Vicente et al 2019c - Revision history

Rimni: /* 2.1 Non-ideal system */

2021-05-27T12:34:26Z

‎2.1 Non-ideal system

Rimni at 11:06, 19 March 2020

2020-03-19T11:06:30Z

Rimni at 11:04, 19 March 2020

2020-03-19T11:04:38Z

Rimni at 10:28, 19 March 2020

2020-03-19T10:28:59Z

Rimni at 10:27, 19 March 2020

2020-03-19T10:27:41Z

Rimni at 10:26, 19 March 2020

2020-03-19T10:26:38Z

Rimni at 10:25, 19 March 2020

2020-03-19T10:25:22Z

Rimni at 10:24, 19 March 2020

2020-03-19T10:24:26Z

Fabioch at 12:00, 4 September 2019

2019-09-04T12:00:50Z

Fabioch at 11:56, 4 September 2019

2019-09-04T11:56:58Z

@@ Line 45: / Line 45: @@
 ===2.1 Non-ideal system===
-A system is said to be ideal when the excitation does not influence the response of the system, ie, the ideal excitation has an ideal power source [8,9,10,11,12]. On the other hand, when the excitation is influenced by the response of the system, the excitation is said to be non-ideal [19]. A non-ideal excitation can be modeled as shown in  Figure [[#img-1|1]] [4,11].
+A system is said to be ideal when the excitation does not influence the response of the system, ie, the ideal excitation has an ideal power source [8,9,10,11,12]. On the other hand, when the excitation is influenced by the response of the system, the excitation is said to be non-ideal [19]. A non-ideal excitation can be modeled as shown in   [[#img-1|Figure 1]] [4,11].
 <div id='img-1'></div>
@@ Line 132: / Line 132: @@
 |}
-Figure&nbsp;[[#img-4|4]] illustrates the six Lyapunov exponents, <math display="inline">\lambda _1=8.9985</math>; <math display="inline">\lambda _2=8.9539</math>; <math display="inline">\lambda _3=-6.4645</math>; <math display="inline">\lambda _4=-1.4999</math> and <math display="inline">\lambda _5=-3.203</math>, <math display="inline">\lambda _6=-3.2329</math>, demonstrating the presence of chaos with one positive Lyapunov exponent. The total time for computing the Lyapunov exponents was <math display="inline">\Delta \tau  =100.000</math> with a time-step of <math display="inline">\tau=1</math> [25].
+[[#img-4|Figure 4]] illustrates the six Lyapunov exponents, <math display="inline">\lambda _1=8.9985</math>; <math display="inline">\lambda _2=8.9539</math>; <math display="inline">\lambda _3=-6.4645</math>; <math display="inline">\lambda _4=-1.4999</math> and <math display="inline">\lambda _5=-3.203</math>, <math display="inline">\lambda _6=-3.2329</math>, demonstrating the presence of chaos with one positive Lyapunov exponent. The total time for computing the Lyapunov exponents was <math display="inline">\Delta \tau  =100.000</math> with a time-step of <math display="inline">\tau=1</math> [25].
 <div id='img-4'></div>

