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a/figs/gravimeter_svd_low_damping.png b/figs/gravimeter_svd_low_damping.png new file mode 100644 index 0000000..4a263b7 Binary files /dev/null and b/figs/gravimeter_svd_low_damping.png differ diff --git a/index.html b/index.html index f650db1..ebea6ae 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Diagonal control using the SVD and the Jacobian Matrix @@ -30,40 +30,41 @@

Table of Contents

@@ -79,58 +80,58 @@ Then, a diagonal controller is used. These two methods are tested on two plants:

    -
  • In Section 1 on a 3-DoF gravimeter
  • -
  • In Section 2 on a 6-DoF Stewart platform
  • +
  • In Section 1 on a 3-DoF gravimeter
  • +
  • In Section 2 on a 6-DoF Stewart platform
-
-

1 Gravimeter - Simscape Model

+
+

1 Gravimeter - Simscape Model

- +

-
-

1.1 Introduction

+
+

1.1 Introduction

In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model:

    -
  • Section 1.2: the model is described and its parameters are defined.
  • -
  • Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
  • -
  • Section 1.4: the plant is decoupled using the Jacobian matrices.
  • -
  • Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
  • -
  • Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
  • -
  • Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
  • -
  • Section 1.8: the obtained decoupled plants are compared
  • -
  • Section 1.9: the diagonal controller is developed
  • -
  • Section 1.10: the obtained closed-loop performances for the two methods are compared
  • +
  • Section 1.2: the model is described and its parameters are defined.
  • +
  • Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
  • +
  • Section 1.4: the plant is decoupled using the Jacobian matrices.
  • +
  • Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
  • +
  • Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
  • +
  • Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
  • +
  • Section 1.8: the obtained decoupled plants are compared
  • +
  • Section 1.9: the diagonal controller is developed
  • +
  • Section 1.10: the obtained closed-loop performances for the two methods are compared
-
-

1.2 Gravimeter Model - Parameters

+
+

1.2 Gravimeter Model - Parameters

- +

-The model of the gravimeter is schematically shown in Figure 1. +The model of the gravimeter is schematically shown in Figure 1.

-
+

gravimeter_model.png

Figure 1: Model of the gravimeter

-
+

leg_model.png

Figure 2: Model of the struts

@@ -160,11 +161,11 @@ g = 0; % Gravity [m/s2]
-
-

1.3 System Identification

+
+

1.3 System Identification

- +

@@ -183,12 +184,12 @@ io(io_i) = linio([mdl, '/Acc_top'], 2, 'F1', 'F2', 'F3'}; -G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'}; +G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};

-The inputs and outputs of the plant are shown in Figure 3. +The inputs and outputs of the plant are shown in Figure 3.

@@ -201,11 +202,11 @@ More precisely there are three inputs (the three actuator forces): And 4 outputs (the two 2-DoF accelerometers):

\begin{equation} - \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix} + \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix} \end{equation} -
+

gravimeter_plant_schematic.png

Figure 3: Schematic of the gravimeter plant

@@ -261,11 +262,11 @@ State-space model with 4 outputs, 3 inputs, and 6 states.

-The bode plot of all elements of the plant are shown in Figure 4. +The bode plot of all elements of the plant are shown in Figure 4.

-
+

open_loop_tf.png

Figure 4: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

@@ -273,42 +274,42 @@ The bode plot of all elements of the plant are shown in Figure -

1.4 Decoupling using the Jacobian

+
+

1.4 Decoupling using the Jacobian

- +

-Consider the control architecture shown in Figure 5. +Consider the control architecture shown in Figure 5.

The Jacobian matrix \(J_{\tau}\) is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass:

\begin{equation} - \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_z \\ M_y \end{bmatrix} + \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix} \end{equation}

The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass:

\begin{equation} - \begin{bmatrix} a_x \\ a_z \\ a_{R_y} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{z1} \\ a_{x2} \\ a_{z2} \end{bmatrix} + \begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix} \end{equation}

-We thus define a new plant as defined in Figure 5. +We thus define a new plant as defined in Figure 5. \[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \]

-\(\bm{G}_x(s)\) correspond to the $3 × 3$transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5). +\(\bm{G}_x(s)\) correspond to the $3 × 3$transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5).

