UNCO

RREC

TED

PRO

OF

RNC 1696pp: 1–13 (col.fig.: Nil)

PROD. TYPE: COM

ED: Arun

PAGN: Gayathri MK -- SCAN:

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control. (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1696

1

Generating oscillations in inertia wheel pendulumvia two-relay controller3

Luis T. Aguilar1,∗,†, Igor M. Boiko2, Leonid M. Fridman3 and Leonid B. Freidovich4

1Instituto Politecnico Nacional, Ave. del parque 1310 Mesa de Otay Tijuana Mexico 22510, Mexico52University of Calgary, 2500 University Dr. N.W., Calgary, AB, Canada

3Universidad Nacional Autonoma de Mexico (UNAM), Department of Control,7Engineering Faculty, C.P. 04510, Mexico D.F.

4Department of Applied Physics and Electronics, Umeå University, SE-901 87 Umeå, Sweden9

SUMMARY

The problem of generating oscillations of the inertia wheel pendulum is considered. We combine exact11feedback linearization with two-relay controller, tuned using frequency-domain tools, such as computingthe locus of a perturbed relay system. Explicit expressions for the parameters of the controller in terms13of the desired frequency and amplitude are derived. Sufficient conditions for orbital asymptotic stabilityof the closed-loop system are obtained with the help of the Poincare map. Performance is validated via15experiments. The approach can be easily applied for a minimum phase system, provided the behavior ofthe states of the zero dynamics is of no concern. Copyright � 2011 John Wiley & Sons, Ltd.17

Received 7 September 2009; Revised 29 November 2010; Accepted 8 December 2010

KEY WORDS: variable structure controller; inertia wheel pendulum; periodic motion; Poincare map

1. INTRODUCTION19

Overview. The inertia (reaction) wheel pendulum (see Figure 1 below) is a planar mechanicalsystem consisting of a free rotational pendulum with a symmetric disk, attached to its end and21directly controlled by a DC motor. This underactuated mechanical system is a standard benchmarkset-up [1]. The equilibrium stabilization problem has been considered, in particular, in [2–5].23

In this paper, we consider the problem of generation of a periodic motion. An alternativetechnique based on an accurate design of a smooth trajectory by imposing a virtual holonomic25constraint and orbital stabilization via computation of a controlled transversal linearization formechanical systems is presented in Shiriaev et al. [6]. Results for the inertia wheel pendulum27are presented in Freidovich et al. [7]. The controller suggested that there requires solving aperiodic matrix differential Riccati equation, which is a numerically challenging task. Below we29propose another procedure for inducing stable oscillations. For our approach, no challenging offlinecomputations are needed; however, the amplitude of oscillations of the disk and the pendulum can31not be adjusted independently.

It is of interest to note that our approach can be easily applied to an arbitrary minimum-phase33system, that can be transformed into the normal form [8], provided the behavior of the states ofthe zero dynamics is of no concern.35

∗Correspondence to: Luis T. Aguilar, Instituto Politécnico Nacional, Ave. del parque 1310 Mesa de Otay TijuanaMéxico 22510, Mexico.

†E-mail: laguilarb@ipn.mx

Copyright � 2011 John Wiley & Sons, Ltd.

UNCO

RREC

TED

PRO

OF

2

RNC 1696

L. T. AGUILAR ET AL.

We concentrate on the problem of creating oscillation with time-independent feedback. More1specifically, we use the two-relay controller, originally proposed as a generalization of the second-order sliding-mode (SOSM) twisting controller [9] widely used for finite-time stabilization (see,3for example [10, 11]). It has been recently generalized in [12] to generate self-excited oscillationsin linear systems putting to work the well-known chattering effect. The desired frequencies and5amplitudes of periodic motions are produced without tracking of precomputed trajectories. Notethat the modification allows a wider range of frequencies than for the original twisting algorithm7and encompassing a variety of plant dynamics.

Below we will exploit the fact that the dynamics of the inertial wheel pendulum can be trans-9formed into the exactly linearized one with stable internal (zero) dynamics [2]. The specific featureof the considered system is that the dynamics of interest are of relative degree 3 and the zero11dynamics are stable and linear, although linearity is not exploited.