@@ Line 230: / Line 230: @@
 We will obtain
-<math display="inline"> B = \begin{pmatrix} 1 \\ 1 \\1 \\1 \\1 \\1 \end{pmatrix}</math>, <math display="inline">y = \begin{pmatrix} x_1-\tilde{x}_1 \\ x_2-\tilde{x}_2 \\ x_3-\tilde{x}_3 \\ x_4-\tilde{x}_4 \\ x_5-\tilde{x}_5 \\ x_6-\tilde{x}_6     \end{pmatrix}</math>, <math display="inline">\tilde{x}=\begin{pmatrix} 0 \\ 0 \\ 0 \\0 \\0\\0 \end{pmatrix}</math>,<math display="inline">Q=I_6</math>, <math display="inline">A=\begin{pmatrix} 0&1&0&0&0&0 \\ -91.77&-0.01&91.10&0&0&0\\9.50&0&0&1&0&0 \\ -0.09&0&0.09&-1.49&0&0 \\0&0&0&1&0&1 \\9.50&0&0&0&-87.00&17.99 \end{pmatrix}</math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 where the controllability matrix <math display="inline">R</math> of the system for the pair [<math display="inline">A,B</math>] is obtained by <math display="inline">R=[B|AB|A^2B|A^3B|A^4B|A^5B]</math>.
@@ Line 236: / Line 243: @@
 Thus, <math display="inline">R= \begin{pmatrix} 1\end{pmatrix}</math>. Then the Matrix
-<math display="inline">P(t)</math> is given by <math display="inline">P= 10^4\begin{pmatrix}  0.216 & 0.031 &  0.406 & -0.030 & -0.647 &  0.040 \\ 0.031 & 0.004 &  0.058 & -0.004 & -0.093 &  0.005 \\ 0.406 & 0.058 &  0.767 & -0.058 & -1.218 &  0.071 \\ -0.030& -0.004& -0.058 &  0.004 &  0.092 & -0.005 \\ -0.647& -0.093& -1.218 &  0.092 &  1.944 & -0.123 \\ 0.040 & 0.005 &  0.076 & -0.005 & -0.123 &  0.008  \end{pmatrix}</math>
+{| class="formulaSCP" style="width: 100%; text-align: left;"
+|-
 and (an optimal control)
-<math display="inline">u= 10^2(1.6765 x_1 + 0.2493 x_2 + 3.2667 x_3 - 0.2353 x_4 - 4.6331 x_5 + 0.2116 x_6</math>).
+{| class="formulaSCP" style="width: 100%; text-align: left;"
 The trajectories of the system with control can be seen in Figures [[#img-5|5]] and [[#img-6|6]]. According to the optimal control verification [23], the function   is numerically calculated by <math display="inline">L(t)=y^T\tilde{Q}y</math>, where <math display="inline">L(t)</math> is positive definite.

@@ Line 245: / Line 245: @@
 <div id='img-5'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |[[Image:Draft_Chavarette_796969493-fhistcontrol.png|600px|Time History - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a)  displacement for the Mathieu system; (b)  velocity for the Mathieu system; (c)displacement for the non-ideal excitation; (d) velocity for the non-ideal excitation; (e) displacement for Van der Pol system, and (f) velocity for the  Van der Pol system   ]]
@@ Line 253: / Line 253: @@
 <div id='img-6'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |[[Image:Draft_Chavarette_796969493-fdisplacontrol.png|600px|Phase Portrait - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a) Mathieu system, (b) Non-ideal excitation, and (c) Van der Pol system   ]]

@@ Line 148: / Line 148: @@
 The real unknown parameters of system are supposed to be as follows: <math display="inline">\bar{a}=73.36+36.68r(t)</math>, <math display="inline">\bar{b}=4.024+2.012(t)</math>, <math display="inline">\bar{c}=0.008+0.002(t)</math>, <math display="inline">\bar{d}=72.8+36.4r(t)</math>, <math display="inline">\bar{e}=69.6008+34.8004r(t)</math>, <math display="inline">\bar{f}=14.4+7.2r(t)</math>, <math display="inline">\bar{g}=7.5928+3.8288r(t)</math>, <math display="inline">\bar{\alpha }=0.64+0.32r(t)</math>, <math display="inline">\bar{\mu }=0.16+0.08r(t)</math>, <math display="inline">\bar{\beta }=1.2+0.6r(t)</math> and <math display="inline">\bar{eta}=0.24+0.12r(t)</math>,  where <math display="inline">r(t)</math> are normally distributed random functions [20]. Numerical simulation results are shown in the next section, where the robustness of the methods is assessed considering that the system parameters present uncertainties.
-==3. Control Design==
+==3. Control design==
 The Optimal Linear Control design for the Non-ideal Mathieu-Van der Pol system reduces the oscillatory movement to a small stable orbit. Due to its simplicity in terms of configuration and implementation, the linear state feedback control is especially attractive [18].
@@ Line 199: / Line 199: @@
 Using the theorem [20] the dynamic error y can be minimized (<math display="inline">y \rightarrow 0</math>).
-===3.1 Application of the Optimal Linear Control to the System===
+===3.1 Application of the optimal linear control to the system===
 From equation [[#eq-4|(4)]]:
@@ Line 249: / Line 249: @@
 |[[Image:Draft_Chavarette_796969493-fhistcontrol.png|600px|Time History - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a)  displacement for the Mathieu system; (b)  velocity for the Mathieu system; (c)displacement for the non-ideal excitation; (d) velocity for the non-ideal excitation; (e) displacement for Van der Pol system, and (f) velocity for the  Van der Pol system   ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 5'''. Time History - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a)  Displacement for the Mathieu system. (b) Velocity for the Mathieu system. (c) Displacement for the non-ideal excitation. (d) Velocity for the non-ideal excitation. (e) Displacement for Van der Pol system. (f) Velocity for the  Van der Pol system
+| colspan="1" style="padding:10px;"| '''Figure 5'''. Time History - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a)  Displacement for the Mathieu system. (b) Velocity for the Mathieu system. (c) Displacement for the non-ideal excitation. (d) Velocity for the non-ideal excitation. (e) Displacement for Van der Pol system. (f) Velocity for the  Van der Pol system
 |}
@@ Line 257: / Line 257: @@
 |[[Image:Draft_Chavarette_796969493-fdisplacontrol.png|600px|Phase Portrait - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a) Mathieu system, (b) Non-ideal excitation, and (c) Van der Pol system   ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6'''. Phase Portrait - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a) Mathieu system. (b) Non-ideal excitation. (c) Van der Pol system
+| colspan="1" style="padding:10px;"| '''Figure 6'''. Phase Portrait - the blue line is related to the system with uncertain parameters; the red line, with fixed parameters; the black line, in turn, is related to the controlled system. (a) Mathieu system. (b) Non-ideal excitation. (c) Van der Pol system
 |}