-
+

gravimeter_decouple_jacobian.png

Figure 5: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

@@ -334,8 +335,8 @@ And the plant \(\bm{G}_x\) is computed:

Gx = pinv(Ja)*G*pinv(Jt');
-Gx.InputName  = {'Fx', 'Fz', 'My'};
-Gx.OutputName  = {'Dx', 'Dz', 'Ry'};
+Gx.InputName  = {'Fx', 'Fy', 'Mz'};
+Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
 
@@ -346,11 +347,11 @@ State-space model with 3 outputs, 3 inputs, and 6 states.

-The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6. +The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6.

-
+

gravimeter_jacobian_plant.png

Figure 6: Diagonal and off-diagonal elements of \(G_x\)

@@ -358,11 +359,11 @@ The diagonal and off-diagonal elements of \(G_x\) are shown in Figure -

1.5 Decoupling using the SVD

+
+

1.5 Decoupling using the SVD

- +

@@ -511,11 +512,11 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7.

-
+

gravimeter_decouple_svd.png

Figure 7: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

@@ -546,10 +547,10 @@ The 4th output (corresponding to the null singular value) is discarded, and we o

-The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8. +The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8.

-
+

gravimeter_svd_plant.png

Figure 8: Diagonal and off-diagonal elements of \(G_{svd}\)

@@ -557,11 +558,11 @@ The diagonal and off-diagonal elements of the “SVD” plant are shown
-
-

1.6 Verification of the decoupling using the “Gershgorin Radii”

+
+

1.6 Verification of the decoupling using the “Gershgorin Radii”

- +

@@ -574,7 +575,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:

-
+

gravimeter_gershgorin_radii.png

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

@@ -582,11 +583,11 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:
-
-

1.7 Verification of the decoupling using the “Relative Gain Array”

+
+

1.7 Verification of the decoupling using the “Relative Gain Array”

- +

@@ -600,11 +601,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an

-The obtained RGA elements are shown in Figure 10. +The obtained RGA elements are shown in Figure 10.

-
+

gravimeter_rga.png

Figure 10: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

@@ -618,7 +619,7 @@ The RGA-number is also a measure of diagonal dominance: \end{equation} -
+

gravimeter_rga_num.png

Figure 11: RGA-Number for the Gravimeter

@@ -626,30 +627,30 @@ The RGA-number is also a measure of diagonal dominance:
-
-

1.8 Obtained Decoupled Plants

+
+

1.8 Obtained Decoupled Plants

- +

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12.

-
+

gravimeter_decoupled_plant_svd.png

Figure 12: Decoupled Plant using SVD

-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13.

-
+

gravimeter_decoupled_plant_jacobian.png

Figure 13: Gravimeter Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

@@ -657,12 +658,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
-

1.9 Diagonal Controller

+
+

1.9 Diagonal Controller

- -The control diagram for the centralized control is shown in Figure 14. + +The control diagram for the centralized control is shown in Figure 14.

@@ -671,19 +672,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control_gravimeter.png

Figure 14: Control Diagram for the Centralized control

-The SVD control architecture is shown in Figure 15. +The SVD control architecture is shown in Figure 15. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

-
+

svd_control_gravimeter.png

Figure 15: Control Diagram for the SVD control

@@ -721,11 +722,11 @@ G_svd = feedback(G, inv(V')

-The obtained diagonal elements of the loop gains are shown in Figure 16. +The obtained diagonal elements of the loop gains are shown in Figure 16.