In the case when the zero dynamics are linear, other approaches for generating periodic motions13can be used, see e.g. [12, 13] for frequency domain-based methods. Contrary to these papers, herewe will follow exact methods that allow accurate characterization of the periodic solution for the15case of autonomous closed-loop system. A possible alternative might be a conventional designof a tracking controller for a periodic input. However, tracking of a certain harmonic (or of an17arbitrary periodic) input requires that the phase lag of the closed-loop system at the frequency ofthe input should be small. If the frequency of the input signal is high enough, so that the phase lag19of the plant at this frequency is close to −180◦, then, because the stability of the system is definedby the point on the frequency response at the frequency corresponding to −180◦ phase response21(Nyquist criterion) and the existence of the stability constraint, the open-loop gain of the systemat this frequency is always smaller than 1 (usually it is smaller than 0.5). As a result, the phase23response of the closed-loop system at this frequency is also close to −180◦. The latter followsfrom the formula relating transfer functions of the open-loop and closed-loop systems. Therefore,25tracking of periodic inputs of high enough frequency cannot be provided.

Contributions. Following our previous work [12], we use the two-relay controller to generate27oscillations of a desired amplitude and frequency. However, instead of using approximate methodof describing functions, we suggest two exact approaches. They are based on computing the first-29return Poincare map and the Locus of a Perturbed Relay System (LPRS) [14]. More precisely, wedo the following:31

• Poincare map-based analysis is successfully applied to obtain a system of nonlinear algebraicequations, solution of which gives the gains of the controller to achieve a desired periodic33solutions.

• As a possible alternative, explicit formulas for these gains are derived via the LPRS method35[14].

• Necessary and sufficient conditions for orbital exponential stability of the cycle are derived37from the analysis of the linearization of the Poincare map.

• The results are validated by experiments.39

2. PROBLEM FORMULATION AND PRELIMINARIES

Dynamics of an inertia wheel pendulum can be described as follows [1]:41 [J1 J2

J2 J2

][q1

q2

]+

[h sinq1

0

]=

[0

1

]�, (1)

where q1∈R is the absolute angle of the pendulum, counted clockwise from the vertical downward43position; q2∈R is the absolute angle of the disk; J1, J2, and h are positive physical parameters,which depends on the geometric dimensions and the inertia-mass distribution; and � is the controlled45torque applied to the disk (see Figure 1). It should be noted that the system (1) is nonlinear and

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 3

Figure 1. Inertia wheel pendulum.

underactuated. The purpose of the control is to produce a periodic motion of the indirectly actuated1link with desired frequency and amplitude.

In contrast to previous works [14, 15], where the LPRS method was used for analysis of relay3feedback systems, with linear plants, now we are dealing with a nonlinear plant. Fortunately,feedback linearization can be used as a preliminary step.5

The inertia wheel pendulum has underactuation degree one and satisfies certain structural prop-erty noted in Grizzle et al. [2]. For this system, it is possible to find a feedback transformation7achieving exact linearization with stability of the zero dynamics. Following [2], let us take

z=q1−�+ J−11 J2q2 and �= J1q1+ J2q2+K z,9

where �=3.14 . . . and K>0 is a constant. It is easy to verify that

J1 z=�−K z,11

while

� = K J−11 J2q2−h sin(q1)+K q1,

� = −h cos(q1)q1−K J−11 h sin(q1),

� = R(q1, q1)+H (q1)�,13

where

H (q1) = h cos(q1)

J1− J2,

R(q1, q1) = h

((q21 +H (q1)

)sin(q1)− K

J1q1 cos(q1)

).

(2)

15

Hence, we can take

�=H−1(q1) (u−a0�−a1�−a2�−R(q1, q1)) , (3)17

where H (q) is nonsingular around the equilibrium point (q�1 , q

�1)= (�,0), a0, a1, and a2 are positive

constants.19Finally, introducing the new state coordinates x= [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,

z = − K

J1z+ 1

J1y, y= [1 0 0]︸ ︷︷ ︸

C

x .

