PLEASE SCROLL DOWN FOR ARTICLE

Transcription

1 This article was downloaded by: [UNAM Ciudad Universitaria] On: 8 July 9 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 7954 Registered office: Mortimer House, 37-4 Mortimer Street, London WT 3JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: Generating self-excited oscillations for underactuated mechanical systems via two-relay controller Luis T. Aguilar a ; Igor Boiko b ; Leonid Fridman c ; Rafael Iriarte c a Instituto Politécnico Nacional, Centro de Investigación y Desarrollo de Tecnología Digital, San Ysidro, CA, USA b University of Calgary, 5 University Dr. NW, Calgary, Alberta, Canada c Engineering Faculty, Department of Control, Universidad Nacional Autónoma de México (UNAM), Mexico City, Mexico First Published:September9 To cite this Article Aguilar, Luis T., Boiko, Igor, Fridman, Leonid and Iriarte, Rafael(9)'Generating self-excited oscillations for underactuated mechanical systems via two-relay controller',international Journal of Control,8:9, To link to this Article: DOI:.8/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 International Journal of Control Vol. 8, No. 9, September 9, Generating self-excited oscillations for underactuated mechanical systems via two-relay controller Luis T. Aguilar a *, Igor Boiko b, Leonid Fridman c and Rafael Iriarte c a Instituto Polite cnico Nacional, Centro de Investigacio n y Desarrollo de Tecnologıá Digital, PMB 88, PO Box 4396, San Ysidro, CA, USA; b University of Calgary, 5 University Dr. NW, Calgary, Alberta, Canada; c Engineering Faculty, Department of Control, Universidad Nacional Auto noma de Me xico (UNAM), CP 45, Mexico City, Mexico (Received April 8; final version received 3 November 8) Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed. First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter the hypothesis that usually holds when the oscillations are relatively fast. The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear. Finally, the Poincare map s design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant. The proposed approach is shown by the controller design and experiments on the Furuta pendulum. Keywords: variable structure systems; underactuated systems; periodic solution; frequency domain methods. Introduction. Overview In this article, we consider the control of one of the simplest types of a functional motion: generation of a periodic motion in underactuated mechanical systems which could be of non-minimum-phase. Current representative works on periodic motions in an orbital stabilisation of underactuated systems involve finding and using a reference model as a generator of limit cycles (e.g. Orlov, Riachy, Floquet, and Richard (6)), thus considering the problem of obtaining a periodic motion as a servo problem. Orbital stabilisation of underactuated systems finds applications in the coordinated motion of biped robots (Chevallereau et al. 3), gymnastic robots and others (see, e.g., Grizzle, Moog, and Chevallereau (5), Shiriaev, Freidovich, Robertsson, and Sandberg (7) and references therein).. Methodology In this article, underactuated systems are considered as the systems with internal (unactuated) dynamics with respect to the actuated variables. It allows us to propose a method of generating a periodic motion in an underactuated system where the same behaviour can be seen via second-order sliding mode (SOSM) algorithms, i.e. generating self-excited oscillations using the same mechanism as the one that produces chattering. However, the generalisation of the SOSM algorithms and the treatment of the unactuated part of the plant as additional dynamics result in the oscillations that may not necessarily be fast and of small amplitude. There exist two approaches to analysis of periodic motions in the sliding mode systems due to the presence of additional dynamics: the time-domain approach, which is based on the state-space representation, and the frequency-domain approach. The Poincare maps (Varigonda and Georgiou ) are successfully used to ensure the existence and stability of periodic motions in the relay control systems (see Di Bernardo, Johansson, and Vasca (), Fridman () and references therein). The describing function (DF) method (see, e.g., Atherton (975)) offers finding approximate values of the frequency and the amplitude of periodic motions in the systems with linear plants driven by the sliding mode controllers. The locus of perturbed relay system (LPRS) method (Boiko 5) provides an exact solution of the periodic problem in discontinuous control systems, including finding exact values of the amplitude and the frequency of the self-excited oscillation. *Corresponding author. luis.aguilar@ieee.org ISSN 779 print/issn online ß 9 Taylor & Francis DOI:.8/

