- Email: [email protected]
Dynamic harmonic balance principle and analysis of rocking block motions
Available online at www.sciencedirect.com
Journal of the Franklin Institute 349 (2012) 1198–1212 www.elsevier.com/locate/jfranklin
Dynamic harmonic balance principle and analysis of rocking block motions Igor M. Boiko The Petroleum Institute, P.O. Box 2533, Abu Dhabi, U.A.E Received 18 July 2011; received in revised form 18 October 2011; accepted 10 January 2012 Available online 20 January 2012
Abstract The harmonic balance (HB) principle is a powerful and convenient tool for finding periodic solutions in nonlinear systems. In the present paper, this principle is extended to transient processes in systems with one single-valued odd-symmetric nonlinearity and linear plant not having zeros in the transfer function, and named the dynamic HB. Based on the dynamic HB, first the equations for the amplitude, frequency, and amplitude decay of an oscillatory process in the Lure system are derived. It is then applied to analysis of rocking block decaying motions. An example is provided. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Harmonic balance principle is a convenient tool for finding parameters of self-excited periodic motions. Due to this convenience, it is widely used in many areas of science and engineering. For a system with one nonlinearity and linear dynamics (Lure system), it can be illustrated by drawing the Nyquist plot of the linear dynamics and the plot of the negative reciprocal of the describing function (DF) [1] of the nonlinearity in the complex plane and finding the point of intersection of the two plots, which would correspond to the self-excited periodic motion in the system. Therefore, the harmonic balance principle treats the system as a loop connection of the linear dynamics and of the nonlinearity. It is also possible to reformulate the harmonic balance, so that the format of the system analyzed is not a loop connection but the denominator of the closed-loop system. This would imply a different interpretation of the harmonic balance, which would allow one to extend the harmonic balance principle to analysis of not only self-excited periodic motions but also other types of oscillatory motions. E-mail address: [email protected] 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2012.01.007
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1199
One of the types of the systems that exhibit vanishing oscillatory motion of variable frequency is the conventional and second-order sliding mode (SM) control system. There are a number of second-order SM (SOSM) algorithms available now, the most popular of which are ‘‘twisting’’, ‘‘super-twisting’’, ‘‘twisting-as-a-filter’’ [2,3], ‘‘sub-optimal’’ [4,5]. These algorithms provide finite-time convergence and because of this feature are used in both control and observation [6–8]. An overview of finite-time convergent second-order SM algorithms and algorithms with asymptotic convergence is presented in [9]. Another example of a transient oscillatory motion is the transient processes in oscillatory mechanical systems. A number of examples are given in [10,11]. The same mechanical system, as of the present paper, but in a different setup was considered in [12]. A number of examples from different areas of engineering and physics can be found where transient oscillatory motions of variable frequency occur. Therefore, some common approach to the problem of the convergence rate assessment, including qualitative (finite-time or asymptotic) and quantitative assessment, would be of high importance. The availability of the model of the system revealing transient oscillations might seem to be sufficient for analysis and not requiring development of methods other than simulations. However, the frequency-domain approach to assessment of convergence rate would provide a number of advantages over the direct solution of the system of differential equations. The most important one is the possibility of explanation of the mechanism of the frequency and amplitude change during the transient. This analysis can also lead to the formulation of the criteria of finite-time and asymptotic convergence in terms of frequency-domain characteristics. In the present paper, a frequency-domain approach to analysis of transient oscillations is presented, which is suitable for analysis of high-order nonlinear systems. The harmonic balance principle is extended to the case of transient oscillations and named as the dynamic harmonic balance principle. The paper is organized as follows. At first the harmonic balance principle is considered. Then a Lure system with a high-order plant is analyzed with the use of the dynamic harmonic balance involving quasi-static approach to the frequency of the oscillations. Such characteristics as frequency and amplitude of oscillations as functions of time are derived. After that the condition of the full dynamic harmonic balance is derived. Finally, the proposed approach is applied to analysis of the decaying motions of increasing frequency of the rocking block. Example of analysis and simulations supporting the proposed theory are provided. 2. Harmonic balance for transient oscillations Consider the system that includes linear dynamics given by the following equations: x_ ¼ Ax þ Bu y ¼ Cx,
ð1Þ
where x 2 Rn , y 2 R1 , u 2 R1 , A 2 Rnn , B 2 Rn1 , C 2 R1n , and a single-valued oddsymmetric nonlinearity f(y) with zero mean: u ¼ f ðyÞ,
ð2Þ
We shall refer to (1) as to the linear part of the system. One can see that the system (1), (2) is a Lure system. The transfer function of the linear part is Wl ðsÞ ¼ CðIsAÞ1 B, which can also be presented as a ratio of two polynomials Wl(s)¼ P(s)/Q(s). Assume also an
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1200
autonomous mode, so that the input to the nonlinearity is the output of the linear dynamics, and the output of the nonlinearity is the input to the linear dynamics. Selfexcited periodic motions in the system can be found through the use of the HB principle. The conventional HB condition (for periodic motion) is formulated as Wl ðjOÞNðaÞ ¼ 1,
ð3Þ
where O is the frequency and a is the amplitude of the self-excited periodic motion at the input to the nonlinearity, N(a) is the describing function of the nonlinearity, which does not depend on the frequency because the nonlinearity is a single-valued function. If the linear part has relative degree higher than two, then the Nyquist plot of system (1) has a point of intersection with the real axis at some finite frequency and, therefore, Eq. (3) has a solution. We now find the closed-loop transfer function Wcl(s) of system (1), (2) using the replacement of the nonlinearity with the DF u ¼ NðaÞy: Wcl ðsÞ ¼
Wl ðsÞNðaÞ PðsÞNðaÞ ¼ 1 þ Wl ðsÞNðaÞ QðsÞ þ PðsÞNðaÞ
ð4Þ
Let us note that (3) is equivalent to Rða,jOÞ ¼ QðjOÞ þ PðjOÞNðaÞ ¼ 0,
ð5Þ
which means that the denominator of the closed-loop transfer function turns into zero when the frequency and the amplitude become equal to the frequency and the amplitude of the periodic motion. Eq. (5) is also sometimes used for finding a periodic solution via algebraic methods. However, Eq. (5) usually is not attributed to the denominator of the closed-loop transfer function but considered a direct result of (3). Assuming that R(a,s) can be represented in the following form R(a,s)¼ (s–s1)(s–s2) y (s–sn), where si are roots of the characteristic polynomial, we must conclude that there must be at least one pair of complex conjugate root with zero real parts. It would imply the existence of the conservative component in Wcl(s). Indeed, we can consider the existence of non-vanishing oscillations as a result of the existence of the component (s2þr2) in the denominator of Wcl(s), where r is a parameter that depends on the amplitude a. However, one can notice that even if a damped oscillation occurs, so that there exists a pair of complex conjugate roots si, siþ1 then (ssi)(ssiþ1)¼ 0, and the characteristic polynomial becomes zero, with s¼ s7jO, where s is the decay (Note: strictly speaking, we have a decaying oscillation only if so0; yet we will refer to this variable as to the decay even if sZ0). We can view the harmonic balance condition (3) as the realization of the regeneration principle, according to which y(t)¼ L1[L[u(t)]Wl(s)], and u(t)¼ f(y(t)). We can now use the regeneration principle approach to a transient oscillation, in which we consider both u(t) and y(t) sinusoidal signals with exponentially decaying (or growing) amplitude with constant value of the decay, and formulate and prove the following property. Theorem 1. With the input signal to the linear dynamics given by the transfer function Wl(s) being the harmonic signal with decaying amplitude uðtÞ ¼ est sinðOtÞ, the output of the linear dynamics is also a harmonic signal of the same frequency and decay yðtÞ ¼ aest sinðOt þ fÞ. Proof. It follows from the property of the Laplace transform that L[eaf(t)] ¼ F(sþa). Therefore, for the system input u(t), the Laplace transform will be L[u(t)] ¼ O/[(s–s)2þO2], which will result in the system output (in the Laplace domain) Y(s)¼ OWl(s)/[(s–s)2þO2].
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1201
The substitution s0 ¼ s–s yields Y(s0 )¼ OWl(s0 þs)/[(s0 )2þO2], which means that y0 (t)¼ L1[Y(s0 )] is a sinusoid of frequency O, amplitude 9Wl(sþjO)9, and having the phase shift argWl ðs þ jOÞ. In turn, the output signal is y(t) ¼ esty0 (t), i.e. a decaying sinusoid. & Therefore, for our analysis of propagation of the decaying sinusoids through linear dynamics we can use the same transfer functions, in which the Laplace variable should be replaced with (sþjO). The describing function N in the case of a transient oscillation may become a function of not only amplitude but of its derivatives too [13] (we disregard possible dependence of the DF on the frequency). Considering that conditions (3) and (5) are equivalent, and the equality of the denominator of the closed-loop transfer function to zero (for some s) implies the fulfillment of (3), we can rewrite (3) for the transient oscillation as follows: _ . .ÞWl ðs þ jOÞ ¼ 1, Nða, a,.