(4)

21

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

4

RNC 1696

L. T. AGUILAR ET AL.

It is possible to make another feedback transform of the dynamics (1). For example, if we define1�= J1q1+ J2q2, it is not hard to verify that

�=H (q1)(�+h sin(q1))+hq21 sin(q1).3

Hence, taking x= [x1, x2, x3, x4]= [�, �, �,�], it is possible to define a feedback transformation torewrite the dynamics as5

x= A x+B u, y=C x,

where (A, B,C) are in the canonical Brunovski form. The analysis presented below is readily7applicable for both representations. However, for numerical calculations, it is preferable to havematrix A of smaller dimension. Therefore, we choose to proceed with representation (4), which is9a linear system of dimension 3 with generalized output y and decoupled input-to-state stable zerodynamics. Note that the fact that these dynamics are linear is not essential as well as the fact that11the system can be linearized completely.

Let us design a stabilizing controller for a periodic solution with desired frequency and amplitude13for y(t). It is not hard to show that oscillations in the original coordinates with the same frequencyfollow from the periodicity of y(t) and a few reasonable assumptions. We will demonstrate how15to obtain the estimates for all the amplitudes later. The presented change of coordinates and ofcontrol input transforms the system into the one that has stable zero dynamics, while the remaining17dynamics are forced to have a desired periodic motion.

Following [12] we suggest to apply the two-relay controller19

u=−c1sign(y)−c2sign(y), (5)

where the parameters c1 and c2 should be tuned.21Below, we propose two approaches to complete the design.

3. POINCARE MAP-BASED DESIGN OF THE GAINS OF THE TWO-23RELAY CONTROLLER

Here, the analysis and design objectives are to find two values of scalar parameters c1 and c2 of25the two-relay controller (5) such that the output y(t) is periodic with the desired frequency � andamplitude A in the closed-loop system with plant (4).27

To construct the Poincare map, one has to choose a surface of section S in the state space R4 andconsider the points of successive intersections of a given trajectory with this surface. Switching29occur on the level surfaces defined by

S1 = {x : y=0, y<0}, S2={x : y<0, y=0},S3 = {x : y=0, y>0}, S4={x : y>0, y=0}. (6)

31

The space R4 is divided into four regions by Si , i =1, . . . ,4:

R1 = {x : y<0, y<0}, R2={x : y<0, y>0},R3 = {x : y>0, y>0}, R4={x : y>0, y<0}. (7)

33

Depending on the state, the system is governed by one of the four models defined by

M1 : x = Ax+B(c1+c2),

M2 : x = Ax+B(c1−c2),

M3 : x = Ax−B(c1+c2),

M4 : x = Ax+B(−c1+c2).

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 5

The solution of M1 on the time interval [0; t1], where t1 is the transition time from S1 to S2, subject1to the initial conditions x(0)=�p, where ‘(·)p’ stands for ‘periodic’, such that (without loss ofgenerality)3

y(0)=C x(0)=C �p =0, y(0)=C (A x(0)+B u)=C A�p<0 (8)

is given by5

x(t)=eAt x(0)+∫ t

0eA� d�Bu,

where7 ∫ t

0eA�d�=

∞∑i=1

Ai−1t i/ i!= A−1(eAt − I )

and u=c1+c2. The transition to S2 and switching to u=c1−c2 is ensured under the technical9transversality condition

y(t1)=C A2 �k>0. (9)11

Under this condition, the trajectory will enter the region R2, and since the matrix A is Hurwitzit will reach either S3 or return back to S2. We will assume for now that the latter does not happen.13

Analogously, for the case of the twisting projection of the motion onto the (y, y)-plane, the fourstate transitions initiated at �k =�p are given by15

�k = eAt1�k+A−1(eAt1 − I )B(c1+c2),

�−k = eAt2�k +A−1(eAt2 − I )B(c1−c2),

�−k = eAt3�−

k −A−1(eAt3 − I )B(c1+c2),

�k+1 = eAt4�−k −A−1(eAt4 − I )B(c1−c2),

(10)

where t2 is the time interval between S2 and S3, t3 is the time interval between S3 and S4, and t417is the time interval between S4 and S1.