3 International Journal of Control 679 Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9.3 Results of this article The proposed approach is based on the fact that all SOSM algorithms (Boiko, Fridman, and Castellanos 4; Boiko, Fridman, Pisano, and Usai 7) produce chattering (periodic motions of relatively small amplitude and high frequency) in the presence of unmodelled dynamics. In sliding mode control, chattering is usually considered an undesirable component of the motion. In this article, we aim to use this property of SOSM for the purpose of generating a relatively slow motion with a significantly higher amplitude and lower frequency than respectively the amplitude and frequency of chattering. The twisting algorithm (Levant 993) originally created as a SOSM controller to ensure the finitetime convergence is generalised, so that it can generate self-excited oscillations in the closed-loop system containing an underactuated plant. The required frequencies and amplitudes of periodic motions are produced without tracking of precomputed trajectories. It allows for generating a wider (than the original twisting algorithm with additional dynamics) range of frequencies and encompassing a variety of plant dynamics. A systematic approach is proposed to find the values of the controller parameters allowing one to obtain the desired frequencies and the output amplitude, which includes:. an approximate approach based on the DF method that requires for the plant to be a lowpass filter;. a design methodology based on LPRS that gives exact values of controller parameters for the linear plants;. an algorithm that uses Poincare maps and provides the values of the controller parameters ensuring the existence of the locally orbitally stable periodic motions for an arbitrary mechanical plant. The theoretical results are validated experimentally via the tests on the laboratory Furuta pendulum. The periodic motion is generated around the upright position (which gives the non-minimum phase system case)..4 Organisation of this article This article is structured as follows. Section introduces the problem statement. In Section 3, the idea of the two-relay controller is explained via the frequency domain methods. In Section 4 the approximate values of controller parameters are computed via the DF method and stability of the periodic solution will be provided. In Section 5, the LPRS method is provided to compute the exact values of the controller parameters. In Section 6, the Poincare method will be used as a design tool. An example is provided in Section 7 in order to illustrate the design methodologies given in Sections 4 6. In Section 8, the design methodology is validated via periodic motion design for the experimental Furuta pendulum. Section 9 provides final conclusions.. Problem statement Let the underactuated mechanical system, which is a plant in the system where a periodic motion is supposed to occur, be given by the Lagrange equation: MðqÞ q þ Hðq, _qþ ¼B u where q IR m is the vector of joint positions; u IR is the vector of applied joint torques where m5n; B ¼ [ (m ),] T is the input that maps the torque input to the joint coordinates space; MðqÞ IR mm is the symmetric positive-definite inertia matrix; and Hðq, _qþ IR m is the vector that contains the Coriolis, centrifugal, gravity and friction torques. The following two-relay controller is proposed for the purpose of exciting a periodic motion: u ¼ c signð yþ c signð _yþ, where c and c are parameters designed such that the scalar output of the system (the position of a selected link of the plant) y ¼ hðqþ has a steady periodic motion with the desired frequency and amplitude. Let us assume that the two-relay controller has two independent parameters c C IR and c C IR, so that the changes to those parameters result in the respective changes of the frequency WIR and the amplitude A AIR of the self-excited oscillations. Then we can note that there exist two mappings F : C C W and F : C C A, which can be rewritten as F : C C WAIR. Assume that mapping F is unique. Then there exists an inverse mapping G : WA C C. The objective is, therefore, (a) to obtain mapping G using a frequencydomain method for deriving the model of the periodic process in the system, (b) to prove the uniqueness of mappings F and G for the selected controller and (c) to find the ranges of variation of and A that can be achieved by varying parameters c and c. The analysis and design objectives are formulated as follows: find the parameter values c and c in () such that the system () has a periodic motion with the desired frequency and desired amplitude of the ðþ ðþ ð3þ

4 68 L.T. Aguilar et al. Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 output signal A. Therefore, the main objective of this research is to find mapping G to be able to tune c and c values. 3. The idea of the method The idea of the method is to provide the mapping from a set of desired frequencies WIR and amplitudes AIR into a set of gain values CIR, that is G : WA C. To achieve the objective, let us start with the design via the DF method which is a useful frequency-domain tool for time-invariant linear plants to predict the existence or absence of limit cycles and estimate the frequency and amplitude when it exists. In the scenario introduced in Boiko et al. (4), the DF method was used for analysis of chattering for the closed-loop system with the twisting algorithm where the inverse of this mapping was derived. 3. Some specific features of underactuated systems To begin with, let us consider the following underactuated system called cart-pendulum (Fantoni and Lozano, p. 6): M þ m mlcos " þ x F v _ # ml sin _ ml cos ml ¼ u mgl sin q l ð4þ 8 x where x IR is the linear position of the cart along the horizontal axis, IR is the rotational angle of the pendulum, M ¼.35 kg and m ¼.65 kg are the masses of the cart and the inverted pendulum, respectively; l ¼.45 m is the distance from the centre of gravity of the link to its attachment point, F v ¼. N m s rad is the viscous friction coefficient, and g ¼ 9.8 m s is the gravitational acceleration constant (see Figure ). Linearising around the unstable equilibrium point ( ¼ ) and substituting the value of the parameters, the transfer function from the angle of the pendulum to the input u is WðsÞ ¼ ðsþ UðsÞ ¼ :s þ :5 ðs þ s 887Þ : ð5þ Note that the Nyquist plot of the above transfer function is located in the second quadrant of the complex plane. 4. DF of the two-relay controller Let first, the linearised plant be given by: _x ¼ Ax þ Bu, x IR n, y IR, n ¼ m ð6þ y ¼ Cx mg 5 ImW(jw) 4 3 M u x x 3 Re W(jw) Figure. The cart-pendulum system and its corresponding Nyquist plot of transfer function from the angle of the pendulum to the input of the actuator.