ð6Þ
The use of the derivatives of the amplitude as arguments of the DF is inconvenient because it results in the necessity of consideration of additional variables (derivatives of the amplitude), which are not present otherwise. It is more convenient to consider s and its derivatives than the derivatives of the amplitude. Also, we limit our consideration of the describing function arguments to the first derivative of the amplitude (or equivalently, to s) only, which will correspond to the use of the regeneration principle for decaying sinusoids. We show below that for many nonlinearities the DF is a function of the amplitude only—like in the conventional DF analysis. Therefore, we can write the condition of the existence of a transient or steady oscillation as follows: Nða,sÞWl ðs þ jOÞ ¼ 1,
ð7Þ
where the decay and frequency are considered constant on some small interval of time (theoretically on infinitely small interval of time, and practically on one step of integration when using a numeric method), while the amplitude is considered exponentially changing on this interval (due to the constant decay). Joining solutions for these intervals, we obtain the frequency and the decay as functions of time. Therefore, we assume that the decay and the frequency are slowly or quasi-statically changing. Eq. (7) is referred to in [14] as the dynamic harmonic balance condition (equation). However, it does not account for the derivative of the frequency and we shall further refer to (7) as to the dynamic harmonic balance quasi-static w.r.t. frequency. Assume now that the characteristic polynomial of the closed-loop system (with parametric dependence on the amplitude of the oscillations) has a pair of complex conjugate roots with negative real parts. Then a vanishing oscillation of certain frequency and amplitude occurs. The idea of considering equations of vanishing oscillations is similar to the one of the Krylov–Bogoliubov method [15]. However, the latter can only deal with small ‘‘deviations’’ from the harmonic oscillator and is limited to second-order systems. In the present approach, the ‘‘equivalent damping’’ is not limited to small values. Let us consider instantaneous values of the frequency, amplitude and decay and formulate the dynamic HB quasi-static w.r.t. frequency and decay as follows. At every time, a single-frequency mode transient oscillation can be described as a process of variable (instantaneous) frequency, amplitude and decay, which must satisfy Eq. (7). Note: In (7) and the formulation given above, we consider only transient oscillations with zero mean and single-frequency mode when the characteristic polynomial (5) has only one pair of complex conjugate roots.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1202
The describing function of an arbitrary nonlinearity for a decaying amplitude harmonic input signal should be computed as suggested in [16] (see also [17]) ss , ð8Þ Nða,s,oÞ ¼ qða,s,oÞ þ q0 ða,s,oÞ o where s¼
d , dt
1 q ¼ 2p 0
q¼ Z
1 2p
Z
2p
f ½asin C,aðssin C þ o cosCÞsin CdC, 0
2p
f ½asin C,aðssin C þ o cosCÞcos CdC, 0
which accounts for the dependence on the frequency also. It should be noted that for single-valued or hysteretic symmetric nonlinearities formula (8) produces conventional describing function expressions that are commonly used for harmonic inputs. The overall motion can now be obtained from the dynamics HB as follows: yðtÞ ¼ aðtÞ esðtÞt sin CðtÞ,
ð9Þ
where a(t), s(t) are obtained from the following differential equation (s(t) is supposed to be expressed through a(t) using other equations, which is given in details below): _ ¼ aðtÞsðtÞ, aðtÞ
að0Þ ¼ a0 ,
and C(t) is the phase computed as follows: CðtÞ ¼ from (7), f is selected to satisfy initial conditions.
ð10Þ Rt 0
OðtÞdt þ f, where O(t) is obtained
3. Analysis of motions in the vicinity of a periodic solution Carry out frequency-domain analysis of the transient process of the convergence to the periodic motion in the vicinity of a periodic solution in system (1), (2), using the dynamic harmonic balance condition (7). We can write the conventional harmonic balance condition, which can also be obtained from (7) when s ¼ 0, as follows: Nða0 ÞWl ðjO0 Þ ¼ 1,
ð11Þ
where O0 and a0 are the frequency and the amplitude of the periodic solution. Write the dynamics harmonic balance condition for the increments from the periodic solution: Nða0 þ Da,sÞWl ðs þ jðO0 þ DOÞÞ ¼ 1,
ð12Þ
We now take the derivative from both sides of (12) with respect to Da (or a) in the point a¼ a0: ! ds @Nða,sÞ @Nða,sÞ ds dWl ðsÞ þ ðjO Þ þ Nða Þ ¼ 0, W 0 0 l @a a ¼ a0 @s s ¼ 0 daa ¼ a0 ds s ¼ jO0 daa ¼ a0 ð13Þ At first limit our analysis only to the nonlinearities the describing function of which does not depend on s(for example, the ideal relay nonlinearity). Later the same analysis can be applied to nonlinearities that depend on s. Express the derivative ds=daja ¼ a0
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1203
from Eq. (13):
dNðaÞ da a ¼ a Wl ðjO0 Þ ds 0 ¼ : dWl ðsÞ daa ¼ a0 Nða Þ 0
ds
ð14Þ
s ¼ jO0
Considering that s¼ sþjO, we can rewrite Eq. (14) as follows: dNðaÞ Wl ðjO0 Þ da ds dO a ¼ a0 þj ¼ : da a ¼ a0 da a ¼ a0 Nða0 ÞdWdsl ðsÞ
ð15Þ
s ¼ jO0
Eq. (15) is a complex equation. It can be split into two equations for the real and imaginary parts. However, only real parts of (15) give an equation that has a solution. Once it is solved and a(t) is found, O(t) can be found too. Considering that dlnWl ðjoÞ 1 dWl ðsÞ dlnWl ðsÞ dargWl ðjoÞ ¼ ¼ j , Wl ðsÞ ds s ¼ jO0 ds s ¼ jO0 do do o ¼ jO0 o ¼ jO0 and
dlnN~ ðaÞ 1 dNðaÞ dlnN~ ðaÞ ¼ ¼ NðaÞ da a ¼ a0 da a ¼ a0 da
a ¼ a0
dargN~ ðaÞ þj , da a ¼ a0
where N~ ðaÞ ¼ N ðaÞ, we can write for the real part of (15): 8 9 dlnjN~ ðaÞj dargN~ ðaÞ = < þ j da ds da ¼ Re , :dargWl ðjoÞ j dlnjWl ðjoÞj; da 1
do
do
which can be rewritten as follows (we skip for brevity the notation of the point in which the derivative is taken): ds ¼ da
~ dlnjN~ ðaÞj dargWl ðjoÞ dln W ðjoÞ dargdaN ðaÞ j dol j do da 2 dlnjWl ðjoÞj 2 dargWl ðjoÞ þ do do
ð16Þ
As a ‘‘side’’ product of our analysis, stability of a periodic solution can be assessed as follows: ds=daja ¼ a0 o0.The left-hand side of (16) can be rewritten as follows: ds 1 a€ a_ ¼ da a a_ a and, therefore, Eq. (16) is a second-order differential equation. Because the right-hand side of (16) contains only one unknown variable a, this equation can be numerically solved and convergence of the amplitude to the constant value can be obtained as a function of time. 4. Dynamic harmonic balance including frequency rate of change In many cases, such as analysis of motions in the vicinity of a periodic solution, the dynamic HB quasi-static w.r.t. to frequency derivative is quite capable of providing a precise result because the frequency changes insignificantly and the derivative of the
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1204
frequency can be neglected. In some other cases, such as analysis of convergence of systems with second-order sliding modes [18] and finite-time convergence, the system experiences significant changes of the instantaneous frequency of the oscillations. In this situation the quasi-static approach to the account of the oscillations frequency may result in the loss of accuracy of analysis. Therefore, inclusion of the frequency derivative would be desirable in such cases. Consider the following illustrative example. Let the linear part be the second-order dynamics and the controller be the ideal relay with the amplitude h. The describing function of the relay is N(a)¼ 4h/pa. If we assume that the input to the nonlinearity is a decaying sinusoid of constant frequency yðtÞ ¼ a0 est sinðotÞ then the control amplitude should be au ¼ aN(a), the _ _ derivative of the control amplitude a_ u ¼ @N @a aa þ N a, and the decay su ¼
4h _ 4h _ a pa a_ u 2 a þ pa a ¼ 0: ¼ 4h au pa a
