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Bifurcation Analysis and Chaos Control of a Fractional Order Portal Frame with Nonideal Loading Using Adaptive Sliding Mode Control.

1. Introduction

Many recent researches have been dealing with the study of nonideal vibration systems when the excitation has an influence from the system response. These kind of nonideal vibrations systems are considered as major challenge in engineering research [1, 2]. Chaotic vibrations of a portal frame system with nonideal loading and its control are presented by Tusset et al. [3]. Early researches on nonlinear vibration of frames under support motion do not consider the nonlinear elastic forces [4]. After two decades Brasil and Mazzilli studied the framed machine foundation considering both initial and elastic nonlinear effects including stiffness of columns and elastic deformations of the frames [5]. A simple portal frame structure with its nonlinear behavior under internal resonance conditions is analyzed for several energy levels [6].

Optimal linear feedback control for suppression chaotic oscillations was proposed by Rafikov et al., [7]. Nonlinear oscillations of a portal frame excited by a nonideal motor with limited power output are considered and, with slow increase of power levels, the possibilities of occurrence of Sommerfeld effect are investigated [8]. Dynamical behavior of an elastic nonideal portal frame with fractional nonlinearities taken into account the full interactions of the vibration systems with an energy source of limited supply is investigated [9]. A nonlinear control method based on the saturation phenomenon ofsystems coupled with quadratic nonlinearities in a shear building portal frame is studied [10]. A portal frame structure based on energy harvesters with piezo electric coupling exhibiting chaotic behavior is investigated and an optimal control scheme to regulate the energy captured to a designed operating frequency is preferred [11].

Recently many researchers have discussed fractional order calculus and its applications [12-14]. Fractional order nonlinear systems with different control approaches are investigated [15-17]. Numerical analysis and methods for simulating fractional order nonlinear system are proposed by Petras [18] and MATLAB solutions for fractional order chaotic systems, discussed by Trzaska Zdzislaw [19]. A fractional order control based approach is proposed for piezo-actuated nanopositioning stage to suppress the vibration of the low-damped resonant mode and also to minimize the tracking error for nanopositioning applications and it is proposed [20]. A fractional order proportional-integral (FOPI) controller for a mass-spring-damper system which is poorly damped is proposed and it is proved that FOPI controllers are effective compared to the classical PI controllers [21]. A fractional order (FO) controller is proposed for solving the vibration suppression problem in civil structures experimented on a laboratory scaled steel structure, with one floor, modeled as a single degree-of-freedom system proposed and investigated [22]. The comparisons of the step responses of the integer order and the three types of fractional order damping systems are studied and it is shown that the optimal fractional order damping systems achieve much better step responses than optimal integer order systems [23]. The stability of fractional order systems using Lyapunov stability theory has been investigated in the literature [24, 25]. A fractional order controller to stabilize the unstable fixed points of an unstable open-loop system was proposed by Tavazoei and Haeri [26]. A bifurcation diagram shows the long term qualitative changes (equilibria or periodic orbits) of a system as a function of a bifurcation parameters of the system. The complete dynamics of the system with the variation of the parameters can be studied with the help of bifurcation diagram [27-29]. Nonlinear dynamical system undergoes abrupt qualitative changes when crossing bifurcation points [30]. For a more exhaustive qualitative analysis of a nonlinear dynamic system, it is compulsory to identify both singularities of the parameter plane and singularities of the phase plane [31].

Implementation of chaotic and hyperchaotic system using Field Programmable Gate Arrays (FPGA) is widely investigated [32-34]. Chaotic random number generators are implemented in FPGA for applications in image cryptography [35]. FPGA implemented Duffing oscillator based signal detectors are proposed by Rashtchi and Nourazar [36]. Digital implementation of chaotic multiscroll attractors is extensively investigated [32, 37]. Memristor based chaotic system and its FPGA circuits are proposed by Xu et al. [38]. A FPGA implementation of fractional order chaotic system using approximation method is investigated by Rajagopal et al. [39, 40].