5 International Journal of Control 68 Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 which can be represented in the transfer function form as follows: WðsÞ ¼CðsI AÞ B: Let us assume that matrix A has no eigenvalues at the imaginary axis and the relative degree of (6) is greater than. The DF, N, of the variable structure controller () is the first harmonic of the periodic control signal divided by the amplitude of y(t) (Atherton 975): N ¼! A Z =!! uðtþ sin!t dt þ j A Z =! uðtþ cos!t dt where A is the amplitude of the input to the nonlinearity (of y(t) in our case) and! is the frequency of y(t). However, the algorithm () can be analysed as the parallel connection of two ideal relays where the input to the first relay is the output variable and the input to the second relay is the derivative of the output variable (Figure ). For the first relay the DF is: N ¼ 4c A, and for the second relay it is (Atherton 975): N ¼ 4c A, where A is the amplitude of dy/dt. Also, take into account the relationship between y and dy/dt in the Laplace domain, which gives the relationship between the amplitudes A and A : A ¼ A, where is the frequency of the oscillation. Using the notation of the algorithm () we can rewrite this equation as follows: ð7þ with the amplitude of the oscillation of the input to the first relay this equation can be rewritten as follows: Wð jþ ¼ NðA Þ, ðþ where the function at the right-hand side is given by: NðA Þ ¼ A c þ jc 4ðc þ c Þ : Equation (9) is equivalent to the condition of the complex frequency response characteristic of the open-loop system intersecting the real axis in the point (, j). The graphical illustration of the technique of solving (9) is given in Figure. The function /N is a straight line the slope of which depends on c /c ratio. The point of intersection of this function and of the Nyquist plot W( j!) provides the solution of the periodic problem (Figure 3). f s s c c c u c Figure. Relay feedback system. Q ṡ + u ImW ẋ=ax+bu y=cx W(s) Q y N ¼ N þ sn ¼ 4c A þ j 4c A ¼ 4 A ðc þ jc Þ, ð8þ where s ¼ j. Let us note that the DF of the algorithm () depends on the amplitude value only. This suggests the technique of finding the parameters of the limit cycle via the solution of the harmonic balance equation (Atherton 975): ψ ReW Wð jþnðaþ ¼, ð9þ N where a is the generic amplitude of the oscillation at the input to the non-linearity, and W( j!) is the complex frequency response characteristic (Nyquist plot) of the plant. Using the notation of the algorithm () and replacing the generic amplitude W(jω) Q 3 Q 4 Figure 3. Example of a Nyquist plot of the open-loop system Wð j!þ with two-relay controller.

6 68 L.T. Aguilar et al. Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 4. Tuning the parameters of the controller Here, we summarise the steps to tune c and c : (a) Identify the quadrant in the Nyquist plot where the desired frequency is located, which falls into one of the following categories (sets): Q ¼f! IR : RefWð j!þg 4, ImfWð j!þg g Q ¼f! IR : RefWð j!þg, ImfWð j!þg g Q 3 ¼f! IR : RefWð j!þg, ImfWð j!þg 5 g Q 4 ¼f! IR : RefWð j!þg 4, ImfWð j!þg 5 g: (b) The frequency of the oscillations depends only on the c /c ratio, and it is possible to obtain the desired frequency by tuning the ¼ c /c ratio: ¼ c ImfWð jþg ¼ c RefWð jþg : ðþ Since the amplitude of the oscillations is given by A ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi jwð jþj c þ c, ðþ then the c and c values can be computed as follows: 8 4 A jwð jþj pffiffiffiffiffiffiffiffiffiffiffiffi >< þ if Q [ Q 3 c ¼ 4 A jwð jþj pffiffiffiffiffiffiffiffiffiffiffiffi >: þ elsewhere c ¼ c : ð3þ ð4þ 4. Stability of periodic solutions We shall consider that the harmonic balance condition still holds for small perturbations of the amplitude and the frequency with respect of the periodic motion. In this case the oscillation can be described as a damped one. If the damping parameter will be negative at a positive increment of the amplitude and positive at a negative increment of the amplitude then the perturbation will vanish, and the limit cycle will be asymptotically stable. Theorem : Suppose that for the values of the c and c given by (3) and (4) there exists a corresponding periodic solution to the systems () and (6). If Re d arg W dln! c c!¼ c þ ð5þ c then the above-mentioned periodic solutions to the systems () and (6) is orbitally asymptotically stable. Proof: The proof of this theorem is given in Aguilar, Boiko, Fridman, and Iriarte (9). œ 5. Locus of a perturbed relay system design The LPRS proposed in Boiko (5) provides an exact solution of the periodic problem in a relay feedback system having a plant (6) and the control given by the hysteretic relay. The LPRS is defined as a characteristic of the response of a linear part to an unequally spaced pulse control of variable frequency in a closed-loop system (Boiko 5). This method requires a computational effort but will provide an exact solution. The LPRS can be computed as follows: Jð!Þ ¼ X ð Þ kþ RefWðk!Þg k¼ þ j X k¼ Im W½ðk Þ!Š : ð6þ k The frequency of the periodic motion for the algorithm () can be found from the following equation (Boiko 5) (Figure 4): Im JðÞ ¼ In effect, we are going to consider the plant being non-linear, with the second relay transposed to the feedback in this equivalent plant. Introduction of the following function will be instrumental in finding a response of the non-linear plant to the periodic square-wave pulse control. Lð!, Þ ¼ X ðsin½ðk ÞŠRefW½ðk Þ!Šg k k¼ þ cos½ðk ÞŠImfW½ðk Þ!ŠgÞ: ð7þ The function L(!, ) denotes a linear plant output (with a coefficient) at the instant t ¼ T (with T being πb 4c ω=ω J(ω) Figure 4. LPRS and oscillation analysis. Im Kn Re