The result showing that su ¼ 0 is quite predictable because the output of the relay controller is an oscillation of constant amplitude, which features zero decay. However, it is well, known (see [1,13], for example) that the output of the linear part is oscillations of decaying amplitude and growing frequency. Because the input to the linear dynamics has zero decay, we can conclude that the decay in the signal y(t) is not a result of the decay in the control u(t) but exists because of the variable frequency of u(t). Moreover, the nonlinearity may change the decay according to @N _ _ @lnN @lnN a_ u @a aa þ N a su ¼ þ 1 s, ð17Þ ¼ ¼ a_ þ s ¼ @a @lna Na au and this change is offset by the decay change due to the frequency variation at the signal propagation through the linear part. This situation is not covered by the Eq. (7) that assumes constant frequency, which requires the development of a different model. Another example is the example of the propagation of a sinusoid of nearly constant amplitude and increasing frequency through the integrator. We assume that the signal of ideally constant amplitude is 1 un ðtÞ ¼ cos o0 t þ o_ 0 t2 , 2 propagation of which through the integrator produces signal 1 1 1 n 2 _ 0 t ¼ sin C sin o0 t þ o y ðtÞ yðtÞ ¼ o0 þ o_ 0 t 2 o _ 0 is small enough). We note that the instantaneous (the approximate equality is valid if o _ 0 t2 . Therefore, the instantaneous frequency is phase of the signal is CðtÞ ¼ o0 t þ ð1=2Þo _ € ¼ oðtÞ ¼ CðtÞ ¼ o0 þ o_ 0 t, and the instantaneous derivative of the frequency is CðtÞ _ 0 ¼ const. Find the derivative of y(t) as follow: o _0 1 1 o o_ 0 2 2 _ _ _ ¼ cos o0 t þ o 0 t þ sin o0 t þ o 0 t ¼ cos C þ 2 sin C yðtÞ 2 2 2 o _ ðo0 þ o 0 tÞ Therefore, the integrator introduces the phase lag less than 901 and the equations of the integrator can be represented by the conventional integrator having an additional feedback, which accounts for the effect of the variable frequency.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1205
Therefore, the frequency response of the integrator to a decaying sinusoid of variable frequency is W ðs þ jo, o_ 0 Þ ¼
1 sþ
o_ 0 o2
þ jo
:
_ 0 =o2 then the phase lag introduced by the From this formula, one can see that if s ¼ o _ 0 Þ is 901. The amplitude characteristic of the integrator is integrator f ¼ argW ðs þ jo, o given by M ¼ jW ðs þ jo, o_ 0 Þj, and the decay introduced by the integrator due to variable _ ¼ o_ 0 =o (provided the amplitude of y(t) is a ¼ 1=o0 þ o_ 0 t). frequency is sðtÞ ¼ a=a To analyze propagation of a decaying sinusoid of variable frequency through higherorder linear dynamics we shall use the notion of the rotating phasor. Let the output of the linear dynamics y(t) be given by yðtÞ ¼ aðtÞejCðtÞ ¼ elnaðtÞ ejCðtÞ ¼ elnaðtÞþjCðtÞ ,
ð18Þ
where a represents the length of the phasor and C represents the angle between the phasor and the real axis. We can associate either the real or the imaginary part of yðtÞ with the real signal y(t). We assume that the transfer function of the linear part does not have any zeros: Wl(s)¼ 1/Q(s)¼ 1/(a0þa1sþa2s2þyþansn). In the present paper we limit our analysis to plants not containing zeros in the transfer function. Given this transfer function, we can write for y(t): u ¼ a0 y þ a1 y_ þ a2 y€ þ . . . þ an yðnÞ
ð19Þ
We can find the derivatives of y(t) as follows: _ ¼ elnaðtÞþjCðtÞ 1 a_ þ j C _ _ ¼ elnaðtÞþjCðtÞ dlna þ j C yðtÞ dt a ¼ elnaðtÞþjCðtÞ ðs þ joÞ ¼ ðs þ joÞy,
ð20Þ
€ ¼ ðs_ þ j oÞy _ þ ðs þ joÞy_ ¼ ½ðs_ þ j oÞ _ þ ðs þ joÞ2 y yðtÞ Considering that we disregard all derivatives of the decay and higher than first derivatives of the frequency in our model we can rewrite the last formula as follows: € ½ðs þ joÞ2 þ j oy _ yðtÞ
ð21Þ
and the
formula for the third and forth derivatives as follows: & _ þ j oy € þ ½ðs þ joÞ2 þ j oðs _ yðtÞ ½2ðs þ joÞðs_ þ j oÞ þ joÞy 2 3 _ þ joÞy _ þ ðs þ joÞ ðs þ joÞy ¼ ½ðs þ joÞ þ j3oðs ½j3o
ð22Þ
_ þ joÞ3o _ 2 y yð4Þ ðtÞ ½ðs þ joÞ4 þ j6oðs We can continue with taking further derivatives. It is worth noting that the formulas above are organized the way that they have a term (sþjo) to the respective power and the term which is the product of o_ and another multiplier. Therefore, we can write for y(t): _ u ¼ ½a0 þ a1 ðs þ joÞ þ a2 ðs þ joÞ2 þ . . . þ an ðs þ joÞn y þ Sðs,o, oÞy,
ð23Þ
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1206
_ includes all terms containing o. _ This component can be accounted for as where Sðs,o, oÞ an additional feedback—in the same way as it was done for the integrator. If we introduce certain modified frequency response as _ ¼ Wln ðs,o, oÞ
y 1 Wl ðs þ joÞ ¼ ¼ _ _ Qðs þ joÞ þ Sðs,o, oÞ 1 þ Wl ðs þ joÞSðs,o, oÞ u
ð24Þ
we can write the dynamic harmonic balance equation as follows: _ ¼ 1 NðaÞWln ðs,o, oÞ
ð25Þ
Obviously, Eq. (25) can be split into two equations: for real and imaginary parts, or for equations of the magnitude balance and phase balance. Eq. (25) must be complemented with an equation that relates the difference of the decays at the input and the output of the linear part and the frequency rate of change—in the same way as it was done in the example of the integrator analysis. We note that the _ and, amplitude of the control (first harmonic) is as follows au ¼ ajQðs þ joÞ þ Sðs,o, oÞj, therefore, its time derivative is _ dQðs þ joÞ þ Sðs,o, oÞ _ þa _ : a_ u ¼ ajQðs þ joÞ þ Sðs,o, oÞj dt The decay of signal u(t) is computed as _
oÞj _ þ a djQðsþjoÞþSðs,o, _ ajQðs þ joÞ þ Sðs,o, oÞj a_ u dt ¼ _ au ajQðs þ joÞ þ Sðs,o, oÞj _ dlnjQðs þ joÞ þ Sðs,o, oÞj ¼sþ dt € we can rewrite the last formula as follows: Because we disregard s_ and o
su ¼
su ¼ s þ
_ _ @lnjWln ðs,o, oÞj @lnjQðs þ joÞ þ Sðs,o, oÞj _ o_ ¼ s o @o @o
The derivative in the last formula defines the slope of the magnitude-frequency _ Considering also the formula for the decay of u(t) derived characteristic of Wln ðs,o, oÞ. above through the DF, we can now write the condition of the balance of the decays in the closed-loop system as follows: _ o _ @lnjWln ðs,o, oÞj @lnNðaÞ s ¼ @lna @lno o
ð26Þ
We can formulate the dynamic harmonic balance principle as theorem. At every time during a single-frequency mode transient oscillation, the oscillation can be described as a process of variable instantaneous frequency, amplitude, decay and frequency rate of change (time derivative), which must satisfy Eqs. (25) and (26). It is worth noting that _ @lnjWln ðs,o, oÞj dMðoÞ , ¼ 0:05 dlogo @lno where M(o) is the magnitude frequency response [dB] (Bode plot) for the transfer function _ Therefore, this term gives the slope of the Bode plot. Both o=o _ _ Wln ðs,o, oÞ. and s ¼ a=a are relative rates of change of the frequency and the amplitude, respectively. Eq. (26),
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1207
therefore, is establishing the balance between those two rates of change. The use of the dynamic HB principle is illustrated by the following example. Overall solution is obtained via integration of the first-order differential Eq. (10) with s expressed through a using the three algebraic equations presented above. 5. Analysis of decaying oscillations of rocking block Consider planar rocking motion of the 2-dimensional block (Fig. 1). The block of width c and height b and the base on which the block rests are assumed to be rigid. The possible considered motion includes only rocking (we assume that sliding and liftoff do not take place). However, we consider that friction is present in the form of a viscous friction, so that the resulting motion is alternating rocking around pivots A and B decaying due to the energy dissipation. Rocking around pivot point A can be described by the following equation: mR2 g€ A þ Z_g A mgsin gA ¼ 0, where jgA joarcsinb=2R, gA is the angle between the line connecting pivot A and the center of mass of the block and the vertical line. And rocking around pivot B is given by: mR2 g€ B þ Z_g B mgsin gB ¼ 0, where jgB joarcsinb=2R, gB is the angle between the line connecting pivot B and the center of mass of the block and the vertical line. Introduce the variable x that describes the angular position of the block in both motions, which is the angle between the vertical axis of the block and the true vertical line: x ¼ arcsin
b b g ¼ gA arcsin 2R B 2R
Fig. 1. Rocking block.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1208
Now the motions are described as follows. Rocking around pivot A is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 b 2 _ cosx, if xo0, mR x€ þ Zxmg 1 2 sin x ¼ mg 2R 4R And rocking around pivot B is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 b 2 _ mR x€ þ Zxmg cos x, 1 2 sin x ¼ mg 2R 4R
if x40
ð27Þ
ð28Þ
Attributing the right-hand side of (27) and (28) to control, we can rewrite (27) and (28) as an equation of a system with a discontinuous control: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 _ mR x€ þ Zxmg ð29Þ 1 2 sin x ¼ u 4R 8 b > > cosx if xo0 < mg 2R u¼ b > > cos x if x40 : mg 2R One can see that the control can be presented as u ¼ mg
b sgnx cosx 2R
ð30Þ
We now carry out analysis of motions in the system using the dynamic harmonic balance principle introduced above. We note that the system has two nonlinearities: f1 ðxÞ ¼ sin x, and f2 ðxÞ ¼ sgn x cos x. Both nonlinearities are odd-symmetric. The second nonlinearity is depicted in Fig. 2.