@@ Line 129: / Line 129: @@
 |[[Image:Draft_Chavarette_796969493-fdisplac.png|500px|Phase Portrait. (a) Mathieu system, (b) Non-ideal excitation, and (c) Van der Pol system   ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 3'''. Phase Portrait. (a) Mathieu system. (b) Non-ideal excitation. (c) Van der Pol system
+| colspan="1" style="padding-bottom:10px;"| '''Figure 3'''. Phase Portrait. (a) Mathieu system. (b) Non-ideal excitation. (c) Van der Pol system
 |}
@@ Line 139: / Line 139: @@
 |[[Image:Draft_Chavarette_796969493-flyapunov.png|500px|Dynamics of Lyapunov expoents   ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 4'''. Dynamics of Lyapunov expoents
+| colspan="1" style="padding-bottom:10px;"| '''Figure 4'''. Dynamics of Lyapunov expoents
 |}

@@ Line 117: / Line 117: @@
 <div id='img-2'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
 |[[Image:Draft_Chavarette_796969493-fhistorico.png|500px|Time History. (a)  displacement for the Mathieu system; (b)  velocity for the Mathieu system; (c)displacement for the non-ideal excitation; (d) velocity for the non-ideal excitation; (e) displacement for Van der Pol system, and (f) velocity for the Van der Pol system   ]]
@@ Line 125: / Line 125: @@
 <div id='img-3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
 |[[Image:Draft_Chavarette_796969493-fdisplac.png|500px|Phase Portrait. (a) Mathieu system, (b) Non-ideal excitation, and (c) Van der Pol system   ]]
@@ Line 135: / Line 135: @@
 <div id='img-4'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
 |[[Image:Draft_Chavarette_796969493-flyapunov.png|500px|Dynamics of Lyapunov expoents   ]]

@@ Line 19: / Line 19: @@
 Nowadays, the applications in mechanical engineering require that the systems operate with optimal parameters under certain operation conditions, which demonstrates the need for developing realistic numerical models which consider uncertainties in the parameters and inputs of the system properly. In mechanical engineering, the uncertainties have been considered in several areas, such as structure dynamics [13], rotor dynamics [14], robot manipulators  [15], among others. In the process of modeling dynamical systems, uncertainties may be considered based on a probabilistic approach, through predefined intervals, or even the possibility theory [16].
-==2. Mathematical Model==
+==2. Mathematical model==
 The Mathieu-Van der Pol autonomous system is a model with four state variables and three positive Lyaponuv exponents, proposed by Shih-Yu ''et al.'' [3]. It is a hyper-chaotic system that was modeled via linear coupling of the non-linear and non-autonomous systems of Mathieu and Van der Pol [3], whose equations of motion are as follows:
@@ Line 43: / Line 43: @@
 The next section deals with non-ideal excitations, as well as presents a proposal for a new model called Non-Ideal Mathieu-Van der Pol System.
-===2.1 Non-ideal System===
+===2.1 Non-ideal system===
 A system is said to be ideal when the excitation does not influence the response of the system, ie, the ideal excitation has an ideal power source [8,9,10,11,12]. On the other hand, when the excitation is influenced by the response of the system, the excitation is said to be non-ideal [19]. A non-ideal excitation can be modeled as shown in  Figure [[#img-1|1]] [4,11].