Motivated by the above, in this paper we investigate the fractional order model of a portal frame (FOPF) under a nonideal excitation. Bifurcation plots of the FOPF system are investigated. To study the quadratic nonlinearities in the FOPF system, bicoherence contours along with the higher dimension power spectrum are presented. An adaptive sliding mode controller is designed to suppress chaotic oscillations and finally the proposed controller is implemented in FPGA for real-time implementation.

2. Preliminaries and Problem Formulation

We consider the horizontal motion of a portal frame with nonideal excitation [3] as described in Figure 1(a). The portal frame shown in Figure 1(a) is approximated with coupled oscillators [3] as shown in Figure 1(b).

As described in [3], the parameters of this dynamical system consist of [m.sub.0], [m.sub.1], the mass and unbalanced mass, [k.sub.l], [], linear and nonlinear stiffness, [c.sub.1], linear damping, [x.sub.1], [phi], displacement and angular displacement, J, inertial moment, and r, eccentricity of the unbalanced mass. d is related to the voltage applied across the armature of the DC motor and s is a constant for each model of the DC motor considered. The resulting mathematical model of the structure is given by

[mathematical expression not reproducible]. (1)

Using [tau] = [omega]t, x = x/[x.sup.*], and z = f/([[phi].sup.*], the dimensionless form of (1) can be derived as,

[mathematical expression not reproducible], (2)

[mathematical expression not reproducible], (3)

System (2) shows chaotic oscillations when a = 0.1, b = 1, c = 2, d = 8.473 with initial conditions [0.1,0.1,0.1,0.1]. Figure 2 shows the 3D phase portraits of system (2).

3. Fractional Order Portal Frame (FOPF)

In this section we derive the fractional order model of the portal frame (FOPF) from the integer order dimensionless model discussed in (2). There are three commonly used definitions of the fractional order differential operator, namely, Grunwald-Letnikov, Riemann-Liouville, and Caputo [1214]. We use the fractional order system derived from the portal frame system (1) with the Grunwald-Letnikov (GL) definition, which is defined as

[mathematical expression not reproducible], (4)

where a and t are limits of the fractional order equation, Aqhf(t) is generalized difference, h is the step size, and q is the fractional order of the differential equation.

For numerical calculations the above equation is modified as

[mathematical expression not reproducible]. (5)

Theoretically fractional order differential equations use infinite memory. Hence when we want to numerically calculate or simulate the fractional order equations we have to use finite memory principal, where L is the memory length and h is the time sampling.

N(t) = min {[t/h], [L/h]}. (6)

The binomial coefficients required for the numerical simulation are calculated as

[b.sub.j] = (1 - a + q/j) [b.sub.j-1]. (7)

Using the relations (4) and (5), the fractional order portal frame dimensionless model can be derived as

[mathematical expression not reproducible]. (8)

The parameter values are a = 0.1, b = 1, c = 2, d = 8.473 and the initial conditions are [0.1, 0.1, 0.1, 0.1]. The system shows its largest Lyapunov exponent (0.08138) when q = 0.998 against its integer order Lyapunov exponent of 0.075 [3], confirming that the chaotic oscillations are more in fractional order close to 1 compared to the integer order. Figure 3 shows the 3D phase portraits of the FOPF system.

4. Dynamic Analysis of FOPF System

4.1. Bifurcation Analysis with Parameters. The FOPF system described in (8) has four parameters a, b, c, d and to study the impact of these parameters of the FOPF system, we derive the bifurcation plots. Figure 4 shows the bifurcation plots of FOPF system for the parameter a. As can be seen from Figure 4, the FOPF system shows multiple chaotic regions for a. There exists a wide band chaotic region for 0 [less than or equal to] a [less than or equal to] 0.12 and the systems maximum Lyapunov exponent (0.08138) is shown when a = 0.1. The system shows quasi-periodic state when positive Lyapunov exponent becomes zero for 0.12 [less than or equal to] a [less than or equal to] 0.14. The second chaotic region of the FOPF system is seen for 0.14 [less than or equal to] a [less than or equal to] 0.18 and the Lyapunov exponent of the system lies between [0.0317,0.0453]. For a narrow band of 0.18 [less than or equal to] a [less than or equal to] 0.19 the FOPF systems show multiple stable limit cycles attracting the neighboring trajectories exhibiting self-sustained oscillations. There exists the third chaotic region for 0.21 [less than or equal to] a [less than or equal to] 0.25 and the FOPF system takes period halving route for 0.25 [less than or equal to] a [less than or equal to] 0.27 to exit chaotic oscillations.