Fig. 2. Second nonlinearity.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1209
Find the describing functions for these nonlinearities. For the first nonlinearity the DF is given by [1] N1 ðaÞ ¼ 2J1 ðaÞ=a,
ð31Þ
where J1(a) is the Bessel function of the first order. We shall now derive the DF for the second nonlinearity. According to the DF definition: Z Z 4 p=2 4 p=2 N2 ðaÞ ¼ f2 ða sin cÞsin cdc ¼ cosða sin cÞsin cdc pa 0 pa 0 Considering the fact that we are mostly interested in the value of the DF at small values of the amplitude a, we will use the expansion of the function A(a)¼ aN2(a) that represents the first harmonic of the output signal of the nonlinearity into the Taylor series at a¼ 0 with the account of only first three terms in the series: Að0Þ ¼
4 , p
dAðaÞ ¼ 0, da a ¼ 0 d2 AðaÞ 8 ¼ da2 a ¼ 0 3p Therefore A(a)E4/p8a2/(3p) and the whole DF for the second nonlinearity can be written as N2 ðaÞ
4 8a pa 3p
ð32Þ
We can now carry out analysis of the motions in the system using the formulated dynamic harmonic balance principle. The system can be presented as the block diagram (Fig. 3), with the transfer function of the linear part given as follows: Wl ðsÞ ¼
1 ðmR2 s þ ZÞs
Fig. 3. Block diagram of rocking block dynamics.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1210
The modified frequency response can be written for Wl(s) as follows: _ ¼ Wln ðs,o, oÞ
mR2 ðs
1 þ joÞ þ Zðs þ joÞ þ j o_ 2
ð33Þ
The nonlinearity in the Lure representation of the system can be written as the sum of the two nonlinearities. Therefore, the DF of the combined nonlinearity is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 b 4 8a b2 2J1 ðaÞ N2 ðaÞmg 1 2 N1 ðaÞ ¼ mg NðaÞ ¼ mg mg 1 2 2R 2R pa 3p a 4R 4R ð34Þ Complex Eq. (25) can be rewritten as two equations: for the real and imaginary parts as follows: _ ¼0 2mR2 so þ Zo þ o
ð35Þ
mR2 s2 o2 þ Zs ¼ NðaÞ
ð36Þ
And the third algebraic equation is obtained per (26), considering that _ @lnjWln ðs,o, oÞj 2omR2 ¼ , @o mR2 ðs2 o2 Þ þ Zs
dJ1 ðaÞ J0 ðaÞJ2 ðaÞ ¼ , da 2
with J0(a) and J2(a) being the Bessel functions of zero and second order [19], respectively, and " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # @lnN 1 b 4 8a b2 2J1 ðaÞ ¼ mg þ J0 ðaÞ þ J2 ðaÞ þ mg 1 2 @lna NðaÞ 2R pa 3p a 4R as follows: s
2omR2 mR2 ðs2 o2 Þ (
¼
"
N
1
þ Zs
_ o
) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # b 4 8a b2 2J1 ðaÞ þ J0 ðaÞ þ J2 ðaÞ ðaÞ mg þ1 s þ mg 1 2 2R pa 3p a 4R ð37Þ
_ from (35) and substituting into (37), we obtain two equations with three Expressing o unknown variables. We also denote rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 4 8a b2 2J1 ðaÞ g1 ðaÞ ¼ NðaÞ ¼ mg þ mg 1 2 2R pa 3p a 4R and "
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # b 4 8a b2 2J1 ðaÞ þ J0 ðaÞ þ J2 ðaÞ g2 ðaÞ ¼ N ðaÞ mg þ1 þ mg 1 2 2R pa 3p a 4R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 1 ¼ N ðaÞmg 1 2 ðJ2 ðaÞJ0 ðaÞÞ: 4R 1
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1211
With this notation the set of two equations can be written as 8 2 2 2 > < mR ðs o Þ þ Zs ¼ g1 ðaÞ 2 2o mR2 ð2mR2 s þ ZÞ ¼ g2 ðaÞs s þ > : mR2 ðs2 o2 Þ þ Zs
ð38Þ
We shall solve (38) as a set of equations for s and o, considering amplitude a a given parameter. With this approach, if we express o from the first equation of (38) and substitute in the second equation, we arrive at one algebraic equation with one unknown variable s 2mR2 ð2mR2 s þ ZÞ s2 g1 ðaÞZs 2 mR ¼ g2 ðaÞ1, sg1 ðaÞ which can be transformed into third-order algebraic equation for s: c0 þ c1 s þ c2 s2 þ c3 s3 ¼ 0, 2
ð39Þ 2
2
2
4
where c0 ¼ 2g1(a)Z, c1 ¼ 4mR g1(a)þ2Z –g1(a)(g2(a)–1), c2 ¼ 6mR Z, c3 ¼ 4m R Solution of the cubic Eq. (39) is well known (Cardano’s formula and other methods [19]). In the considered example we can use the Cardano’s formula, which allows one to analytically find the real root of the Eq. (39). This is possible due to the fact that s is a real value. With a and s available, we can numerically integrate the differential Eq. (10). After that o is computed through the formula that was used for the substitution (based on (36)). Note: Eqs. (35)–(37) are solved at every step of the integration of (10). The transient processes of the rocking motion for the block parameters R ¼ 1, b ¼ 0.2, m ¼ 1, Z ¼ 0.4 and initial amplitude a(0) ¼ 0.05 are presented in Fig. 4 for the presented dynamic HB-based solution and simulations based on the original differential equations. One can see that the proposed approach provides a good estimate of the transient dynamics of the rocking block. In fact there is only some phase mismatch accumulated at the beginning of the transient; later this phase shift does not change.
Fig. 4. Rocking block motions (dynamic harmonic balance-based analysis and simulations).
1212
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
6. Conclusion The dynamic harmonic balance condition is formulated, and a frequency-domain approach to analysis of transient oscillatory processes is developed. The previously developed approach that was quasi-static with respect to frequency, i.e. did not account for the rate of change of the frequency, is improved providing now equations of the full dynamic harmonic balance. The proposed method is applied to analysis of the decaying motions of increasing frequency of the block alternately rocking about its two pivots. The proposed approach may find numerous applications in various areas of engineering such as in solving the problems that involve estimation of the dynamics of oscillations near the periodic solution, converging or diverging oscillations, etc. References [1] D.P. Atherton., Nonlinear Control Engineering—Describing Function Analysis and Design, Van Nostrand Company Limited, Workingham, Berks, UK, 1975. [2] A. Levant (Levantovsky, L.V.), Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (6) (1993) 1247–1263. [3] A. Levant, Universal SISO sliding-mode controllers with finite-time convergence, IEEE transactions on Automatic Control 46 (9) (2001) 1447–1451. [4] G. Bartolini, A. Ferrara, E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Transactions on Automatic Control 43 (2) (1998) 241–246. [5] G. Bartolini, A. Ferrara, A. Levant, E. Usai, On second order sliding mode controllers, in: K.D. Young, U. Ozguner (Eds.), Variable Structure Systems, Sliding Mode and Nonlinear (Lecture Notes in Control and Information Science, 247), Springer-Verlag, London, 1999, pp. 329–350. [6] Y. Shtessel, P. Kaveh, A. Ashrafi, Harmonic oscillator utilizing double-fold integral, traditional and secondorder sliding mode control, Journal of the Franklin Institute 346 (9) (2009) 872–888. [7] N. Orani, A. Pisano, E. Usai, Fault diagnosis for the vertical three-tank system via high-order sliding-mode observation, Journal of the Franklin Institute (2010). doi:10.1016/j.jfranklin.2009.11.010. [8] V.Y. Glizer, V. Turetsky, L. Fridman, J. Shinar, History-dependent modified sliding mode interception strategies with maximal capture zone, Journal of the Franklin Institute (2011). doi:10.1016/j.jfranklin. 2011.07.021. [9] L Fridman, A. Levant, Higher order sliding modes, in: J.P Barbot, W Perruguetti (Eds.), Sliding Mode Control in Engineering, Marcel Dekker, New York, 2002, pp. 53–101. [10] K. Ogata, System Dynamics, 4th ed., Prentice-Hall, 2004. [11] L. Meirovitch, Fundamentals of Vibrations, McGraw-Hill, Boston, 2001. [12] S.J. Hogan, On the dynamics of rigid block motion under harmonic forcing, Proceedings of the Royal Society of London, Series A 425 (1989) 441–476. [13] A. Gelb, W.E. Vander Velde., Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, 1968. [14] I. Boiko, ‘‘Dynamic harmonic balance and its application to analysis of convergence of second-order sliding mode control algorithms,’’ in: Proceedings of the 2011 American Control Conference, San Francisco, USA, pp. 208–213. [15] N.M. Krylov, N.N. Bogoliubov., Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1947. [16] E.P. Popov, I.P. Paltov., Approximate Methods of Analysis of Nonlinear Control Systems, Fizmatgiz, Moscow, 1960 (in Russian). [17] Z. Vukic, L. Kuljaca, D. Donlagic, S. Tesnjak., Nonlinear Control Systems, Marcel Dekker, New York, Basel, 2003. [18] I. Boiko, On frequency-domain criterion of finite-time convergence of second-order sliding mode control algorithms, Automatica 47 (9) (2011) 1969–1973. [19] G.A. Korn, T.M. Korn., Mathematical Handbook for Scientists and Engineers, 2nd ed., McGraw-Hill, New York, 1968.