7 International Journal of Control 683 Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 the period: T ¼ /!) if a periodic square-wave pulse signal of amplitude /4 is applied to the plant: Lð!, Þ ¼ yðtþ 4c t¼=! with ½ :5, :5Š and! ½, Š, where t ¼ corresponds to the control switch from toþ. With L(!, ) available, we obtain the following expression for ImfJð!Þg of the equivalent plant: ImfJð!Þg ¼ Lð,Þþ c Lð, Þ: ð8þ c The value of the time shift between the switching of the first and second relay can be found from the following equation: _yðþ ¼: As a result, the set of equations for finding the frequency and the time shift is as follows: c Lð,Þþc Lð, Þ ¼ c L ð, Þþc L ð,þ¼: ð9þ The amplitude of the oscillations can be found as follows. The output of the system is: yðtþ ¼ 4 X c sin½ðk Þ þ L ððk ÞÞŠ i¼ þ c sin½ðk Þt þ L ððk ÞÞ þðk ÞŠ A L ððk ÞÞ ðþ where L (!) ¼ arg W(!), which is a response of the plant to the two square pulse-wave signals shifted with respect to each other by the angle. Therefore, the amplitude is A ¼ max yðtþ: t½;=!š ðþ Yet, instead of the true amplitude we can use the amplitude of the fundamental frequency component (first harmonic) as a relatively precise estimate. In this case, we can represent the input as the sum of two rotating vectors having amplitudes 4c / and 4c /, with the angle between the vectors. Therefore, the amplitude of the control signal (first harmonic) is A u ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ c þ c c cosðþ, ðþ and the amplitude of the output (first harmonic) is A ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ c þ c c cosðþa L ðþ, ð3þ where A L ð!þ ¼jWðj!Þj. We should note that despite using approximate value for the amplitude in (3), the value of the frequency is exact. c =ξc Ω Ω a 4 a c Expressions (9) and (3) if considered as equations for and A provide one with mapping F. This mapping is depicted in Figure 5 as curves of equal values of and A in the coordinates (c, c ). From (9) one can see that the frequency of the oscillations depends only on the ratio c /c ¼. Therefore, is invariant with respect to c /c : (c, c ) ¼ (c, c ). It also follows from (3) that there is the following invariance for the amplitude: A (c, c ) ¼ A (c, c ). Therefore, and A can be manipulated independently in accordance with mapping G considered below. Mapping G (inverse of F) can be derived from (9) and (3) if c, c and are considered unknown parameters in those equations. For any given, from Equation (9) the ratio c /c ¼ can be found (as well as ). Therefore, we can find first ¼ c /c ¼ h(), where h() is an implicit function that corresponds to (9). After that c and c can be computed as per the following formulas: c ¼ A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4þ 4 A L ðþ þ cosðþþ c ¼ A p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð5þ 4 A L ðþ þ cosðþþ 6. Poincare design Poincare map is a recognised tool for analysis of the existence of limit cycles for non-linear systems. Therefore, this tool is appropriate to satisfy the goal defined in Section. To begin with, let us consider that the actuated degrees of freedom are represented by the elements of ¼ð, ÞIR and the unactuated a 6 Ω 3 a 8 Ω 4 Ω 5 a Figure 5. Plot of c vs c for arbitrary frequencies and amplitudes a 5 A 5 a.