Figure 5 shows the bifurcation of the FOPF system for b. The FOPF system takes a routine period doubling route to chaos and shows multiple period doubling for 0.15 [less than or equal to] b < 0.25. Figure 6 shows the bifurcation of FOPF system with parameter c. The FOPF system shows multiple chaotic regions for c. For 0.2 [less than or equal to] c [less than or equal to] 0.3 the FOPF systems show a chaotic region. Stable and unstable limit cycles are seen for 0.41 [less than or equal to] c [less than or equal to] 0.63 and 0.63 [less than or equal to] c [less than or equal to] 1.27, respectively. The system takes a routine period doubling rote to chaos for 1.35 [less than or equal to] c [less than or equal to] 1.57 and 1.63 [less than or equal to] c [less than or equal to] 1.94. An arrow band of quasi-chaotic state exists for 1.94 [less than or equal to] c [less than or equal to] 1.97 where the systems only positive Lyapunov exponent becomes zero. The third chaotic region is seen for 1.97 [less than or equal to] c [less than or equal to] 2.4 and the systems maximum Lyapunov exponent (0.08138) exists when c = 2.

Figure 7 shows the bifurcation of FOPF system with d. The system enters in to chaos with period doubling for 6.50 [less than or equal to] d [less than or equal to] 6.92. The first chaotic region exists for 6.92 [less than or equal to] d [less than or equal to] 7.4. Then the system enters in to the second chaotic region (7.6 [less than or equal to] d [less than or equal to] 8.37) through period doubling (7.4 [less than or equal to] d [less than or equal to] 7.6). There exists a small band of quasi-chaotic region for 8.37 [less than or equal to] d [less than or equal to] 8.39 when the Lyapunov exponent goes to zero. The third chaotic region exists for 8.39 [less than or equal to] d [less than or equal to] 9 and the FOPF systems largest Lyapunov exponent (0.08138) is seen when d = 8.473.

4.2. Bifurcation Analysis with Fractional Order. The bifurcation of the FOPF system with fractional orders is another important topic of investigation. Figure 8 shows the bifurcation plots of the FOPF system for various fractional orders. The FOPF system shows chaotic oscillations for the commensurate orders qt > 0.99 and the largest positive Lyapunov exponent (0.08138) of the nonideal portal frame system exists in the fractional order q = 0.998 against the integer order Lyapunov exponent of 0.075 [3]. This clearly confirms that fractional order chaos control is effective compared to the integer order control as discussed in [23].

4.3. Bicoherence. Higher order spectra have been used to study the nonlinear interactions between frequency modes [41]. Let x(t) be a stationary random process defined as

[mathematical expression not reproducible], (9)

where [omega] is the angular frequency, n is the frequency modal index, and [A.sub.n] are the complex Fourier coefficients. The power spectrum can be defined as

[mathematical expression not reproducible] (10)

and discrete bispectrum can be defined as

[mathematical expression not reproducible]. (11)

If the modes are independent then the average triple products of Fourier components are zero resulting in a zero bispectrum [41]. The study of bicoherence is to give an indication of the relative degree of phase coupling between triads of frequency components. The motivation to study the bicoherence is twofold. First, the bicoherence can be used to extract information due to deviations from Gaussianity and suppress additive (colored) Gaussian noise. Second, the bicoherence can be used to detect and characterize asymmetric nonlinearity in signals via quadratic phase coupling or identify systems with quadratic nonlinearity. The bicoherence is the third-order spectrum. Whereas the power spectrum is second-order statistics, formed from X'(f) * X(f), where X(f) is the Fourier transform of x(t), the bispectrum is third-order statistics formed from X([f.sub.j]) * X([f.sub.k]) * X'([f.sub.j] + [f.sub.k]). The bispectrum is therefore a function of a pair of frequencies ([f.sub.j], [f.sub.k]). It is also a complex-valued function. The (normalized) square amplitude is called the bicoherence (by analogy with the coherence from the cross-spectrum). The bispectrum is calculated by dividing the time series into M segments of length N_seg, calculating their Fourier transforms and biperiodogram and then averaging over the ensemble. Although the bicoherence is a function of two frequencies the default output of this function is a one-dimensional output, the bicoherence refined as a function of only the sum of the two frequencies. The autobispectrum of a chaotic system is given by Pezeshki [42]. He derived the autobispectrum with the Fourier coefficients.