Journal of the Franklin Institute 349 (2012) 1198–1212 www.elsevier.com/locate/jfranklin
Dynamic harmonic balance principle and analysis of rocking block motions Igor M. Boiko The Petroleum Institute, P.O. Box 2533, Abu Dhabi, U.A.E Received 18 July 2011; received in revised form 18 October 2011; accepted 10 January 2012 Available online 20 January 2012
Abstract The harmonic balance (HB) principle is a powerful and convenient tool for finding periodic solutions in nonlinear systems. In the present paper, this principle is extended to transient processes in systems with one single-valued odd-symmetric nonlinearity and linear plant not having zeros in the transfer function, and named the dynamic HB. Based on the dynamic HB, first the equations for the amplitude, frequency, and amplitude decay of an oscillatory process in the Lure system are derived. It is then applied to analysis of rocking block decaying motions. An example is provided. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Harmonic balance principle is a convenient tool for finding parameters of self-excited periodic motions. Due to this convenience, it is widely used in many areas of science and engineering. For a system with one nonlinearity and linear dynamics (Lure system), it can be illustrated by drawing the Nyquist plot of the linear dynamics and the plot of the negative reciprocal of the describing function (DF) [1] of the nonlinearity in the complex plane and finding the point of intersection of the two plots, which would correspond to the self-excited periodic motion in the system. Therefore, the harmonic balance principle treats the system as a loop connection of the linear dynamics and of the nonlinearity. It is also possible to reformulate the harmonic balance, so that the format of the system analyzed is not a loop connection but the denominator of the closed-loop system. This would imply a different interpretation of the harmonic balance, which would allow one to extend the harmonic balance principle to analysis of not only self-excited periodic motions but also other types of oscillatory motions. E-mail address: [email protected] 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2012.01.007
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1199
One of the types of the systems that exhibit vanishing oscillatory motion of variable frequency is the conventional and second-order sliding mode (SM) control system. There are a number of second-order SM (SOSM) algorithms available now, the most popular of which are ‘‘twisting’’, ‘‘super-twisting’’, ‘‘twisting-as-a-filter’’ [2,3], ‘‘sub-optimal’’ [4,5]. These algorithms provide finite-time convergence and because of this feature are used in both control and observation [6–8]. An overview of finite-time convergent second-order SM algorithms and algorithms with asymptotic convergence is presented in [9]. Another example of a transient oscillatory motion is the transient processes in oscillatory mechanical systems. A number of examples are given in [10,11]. The same mechanical system, as of the present paper, but in a different setup was considered in [12]. A number of examples from different areas of engineering and physics can be found where transient oscillatory motions of variable frequency occur. Therefore, some common approach to the problem of the convergence rate assessment, including qualitative (finite-time or asymptotic) and quantitative assessment, would be of high importance. The availability of the model of the system revealing transient oscillations might seem to be sufficient for analysis and not requiring development of methods other than simulations. However, the frequency-domain approach to assessment of convergence rate would provide a number of advantages over the direct solution of the system of differential equations. The most important one is the possibility of explanation of the mechanism of the frequency and amplitude change during the transient. This analysis can also lead to the formulation of the criteria of finite-time and asymptotic convergence in terms of frequency-domain characteristics. In the present paper, a frequency-domain approach to analysis of transient oscillations is presented, which is suitable for analysis of high-order nonlinear systems. The harmonic balance principle is extended to the case of transient oscillations and named as the dynamic harmonic balance principle. The paper is organized as follows. At first the harmonic balance principle is considered. Then a Lure system with a high-order plant is analyzed with the use of the dynamic harmonic balance involving quasi-static approach to the frequency of the oscillations. Such characteristics as frequency and amplitude of oscillations as functions of time are derived. After that the condition of the full dynamic harmonic balance is derived. Finally, the proposed approach is applied to analysis of the decaying motions of increasing frequency of the rocking block. Example of analysis and simulations supporting the proposed theory are provided. 2. Harmonic balance for transient oscillations Consider the system that includes linear dynamics given by the following equations: x_ ¼ Ax þ Bu y ¼ Cx,
ð1Þ
where x 2 Rn , y 2 R1 , u 2 R1 , A 2 Rnn , B 2 Rn1 , C 2 R1n , and a single-valued oddsymmetric nonlinearity f(y) with zero mean: u ¼ f ðyÞ,
ð2Þ
We shall refer to (1) as to the linear part of the system. One can see that the system (1), (2) is a Lure system. The transfer function of the linear part is Wl ðsÞ ¼ CðIsAÞ1 B, which can also be presented as a ratio of two polynomials Wl(s)¼ P(s)/Q(s). Assume also an
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1200
autonomous mode, so that the input to the nonlinearity is the output of the linear dynamics, and the output of the nonlinearity is the input to the linear dynamics. Selfexcited periodic motions in the system can be found through the use of the HB principle. The conventional HB condition (for periodic motion) is formulated as Wl ðjOÞNðaÞ ¼ 1,
ð3Þ
where O is the frequency and a is the amplitude of the self-excited periodic motion at the input to the nonlinearity, N(a) is the describing function of the nonlinearity, which does not depend on the frequency because the nonlinearity is a single-valued function. If the linear part has relative degree higher than two, then the Nyquist plot of system (1) has a point of intersection with the real axis at some finite frequency and, therefore, Eq. (3) has a solution. We now find the closed-loop transfer function Wcl(s) of system (1), (2) using the replacement of the nonlinearity with the DF u ¼ NðaÞy: Wcl ðsÞ ¼
Wl ðsÞNðaÞ PðsÞNðaÞ ¼ 1 þ Wl ðsÞNðaÞ QðsÞ þ PðsÞNðaÞ
ð4Þ
Let us note that (3) is equivalent to Rða,jOÞ ¼ QðjOÞ þ PðjOÞNðaÞ ¼ 0,
ð5Þ
which means that the denominator of the closed-loop transfer function turns into zero when the frequency and the amplitude become equal to the frequency and the amplitude of the periodic motion. Eq. (5) is also sometimes used for finding a periodic solution via algebraic methods. However, Eq. (5) usually is not attributed to the denominator of the closed-loop transfer function but considered a direct result of (3). Assuming that R(a,s) can be represented in the following form R(a,s)¼ (s–s1)(s–s2) y (s–sn), where si are roots of the characteristic polynomial, we must conclude that there must be at least one pair of complex conjugate root with zero real parts. It would imply the existence of the conservative component in Wcl(s). Indeed, we can consider the existence of non-vanishing oscillations as a result of the existence of the component (s2þr2) in the denominator of Wcl(s), where r is a parameter that depends on the amplitude a. However, one can notice that even if a damped oscillation occurs, so that there exists a pair of complex conjugate roots si, siþ1 then (ssi)(ssiþ1)¼ 0, and the characteristic polynomial becomes zero, with s¼ s7jO, where s is the decay (Note: strictly speaking, we have a decaying oscillation only if so0; yet we will refer to this variable as to the decay even if sZ0). We can view the harmonic balance condition (3) as the realization of the regeneration principle, according to which y(t)¼ L1[L[u(t)]Wl(s)], and u(t)¼ f(y(t)). We can now use the regeneration principle approach to a transient oscillation, in which we consider both u(t) and y(t) sinusoidal signals with exponentially decaying (or growing) amplitude with constant value of the decay, and formulate and prove the following property. Theorem 1. With the input signal to the linear dynamics given by the transfer function Wl(s) being the harmonic signal with decaying amplitude uðtÞ ¼ est sinðOtÞ, the output of the linear dynamics is also a harmonic signal of the same frequency and decay yðtÞ ¼ aest sinðOt þ fÞ. Proof. It follows from the property of the Laplace transform that L[eaf(t)] ¼ F(sþa). Therefore, for the system input u(t), the Laplace transform will be L[u(t)] ¼ O/[(s–s)2þO2], which will result in the system output (in the Laplace domain) Y(s)¼ OWl(s)/[(s–s)2þO2].
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1201
The substitution s0 ¼ s–s yields Y(s0 )¼ OWl(s0 þs)/[(s0 )2þO2], which means that y0 (t)¼ L1[Y(s0 )] is a sinusoid of frequency O, amplitude 9Wl(sþjO)9, and having the phase shift argWl ðs þ jOÞ. In turn, the output signal is y(t) ¼ esty0 (t), i.e. a decaying sinusoid. & Therefore, for our analysis of propagation of the decaying sinusoids through linear dynamics we can use the same transfer functions, in which the Laplace variable should be replaced with (sþjO). The describing function N in the case of a transient oscillation may become a function of not only amplitude but of its derivatives too [13] (we disregard possible dependence of the DF on the frequency). Considering that conditions (3) and (5) are equivalent, and the equality of the denominator of the closed-loop transfer function to zero (for some s) implies the fulfillment of (3), we can rewrite (3) for the transient oscillation as follows: _ . .ÞWl ðs þ jOÞ ¼ 1, Nða, a,.