8 684 L.T. Aguilar et al. Figure 6. Partitioning of the state space and the Poincare map. Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 degrees of freedom are represented by the elements of ¼ð, ÞIR m. Let us define the output y ¼ The Lagrange equation () can be represented in the state-space form by: 3 3 _ m _ fm ð, Þ½u N ð, ÞŠ _ 5 ¼ þm ð, ÞN ð, Þg 6 _ m f M 7 4 ð, Þ½u N ð, ÞŠ 5 M ð, ÞN ð, Þg 3 f ð,, uþ ¼ 6 7 ð6þ 4 5 f ð,, uþ where u is given in (), and m ¼ M (, ) M (, ) M (, )M (, ). Control law () switches on the surface ¼ and ¼. Consider the sets (Figure 6): S ¼fð,,, Þ : 4, ¼ g S ¼fð,,, Þ : ¼, 5 g S 3 ¼fð,,, Þ : 5, ¼ g S 4 ¼fð,,, Þ : ¼, 4 g: ð7þ The space IR n is divided by S i, into four regions i ¼,..., 4, namely R ¼fð,,, Þ : 4, 4 g, R ¼fð,,, Þ : 4, 5 g, R 3 ¼fð,,, Þ : 5, 5 g, R 4 ¼fð,,, Þ : 5, 4 g: ð8þ with f 5 for all,, R [ R ; and f 4 for all,, R 3 [ R 4. Assume that f and f are differentiable in the set R i, i ¼,..., 4. Moreover, suppose that the values of the functions of f k, k ¼, in the sets R i could be smoothly extended till their closures R i. Considering (, ), let us derive the Poincaré map from () ¼ (, ), where 4, into () ¼ (, ), where 5 (see region R in Figure 6). Let 4 and denote as þ ðt,,, c, c Þ, þ ðt,,, c, c Þ þ ðt,,, c, c Þ, þ ðt,, ð9þ, c, c Þ the solution of the system (6) with the initial conditions þ ð,,, c, c Þ¼, þ ð,,, c, c Þ¼ : þ ð,,, c, c Þ¼, ð3þ Let T sw (,, c, c ) be the smallest positive root of the equation þ ðt sw,,, c, c Þ¼ ð3þ and such that ðd þ =dtþðt sw,,, c, c Þ¼ þ ðt sw,,, c, c Þ 5, i.e. the functions T sw ð,, c, c Þ, þ ðt sw,,, c, c Þ, þ ðt sw,,, c, c Þ, þ ðt sw,,, c, c Þ, smoothly depend on their arguments. Now, let us derive the Poincaré map from the sets ðþ ¼ ð,, Þ, where 5, into the sets 3 ðþ ¼ ð,, Þ where 5 (see region R 3 in Figure 6). To this end, denote as þ p ðt,,,c,c Þ, þ p ðt,,,c,c Þ, þ p ðt,,,c,c Þ, ð3þ

9 International Journal of Control 685 the solution of the system (6) with the initial conditions þ p ðt þ sw ð,, c, c Þ,,, c, c Þ¼, þ p ðt þ sw ð,, c, c Þ,,, c, c Þ.5 ¼ þ ðt þ sw ð,, c, c Þ,,, c, c Þ, ð33þ η 3 þ p ðt þ sw ð,, c, c Þ,,, c, c Þ ¼ þ ðt þ sw ð,, c, c Þ,,, c, c Þ, Let Tp þð,, c, c Þ be the smallest root satisfying the restrictions Tp þ 4 T sw þ 4 of the equation.5.5 η.5 η Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 þ p ðt þ p,,, c, c Þ¼ ð34þ and such that ðd þ =dtþðt þ p Þ¼f ðt þ p,,,, c, c Þ 5, i.e. the functions T p ð,, c, c Þ, þ ðt p,,, c, c Þ, þ ðt p,,, c, c Þ, þ ðt p,,, c, c Þ, þ ðt p,,, c, c Þ smoothly depend on their arguments. Therefore, we have designed the map þ þ ð, ðt p þð,, c, c Þ,, 3, c, c Þ, c, c Þ¼4 5 þ ðtp þð,, c, c Þ,,, c, c Þ ð35þ The map ð,, c, c Þ of 3 () ¼ (,, ), 5 starting at the point þ ð,, c, c Þ into () ¼ (, ), 4 together with the time constant Tp þ 5 T sw 5 T p can be defined by the similar procedure. Therefore the desired periodic solution corresponds to the fixed point of the Poincare map (Figure 7)?? ðtp,?,?, c, c Þ¼: ð36þ Finally, to complete the design of periodic solution with desired period Tp ¼ = and amplitude? ¼ A the set of algebraic equations needs to be solved with respect to c, c, and : " # A ð=, A,?, c, c Þ¼,? ð37þ p ð=, A,?, c, c Þ¼, where c and c, are unknown parameters. Theorem : Suppose that for the given value of amplitude A and value of frequency there exist c Figure 7. Evolution of the states, of the closed-loop system. and c such the Poincare map ð,, c, c Þ has a fixed point ½?,? Š such that T p ¼ /,? ¼ ð,, c, c Þ ða,? Þ 5 ð38þ holds. Then, the system (6) has an orbitally asymptotically stable limit cycle with a desired period / and amplitude A. 7. Illustrative example Let us consider the transfer function WðsÞ ¼ ðjs þ Þðs þ F v s a Þ, J 4 ð39þ and its corresponding linear state-space representation d dt ¼ d dt ¼ d dt ¼ J a ðf v Ja Þ ðþjf v Þ þ u p where J ¼ 4/3, F v ¼ /4, a ¼ = ffiffiffi 8, and u ¼ c signð Þ c signð Þ ð4þ ð4þ is the two-relay controller which forces the output y ¼ to have a periodic motion. In the example, we set ¼ rad s and A ¼.7.