B ([[omega].sub.1][[omega].sub.2]) = E[A([[omega].sub.1])A([[omega].sub.2])[A.sup.*]([[omega].sub.1] + [[omega].sub.2])], (12)

where ton is the radian frequency and A are the Fourier coefficients of the time series. The normalized magnitude spectrum of the bispectrum known as the squared bicoherence is given by

b([[omega].sub.1], [[omega].sub.2]) = [[absolute value of (B([[omega].sub.1], [[omega].sub.2])].sup.2]/P([[omega].sub.1])P([[omega].sub.2])P([[omega].sub.1] + [[omega].sub.2]), (13)

where P([[omega].sub.1]) and P([[omega].sub.2]) are the power spectra at [f.sub.1] and [f.sub.2].

Figures 9(a) and 9(b) show the bicoherence contours of the FOPF system for state x and all states together, respectively. Shades in yellow represent the multifrequency components contributing to the power spectrum. From Figures 9(a) and 9(b) the cross-bicoherence is significantly nonzero and nonconstant, indicating a nonlinear relationship between the states. As can be seen from Figure 9(a), the spectral power is very low as compared to the spectral power of all states together (Figure 9(b)) indicating the existence of multifrequency nodes. Also Figure 9(b) shows the nonlinear coupling (straight lines connecting multiple frequency terms) between the states. The yellow shades/lines and nonsharpness of the peaks, as well as the presence of structure around the origin in figures (cross-bicoherence), indicate that the nonlinearity between the states x, y, z, w is not of the quadratic nonlinearity and hence maybe because of nonlinearity of higher dimensions. The most two dominant frequencies ([f.sub.1], [f.sub.2]) are taken for deriving the contour of bicoherence. The sampling frequency ([f.sub.s]) is taken as the reference frequency. Direct FFT is used to derive the power spectrum for individual frequencies and Hankel operator is used as the frequency mask. Hanning window is used as the FIR filter to separate the frequencies [40].

5. Fractional Order Adaptive Sliding Mode Control (FOASMC)

In this section we derive the fractional order adaptive sliding mode controllers for suppressing the chaotic oscillations in the FOPF system. As discussed in [3], it is sufficient to control states x and y; hence we include two controllers ux and uy. To include uncertainties, we assume the system parameters (a, b, c, d) are unknown. We redefine the FOPF system with the FOASMC controllers as

[mathematical expression not reproducible], (14)

where [u.sub.i] is the adaptive sliding mode controller with i = x, y The sliding surfaces [39, 40, 43] are defined as

[s.sub.x] = x + [k.sup.x] [[integral].sup.t.sub.0] x([tau])d[tau], [s.sub.y] = y + [k.sub.y] [[integral].sup.t.sub.0] y([tau]) d[tau] (15)

The Fractional derivative of sliding surfaces (15) is given by

[mathematical expression not reproducible]. (16)

To include uncertainties in the FOPF system, the parameters a, b, c, d are assumed unknown and hence the parameter estimation errors are defined as

[mathematical expression not reproducible]. (17)

The fractional derivative of the parameter estimation errors (17) is

[mathematical expression not reproducible]. (18)

We define the Lyapunov candidate function

[mathematical expression not reproducible] (19)

The first derivative of (19) is

[mathematical expression not reproducible]. (20)

By definition of fractional calculus [12-14],

[??](t) = [D.sup.1-q.sub.t] x [D.sup.q.sub.t]x(t). (21)

Using (21) in (20),

[mathematical expression not reproducible], (22)

where i = x, y and q is the commensurate order of the system. Finding the sign of the Lyapunov first derivative using (22) seems difficult and hence we use the modified fractional order Lyapunov method defined by Rajagopal et al. [39, 40, 44] as