The fixed point of the Poincare map, corresponding to an isolated periodic solution of system19(4) driven by the two-relay controller, is determined by the equation �k+1=�k =�p . Skipping thesequential numbers of switching in (10) and using the principle of symmetry one can write the21following: �−

p =−�p . For the T -periodic (symmetric) solution we will use the following notation:t1= t3=�1, t2= t4=�2=T/2−�1.23

The equation for the fixed point together with the switching conditions can be rewritten asfollows:25

−�p =eA�2�p+A−1(eA�2 − I )B(c1−c2) (11)

and, with the help of y(0)= y(�1)=0 and CB=0,27

�p = eA�1�p+A−1(eA�1 − I )B(c1+c2)

C�p = 0, CA�p =0, CA�p<0, CA2�p>0.(12)

We assume in (11) and (12) that there are no additional switches on intervals t ∈ (0; t1) and29t ∈ (t1; t2), respectively, since y<0 initially and y monotonically decreases from 0 and cannot cross0 before y changes sign at t= t1. This condition can easily be verified after parameters �1 and �231are determined.

It is left to formalize the condition ensuring transition from S2 to S3 without leaving R2. Defining33two hypothetical (for the fixed control input u=c1−c2) boundary crossing times as t2 and t2, wehave35

t2=min{t>0 :C(eAt�p+A−1(eAt − I )B(c1−c2))=0}

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

6

RNC 1696

L. T. AGUILAR ET AL.

and1

t2=min{t>0 :CA(eAt�p+A−1(eAt − I )B(c1−c2))=0}.Hence, we require3

t2<t2 (13)

to ensure that our analysis of the limit cycle with exactly four switches is correct. In the case when5the transition time is sufficiently small, dropping smaller order terms in the definitions of t2 andt2, one can derive the following simplified approximate algebraic assumption‡:7

0<t2≈− 2CA2�p

C A3�p+c2−c1<

√2C�p

−CA2�p≈ t2.

Let us move onto defining the amplitude and frequency of the oscillations.9The system s(11) and (12) can be considered as a system of algebraic equations for the design

of the two-relay controller providing for the system (4) the desired periodic solution with a given11frequency � and amplitude A. Taking into account that

y(�1)=C�p =A, �1+�2=�/�=T/2 (14)13

(11), (12), and (14) can be reduced to a system of five nonlinear algebraic equations with respectto five variables: c1,c2,�1, and the first and the second coordinates of the vector �p. Once the15resolving set of parameters is found, the two-relay controller gains that provide the periodic solutionof the system (4) with the desired amplitude and frequency are designed unless the corresponding17solution passes through the singularity of the control transformation (3). This can be summarizedas follows.19

Theorem 1Suppose the systems (11), (12), and (14) possess a solution satisfying (13). If the desired amplitude21A is sufficiently small to avoid singularity in the matrix H (q1) defined in (2) and assuming thatthere are no additional switches on intervals t ∈ (0; t1) and t ∈ (t1; t2), then the closed-loop system23(1), (2), (3), (5) has the desired periodic solution.

Note however that (11), (12), and (14) are a systems of nonlinear algebraic equations and might25be hard to solve and even have no solutions for particular values of A and �.

It turns out that the linearity of the (transformed) plant and the fact that the control in the periodic27motion can be represented as a sum of two relay controls, with finding the response of the plant asa linear combination (sum) of the two periodic relay controls of amplitudes c1 and c2, allows for29a reduction of complexity of the original problem. Let us develop an approach that might simplifyfinding fixed points of the Poincare map utilizing the concepts of the LPRS method.31

4. LPRS-BASED ANALYSIS

The LPRS method was developed in [15] as a method of analysis and design of relay systems. This33method cannot be applied to the system directly, since the two-relay control assumes a four-levelversus two-level relay control of the conventional relay system. However, after some modifications,35the methodology of [15, 14] can be used here as well.