10 686 L.T. Aguilar et al. Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 7. DF and LPRS First, we compute the value of c and c through DF by using the set of equations: 8 4 A jwð jþj pffiffiffiffiffiffiffiffiffiffiffiffi >< þ if Q [ Q 3 c ¼ 4 A jwð jþj pffiffiffiffiffiffiffiffiffiffiffiffi >: þ elsewhere c ¼ c : where ¼ c /c, obtaining c ¼.88 and c ¼.78. To check if the periodic solution is stable find the derivative of the phase characteristic of the plant with respect to the frequency dargw dln! ¼ darctanðj!þ F v! darctanð!¼ dln! þ! þ a Þ!¼ dln! ¼ J J þ þ F vða Þ F v þða þ Þ : The stability condition (5) for the system becomes: J J þ þ F vða Þ F v þða þ Þ þ :!¼ ð4þ ð43þ Notice that the left-hand side of (4) is.6447 and the right-hand side is Therefore, the system is orbitally asymptotically stable. Now, let us compute the exact values of c and c through LPRS by using the following formulas: c ¼ A p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 4 A L ðþ þ cosðþþ c ¼ A p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 4 A L ðþ þ cosðþþ obtaining c ¼.77 and c ¼.65, with simulation results given in Figure Poincare map design Let us begin with the mapping from into the set (region R ) where the system (4) takes the form: d dt ¼, d dt ¼, d dt ¼ c þ 3 4 c : ð44þ x (rad) The solution of (44) on the time interval [, T sw ] subject to the initial condition: þ ð,, c, c Þ¼ 4, þ ð,, c, c Þ¼, þ ð,, c, c Þ¼ results in þ ¼ fflfflfflfflffl{zfflfflfflfflffl} 8c 8c þ 8c 8c þ 6 3 e t= fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð,,c,c Þ þ 4c 4c þ þ 4 3 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 3 ð,,c,c Þ þ 4c 4c þ 4 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 ð,,c,c Þ e t=4 e 3t=4 þ ¼ ð,, c, c Þe t= þ 4 3ð,, c, c Þe t= ð,, c, c Þe 3t=4 þ ¼ 4 ð,, c, c Þe t= þ 6 3ð,, c, c Þe t=4 where Time (s) Figure 8. Simulation results for the systems () and (4) under c and c computed through LPRS method. þ 9 6 4ð,, c, c Þe 3t=4, ð45þ ð46þ ð47þ ð48þ T sw ð,, c, c Þ¼4lnz, z ¼ e t=4 ð49þ

11 International Journal of Control 687 Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 is obtained as the smallest positive root of þ ðt sw,,, c, c Þ¼ 3 z 4 þ z 3 þ z þ 4 ¼, ð5þ where ¼ ð,, c, c Þ, ¼ ð,, c, c Þ, and 3 ¼ 3 ð,, c, c Þ. Let us proceed with the mapping from into 3 (region R 3 ) where the system (39) takes the form: d dt ¼, d dt ¼, d dt ¼ þ 3 4 c þ 3 4 c : ð5þ The solution of (5) on the time interval [T sw, T p ] subject to the initial condition: þ sw ¼ þ p ðt sw,, c, c Þ¼, þ sw ¼ þ p ðt sw,, c, c Þ¼ e T sw= þ 4 3e T sw= e 3T sw=4 results in þ p ¼ 8c 8c þ 8 c þ c 3 þ sw fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 c þ c 4 þ sw fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} p ð,, c, c Þ þ 4 c þ c þ 5 þ sw fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 3p ð,, c, c Þ p ð,, c, c Þ e 3ðt T swþ=4 e ðt T swþ=4 e ðt T swþ= þ p ¼ 4 pe ðt T swþ= þ 3 p e 3ðt T swþ=4 þ 3p e ðt T swþ=4 þ 3p ¼ pe ðt T swþ= 9 4 pe 3ðt T swþ=4 þ 4 3pe ðt T swþ=4 ð5þ ð53þ ð54þ where p ¼ p ð,, c, c Þ, p ¼ p ð,, c, c Þ, 3p ¼ 3p ð,, c, c Þ and T p ð,, c, c Þ¼4lnz p þ T sw, z p ¼ e ðt T swþ=4 þ ðð Þ?, ð Þ?, c, c Þ ð Þ?,ð þ ðð Þ?, ð Þ?, c, c þ ðð Þ?, ð Þ? 6, c, c Þ ð Þ?,ð Þ? ð Þ?,ð Þ? results from the the smallest positive root of þ p ðt p,,, c, c Þ¼ 3p z 4 p 4 pz p þ 3 p ¼ : ð56þ Then, the Poincare map is þ ðt pð,, c, c Þ,,, c, c Þ " # ¼ þ e Tp= þ 3 e Tp=4 þ 4 e 3T p=4 4 e Tp= þ 6 3e Tp=4 þ 9 6 4e 3T p=4 ð57þ and the fixed point that is the solution of ¼ þ ðt pð,, c, c Þ,,, c, c Þ which results in ( )? ¼ þ 4 3 ŠeTp=4 þ½4c 4c þ 4 Še 3T p=4 þ½8c 8c 6 3 Še Tp= þ½4c 4c þ e T= þ et=4 ( e 3T=4 4 8c 8c þ e T p = þ 6 ½4c ) 4c? þ ŠeTp=4 þ 6 9 4c 4c þ e 3T p =4 ¼ 4 3 e Tp= etp=4 9 4 e 3T p=4 where T ¼ T p T sw. To complete the design it remains to provide the set of equations to find c and c in terms of the known parameters T p and. Towards this end, we obtain from above equations that c and c are solution of the following set of equations: ( ) ½ þ e Tp= þ etp=4 4 e 3Tp=4 Šð Þ? c c ¼ 4 þð 6 3 e Tp= 4 3 etp=4 4e 3Tp=4 Þ e Tp= þ e Tp=4 e 3T p=4 ð58þ ( 4 3 e Tp= etp=4 9 4 e 3T p=4 ð Þ )? 4 c þ c ¼ e Tp= þ 3 etp=4 þ 9 3 e 3T p=4 e Tp= þ 4 etp=4 9 : ð59þ 4 e 3T p=4 Then, for a given frequency T þ T ¼ / and amplitude ¼ A we obtain that c ¼.77 and c ¼.67. Finally, we need to check the stability, ðð Þ?, ð Þ?, c, ðð Þ?, ð Þ?, c, c ð Þ?,ð Þ? ð Þ?,ð Þ? 3 5, ð6þ 7 5

