp.893–906 (2010).21
11. Aguilar, L., I. Boiko, L. Freidovich, and L. Fridman,“Inducing Oscillations in an Inertia Wheel23Pendulum via Two-Relays Controller: Theory andExperiments,” 2009 Amer. Control Conf., St. Louis,25Missouri, pp. 65–70 (2009).
12. Berkemeier,M. D., and R. S. Fearing, “Tracking fast27inverted trajectories of the underactuated Acrobot,”IEEE Trans. Robot. Automat., Vol. 15, No. 4, pp.29740–750 (1999).
13. Boiko, I., Discontinuous Control Systems:31Frequency-domain Analysis and Design.Birkhauser, Boston (2009).33
14. Aguilar, L., I. Boiko, L. Fridman, and A. Ferreira,“Identification based generation of self-excited35oscillations for underactuated mechanical systemsvia two-relays algorithm,” 10th Int. Workshop37Variable Structure Systems, Antalya, Turkey, June8–10, pp. 41–46 (2008).39
15. Estrada, A. and L. Fridman, “Exact compensationof unmatched perturbation via quasi-continuous41HOSM,” 47th IEEE Conf. Decis. Control, pp. 2202–2207 (2008).43
16. Astrom, K., D. Block, andM. Spong, “The ReactionWheel Pendulum,” Lecture Notes for the Reaction45Wheel Pendulum (Part of the Mechatronics ControlKit), Urbana-Champaigne, IL (2001).47
17. Atherton, D., Nonlinear Control Engineering:Describing Function Analysis and Design. Van49Nostrand, Workingham (1975).
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19. Levant, A., “High-order sliding modes:differentiation and output feedback control,” Int. J. 55Control, Vol. 76, No. 11, pp. 924–941 (2003).
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21. Kelly, R., J. Llamas, and R. Campa, “Ameasurement procedure for viscous and Coulomb 61friction,” IEEE Trans. Instrum. Meas., Vol. 49,No. 4, pp. 857–861 (2000). 63
Antonio Estrada was born in 65Mexico City. He received his BSdegree in Electrical and Elec- 67tronic Engineering and his MSdegree in automatic control from 69National Autonomous Universityof Mexico (UNAM), in 2007 71and 2008, where he is currentlypursuing his PhD degree. His 73research interest include sliding
mode control, robust control and unmatched perturba- 75tions compensation in nonlinear systems.
Luis T. Aguilar received the 77Industrial Electronics Engineerdiploma from Instituto Tecno- 79logico de Tijuana, in 1994. Hereceived the MSc degree in Digital 81Systems from Centro de Investi-gacion y Desarrollo de Tecnologıa 83Digital (CITEDI-IPN), Tijuana,Mexico and the PhD degree in 85Electronics and Telecommuni-
cations from CICESE Research Center, Ensenada, 87Mexico, in 1998 and 2003, respectively. Since 2004,he has been a full-time professor at the CITEDI-IPN 89Research Center. His current research interests includevariable structure systems, nonlinear H-infinity control, 91and control of electromechanical systems.
Rafael Iriarte was born in 93Mexico City. He received his BSdegree in Electrical and Elec- 95tronic Engineering from NationalAutonomousUniversity of Mexico 97(UNAM) in 1976 and his MSdegree in Control Educational 99Techniques. He is currently a PhDstudent at National Polithecnic 101Institute of Mexico (IPN) as well
q 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
UNCO
RREC
TED
PRO
OF
ASJC 339
A. Estrada et al.: Two Relay Controller for Real Time Trajectory Generation 9
as a professor at the Electrical Engineering Department1of UNAM since 1995. His research and educationalinterest are frequency domain analysis of variable3structure systems and its applications. He is theauthor of a text book in basic numerical analysis5techniques.
Leonid M. Fridman (M’98)7received the MS degree in math-ematics from Kuibyshev State9University, Samara, Russia, in1976, the PhD degree in applied11mathematics from the Institute ofControl Science, Moscow, Russia,13in 1988, and the Dr Sci degreein control science from Moscow15State University of Mathematics
and Electronics, Moscow, Russia, in 1998. From 1976
to 1999, he was with the Department of Mathematics, 17Samara State Architecture and Civil EngineeringAcademy. From 2000 to 2002, he was with the Depart- 19ment of Postgraduate Study and Investigations atthe Chihuahua Institute of Technology, Chihuahua, 21Mexico. In 2002, he joined the Department of Control,Division of Electrical Engineering of Engineering 23Faculty, National Autonomous University of Mexico(UNAM), Mexico. He is an Editor of three books 25and five special issues on sliding mode control. Hehas published over 200 technical papers. His research 27interests include variable structure systems and singularperturbations. Dr Fridman is an Associate Editor 29of the International Journal of System Science andConference Editorial Board of IEEE Control Systems 31Society, Member of TC on Variable Structure Systemsand Sliding mode control of IEEE Control Systems 33Society.
q 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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