ð6Þ
The use of the derivatives of the amplitude as arguments of the DF is inconvenient because it results in the necessity of consideration of additional variables (derivatives of the amplitude), which are not present otherwise. It is more convenient to consider s and its derivatives than the derivatives of the amplitude. Also, we limit our consideration of the describing function arguments to the first derivative of the amplitude (or equivalently, to s) only, which will correspond to the use of the regeneration principle for decaying sinusoids. We show below that for many nonlinearities the DF is a function of the amplitude only—like in the conventional DF analysis. Therefore, we can write the condition of the existence of a transient or steady oscillation as follows: Nða,sÞWl ðs þ jOÞ ¼ 1,
ð7Þ
where the decay and frequency are considered constant on some small interval of time (theoretically on infinitely small interval of time, and practically on one step of integration when using a numeric method), while the amplitude is considered exponentially changing on this interval (due to the constant decay). Joining solutions for these intervals, we obtain the frequency and the decay as functions of time. Therefore, we assume that the decay and the frequency are slowly or quasi-statically changing. Eq. (7) is referred to in [14] as the dynamic harmonic balance condition (equation). However, it does not account for the derivative of the frequency and we shall further refer to (7) as to the dynamic harmonic balance quasi-static w.r.t. frequency. Assume now that the characteristic polynomial of the closed-loop system (with parametric dependence on the amplitude of the oscillations) has a pair of complex conjugate roots with negative real parts. Then a vanishing oscillation of certain frequency and amplitude occurs. The idea of considering equations of vanishing oscillations is similar to the one of the Krylov–Bogoliubov method [15]. However, the latter can only deal with small ‘‘deviations’’ from the harmonic oscillator and is limited to second-order systems. In the present approach, the ‘‘equivalent damping’’ is not limited to small values. Let us consider instantaneous values of the frequency, amplitude and decay and formulate the dynamic HB quasi-static w.r.t. frequency and decay as follows. At every time, a single-frequency mode transient oscillation can be described as a process of variable (instantaneous) frequency, amplitude and decay, which must satisfy Eq. (7). Note: In (7) and the formulation given above, we consider only transient oscillations with zero mean and single-frequency mode when the characteristic polynomial (5) has only one pair of complex conjugate roots.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1202
The describing function of an arbitrary nonlinearity for a decaying amplitude harmonic input signal should be computed as suggested in [16] (see also [17]) ss , ð8Þ Nða,s,oÞ ¼ qða,s,oÞ þ q0 ða,s,oÞ o where s¼
d , dt
1 q ¼ 2p 0
q¼ Z
1 2p
Z
2p
f ½asin C,aðssin C þ o cosCÞsin CdC, 0
2p
f ½asin C,aðssin C þ o cosCÞcos CdC, 0
which accounts for the dependence on the frequency also. It should be noted that for single-valued or hysteretic symmetric nonlinearities formula (8) produces conventional describing function expressions that are commonly used for harmonic inputs. The overall motion can now be obtained from the dynamics HB as follows: yðtÞ ¼ aðtÞ esðtÞt sin CðtÞ,
ð9Þ
where a(t), s(t) are obtained from the following differential equation (s(t) is supposed to be expressed through a(t) using other equations, which is given in details below): _ ¼ aðtÞsðtÞ, aðtÞ
að0Þ ¼ a0 ,
and C(t) is the phase computed as follows: CðtÞ ¼ from (7), f is selected to satisfy initial conditions.
ð10Þ Rt 0
OðtÞdt þ f, where O(t) is obtained
3. Analysis of motions in the vicinity of a periodic solution Carry out frequency-domain analysis of the transient process of the convergence to the periodic motion in the vicinity of a periodic solution in system (1), (2), using the dynamic harmonic balance condition (7). We can write the conventional harmonic balance condition, which can also be obtained from (7) when s ¼ 0, as follows: Nða0 ÞWl ðjO0 Þ ¼ 1,
ð11Þ
where O0 and a0 are the frequency and the amplitude of the periodic solution. Write the dynamics harmonic balance condition for the increments from the periodic solution: Nða0 þ Da,sÞWl ðs þ jðO0 þ DOÞÞ ¼ 1,
ð12Þ
We now take the derivative from both sides of (12) with respect to Da (or a) in the point a¼ a0: ! ds @Nða,sÞ @Nða,sÞ ds dWl ðsÞ þ ðjO Þ þ Nða Þ ¼ 0, W 0 0 l @a a ¼ a0 @s s ¼ 0 daa ¼ a0 ds s ¼ jO0 daa ¼ a0 ð13Þ At first limit our analysis only to the nonlinearities the describing function of which does not depend on s(for example, the ideal relay nonlinearity). Later the same analysis can be applied to nonlinearities that depend on s. Express the derivative ds=daja ¼ a0
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1203
from Eq. (13):
dNðaÞ da a ¼ a Wl ðjO0 Þ ds 0 ¼ : dWl ðsÞ daa ¼ a0 Nða Þ 0
ds
ð14Þ
s ¼ jO0
Considering that s¼ sþjO, we can rewrite Eq. (14) as follows: dNðaÞ Wl ðjO0 Þ da ds dO a ¼ a0 þj ¼ : da a ¼ a0 da a ¼ a0 Nða0 ÞdWdsl ðsÞ
ð15Þ
s ¼ jO0
Eq. (15) is a complex equation. It can be split into two equations for the real and imaginary parts. However, only real parts of (15) give an equation that has a solution. Once it is solved and a(t) is found, O(t) can be found too. Considering that dlnWl ðjoÞ 1 dWl ðsÞ dlnWl ðsÞ dargWl ðjoÞ ¼ ¼ j , Wl ðsÞ ds s ¼ jO0 ds s ¼ jO0 do do o ¼ jO0 o ¼ jO0 and
dlnN~ ðaÞ 1 dNðaÞ dlnN~ ðaÞ ¼ ¼ NðaÞ da a ¼ a0 da a ¼ a0 da
a ¼ a0
dargN~ ðaÞ þj , da a ¼ a0
where N~ ðaÞ ¼ N ðaÞ, we can write for the real part of (15): 8 9 dlnjN~ ðaÞj dargN~ ðaÞ = < þ j da ds da ¼ Re , :dargWl ðjoÞ j dlnjWl ðjoÞj; da 1
do
do
which can be rewritten as follows (we skip for brevity the notation of the point in which the derivative is taken): ds ¼ da
~ dlnjN~ ðaÞj dargWl ðjoÞ dln W ðjoÞ dargdaN ðaÞ j dol j do da 2 dlnjWl ðjoÞj 2 dargWl ðjoÞ þ do do
ð16Þ
As a ‘‘side’’ product of our analysis, stability of a periodic solution can be assessed as follows: ds=daja ¼ a0 o0.The left-hand side of (16) can be rewritten as follows: ds 1 a€ a_ ¼ da a a_ a and, therefore, Eq. (16) is a second-order differential equation. Because the right-hand side of (16) contains only one unknown variable a, this equation can be numerically solved and convergence of the amplitude to the constant value can be obtained as a function of time. 4. Dynamic harmonic balance including frequency rate of change In many cases, such as analysis of motions in the vicinity of a periodic solution, the dynamic HB quasi-static w.r.t. to frequency derivative is quite capable of providing a precise result because the frequency changes insignificantly and the derivative of the
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1204
frequency can be neglected. In some other cases, such as analysis of convergence of systems with second-order sliding modes [18] and finite-time convergence, the system experiences significant changes of the instantaneous frequency of the oscillations. In this situation the quasi-static approach to the account of the oscillations frequency may result in the loss of accuracy of analysis. Therefore, inclusion of the frequency derivative would be desirable in such cases. Consider the following illustrative example. Let the linear part be the second-order dynamics and the controller be the ideal relay with the amplitude h. The describing function of the relay is N(a)¼ 4h/pa. If we assume that the input to the nonlinearity is a decaying sinusoid of constant frequency yðtÞ ¼ a0 est sinðotÞ then the control amplitude should be au ¼ aN(a), the _ _ derivative of the control amplitude a_ u ¼ @N @a aa þ N a, and the decay su ¼
4h _ 4h _ a pa a_ u 2 a þ pa a ¼ 0: ¼ 4h au pa a