-
+

gravimeter_comp_loop_gain_diagonal.png

Figure 16: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

@@ -733,11 +734,11 @@ The obtained diagonal elements of the loop gains are shown in Figure -

1.10 Closed-Loop system Performances

+
+

1.10 Closed-Loop system Performances

- +

@@ -768,18 +769,18 @@ ans =

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17.

-
+

gravimeter_platform_simscape_cl_transmissibility.png

Figure 17: Obtained Transmissibility

-
+

gravimeter_cl_transmissibility_coupling.png

Figure 18: Obtain coupling terms of the transmissibility matrix

@@ -788,8 +789,8 @@ The obtained transmissibility in Open-loop, for the centralized control as well
-
-

1.11 Robustness to a change of actuator position

+
+

1.11 Robustness to a change of actuator position

Let say we change the position of the actuators: @@ -809,7 +810,7 @@ The closed-loop system are still stable, and their

-
+

gravimeter_transmissibility_offset_act.png

Figure 19: Transmissibility for the initial CL system and when the position of actuators are changed

@@ -817,8 +818,8 @@ The closed-loop system are still stable, and their
-
-

1.12 Combined / comparison of K and M decoupling

+
+

1.12 Combined / comparison of K and M decoupling

If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobians at a point where the stiffness matrix is diagonal. @@ -839,11 +840,11 @@ To do so, the actuators (springs) should be positioned such that the stiffness m

-
-

1.12.1 Decoupling of the mass matrix

+
+

1.12.1 Decoupling of the mass matrix

-
+

gravimeter_model_M.png

Figure 20: Choice of {O} such that the Mass Matrix is Diagonal

@@ -871,7 +872,7 @@ io(io_i) = linio([mdl, '/Acc_top'], 2, 'F1', 'F2', 'F3'}; -G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'}; +G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
@@ -892,24 +893,25 @@ JMt = [1 0 -ha
GM = pinv(JMa)*G*pinv(JMt');
-GM.InputName  = {'Fx', 'Fz', 'My'};
-GM.OutputName  = {'Dx', 'Dz', 'Ry'};
+GM.InputName  = {'Fx', 'Fy', 'Mz'};
+GM.OutputName  = {'Dx', 'Dy', 'Rz'};
 
-
+

jac_decoupling_M.png

+

Figure 21: Diagonal and off-diagonal elements of the decoupled plant

-
-

1.12.2 Decoupling of the stiffness matrix

+
+

1.12.2 Decoupling of the stiffness matrix

-
+

gravimeter_model_K.png

Figure 22: Choice of {O} such that the Stiffness Matrix is Diagonal

@@ -935,24 +937,25 @@ And the plant \(\bm{G}_x\) is computed:

GK = pinv(JKa)*G*pinv(JKt');
-GK.InputName  = {'Fx', 'Fz', 'My'};
-GK.OutputName  = {'Dx', 'Dz', 'Ry'};
+GK.InputName  = {'Fx', 'Fy', 'Mz'};
+GK.OutputName  = {'Dx', 'Dy', 'Rz'};
 
-
+

jac_decoupling_K.png

+

Figure 23: Diagonal and off-diagonal elements of the decoupled plant

-
-

1.12.3 Combined decoupling of the mass and stiffness matrices

+
+

1.12.3 Combined decoupling of the mass and stiffness matrices

-
+

gravimeter_model_KM.png

Figure 24: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal

@@ -984,7 +987,7 @@ io(io_i) = linio([mdl, '/Acc_top'], 2, 'F1', 'F2', 'F3'}; -G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'}; +G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
@@ -1002,21 +1005,22 @@ JMt = [1 0 -ha
GKM = pinv(JMa)*G*pinv(JMt');
-GKM.InputName  = {'Fx', 'Fz', 'My'};
-GKM.OutputName  = {'Dx', 'Dz', 'Ry'};
+GKM.InputName  = {'Fx', 'Fy', 'Mz'};
+GKM.OutputName  = {'Dx', 'Dy', 'Rz'};
 