1/2 [D.sup.q.sub.t][x.sup.2] (t) [less than or equal to] x(t) 1/2 [D.sup.q.sub.t](t), q [member of] (0,1). (23)

Using (14), (16), (18), and (23) in (20)

[mathematical expression not reproducible]. (24)

The adaptive sliding mode controllers are defined as

[mathematical expression not reproducible], (25)

where [[eta].sub.i], [[rho].sub.i] and [k.sub.i] are positive constants for i = x, y. The parameter estimation laws are derived as

[mathematical expression not reproducible]. (26)

Using (25) and (26) in (24), we simplify the Lyapunov candidate function as

[??] [less than or equal to] -[[eta].sub.x] [absolute value of ([s.sub.x])] - [[eta].sub.y][absolute value of ([s.sub.y])] - [[rho].sub.x][s.sup.2.sub.x] - [[rho].sub.y][s.sup.2.sub.y] (27)

as [[eta].sub.i] and [[rho].sub.i] are all positive for i = x, y; [??] is negative definite. Using Barbalat's lemma [45], we conclude that [e.sub.i](t) [right arrow] 0 as t [right arrow] [infinity]. Figures 10 and 11 show the time history of the states (x, y) and parameter estimates, respectively. The controller is switched on at time t = 70 s and the initial conditions of the FOPF system are taken as [0.1, 0.02, 0.3, 0.04] and parameter estimates as [1, 4, 5, 6].

6. FPGA Implementation of the FOPF System

For numerically simulating the FOPF control scheme, we first implement the FOPF model in FPGA [39,40] using the Xilinx (Vivado) System Generator toolbox in Simulink. Firstly we configure the available built in blocks of the System Generator toolbox. The Add/Sub blocks are configured with zero latency and 32/16 bit fixed point settings. The output of the block is configured to rounded quantization in order to reduce the bit latency. For the multiplier block a latency of 1 is configured and the other settings are same as in Add/Sub block. Next we will have to design the fractional order integrator which is not a readily available block in the System Generator [39,40]. Hence we implement the integrators using the mathematical relation discussed in Section 3 and the value of h is taken as 0.001 and the initial conditions are fed in to the forward register with fractional order taken as q = 0.998 for FOPF system. Figure 12 shows the Xilinx RTL schematics of the FOPF system implemented in Kintex 7 (Device = 7k160t Package = fbg484 S) and Figure 13 shows the 3D phase portraits of the FOPF system implemented in FPGA. Here we used a sampling period of 0.01 s. Increasing the sampling time period in some implementation may lead to a clock frequency mismatch and hence plays a critical role in implementation. Also negative time slack may also create problems while implementing the design and hence choosing constraints may also be critical in cases where the number of logical operations is more. Avoiding DDR clocks help in reducing the route delays. Table 1 shows the resources consumed and for analyzing the power consumed by the controllers, we use the approximation methodology discussed in [39, 40]. Figures 14(a) and 14(b) show the power utilization for fractional order q = 0.998 and power utilization for various fractional orders. It confirms that larger power will be consumed by the system when the FOPF system shows the largest Lyapunov exponents (FOPF q = 0.998) as shown in Figure 14(b).

6.1. FPGA Implementation of FOASMC Synchronisation. For real-time implementation of the proposed control scheme, the FOASMC controller can be implemented in FPGA and the output voltages from the respective control pins can be configured with an active magnetorheological damper. In this section we implement the proposed fractional order adaptive sliding mode controllers (FOASMC) derived in (23) along with the fractional parameter update laws (24) and sliding surfaces (13). For implementation of the entire control scheme we use Virtex 7 (Device: xc7vx980t-1ffg1926). The fractional order of the FOPF master system, slave system, and the FOASMC is kept as q = 0.998. For analyzing the power consumed by the controllers, we use the approximation methodology discussed in [43]. It confirms that larger power will be consumed by the controller when the master and the slave system show largest Lyapunov exponents. Figure 15 shows the RTL schematics of the fractional order ASMC controllers implemented in Virtex 7 (Device: xc7vx980t1ffg1926). Figures 16(a) and 16(b) show the power utilization of the controller and power utilization with change in fractional orders, respectively. Figures 17 and 18 show the controlled states and estimated parameters of the FOPF system, respectively. To utilize the power of FPGA, the computation needs to be divided into several independent blocks of threads that can be executed simultaneously. The performance on FPGA is directly related to the number of threads and its performances decrease when number of branching instructions increases. The fractional order operators are implemented as building blocks and the so-called "frame delay" is not noticeable in the FPGA hardware implementation due to its parallel data structure, unlike a microprocessor-based implementation. While FPGA implementation has a reputation for being difficult to design, with the help of systematic methodology, a system can be put together with less work than is required for more traditional software-based realizations.

