Update Content - 2021-08-26

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parent 480f5b9cfe
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--- a/content/article/stoev17_tensor_method_mimo_decoup_contr.md
+++ b/content/article/stoev17_tensor_method_mimo_decoup_contr.md
@@ -8,7 +8,7 @@ Tags
 : [Decoupled Control](decoupled_control.md), [Multivariable Control](multivariable_control.md)

 Reference
-: ([Stoev et al. 2017](#orgc430b4f))
+: ([Stoev et al. 2017](#org29c5c20))

 Author(s)
 : Stoev, J., Ertveldt, J., Oomen, T., & Schoukens, J.
@@ -16,10 +16,27 @@ Author(s)
 Year
 : 2017

-Step by step approach:

-   [Interaction Analysis](interaction_analysis.md)
-   Decoupling transformations
+## Introduction {#introduction}
+
+By appropriate system design, most systems are either decoupled or can be decoupled using static input-output transformations.
+Hence, most motion system and their motion software architecture use SISO control design method and solutions.
+
+The first step typically involves a FRF identification using specific excitation signals.
+Once the FRF is available, the controller \\(K\\) can be designed directly based on the FRF data.
+Many classical MIMO control design methods aim at decoupling the open loop function at some location in the feedback loop.
+Because their are strong non-intuitive aspect of MIMO loop-shaping, the following step-by-step approach is proposed, in which the design complexity is only increased if justified by the problem at hand:
+
+-   **[Interaction Analysis](interaction_analysis.md)**.
+    The goal is to identify two sided interactions in the plant dynamics.
+    If there is no two sided interaction, then feedback design becomes a standard multi-loop SISO design problem.
+    Two measured of the plant interaction are [Relative Gain Array](relative_gain_array.md) and [Structured Singular Value](structured_singular_value.md).
+-   **Decoupling transformations**.
+    To reduce interaction, one may redefine the input and output of the plant using a decoupling transformation.
+    For motion systems, most transformations are found on the basis of **kinematic model**.
+    Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
+    Similarly, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
+    This, this basically amounts to the **inversion of a kinematic model** of the plant.
 -   Independent feedback control design
 -   Sequential feedback control design
 -   Norm based control design
@@ -27,7 +44,159 @@ Step by step approach:
 All steps, except for the last, can be performed with a non-parametric model of the plant (i.e. an identified FRF).