-
+

jac_decoupling_KM.png

+

Figure 25: Diagonal and off-diagonal elements of the decoupled plant

-
-

1.12.4 Conclusion

+
+

1.12.4 Conclusion

Ideally, the mechanical system should be designed in order to have a decoupled stiffness matrix at the CoM of the solid. @@ -1029,16 +1033,59 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C

+ +
+

1.13 SVD decoupling performances

+
+

+As the SVD is applied on a real approximation of the plant dynamics at a frequency \(\omega_0\), it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation. +

+ +

+Let’s do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping). +

+ +

+Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26. +

+
+
c = 2e1; % Actuator Damping [N/(m/s)]
+
-
-

2 Stewart Platform - Simscape Model

+ +
+

gravimeter_svd_low_damping.png +

+

Figure 26: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping

+
+ +

+Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27. +

+
+
c = 5e2; % Actuator Damping [N/(m/s)]
+
+
+ + +
+

gravimeter_svd_high_damping.png +

+

Figure 27: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping

+
+
+
+
+ +
+

2 Stewart Platform - Simscape Model

- +

-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 26. +In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 28.

@@ -1051,33 +1098,33 @@ Some notes about the system: -

+

SP_assembly.png

-

Figure 26: Stewart Platform CAD View

+

Figure 28: Stewart Platform CAD View

The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections:

    -
  • Section 2.1: The parameters of the Simscape model of the Stewart platform are defined
  • -
  • Section 2.2: The plant is identified from the Simscape model and the system coupling is shown
  • -
  • Section 2.3: The plant is first decoupled using the Jacobian
  • -
  • Section 2.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
  • -
  • Section 2.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
  • -
  • Section 2.6:
  • -
  • Section 2.7: The dynamics of the decoupled plants are shown
  • -
  • Section 2.8: A diagonal controller is defined to control the decoupled plant
  • -
  • Section 2.9: Finally, the closed loop system properties are studied
  • +
  • Section 2.1: The parameters of the Simscape model of the Stewart platform are defined
  • +
  • Section 2.2: The plant is identified from the Simscape model and the system coupling is shown
  • +
  • Section 2.3: The plant is first decoupled using the Jacobian
  • +
  • Section 2.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
  • +
  • Section 2.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
  • +
  • Section 2.6:
  • +
  • Section 2.7: The dynamics of the decoupled plants are shown
  • +
  • Section 2.8: A diagonal controller is defined to control the decoupled plant
  • +
  • Section 2.9: Finally, the closed loop system properties are studied
-
-

2.1 Simscape Model - Parameters

+
+

2.1 Simscape Model - Parameters

- +

open('drone_platform.slx');
@@ -1133,30 +1180,30 @@ Kc = tf(zeros(6));
 
-
+

stewart_simscape.png

-

Figure 27: General view of the Simscape Model

+

Figure 29: General view of the Simscape Model

-
+

stewart_platform_details.png

-

Figure 28: Simscape model of the Stewart platform

+

Figure 30: Simscape model of the Stewart platform

-
-

2.2 Identification of the plant

+
+

2.2 Identification of the plant

- +

-The plant shown in Figure 29 is identified from the Simscape model. +The plant shown in Figure 31 is identified from the Simscape model.

@@ -1172,10 +1219,10 @@ The outputs are the 6 accelerations measured by the inertial unit.

-
+

stewart_platform_plant.png

-

Figure 29: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

+

Figure 31: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

@@ -1214,7 +1261,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.

-The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 30. +The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 32.

@@ -1222,20 +1269,20 @@ One can easily see that the system is strongly coupled.

-
+

stewart_platform_coupled_plant.png

-

Figure 30: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

+

Figure 32: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

-
-

2.3 Decoupling using the Jacobian

+
+

2.3 Decoupling using the Jacobian

- -Consider the control architecture shown in Figure 31. + +Consider the control architecture shown in Figure 33. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

@@ -1317,10 +1364,10 @@ The Jacobian matrix is computed from the geometry of the platform (position and -
+

plant_decouple_jacobian.png

-

Figure 31: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

+

Figure 33: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

@@ -1340,11 +1387,11 @@ Gx.InputName = {'Fx',

-
-

2.4 Decoupling using the SVD

+
+

2.4 Decoupling using the SVD

- +

@@ -1680,14 +1727,14 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 32. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 34.