The result showing that su ¼ 0 is quite predictable because the output of the relay controller is an oscillation of constant amplitude, which features zero decay. However, it is well, known (see [1,13], for example) that the output of the linear part is oscillations of decaying amplitude and growing frequency. Because the input to the linear dynamics has zero decay, we can conclude that the decay in the signal y(t) is not a result of the decay in the control u(t) but exists because of the variable frequency of u(t). Moreover, the nonlinearity may change the decay according to @N _ _ @lnN @lnN a_ u @a aa þ N a su ¼ þ 1 s, ð17Þ ¼ ¼ a_ þ s ¼ @a @lna Na au and this change is offset by the decay change due to the frequency variation at the signal propagation through the linear part. This situation is not covered by the Eq. (7) that assumes constant frequency, which requires the development of a different model. Another example is the example of the propagation of a sinusoid of nearly constant amplitude and increasing frequency through the integrator. We assume that the signal of ideally constant amplitude is 1 un ðtÞ ¼ cos o0 t þ o_ 0 t2 , 2 propagation of which through the integrator produces signal 1 1 1 n 2 _ 0 t ¼ sin C sin o0 t þ o y ðtÞ yðtÞ ¼ o0 þ o_ 0 t 2 o _ 0 is small enough). We note that the instantaneous (the approximate equality is valid if o _ 0 t2 . Therefore, the instantaneous frequency is phase of the signal is CðtÞ ¼ o0 t þ ð1=2Þo _ € ¼ oðtÞ ¼ CðtÞ ¼ o0 þ o_ 0 t, and the instantaneous derivative of the frequency is CðtÞ _ 0 ¼ const. Find the derivative of y(t) as follow: o _0 1 1 o o_ 0 2 2 _ _ _ ¼ cos o0 t þ o 0 t þ sin o0 t þ o 0 t ¼ cos C þ 2 sin C yðtÞ 2 2 2 o _ ðo0 þ o 0 tÞ Therefore, the integrator introduces the phase lag less than 901 and the equations of the integrator can be represented by the conventional integrator having an additional feedback, which accounts for the effect of the variable frequency.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1205
Therefore, the frequency response of the integrator to a decaying sinusoid of variable frequency is W ðs þ jo, o_ 0 Þ ¼
1 sþ
o_ 0 o2
þ jo
:
_ 0 =o2 then the phase lag introduced by the From this formula, one can see that if s ¼ o _ 0 Þ is 901. The amplitude characteristic of the integrator is integrator f ¼ argW ðs þ jo, o given by M ¼ jW ðs þ jo, o_ 0 Þj, and the decay introduced by the integrator due to variable _ ¼ o_ 0 =o (provided the amplitude of y(t) is a ¼ 1=o0 þ o_ 0 t). frequency is sðtÞ ¼ a=a To analyze propagation of a decaying sinusoid of variable frequency through higherorder linear dynamics we shall use the notion of the rotating phasor. Let the output of the linear dynamics y(t) be given by yðtÞ ¼ aðtÞejCðtÞ ¼ elnaðtÞ ejCðtÞ ¼ elnaðtÞþjCðtÞ ,
ð18Þ
where a represents the length of the phasor and C represents the angle between the phasor and the real axis. We can associate either the real or the imaginary part of yðtÞ with the real signal y(t). We assume that the transfer function of the linear part does not have any zeros: Wl(s)¼ 1/Q(s)¼ 1/(a0þa1sþa2s2þyþansn). In the present paper we limit our analysis to plants not containing zeros in the transfer function. Given this transfer function, we can write for y(t): u ¼ a0 y þ a1 y_ þ a2 y€ þ . . . þ an yðnÞ
ð19Þ
We can find the derivatives of y(t) as follows: _ ¼ elnaðtÞþjCðtÞ 1 a_ þ j C _ _ ¼ elnaðtÞþjCðtÞ dlna þ j C yðtÞ dt a ¼ elnaðtÞþjCðtÞ ðs þ joÞ ¼ ðs þ joÞy,
ð20Þ
€ ¼ ðs_ þ j oÞy _ þ ðs þ joÞy_ ¼ ½ðs_ þ j oÞ _ þ ðs þ joÞ2 y yðtÞ Considering that we disregard all derivatives of the decay and higher than first derivatives of the frequency in our model we can rewrite the last formula as follows: € ½ðs þ joÞ2 þ j oy _ yðtÞ
ð21Þ
and the
formula for the third and forth derivatives as follows: & _ þ j oy € þ ½ðs þ joÞ2 þ j oðs _ yðtÞ ½2ðs þ joÞðs_ þ j oÞ þ joÞy 2 3 _ þ joÞy _ þ ðs þ joÞ ðs þ joÞy ¼ ½ðs þ joÞ þ j3oðs ½j3o
ð22Þ
_ þ joÞ3o _ 2 y yð4Þ ðtÞ ½ðs þ joÞ4 þ j6oðs We can continue with taking further derivatives. It is worth noting that the formulas above are organized the way that they have a term (sþjo) to the respective power and the term which is the product of o_ and another multiplier. Therefore, we can write for y(t): _ u ¼ ½a0 þ a1 ðs þ joÞ þ a2 ðs þ joÞ2 þ . . . þ an ðs þ joÞn y þ Sðs,o, oÞy,
ð23Þ
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1206
_ includes all terms containing o. _ This component can be accounted for as where Sðs,o, oÞ an additional feedback—in the same way as it was done for the integrator. If we introduce certain modified frequency response as _ ¼ Wln ðs,o, oÞ
y 1 Wl ðs þ joÞ ¼ ¼ _ _ Qðs þ joÞ þ Sðs,o, oÞ 1 þ Wl ðs þ joÞSðs,o, oÞ u
ð24Þ
we can write the dynamic harmonic balance equation as follows: _ ¼ 1 NðaÞWln ðs,o, oÞ
ð25Þ
Obviously, Eq. (25) can be split into two equations: for real and imaginary parts, or for equations of the magnitude balance and phase balance. Eq. (25) must be complemented with an equation that relates the difference of the decays at the input and the output of the linear part and the frequency rate of change—in the same way as it was done in the example of the integrator analysis. We note that the _ and, amplitude of the control (first harmonic) is as follows au ¼ ajQðs þ joÞ þ Sðs,o, oÞj, therefore, its time derivative is _ dQðs þ joÞ þ Sðs,o, oÞ _ þa _ : a_ u ¼ ajQðs þ joÞ þ Sðs,o, oÞj dt The decay of signal u(t) is computed as _
oÞj _ þ a djQðsþjoÞþSðs,o, _ ajQðs þ joÞ þ Sðs,o, oÞj a_ u dt ¼ _ au ajQðs þ joÞ þ Sðs,o, oÞj _ dlnjQðs þ joÞ þ Sðs,o, oÞj ¼sþ dt € we can rewrite the last formula as follows: Because we disregard s_ and o
su ¼
su ¼ s þ
_ _ @lnjWln ðs,o, oÞj @lnjQðs þ joÞ þ Sðs,o, oÞj _ o_ ¼ s o @o @o
The derivative in the last formula defines the slope of the magnitude-frequency _ Considering also the formula for the decay of u(t) derived characteristic of Wln ðs,o, oÞ. above through the DF, we can now write the condition of the balance of the decays in the closed-loop system as follows: _ o _ @lnjWln ðs,o, oÞj @lnNðaÞ s ¼ @lna @lno o
ð26Þ
We can formulate the dynamic harmonic balance principle as theorem. At every time during a single-frequency mode transient oscillation, the oscillation can be described as a process of variable instantaneous frequency, amplitude, decay and frequency rate of change (time derivative), which must satisfy Eqs. (25) and (26). It is worth noting that _ @lnjWln ðs,o, oÞj dMðoÞ , ¼ 0:05 dlogo @lno where M(o) is the magnitude frequency response [dB] (Bode plot) for the transfer function _ Therefore, this term gives the slope of the Bode plot. Both o=o _ _ Wln ðs,o, oÞ. and s ¼ a=a are relative rates of change of the frequency and the amplitude, respectively. Eq. (26),
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1207
therefore, is establishing the balance between those two rates of change. The use of the dynamic HB principle is illustrated by the following example. Overall solution is obtained via integration of the first-order differential Eq. (10) with s expressed through a using the three algebraic equations presented above. 5. Analysis of decaying oscillations of rocking block Consider planar rocking motion of the 2-dimensional block (Fig. 1). The block of width c and height b and the base on which the block rests are assumed to be rigid. The possible considered motion includes only rocking (we assume that sliding and liftoff do not take place). However, we consider that friction is present in the form of a viscous friction, so that the resulting motion is alternating rocking around pivots A and B decaying due to the energy dissipation. Rocking around pivot point A can be described by the following equation: mR2 g€ A þ Z_g A mgsin gA ¼ 0, where jgA joarcsinb=2R, gA is the angle between the line connecting pivot A and the center of mass of the block and the vertical line. And rocking around pivot B is given by: mR2 g€ B þ Z_g B mgsin gB ¼ 0, where jgB joarcsinb=2R, gB is the angle between the line connecting pivot B and the center of mass of the block and the vertical line. Introduce the variable x that describes the angular position of the block in both motions, which is the angle between the vertical axis of the block and the true vertical line: x ¼ arcsin
b b g ¼ gA arcsin 2R B 2R
Fig. 1. Rocking block.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1208
Now the motions are described as follows. Rocking around pivot A is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 b 2 _ cosx, if xo0, mR x€ þ Zxmg 1 2 sin x ¼ mg 2R 4R And rocking around pivot B is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 b 2 _ mR x€ þ Zxmg cos x, 1 2 sin x ¼ mg 2R 4R
if x40
ð27Þ
ð28Þ
Attributing the right-hand side of (27) and (28) to control, we can rewrite (27) and (28) as an equation of a system with a discontinuous control: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 _ mR x€ þ Zxmg ð29Þ 1 2 sin x ¼ u 4R 8 b > > cosx if xo0 < mg 2R u¼ b > > cos x if x40 : mg 2R One can see that the control can be presented as u ¼ mg
b sgnx cosx 2R
ð30Þ
We now carry out analysis of motions in the system using the dynamic harmonic balance principle introduced above. We note that the system has two nonlinearities: f1 ðxÞ ¼ sin x, and f2 ðxÞ ¼ sgn x cos x. Both nonlinearities are odd-symmetric. The second nonlinearity is depicted in Fig. 2.