@@ Line 226: / Line 226: @@
 |}
-where the function of control <math display="inline">U</math> is defined in equation [[#eq-5|(5)]], and then
+where the function of control <math display="inline">U</math> is defined in equation [[#eq-5|(5)]].
 <math display="inline"> B = \begin{pmatrix} 1 \\ 1 \\1 \\1 \\1 \\1 \end{pmatrix}</math>, <math display="inline">y = \begin{pmatrix} x_1-\tilde{x}_1 \\ x_2-\tilde{x}_2 \\ x_3-\tilde{x}_3 \\ x_4-\tilde{x}_4 \\ x_5-\tilde{x}_5 \\ x_6-\tilde{x}_6     \end{pmatrix}</math>, <math display="inline">\tilde{x}=\begin{pmatrix} 0 \\ 0 \\ 0 \\0 \\0\\0 \end{pmatrix}</math>,<math display="inline">Q=I_6</math>, <math display="inline">A=\begin{pmatrix} 0&1&0&0&0&0 \\ -91.77&-0.01&91.10&0&0&0\\9.50&0&0&1&0&0 \\ -0.09&0&0.09&-1.49&0&0 \\0&0&0&1&0&1 \\9.50&0&0&0&-87.00&17.99 \end{pmatrix}</math>

@@ Line 226: / Line 226: @@
 |}
-where the function of control <math display="inline">U</math> is defined in equation [[#eq-5|(5)]].
+where the function of control <math display="inline">U</math> is defined in equation [[#eq-5|(5)]], and then
-−
-Rewriting the dynamical system [[#eq-4|(4)]] in the space-state form:
-−
-{| class="formulaSCP" style="width: 100%; text-align: left;"
-|-
-−
-{| style="text-align: left; margin:auto;width: 100%;"
-|-
-| style="text-align: center;" | <math>\dot{y}_1 = y_2 +U,</math>
-|-
-| style="text-align: center;" | <math> \dot{y}_2 = - \frac{-k_1y_1-k_2(y_1-y_3)+dy_6^2cos(y_5)+dsin(y_5)(a-by_6)}{m_1-dr(sin^2y_5)}</math>
-|-
-| style="text-align: center;" | <math> \dot{y}_3 = y_4,</math>
-|-
-| style="text-align: center;" | <math> \dot{y}_4 = -\frac{y_2}{m_2}(y_3-y_1),</math>
-|-
-| style="text-align: center;" | <math> \dot{y}_5 = y_6,</math>
-|-
-| style="text-align: center;" | <math> \dot{y}_6 = \frac{rsin(y_5)(-k_1y_1-k_2(y_1-y_3))+dy_6^2rcos(y_5)sin(y_5)+drsin(y_5)^2+(m_1-dr(sin^2(y_5)))(a-by_6)}{m_1-dr(sin^2(y_5))}.</math>
-|-
-| style="text-align: center;" |
-|}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
-|}
-−
 and then
 <math display="inline"> B = \begin{pmatrix} 1 \\ 1 \\1 \\1 \\1 \\1 \end{pmatrix}</math>, <math display="inline">y = \begin{pmatrix} x_1-\tilde{x}_1 \\ x_2-\tilde{x}_2 \\ x_3-\tilde{x}_3 \\ x_4-\tilde{x}_4 \\ x_5-\tilde{x}_5 \\ x_6-\tilde{x}_6     \end{pmatrix}</math>, <math display="inline">\tilde{x}=\begin{pmatrix} 0 \\ 0 \\ 0 \\0 \\0\\0 \end{pmatrix}</math>,<math display="inline">Q=I_6</math>, <math display="inline">A=\begin{pmatrix} 0&1&0&0&0&0 \\ -91.77&-0.01&91.10&0&0&0\\9.50&0&0&1&0&0 \\ -0.09&0&0.09&-1.49&0&0 \\0&0&0&1&0&1 \\9.50&0&0&0&-87.00&17.99 \end{pmatrix}</math>