Update Content - 2021-08-24

content/article/aaaaaaaaaaaaaaaaaaaaaaa.md

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+title = "An extended approach of inverted decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
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content/article/garrido11_exten_approac_inver_decoup.md

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+title = "An extended approach of inverted decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
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content/article/garrido12_centr_multiv_contr_by_simpl_decoup.md

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+title = "Centralized Multivariable Control By Simplified Decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Decoupled Control](decoupled_control.md)
+
+Reference
+: ([Garrido, Vázquez, and Morilla 2012](#orgdf49544))
+
+Author(s)
+: Garrido, J., Francisco V\\'azquez, & Morilla, F.
+
+Year
+: 2012
+
+
+## Introduction {#introduction}
+
+Most decoupling approaches use the conventional decoupling scheme in Figure [1](#orge2ae292) with:
+
+-   \\(G(s)\\) the process matrix
+-   \\(D(s)\\) the decoupler matrix
+-   \\(C(s)\\) the diagonal control matrix
+
+The design of the decoupler is obtained from:
+
+\begin{equation}
+D(s) = G^{-1} (s) \cdot Q(s)
+\end{equation}
+
+where \\(Q(s)\\) is the desired apparent process which is a diagonal matrix.
+
+The main problem of this methodology is the fact that the complexity of the decoupler elements increases for high dimensional MIMO processes, which may require model reductions.
+
+An alternative decoupling methods, called _inverted decoupling_, maintains very simple apparent processes and decoupler element independently of the system size.
+However, inverted decoupling cannot be applied to processes with multivariable [Right Half Plane Zeros](right_half_plane_zeros.md).
+
+<a id="orge2ae292"></a>
+
+{{< figure src="/ox-hugo/garrido12_decoupling_control_system.png" caption="Figure 1: Block diagram of a decoupling control system" >}}
+
+This work focuses on one of the most extended forms of conventional decoupling called simplified decoupling, in  which \\(n\\) elements of the decoupler are set to unity.
+When the system has two inputs and two outputs (TITO), the simplified decoupling \\(G(s)\\) is given by:
+
+\begin{equation}
+D(s) = \begin{bmatrix}
+1 & -g\_{12}(s)/g\_{11}(s) \\\\\\
+-g\_{21}(s)/g\_{22}(s) & 1
+\end{bmatrix}
+\end{equation}
+
+And the decoupled apparent process \\(Q(s)\\) is given by:
+
+\begin{equation}
+Q(s) = G(s) \cdot D(s) = \begin{bmatrix}
+g\_{11}(s) - \frac{g\_{21}(s g\_{12}(s))}{g\_{22}(s)} & 0 \\\\\\
+0 & g\_{22}(s) - \frac{g\_{21}(s)g\_{12}(s)}{g\_{11}(s)}
+\end{bmatrix}
+\end{equation}
+
+In cases where the system is larger than 2x2, the decoupler elements set to unity are always the diagonal ones as found using:
+
+\begin{equation}
+D(s) = G(s)^{-1} (\text{diag}(G(s)^{-1}))^{-1}
+\end{equation}
+
+In this work, a simplified decoupling strategy is proposed for stable processes with possibly RHP zeros and time delays.
+
+
+## Methodology {#methodology}
+
+Assuming that the process \\(G(s)\\) may have RHP zeros and time delays, but does not have any unstable poles, the decoupler matrix \\(D(s)\\) is obtained as follows (one of many possible configurations):
+
+\begin{equation}
+D(s) = \begin{bmatrix}
+1 & \frac{\text{adj}G\_{12}}{\text{adj}G\_{22}} & \dots & \frac{\text{adj}G\_{1n}}{\text{adj}\_{nn}} \\\\\\
+\frac{\text{adj}G\_{21}}{\text{adj}G\_{11}} & 1 & \dots & \frac{\text{adj}G\_{2n}}{\text{adj}\_{nn}} \\\\\\
+\vdots & \vdots & \ddots & \vdots \\\\\\
+\frac{\text{adj}G\_{n1}}{\text{adj}G\_{11}} & \frac{\text{adj}G\_{n2}}{\text{adj}G\_{22}} & \dots & 1
+\end{bmatrix}
+\end{equation}
+
+And the decoupled apparent plant is:
+
+\begin{equation}
+A(s) = \begin{bmatrix}
+\frac{|G|}{\text{adj}G\_{11}} & 0 & \dots & 0 \\\\\\
+0 & \frac{|G|}{\text{adj}G\_{22}} & \dots & 0 \\\\\\
+\vdots & \vdots & \ddots & \vdots \\\\\\
+0 & 0 & \dots & \frac{|G|}{\text{adj}G\_{nn}}
+\end{bmatrix}
+\end{equation}
+
+where \\(|G(s)|\\) is the determinant of \\(G(s)\\), \\(\text{adj}G(s)\\) is the adjugate matrix of \\(G(s)\\), that is, the transpose of the cofactor matrix of \\(G(s)\\).
+
+The proposed general simplified decoupling control is performed in three steps:
+
+1.  select a configuration: select the \\(n\\) elements of \\(D(s)\\) to be set to unity, one for each column
+2.  Compose the decoupler elements of \\(D(s)\\)
+3.  Design the \\(n\\) controllers of the diagonal control \\(C(s)\\) for the decoupled processes
+
+The realizability requirement for the decoupler is that all of its elements must be proper, causal and stable.
+For processes with time delays, non-minimum phase zeros or different relative degrees, direct calculation of the decoupler element can lead to elements with RHP poles or negative relative degrees.
+
+Several advice for the proper chose of the configuration are given in the paper.
+
+
+## Design and practical considerations {#design-and-practical-considerations}
+
+It is usually necessary to approximate the expressions of \\(|G(s)|\\) and \\(\text{adj}G(s)\\) as it usually give non-rational expressions.
+
+
+## Bibliography {#bibliography}
+
+<a id="orgdf49544"></a>Garrido, Juan, Francisco Vázquez, and Fernando Morilla. 2012. “Centralized Multivariable Control by Simplified Decoupling.” _Journal of Process Control_ 22 (6):1044–62. <https://doi.org/10.1016/j.jprocont.2012.04.008>.