Fig. 2. Second nonlinearity.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1209
Find the describing functions for these nonlinearities. For the first nonlinearity the DF is given by [1] N1 ðaÞ ¼ 2J1 ðaÞ=a,
ð31Þ
where J1(a) is the Bessel function of the first order. We shall now derive the DF for the second nonlinearity. According to the DF definition: Z Z 4 p=2 4 p=2 N2 ðaÞ ¼ f2 ða sin cÞsin cdc ¼ cosða sin cÞsin cdc pa 0 pa 0 Considering the fact that we are mostly interested in the value of the DF at small values of the amplitude a, we will use the expansion of the function A(a)¼ aN2(a) that represents the first harmonic of the output signal of the nonlinearity into the Taylor series at a¼ 0 with the account of only first three terms in the series: Að0Þ ¼
4 , p
dAðaÞ ¼ 0, da a ¼ 0 d2 AðaÞ 8 ¼ da2 a ¼ 0 3p Therefore A(a)E4/p8a2/(3p) and the whole DF for the second nonlinearity can be written as N2 ðaÞ
4 8a pa 3p
ð32Þ
We can now carry out analysis of the motions in the system using the formulated dynamic harmonic balance principle. The system can be presented as the block diagram (Fig. 3), with the transfer function of the linear part given as follows: Wl ðsÞ ¼
1 ðmR2 s þ ZÞs
Fig. 3. Block diagram of rocking block dynamics.
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1210
The modified frequency response can be written for Wl(s) as follows: _ ¼ Wln ðs,o, oÞ
mR2 ðs
1 þ joÞ þ Zðs þ joÞ þ j o_ 2
ð33Þ
The nonlinearity in the Lure representation of the system can be written as the sum of the two nonlinearities. Therefore, the DF of the combined nonlinearity is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 b 4 8a b2 2J1 ðaÞ N2 ðaÞmg 1 2 N1 ðaÞ ¼ mg NðaÞ ¼ mg mg 1 2 2R 2R pa 3p a 4R 4R ð34Þ Complex Eq. (25) can be rewritten as two equations: for the real and imaginary parts as follows: _ ¼0 2mR2 so þ Zo þ o
ð35Þ
mR2 s2 o2 þ Zs ¼ NðaÞ
ð36Þ
And the third algebraic equation is obtained per (26), considering that _ @lnjWln ðs,o, oÞj 2omR2 ¼ , @o mR2 ðs2 o2 Þ þ Zs
dJ1 ðaÞ J0 ðaÞJ2 ðaÞ ¼ , da 2
with J0(a) and J2(a) being the Bessel functions of zero and second order [19], respectively, and " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # @lnN 1 b 4 8a b2 2J1 ðaÞ ¼ mg þ J0 ðaÞ þ J2 ðaÞ þ mg 1 2 @lna NðaÞ 2R pa 3p a 4R as follows: s
2omR2 mR2 ðs2 o2 Þ (
¼
"
N
1
þ Zs
_ o
) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # b 4 8a b2 2J1 ðaÞ þ J0 ðaÞ þ J2 ðaÞ ðaÞ mg þ1 s þ mg 1 2 2R pa 3p a 4R ð37Þ
_ from (35) and substituting into (37), we obtain two equations with three Expressing o unknown variables. We also denote rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 4 8a b2 2J1 ðaÞ g1 ðaÞ ¼ NðaÞ ¼ mg þ mg 1 2 2R pa 3p a 4R and "
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # b 4 8a b2 2J1 ðaÞ þ J0 ðaÞ þ J2 ðaÞ g2 ðaÞ ¼ N ðaÞ mg þ1 þ mg 1 2 2R pa 3p a 4R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 1 ¼ N ðaÞmg 1 2 ðJ2 ðaÞJ0 ðaÞÞ: 4R 1
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
1211
With this notation the set of two equations can be written as 8 2 2 2 > < mR ðs o Þ þ Zs ¼ g1 ðaÞ 2 2o mR2 ð2mR2 s þ ZÞ ¼ g2 ðaÞs s þ > : mR2 ðs2 o2 Þ þ Zs
ð38Þ
We shall solve (38) as a set of equations for s and o, considering amplitude a a given parameter. With this approach, if we express o from the first equation of (38) and substitute in the second equation, we arrive at one algebraic equation with one unknown variable s 2mR2 ð2mR2 s þ ZÞ s2 g1 ðaÞZs 2 mR ¼ g2 ðaÞ1, sg1 ðaÞ which can be transformed into third-order algebraic equation for s: c0 þ c1 s þ c2 s2 þ c3 s3 ¼ 0, 2
ð39Þ 2
2
2
4
where c0 ¼ 2g1(a)Z, c1 ¼ 4mR g1(a)þ2Z –g1(a)(g2(a)–1), c2 ¼ 6mR Z, c3 ¼ 4m R Solution of the cubic Eq. (39) is well known (Cardano’s formula and other methods [19]). In the considered example we can use the Cardano’s formula, which allows one to analytically find the real root of the Eq. (39). This is possible due to the fact that s is a real value. With a and s available, we can numerically integrate the differential Eq. (10). After that o is computed through the formula that was used for the substitution (based on (36)). Note: Eqs. (35)–(37) are solved at every step of the integration of (10). The transient processes of the rocking motion for the block parameters R ¼ 1, b ¼ 0.2, m ¼ 1, Z ¼ 0.4 and initial amplitude a(0) ¼ 0.05 are presented in Fig. 4 for the presented dynamic HB-based solution and simulations based on the original differential equations. One can see that the proposed approach provides a good estimate of the transient dynamics of the rocking block. In fact there is only some phase mismatch accumulated at the beginning of the transient; later this phase shift does not change.
Fig. 4. Rocking block motions (dynamic harmonic balance-based analysis and simulations).
1212
I.M. Boiko / Journal of the Franklin Institute 349 (2012) 1198–1212
6. Conclusion The dynamic harmonic balance condition is formulated, and a frequency-domain approach to analysis of transient oscillatory processes is developed. The previously developed approach that was quasi-static with respect to frequency, i.e. did not account for the rate of change of the frequency, is improved providing now equations of the full dynamic harmonic balance. The proposed method is applied to analysis of the decaying motions of increasing frequency of the block alternately rocking about its two pivots. The proposed approach may find numerous applications in various areas of engineering such as in solving the problems that involve estimation of the dynamics of oscillations near the periodic solution, converging or diverging oscillations, etc. References [1] D.P. Atherton., Nonlinear Control Engineering—Describing Function Analysis and Design, Van Nostrand Company Limited, Workingham, Berks, UK, 1975. [2] A. Levant (Levantovsky, L.V.), Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (6) (1993) 1247–1263. [3] A. Levant, Universal SISO sliding-mode controllers with finite-time convergence, IEEE transactions on Automatic Control 46 (9) (2001) 1447–1451. [4] G. Bartolini, A. Ferrara, E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Transactions on Automatic Control 43 (2) (1998) 241–246. [5] G. Bartolini, A. Ferrara, A. Levant, E. Usai, On second order sliding mode controllers, in: K.D. Young, U. Ozguner (Eds.), Variable Structure Systems, Sliding Mode and Nonlinear (Lecture Notes in Control and Information Science, 247), Springer-Verlag, London, 1999, pp. 329–350. [6] Y. Shtessel, P. Kaveh, A. Ashrafi, Harmonic oscillator utilizing double-fold integral, traditional and secondorder sliding mode control, Journal of the Franklin Institute 346 (9) (2009) 872–888. [7] N. Orani, A. Pisano, E. Usai, Fault diagnosis for the vertical three-tank system via high-order sliding-mode observation, Journal of the Franklin Institute (2010). doi:10.1016/j.jfranklin.2009.11.010. [8] V.Y. Glizer, V. Turetsky, L. Fridman, J. Shinar, History-dependent modified sliding mode interception strategies with maximal capture zone, Journal of the Franklin Institute (2011). doi:10.1016/j.jfranklin. 2011.07.021. [9] L Fridman, A. Levant, Higher order sliding modes, in: J.P Barbot, W Perruguetti (Eds.), Sliding Mode Control in Engineering, Marcel Dekker, New York, 2002, pp. 53–101. [10] K. Ogata, System Dynamics, 4th ed., Prentice-Hall, 2004. [11] L. Meirovitch, Fundamentals of Vibrations, McGraw-Hill, Boston, 2001. [12] S.J. Hogan, On the dynamics of rigid block motion under harmonic forcing, Proceedings of the Royal Society of London, Series A 425 (1989) 441–476. [13] A. Gelb, W.E. Vander Velde., Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, 1968. [14] I. Boiko, ‘‘Dynamic harmonic balance and its application to analysis of convergence of second-order sliding mode control algorithms,’’ in: Proceedings of the 2011 American Control Conference, San Francisco, USA, pp. 208–213. [15] N.M. Krylov, N.N. Bogoliubov., Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1947. [16] E.P. Popov, I.P. Paltov., Approximate Methods of Analysis of Nonlinear Control Systems, Fizmatgiz, Moscow, 1960 (in Russian). [17] Z. Vukic, L. Kuljaca, D. Donlagic, S. Tesnjak., Nonlinear Control Systems, Marcel Dekker, New York, Basel, 2003. [18] I. Boiko, On frequency-domain criterion of finite-time convergence of second-order sliding mode control algorithms, Automatica 47 (9) (2011) 1969–1973. [19] G.A. Korn, T.M. Korn., Mathematical Handbook for Scientists and Engineers, 2nd ed., McGraw-Hill, New York, 1968.