-
+

plant_decouple_svd.png

-

Figure 32: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

+

Figure 34: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

@@ -1702,11 +1749,11 @@ The decoupled plant is then:

-
-

2.5 Verification of the decoupling using the “Gershgorin Radii”

+
+

2.5 Verification of the decoupling using the “Gershgorin Radii”

- +

@@ -1722,19 +1769,19 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by: This is computed over the following frequencies.

-
+

simscape_model_gershgorin_radii.png

-

Figure 33: Gershgorin Radii of the Coupled and Decoupled plants

+

Figure 35: Gershgorin Radii of the Coupled and Decoupled plants

-
-

2.6 Verification of the decoupling using the “Relative Gain Array”

+
+

2.6 Verification of the decoupling using the “Relative Gain Array”

- +

@@ -1748,55 +1795,55 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an

-The obtained RGA elements are shown in Figure 34. +The obtained RGA elements are shown in Figure 36.

-
+

simscape_model_rga.png

-

Figure 34: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

+

Figure 36: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

-
-

2.7 Obtained Decoupled Plants

+
+

2.7 Obtained Decoupled Plants

- +

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 35. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 37.

-
+

simscape_model_decoupled_plant_svd.png

-

Figure 35: Decoupled Plant using SVD

+

Figure 37: Decoupled Plant using SVD

-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 36. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 38.

-
+

simscape_model_decoupled_plant_jacobian.png

-

Figure 36: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

+

Figure 38: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

-
-

2.8 Diagonal Controller

+
+

2.8 Diagonal Controller

- -The control diagram for the centralized control is shown in Figure 37. + +The control diagram for the centralized control is shown in Figure 39.

@@ -1805,22 +1852,22 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control.png

-

Figure 37: Control Diagram for the Centralized control

+

Figure 39: Control Diagram for the Centralized control

-The SVD control architecture is shown in Figure 38. +The SVD control architecture is shown in Figure 40. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

-
+

svd_control.png

-

Figure 38: Control Diagram for the SVD control

+

Figure 40: Control Diagram for the SVD control

@@ -1854,23 +1901,23 @@ G_svd = feedback(G, inv(V')

-The obtained diagonal elements of the loop gains are shown in Figure 39. +The obtained diagonal elements of the loop gains are shown in Figure 41.

-
+

stewart_comp_loop_gain_diagonal.png

-

Figure 39: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

+

Figure 41: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

-
-

2.9 Closed-Loop system Performances

+
+

2.9 Closed-Loop system Performances

- +

@@ -1901,14 +1948,14 @@ ans =

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 40. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 42.