+## MIMO frequency response decomposition {#mimo-frequency-response-decomposition}
+
+The problem addressed in this paper is to decouple a given set of MIMO FRF.
+Such decoupled representation, if existing, would permit the MIMO FRF to be written as a linear combination of parallel SISO FRFs.
+The existing methods to convert the MIMO FRF into equivalent combination of SISO FRF fall into two groups:
+
+-   **matrix decomposition methods** use linear algebra based on eigen-value, or singular value decomposition which are able to diagonalize the FRF at a single frequency.
+-   **optimization methods** formulate the problem of simultaneous diagonalization of the FRF at multiple frequencies as an optimization problem.
+
+At each frequency \\(\omega\_i, i = 1 \dots N\_f\\), we have a square matrix \\(H(\omega\_i) \in \mathbb{C}^{N \times N}\\) with the complex response of the system relating the inputs and outputs.
+
+**MIMO decoupling of dyadic system**:
+
+\begin{align}
+H(\omega\_i) &= T\_y S(\omega\_i) T\_u + E(\omega\_i), \ i = 1 \dots N\_f \label{eq:decomposition} \\\\\\
+S(\omega\_i) &= \begin{bmatrix}
+S\_1(\omega\_i) & 0 & 0 \\\\\\
+0 & \ddots & 0 \\\\\\
+0 & 0 & S\_N(\omega\_i)
+\end{bmatrix}
+\end{align}
+
+where \\(S(\omega\_i)\\) is a diagonal matrix containing SISO FRFs \\(S\_k(\omega\_i) \in \mathbb{C}\\) on the main diagonal, \\(T\_y \in \mathbb{R}^{N \times N}\\), \\(T\_u \in \mathbb{R}^{N \times N}\\), \\(E(\omega\_i)\\) is the error.
+
+The approximate MIMO system decoupling is shown in Figure [1](#org3f61a67).
+
+In practical cases, the matrix \\(\hat{S}(\omega\_i) = T\_y^{-1} H(\omega\_i) T\_u^{-1}\\) will not be purely diagonal, but rather diagonally dominated.
+
+<a id="org3f61a67"></a>
+
+{{< figure src="/ox-hugo/stoev17_decoupled_system_schematic.png" caption="Figure 1: MIMO FRF decomposition in parallel branches" >}}
+
+The array \\(H(\omega\_i), i = 1 \dots N\_f\\) of complex matrices can be represented as a 3-dimensional sensor \\(\underline{H}\\).
+
+<div class="important">
+  <div></div>
+
+The core result of this paper is that the decomposition can be found by rephrasing \eqref{eq:decomposition} as a "Canonical Polyadic Decomposition" (CPD).
+This is shown in Figure [2](#org58ed38e), where \\(T\_y,T\_u,S\_d\\) can be directly computed using a single Matlab function.
+
+</div>
+
+Mathematically equivalent form of CPD is shown in the lower part of Figure [2](#org58ed38e), where the tensor \\(\underline{S}\\) contains the rows of the matrix \\(S\_d\\) on each of its diagonals in the third dimension, which is exactly the problem of simultaneous diagonalization.
+
+The transformation effectively diagonalises the original frequency response tensor \\(\underline{H}\\) using two transformation matrices \\(T\_y, T\_u\\).
+This operation is closely related to the SVD on a single matrix, however in this case the diagonalisation occurs for a set of matrices, each describing the MIMO FRF at different frequency.
+
+<a id="org58ed38e"></a>
+
+{{< figure src="/ox-hugo/stoev17_decompos_3d_tensor.png" caption="Figure 2: Decomposition of 3D tensor" >}}
+
+The direct application of a CPD procedure on the above complex data tensor would result in complex solutions, including complex matrices \\(T\_y \in \mathbb{C}^{N \times N}\\), \\(T\_u \in \mathbb{C}^{N \times N}\\).
+This is not useful for a practical decoupling of physical systems as we require real solutions for \\(T\_y,T\_u\\).
+The direct solution we use for this is to take the imaginary and real part of the complex tensor \\(\underline{H} \in \mathbb{C}^{N \times N \times N\_f}\\), each of them a real tensor by itself, and stack them one behind the other in the dimension of the frequencies, thus getting an augmented real-valued tensor \\(\underline{\breve{H}} \in \mathbb{R}^{N \times N \times 2N\_f}\\).
+
+
+## Numerical Example {#numerical-example}
+
+Let's now make a Matlab example using the [Tensorlab](https://www.tensorlab.net/) toolbox.
+
+Let's define a 2x2 diagonal system:
+
+```matlab
+S = [4e3/(s^2 + 25*s + 4e3) 0
+     0 4e5/(s^2 + 250*s + 4e5)];
+```
+
+And coupled this system with two random matrices:
+
+```matlab
+Ty = [0.13 0.003
+      0.43 0.51];
+
+Tu = [0.32 0.67
+      0.95 0.006];
+```
+
+The couple system is defined:
+
+```matlab
+H = Ty * S * Tu;
+```
+
+Then, suppose with have the frequency response function of the coupled plant:
+
+```matlab
+freqs = logspace(0,3,1000);
+H_frf = freqresp(H, freqs, 'Hz');
+```
+
+<a id="orgba720df"></a>
+
+{{< figure src="/ox-hugo/stoev17_coupled_diagonal_plants.png" caption="Figure 3: Diagonal and coupled plants" >}}
+
+We take the real and imaginary part of the FRF and concatenate the two along the frequency dimension.
+
+```matlab
+H_frf_real = cat(3, real(H_frf), imag(H_frf));
+```
+
+Then random matrices are initialize the the CPD.
+
+```matlab
+U = cpd_rnd(size(H_frf_real), size(H_frf_real,1));
+```
+
+And the CPD is performed.
+
+```matlab
+[T, ~] = cpd3_sd(H_frf_real, U);
+```
+
+The obtained decoupling matrices are:
+
+```matlab
+Ty_est = T{1};
+```
+
+```text
+Ty_est =
+        -0.289402459385387      -0.00647742171539879
+        -0.957207509228524         -1.10111369041218
+```
+
+```matlab
+Tu_est = T{2};
+```
+
+```text
+Tu_est =
+         0.430893809258741         0.999980044721872
+         0.902402640256826       0.00631744869723862
+```
+
+And the decoupled plant using the estimated optimal decoupling matrices is:
+
+```matlab
+H_dec = inv(Ty_est) * H * inv(Tu_est);
+```
+
+<a id="orgcb9fe2a"></a>
+
+{{< figure src="/ox-hugo/stoev17_results_decoupling_example.png" caption="Figure 4: Diagonal, coupled and decoupled plants" >}}
+
+
+## Conclusion {#conclusion}
+
+The paper presents an application for the tensor decomposition for the design of a static decoupling of a MIMO system.
+The results in this paper are obtained on a _non-parametric_ frequency domain model of the plant and indicate that the procedure is more robust that the eigen-value based decoupling.
+The advantages of this method with respect to some of th existing methods can be found when the FRF data available is disturbed by noise.
+
+

 ## Bibliography {#bibliography}

-<a id="orgc430b4f"></a>Stoev, Julian, Julien Ertveldt, Tom Oomen, and Johan Schoukens. 2017. “Tensor Methods for Mimo Decoupling and Control Design Using Frequency Response Functions.” _Mechatronics_ 45:71–81. <https://doi.org/https://doi.org/10.1016/j.mechatronics.2017.05.009>.
+<a id="org29c5c20"></a>Stoev, Julian, Julien Ertveldt, Tom Oomen, and Johan Schoukens. 2017. “Tensor Methods for Mimo Decoupling and Control Design Using Frequency Response Functions.” _Mechatronics_ 45:71–81. <https://doi.org/https://doi.org/10.1016/j.mechatronics.2017.05.009>.