12 688 L.T. Aguilar et al. þ ð,, c, c ¼ e Tp= þ 4 3e Tp= e p=4 þ e T p= þ et p=4 e 3T p=4 þ p =@ 9:878, p =@ :5876. Using (6), we þ ð, " #, c, c :446 :745 Þ ¼ ð Þ?,ð Þ? :88 :4 :53 p where kak ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi max fa T Ag; therefore, it is verified that the orbit is asymptotically stable. Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July þ ð,, c, c ¼ e Tp= þ 4 3e Tp= e 6 3 e T p= þ 3 4 et p=4 þ 4e 3T p=4 þ ð,, c, c ¼ 8 e Tp= þ 64 3e Tp= e ð6þ þ 4 e T p= þ 6 et p=4 þ 9 6 e 3T p=4 :88 þ ð,, c, c ¼ 8 e Tp= þ 64 3e Tp= e p=4 sw sw 4 3 e T p= þ et p=4 þ 9 4 e 3T p=4 :4, ð64þ ¼ 4 z 4 z 3 þ z þ z3 4 3 z 3 3 z þ 3 z z 4 ¼ 4 4z z 3 4z þ z3 4 3 z 3 þ 3 z þ 3 þ z 3 þ 3 4 z 4 p ¼ 4 þ :4, ¼ 4 z p :468, h 5 z4 p þ 4 3 z p p z 3 p 4 þ 5 z4 p þ 4 3 z p p z 3 p 4 p þ þ 8. Experimental study: the Furuta pendulum 8. Experimental setup In this section, we present experimental results using the laboratory Furuta pendulum, produced by Quanser Consulting Inc., depicted in Figure 9. It consists of a 4 V DC motor that is coupled with an encoder and is mounted vertically in the metal chamber. The L-shaped arm, or hub, is connected to the motor shaft and pivots between 8 degrees. At the end, a suspended pendulum is attached. The pendulum angle is measured by the encoder. As described in Figure 9, the arm rotates about z-axis and its angle is denoted by q while the pendulum attached to the arm rotates about its pivot and its angle is called q. The experimental setup includes a PC equipped with an NI-M series data acquisition card connected to the Educational Laboratory Virtual Instrumentation Suite (NI-ELVIS) workstation from National Instrument. The controller was implemented using Labview programming language allowing debugging, virtual oscilloscope, automation functions, and data storage during the experiments. The sampling frequency for control implementation has been set to 4 Hz. Appendix A gives the dynamic model of the Furuta pendulum. 8. Experimental results Experiments were carried out to achieve the orbital stabilisation of the unactuated link (the pendulum) y ¼ q around the equilibrium point q? ¼ (, ). The equation of motion of the Furuta pendulum (3) is linearised around q? IR and by virtue of the instability of the linearised open-loop system, a state-feedback controller u f ¼ Kx and x ¼ðq q?, _qþ T IR 4, is designed such that the compensated system has an overshoot of 8 and gain crossover frequency at rad s (see Bode diagram in Figure for the open-loop system). Thus, the matrices A, B, and C of the linear

13 International Journal of Control 689 z L p x q y h q r Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 Figure 9. The experimental Furuta pendulum system. system (6) are 3 A ¼ :59 5:685 6:6 5:55 5, 3:3 :48 :879 : T B ¼ :389 5, C ¼ 6 7 : 4 5 5:93 For the experiments, we set initial conditions sufficiently close to the equilibrium point q? IR. The output y ¼ q is driven to a periodic motion for several desired frequencies and amplitudes. The frequencies () and amplitudes (A ) obtained from experiments by using the values of c and c computed by means of the DF and LPRS are given in Tables and, respectively. Inequality (7) holds for the chosen frequencies and amplitudes, thus asymptotical stability of the periodic orbit was established by Theorem. In Figure, experimental oscillations for the output y, for fast ( ¼ 5 rad s ) and slow motion ( ¼ rad s ) are displayed. Note that certain imperfections appear in the slow motion graphics in Figure, which are attributed to the Coulomb magnitude (db) phase (deg) 5 5 Bode diagram Frequency (rad s ) Figure. Bode plot of the open-loop system. friction forces and the dead zone. Also, in some modes natural frequencies of the pendulum mechanical structure are excited, and manifested as higherfrequency vibrations. 9. Conclusions The key feature of the proposed method is that the underactuated system can be considered as a system