The control can be represented as a sum of two relay controls, and the output of the system can37be considered as a superposition of the system reaction to these two relay controls. Therefore, as

‡Here we have used the identities CA�p =0, CB=CAB=0, and CA2B=1, and dropped the third-order terms inthe series expansions for the matrix exponents.

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 7

an auxiliary step, let us find the Poincare map and its fixed point in the system with one relay.1Assume that the control is

u= sign(y). (15)3

Then for the part of the period for which u=1

x(t)=eAt�p+A−1(eAt − I )B. (16)5

Assume that a symmetric periodic process of period T occurs in the system (4), (15). Then at timet=T/2 the state vector is7

x(T/2)=eAT/2�p+A−1(eAT/2− I )B, (17)

which must be equal to −�p to provide a fixed point of the Poincare map for the symmetric motion.9Therefore, solution of the equation −�p = x(T/2), where x(T/2) is given by (17), provides thefixed point:11

�p = (I+eAT/2)−1A−1(I −eAT/2)B. (18)

Now introduce a function, which would provide the value of the system output in a periodicmotion of the frequency =2�/T at the time t=T , where ∈ [− 1

2 ; 12 ], subject to the control

amplitude being �/4 (this value of the amplitude, which is the ratio between the amplitude of thefirst harmonic of the square-pulse signal and the amplitude of the pulses, is used to comply withthe LPRS method [15]). Taking into account (16) and (17), this function can be defined as follows:

L(,)= �

4C{eAT �p+A−1(eAT − I )B}

= �

4C{eA 2�

(I +eA� )−1A−1(I −eA

� )+A−1(eA 2�

− I )}B. (19)

Parameter is related to �1 and �2 in the following way:13

�1=T

and15

�2=T/2−�1= (0.5−)T .

Now consider periodic control u(t) as a sum of two periodic square pulse controls u1(t) and u2(t)17of amplitudes c1 and c2, respectively. Assume that control u2(t) leads with respect to u1(t) bytime t=T , where ∈ [−0.5;0.5]. Then for the system output y(t) at the time of the switch from19−c1 to +c1 of the control u1(t), with y(t) being the system response to the periodic control u(t)of frequency �, we can write the following formula, which is a superposition of the responses to21the two controls:

y(0)= 4c1�

L(�,0)+ 4c2�

L(�,). (20)23

In the exactly same way, we can write the formula for the system derivative output at the time ofthe switch from −c2 to +c2 of the control u2(t):25

y(−T )= 4c1�

L1(�,−)+ 4c2�

L1(�,0), (21)

where function L1 would correspond to the linear plant for the output being the derivative of y(t)27and given by y=Cx=CAx .

Considering the equations of the closed-loop system (4), (5), one would note that the condi-tion y(0)=0 represents the switching condition for the first relay, and the condition y(−T )=0

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

8

RNC 1696

L. T. AGUILAR ET AL.

represents the switching condition for the second relay, which is Equation (12). Therefore, thefixed point of the Poincare map for the system (4), (15) can be written as a set of two algebraicequations with two unknowns � and as follows:

c1L(�,0)+c2L(�,)= 0, (22)

c1L1(�,−)+c2L1(�,0)= 0. (23)

Representing the periodic solution in the format of LPRS can simplify the solution of equations1(22), (23). This simplification comes from considering that the feedback through y(t) is closedand the feedback through y(t) is open, thus giving a SISO plant, and finding the response of this3plant to the discontinuous control of frequencies from a certain frequency range. A methodologyof analysis similar to that of [15] can now be used. With this approach, at the step of computing5of LPRS, the frequency � is known, which reduces the problem to the solution of one nonlinearalgebraic equation for . At the second step, after LPRS is computed, the actual frequency �7is determined via finding the point of intersection of LPRS with the real axis. Considering thedefinition of LPRS [15], at any given , the imaginary part of LPRS can be written in the following9way:

ImJ ()= L(,0)+ c2c1

L(,). (24)11

The value of in (24) is found from Equations (22), (23), which are reduced to one equation