7. Conclusion

Most of the literatures have investigated chaotic oscillations in an integer order portal frame with nonideal loading. In this paper we investigated the chaotic oscillations of a fractional order portal frame. Existence of chaotic oscillations in a fractional order portal frame is investigated with a positive Lyapunov exponent. Bifurcation plots in the parameter space are investigated for the regions of chaotic oscillations. Investigation of bicoherence contours shows the quadratic nonlinearities and its existence because of autocorrelation and cross correlation. To control the chaotic oscillations an adaptive sliding mode control scheme is derived and numerically verified. As can be seen from Figures 9 and 10, the proposed control scheme is effective even in the presence of uncertainties in the parameters. For real-time implementation, the fractional order portal frame system with the adaptive sliding mode control algorithm is implemented in FPGA and as can be seen from Figures 16 and 17, the FPGA implemented controllers are effective in chaos suppression. For real-time chaos suppression, the output voltages from the FPGA pins can be connected to MR dampers.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.


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Caption: FIGURE 1: (a) Portal frame with nonideal excitation. (b) Coupled oscillator approximation.

Caption: FIGURE 2: 3D phase portraits of the portal frame system (2).

Caption: FIGURE 3: 3D phase portraits of the FOPF system.

Caption: FIGURE 4: Bifurcation of FOPF system with a.

Caption: FIGURE 5: Bifurcation plots of FOPF system for b.

Caption: FIGURE 6: Bifurcation of FOPF system for c.

Caption: FIGURE 7: Bifurcation of FOPF system with d.

Caption: FIGURE 8: Bifurcation of FOPF system with fractional order q.

Caption: FIGURE 9: (a) Bicoherence of state x. (b) Bicoherence of all states of FOPF system.

Caption: FIGURE 10: Time history of FOPF states (controller in action at t = 70 s).

Caption: FIGURE 11: Time history of FOPF parameter estimates (controller in action at t = 70 s).

Caption: FIGURE 12: Xilinx RTL schematics of the FOPF system implemented in Kintex 7.

Caption: FIGURE 13: 3D phase portraits of the FOPF system implemented in FPGA with order q = 0.998.

Caption: FIGURE 14: (a) Power utilized and (b) power utilization versus fractional order of FOPF.

Caption: FIGURE 15: Xilinx RTL schematics of the FOASMC controllers implemented in Virtex 7.

Caption: FIGURE 16: (a) Power utilized and (b) power utilization versus fractional order of FOASMC.

Caption: FIGURE 17: Controlled states of FOPF system (controller in action at t = 5s).

Caption: FIGURE 18: Estimated parameters of FOPF system (controller in action at t = 5 s).
TABLE 1: Resources utilized by the FOPF system.

Kintex 7 k160t     Utilization   Available   Utilization%

LUT                    974        101400         0.96
FF                     850        202800         0.42
DSP                    36           600          6.00
IO                     129          285         45.26
BUFG                    1           32           3.13

Kintex 7 k160t        Clock frequency

                    [f.sub.max]    Used

LUT                  500 Mhz     214 Mhz
FF                   300 Mhz     132 Mhz
DSP                  500 Mhz     244 Mhz
IO                   250 Mhz     129 Mhz
BUFG                 300 Mhz     112 Mhz
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Title Annotation:Research Article
Author:Rajagopal, Karthikeyan; Karthikeyan, Anitha; Duraisamy, Prakash
Publication:Shock and Vibration
Article Type:Report
Date:Jan 1, 2017
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