14 69 L.T. Aguilar et al. Table. Computed c and c values for several desired frequencies using DF method. Desired Desired A c c Experimental Experimental A Table. Computed c and c values for several desired frequencies LPRS. Desired Desired A c c Experimental Experimental A Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 (a) q (rad) y=q (rad) Time [s] (b) q (rad) y=q (rad) Time [s] Figure. Steady state periodic motion of each joint where (a) is the periodic motion at ¼ 5 rad s and (b) is the periodic motion at ¼ rad s. with unactuated dynamics with respect to actuated variables. For generation of self-excited oscillations with desired output amplitude and frequencies, a tworelay controller is proposed. The systematic approach for two-relay controller parameter adjustment is proposed. The DF method provides approximate values of controller parameters for the plants with the low-pass filter properties. The LPRS gives exact values of the controller parameters for the linear plants. The Poincare maps provides the values of the controller parameters ensuring the existence of the locally orbitally stable periodic motions for an arbitrary mechanical plant. The effectiveness of the proposed design procedures is supported by experiments carried out on the Furuta pendulum from Quanser. References Aguilar, L., Boiko, I., Fridman, L., and Iriarte, R. (9), Generating Self-excited Oscillations Via Two-relay Controller, IEEE Transactions on Automatic Control, 54(), Atherton, D.P. (975), Nonlinear Control Engineering- Describing Function Analysis and Design, Workingham, UK: Van Nostrand. Boiko, I., Fridman, L., and Castellanos, M.I. (4), Analysis of Second-order Sliding-mode Algorithms in the Frequency Domain, IEEE Transactions on Automatic Control, 49, Boiko, I. (5), Oscillations and Transfer Properties of Relay Servo Systems the Locus of a Perturbed Relay System Approach, Automatica, 4, Boiko, I., Fridman, L., Pisano, A., and Usai, E. (7), Analysis of Chattering in Systems with Second-order

15 International Journal of Control 69 Downloaded By: [UNAM Ciudad Universitaria] At: :55 8 July 9 Sliding Modes, IEEE Transactions on Automatic Control, 5, 85. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Canudas-de-Wit, C., and Grizzle, J.W. (3), RABBIT: A Testbed for Advanced Control Theory, in IEEE Control Systems Magazine, 3, Craig, J. (989), Introduction to Robotics: Mechanics and Control, Massachusetts, MA: Addison-Wesley Publishing. Di Bernardo, M., Johansson, K.H., and Vasca, F. (), Self-oscillations and Sliding in Relay Feedback Systems: Symmetry and Bifurcations, International Journal of Bifurcation and Chaos,, 4. Fantoni, I., and Lozano, R. (), Nonlinear Control for Underactuated Mechanical Systems, London: Springer. Fridman, L. (), An Averaging Approach to Chattering, IEEE Transactions on Automatic Control, 46, Grizzle, J.W., Moog, C.H., and Chevallereau, C. (5), Nonlinear Control of Mechanical Systems with an Unactuated Cyclic Variable, IEEE Transactions on Automatic Control, 5, Levant, A. (993), Sliding Order and Sliding Accuracy in Sliding Mode Control, International Journal of Control, 58, Orlov, Y., Riachy, S., Floquet, T., and Richard, J.-P. (6), Stabilization of the Cart-pendulum System via Quasihomogeneous Switched Control, in Proceedings of the 6 International Workshop on Variable Structure Systems, 5 7 June, Shiriaev, A.S., Freidovich, L.B., Robertsson, A., and Sandberg, A. (7), Virtual-holonomic-constraintsbased Design Stable Oscillations of Furuta Pendulum: Theory and Experiments, IEEE Transactions on Robotics, 3, Varigonda, S., and Georgiou, T.T. (), Dynamics of Relay Relaxation Oscillators, IEEE Transactions on Automatic Control, 46, Appendix A. Dynamic model of Furuta pendulum The equation motion of Furuta pendulum, described by (), was specified by applying the Euler Lagrange formulation (Craig 989), where with MðqÞ ¼ M ðqþ M ðqþ M ðqþ M ðqþ M ðqþ¼j eq þ M p r cos ðq Þ, M ðqþ¼ M prl p cosðq Þcosðq Þ, M ðqþ¼j p þ M p l p,, Hðq, _qþ ¼ H ðq, _qþ H ðq, _qþ H ðq, _qþ¼ M p r cosðq Þsinðq Þ _q þ 4 M prl p cosðq Þsinðq Þ _q H ðq, _qþ¼ M prl p sinðq Þcosðq Þ _q þ M pgl p sinðq Þ where M p ¼.7 Kg is mass of the pendulum, l p ¼.53 m is the length of pendulum centre of mass from pivot, L p ¼.9 m is the total length of pendulum, r ¼.86 m is the length of arm pivot to pendulum pivot, g ¼ 9.8 m s is the gravitational acceleration constant, J p ¼.3 4 Kg m is the pendulum moment of inertia about its pivot axis, and J eq ¼. 4 Kg m is the equivalent moment of inertia about motor shaft pivot axis.