�()= L(,0)L1(,−)−L(,)L1(,0)=0 (25)13

that can be solved via simple numeric algorithms.In the present analysis, the real part of LPRS is not used in calculations, as it reflects the transfer15

properties of relay feedback systems [15], which are not being analyzed. The LPRS analysis ofthe system would include the steps of finding the value of parameter and computing the LPRS17point for every frequency from the range of interest, plotting LPRS in the complex plane andfinding the point of its intersection with the real axis.19

Since function L(,) provides the value of the system output in a periodic motion at time T ,finding the amplitude of the oscillations is equivalent to finding the maximum of L as follows:21

A= maxt∈[0;T ]

{4c1�

L(�, t/T )+ 4c2�

L(�,+ t/T )

}. (26)

However, the problem of finding the amplitude can be simplified if instead of the true amplitude23given by (26) the amplitude of the fundamental frequency (first harmonic) can be used. In thiscase, using the rotating phasor concept, the control can be represented as a sum of two rotating25vectors having amplitudes 4c1/� and 4c2/�, with the angle 2� between them. The amplitude ofthe control vector will be27

au = 4

�

√c21+c22+2c1c2 cos(2�) (27)

and the amplitude of the output (taking into account only the first harmonic) will be29

A≈ 4

�

√c21+c22+2c1c2 cos(2�) |Wp(�)|, (28)

where � is the frequency of the periodic motion and Wp(s)=C(s I −A)−1B is the transfer function31of the plant. It should be noted that this approximation based on the first harmonic is more accuratethan the standard describing the function approach, used in [12], because the frequency of the33oscillations is computed exactly.

We are ready now to propose an alternative design of the gains of (5).35

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 9

5. LPRS-BASED DESIGN OF THE GAINS OF THE TWO-RELAY CONTROLLER1

It follows from formulas (22) and (23) that the frequency of the self-excited periodic motionsin the two-relay controller is invariant to the ratio c1/c2. Therefore, for the desired value of the3oscillations frequency �, coefficients c1 and c2 cannot be determined from (22), (23) uniquely.Using � as the notation for this ratio and applying the desired frequency of oscillations � instead5of generic frequency , we can rewrite (24) as an equation for �:

ImJ (�)= L(�,0)+�L(�,)=0, (29)7

where is found from (25) at =�. With computed numerically and �=−L(�,0)/L(�,), thenecessary values of the relay amplitudes that provide the desired frequency � and amplitude Aof the oscillations are computed as

c1 ≈ �

4

A

|Wp( j�)|1√

1+2�cos(2�)+�2, (30)

c2 ≈ �

4

A

|Wp( j�)|�√

1+2�cos(2�)+�2. (31)

6. LINEARIZED POINCARE MAP-BASED ANALYSIS OF ORBITAL STABILITY9

Let us use (10) to analyze the deviation of a trajectory initiated on the surface S1 at x(0)=�k =�p+�� from a periodic trajectory initiated from some �p for sufficiently small initial deviations��. Using the equation in (10) for �k , the equation in (12) for �p , and the Taylor expansion

eAt1 =eA�1 +eA�1 A�t+O(�t2), �t= t1−�1 one can proceed as follows:

�k = eAt1 (�p+��)+A−1(eAt1 − I )B(c1+c2)

= (eA�1 +eA�1 A�t)(�p+��)+O(�t2)+A−1(eA�1 +(eA�1 − I + I )A�t− I )B(c1+c2)

so that

�k =eA�1 (��+A���t)+(I +A�t)�p+B(c1+c2)�t+O(�t2).11

Now, since CA�k =CA�p =0, premultiplying this equation by CA which results in

CAeA�1 (��+A���t)+CA(A�p+B(c1+c2))�t=O(�t2),13

one immediately concludes that �t=O(��) and obtains an estimate for t1=�1+�t , which can besubstituted back:15

�k =�p+�� =�p+�1��+O(�2�),

where17

�1=(I − v1CA

CAv1

)eA�1, v1= A�p+B(c1+c2). (32)

Following the second equation in (10) and computing t2 using C�−k =C�p =0, one, in a similar19

way, obtains

�−k =−�p+��− =−�p+�2��+O(�2�),21

where

�2=(I − v2C

Cv2

)eA�2, v2= A�p+B(c1−c2). (33)