-
+

stewart_platform_simscape_cl_transmissibility.png

-

Figure 40: Obtained Transmissibility

+

Figure 42: Obtained Transmissibility

@@ -1916,7 +1963,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well

Author: Dehaeze Thomas

-

Created: 2020-12-10 jeu. 13:16

+

Created: 2020-12-10 jeu. 13:51

diff --git a/index.org b/index.org index 231e87e..df72f95 100644 --- a/index.org +++ b/index.org @@ -1346,18 +1346,17 @@ Ideally, the mechanical system should be designed in order to have a decoupled s If not the case, the system can either be decoupled as low frequency if the Jacobian are evaluated at a point where the stiffness matrix is decoupled. Or it can be decoupled at high frequency if the Jacobians are evaluated at the CoM. -** SVD decoupling performances :noexport: +** SVD decoupling performances +As the SVD is applied on a *real approximation* of the plant dynamics at a frequency $\omega_0$, it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation. -#+begin_src matlab - la = l/2; % Position of Act. [m] - ha = 0; % Position of Act. [m] -#+end_src +Let's do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping). +Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_low_damping]]. #+begin_src matlab c = 2e1; % Actuator Damping [N/(m/s)] #+end_src -#+begin_src matlab +#+begin_src matlab :exports none %% Name of the Simulink File mdl = 'gravimeter'; @@ -1374,9 +1373,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C G = linearize(mdl, io); G.InputName = {'F1', 'F2', 'F3'}; G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'}; -#+end_src -#+begin_src matlab wc = 2*pi*10; % Decoupling frequency [rad/s] H1 = evalfr(G, j*wc); D = pinv(real(H1'*H1)); @@ -1385,79 +1382,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C Gsvd = inv(U)*G*inv(V'); #+end_src -#+begin_src matlab - c = 5e2; % Actuator Damping [N/(m/s)] -#+end_src - -#+begin_src matlab - %% Name of the Simulink File - mdl = 'gravimeter'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1; - - G = linearize(mdl, io); - G.InputName = {'F1', 'F2', 'F3'}; - G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'}; -#+end_src - -#+begin_src matlab - wc = 2*pi*10; % Decoupling frequency [rad/s] - H1 = evalfr(G, j*wc); - D = pinv(real(H1'*H1)); - H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); - [U,S,V] = svd(H1); - Gsvdd = inv(U)*G*inv(V'); -#+end_src - -#+begin_src matlab - JMa = [1 0 -h/2 - 0 1 l/2 - 1 0 h/2 - 0 1 0]; - - JMt = [1 0 -ha - 0 1 la - 0 1 -la]; -#+end_src - -#+begin_src matlab - GM = pinv(JMa)*G*pinv(JMt'); - GM.InputName = {'Fx', 'Fy', 'Mz'}; - GM.OutputName = {'Dx', 'Dy', 'Rz'}; -#+end_src - -#+begin_src matlab :exports none - figure; - - % Magnitude - hold on; - for i_in = 1:3 - for i_out = [1:i_in-1, i_in+1:3] - plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end - end - plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); - set(gca,'ColorOrderIndex',1) - for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(GM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); - legend('location', 'southeast'); - ylim([1e-8, 1e0]); -#+end_src - #+begin_src matlab :exports none figure; @@ -1470,18 +1394,58 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C end end plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); + 'DisplayName', '$G_{svd}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); + plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd}(%d,%d)$', i_in_out, i_in_out)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); - legend('location', 'southeast'); + legend('location', 'northwest'); ylim([1e-8, 1e0]); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/gravimeter_svd_low_damping.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:gravimeter_svd_low_damping +#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping +#+RESULTS: +[[file:figs/gravimeter_svd_low_damping.png]] + +Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_high_damping]]. +#+begin_src matlab + c = 5e2; % Actuator Damping [N/(m/s)] +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'gravimeter'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1; + + G = linearize(mdl, io); + G.InputName = {'F1', 'F2', 'F3'}; + G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'}; + + wc = 2*pi*10; % Decoupling frequency [rad/s] + H1 = evalfr(G, j*wc); + D = pinv(real(H1'*H1)); + H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); + [U,S,V] = svd(H1); + Gsvdd = inv(U)*G*inv(V'); +#+end_src + #+begin_src matlab :exports none figure; @@ -1494,18 +1458,27 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C end end plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); + 'DisplayName', '$G_{svd}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gsvdd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); + plot(freqs, abs(squeeze(freqresp(Gsvdd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd}(%d,%d)$', i_in_out, i_in_out)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); - legend('location', 'southeast'); + legend('location', 'northwest'); ylim([1e-8, 1e0]); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/gravimeter_svd_high_damping.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:gravimeter_svd_high_damping +#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping +#+RESULTS: +[[file:figs/gravimeter_svd_high_damping.png]] + * Stewart Platform - Simscape Model :PROPERTIES: :header-args:matlab+: :tangle stewart_platform/script.m