23

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

10

RNC 1696

L. T. AGUILAR ET AL.

Following the third equation in (10) and computing t3 using CA�−k =CA�p =0, one obtains1

�−k =−�p+��− =−�p+�3��−+O(�2�),

where �3=�1.3Following the last equation in (10) and computing t4 using C�k+1=C�p =0, one obtains

�k+1=�p+�4��−+O(�2�),5

where �4=�2.Finally, we have for small �� =�k−�p: �k+1−�p =�·(�k−�p)+O(�2�), with7

�= (�2 ·�1)2. (34)

Since we have just computed a linearization for the Poincare map, we conclude with the following.9

Theorem 2Suppose that the parameters c1 and c2 induce a periodic trajectory for the closed-loop system11controlled by the two-relay algorithm, i.e. (1), (3), and (5). This solution is orbitally exponentiallystable if and only if all the eigenvalues of the matrix �, defined by (34), (32), and (33), are inside13the unit circle.

7. EXPERIMENTAL RESULTS15

In this section, we present experimental results using the laboratory inertial wheel pendulummanufactured by Quanser Inc., depicted in Figure 1 where J1=4.572×10−3, J2=2.495×10−3,17and h=0.3544 (see [1]). Experiments were carried out to achieve the orbital stabilization of theunactuated link (pendulum) y=� around the equilibrium point q� = (�,0). Let us select �=1219[rad/s] and A=0.005 as desired frequency and amplitude. The parameters of the linearizedsystems are K =1×10−4, a0=350, a1=155, and a2=22.21

Since we are interested in presenting the results in the original coordinates q1 and q2, let us showhow to compute an approximation for the amplitude of oscillations for the pendulum itself. Since23z+ J−1

1 K z= J−11 �(t) we know that z exponentially converges to a periodic steady-state function

(the rate of convergence can be regulated by K ). Taking into account only the first harmonic25and letting the steady-state value for � to be �(t)≈Asin(�t), we can compute the approximatesteady-state value for z(t) as27

z(t)≈ A√J 21 �2+K 2

sin(�t+arg{ 1

j J1�+K}).

Now, using �(t)≈�Acos(�t) and the equation h sin(q1)= �−K z one can conclude that q1 expo-29nentially converges to a periodic steady state provided oscillations are small. Finally, for q1 beingclose enough to �, we have sin(q1)≈�−q1, and so31

q1(t)≈�+ �A

hcos(�t)− �A

h√J 21 �2+K 2

sin

(�t+arg{ 1

j J1�+K})

.

This expression gives us an estimate on the amplitude of the achieved oscillations of the pendulum33around � to be

Ar ≈ �A

h

√1+(J 21 �2+K 2)−1.35

Since q2= J1(z−(�−q1))/J2, the steady-state amplitude of q2 can be easily estimated as well.In Table I the measured frequencies and amplitudes denoted by �r and Ar , respectively, are37

depicted; with the values of c1 and c2 computed by means of the LPRS method given in Section 4

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 11

Table I. c1 and c2 values for several desired frequencies and amplitudes.

� A Quadrant c1 c2 �r Ar

1.0 0.3 III −0.333 0.333 1.04 0.282.0 0.2 II 0.3702 2.5917 2.01 0.22

0 2 4 6 8 10

0

2

4

6

8

10

12x 10

Time [s]

y (t

) [r

ad]

10 12 14 16 18 203.1

3.15

3.2

3.25

q 1 [r

ad]

10 12 14 16 18 20805

810

815

820

825

Time [s]

q 2 [r

ad]

(a)

(b)

(c)

Figure 2. (a) Time evolution of the output y(t) and (b) time evolution of the joint positions q1(t) andq2(t).

for =0.1. In Figure 2, the oscillations for the output y and the trajectory of the joint position1vector q1 and q2, where �=12rad/s and A=0.005 are displayed. Finally, Figure 3 illustratesthe achieved trajectory in terms of (q1, q1) and (q1,q2) revealing the limit cycle behavior of the3closed-loop system.

8. CONCLUSIONS5

The problem of generating periodic motions for the inertia wheel pendulum is considered. Poincaremap-based design for the two-relay controller gains is proposed. Simpler explicit formulae for exact7

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

12

RNC 1696

L. T. AGUILAR ET AL.

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

0

1

2

3

q1 [rad]

dq1/

dt [r

ad/s

]

2.6 2.8 3 3.2 3.4 3.6 3.8 4

0

200

400

q1 [rad]

q 2 [r

ad]

(a)

(b)

Figure 3. (a) Phase portrait of the unactuated joint (q1 vs q1) for initial conditions inside and outside thelimit cycle and (b) behavior of the joint position trajectories q1 vs q2.

values of two-relay controller gains are derived via the LPRS method. Necessary and sufficient1conditions for orbital exponential stability of the periodic solution are obtained. The results arevalidated experimentally. The presented analysis is not tailored to the particular example and can3be readily applied to any nonlinear system that can be globally transformed into a normal formwith input-to-state stable zero dynamics. However, in such general settings, we would only ensure5oscillations of the linearizing output with particular amplitude and frequency. Additional analysismay be needed if a periodic trajectory for a particular state variable are of interest. This is also7demonstrated here on the inertia wheel pendulum example.

REFERENCES9

1. Block D, Astrom K, Spong M. The Reaction Wheel Pendulum. Morgan & Claypool Publ.: Princeton, NJ, 2007.2. Grizzle JW, Moog CH, Chevallereau C. Nonlinear control of mechanical systems with an unactuated cyclic11

variable. IEEE Transactions on Automatic Control 2005; 50(5):559–576.3. Ortega R, Spong M, Gomez-Estern F, Blankenstein G. Stabilization of a class of underactuated mechanical systems13

via interconnection and damping assignment. IEEE Transactions on Automatic Control 2002; 47(8):1218–1233.4. Reyhanoglu M, Van der Schaft A, McClamroch NH, Kolmanovsky I. Dynamics and control of a class of15

underactuated mechanical systems. IEEE Transactions on Automatic Control 1999; 44(9):1663–1671.5. Spong M, Corke P, Lozano R. Nonlinear control of the reaction wheel pendulum. Automatica 2001; 37(11):1845–17

1851.

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc

UNCO

RREC

TED

PRO

OF

RNC 1696

GENERATING OSCILLATIONS IN INERTIA WHEEL PENDULUM 13

6. Shiriaev AS, Freidovich LB, Manchester IR. Can we make a robot ballerina perform a pirouette? Orbital1stabilization of underactuated mechanical systems. Annual Reviews in Control 2008; 32(2):200–211.

7. Freidovich L, La Hera P, Mettin U, Robertsson A, Shiriaev A, Johansson R. Shaping stable periodic motions of3inertia wheel pendulum: theory and experiment. Asian Journal of Control 2009; 11(5):548–556.

8. Isidori A. Nonlinear Control Systems (3rd edn). Springer: New York, 1995.59. Levant A. Sliding order and sliding accuracy in sliding mode control. International Journal of Control 1993;

58:1247–1263.710. Orlov Y. Discontinuous Systems—Lyapunov Analysis and Robust Synthesis Under Uncertainty Conditions.

Springer: London, 2009.911. Shtessel YB, Shkolnikov IA, Brown MJ. An asymptotic second-order smooth sliding-mode control. Asian Journal

of Control 2003; 5(4):498–504.1112. Aguilar L, Boiko I, Fridman L, Iriarte R. Generating self-excited oscillations via two-relay controller. IEEE

Transactions on Automatic Control 2009; 54(2):416–420.1313. Boniolo I, Colaneri P, Bolzern P, Corless M, Shorten R. On the design and synthesis of limit cycles using

switching linear systems. International Journal of Control 2010; 83(5):915–927.1514. Boiko I. Discontinuous Control Systems. Frequency-Domain Analysis and Design. Birkhauser: Boston, 2009.15. Boiko I. Oscillations and transfer properties of relay servo systems—The locus of a perturbed relay system17

approach. Automatica 2005; 41(4):677–683.

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2011)DOI: 10.1002/rnc