content/article/heertjes11_minim_cross_talk_high_precis.md

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+title = "Minimizing cross-talk in high-precision motion systems using data-based dynamic decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Decoupled Control](decoupled_control.md)
+
+Reference
+: ([Heertjes and Engelen 2011](#org76c1202))
+
+Author(s)
+: Heertjes, M., & Engelen, A. v.
+
+Year
+: 2011
+
+> In the field of high-precision motion control, a static decoupling control design is generally used to command motion in the directions of an orthogonal basis.
+> Around the center-of-gravity of the system it then usually suffices to apply single-input single-output control in each of these directions separately.
+> Among the advantages are robust stability and performance through straightforward control designs and loop shaping techniques.
+>
+> If the static decoupling part does not fully achieve desired decoupling of the underlying MIMO motion system, a multi-variable controller can be sought to replace the SISO controller part.
+> A more natural approach would therefore be to replace the MIMO static decoupling part by a dynamic part and leave the SISO controller part intact.
+
+<!--quoteend-->
+
+> The aim of the paper is to minimize directly the cross-talk outputs via data-based optimization.
+> The criterion to be optimized consists solely of time-domain signals taken from a performance-relevant time interval.
+
+
+## Bibliography {#bibliography}
+
+<a id="org76c1202"></a>Heertjes, Marcel, and Arjan van Engelen. 2011. “Minimizing Cross-Talk in High-Precision Motion Systems Using Data-Based Dynamic Decoupling.” _Control Engineering Practice_ 19 (12):1423–32. <https://doi.org/10.1016/j.conengprac.2011.07.016>.

content/article/lee03_posit_contr_stewar_platf_using.md

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+title = "Position control of a stewart platform using inverse dynamics control with approximate dynamics"
+author = ["Thomas Dehaeze"]
+draft = false
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content/article/liu19_review_indus_mimo_decoup_contr.md

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+title = "A review of industrial mimo decoupling control"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: ([Liu et al. 2019](#org9f65386))
+
+Author(s)
+: Liu, L., Tian, S., Xue, D., Zhang, T., Chen, Y., & Zhang, S.
+
+Year
+: 2019
+
+-\* Liu, L. et al. (2019): A review of industrial mimo decoupling control :article:ignore:
+
+
+## Bibliography {#bibliography}
+
+<a id="org9f65386"></a>Liu, Lu, Siyuan Tian, Dingyu Xue, Tao Zhang, YangQuan Chen, and Shuo Zhang. 2019. “A Review of Industrial Mimo Decoupling Control.” _International Journal of Control, Automation and Systems_ 17 (5):1246–54. <https://doi.org/10.1007/s12555-018-0367-4>.

content/article/nosova20_review_paral_struc_mechan_with_kinem_decoup.md

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+title = "A review of the parallel structure mechanisms with kinematic decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Parallel Manipulators](parallel_manipulators.md)
+
+Reference
+: ([Nosova 2020](#orgc1a99bc))
+
+Author(s)
+: Nosova, N. Y.
+
+Year
+: 2020
+
+
+## Introduction {#introduction}
+
+Parallel mechanisms can be characterized by high speeds, since the engines are mounted on the base and the links have a relatively small mass.
+The disadvantages are: limited working space, the presence of singularities in the immediate vicinity of the workspace.
+
+The kinematic decoupling for a parallel structure manipulator consists in that one movement of the output platform is provided by only one input link or group of links of the kinematic chain.
+
+
+## Types of Kinematic Decoupling {#types-of-kinematic-decoupling}
+
+There are three different types of decoupling:
+
+1.  **strong coupling**: where each configuration parameter is a function of all joint variable (e.g. Stewart platform)
+2.  **complete decoupling**: each configuration parameter is a function of only one joint variable (e.g. Ortoglide)
+3.  **partial decoupling**: some configuration parameters are in function of only some joint variables
+
+
+
+## Bibliography {#bibliography}
+
+<a id="orgc1a99bc"></a>Nosova, N. Yu. 2020. “A Review of the Parallel Structure Mechanisms with Kinematic Decoupling.” _Advanced Technologies in Robotics and Intelligent Systems_. Springer International Publishing, 247–55. <https://doi.org/10.1007/978-3-030-33491-8%5F30>.

content/article/sdlkfj.md

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+title = "An extended approach of inverted decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
++++

content/article/souleille18_concep_activ_mount_space_applic.md

@@ -5,10 +5,10 @@ draft = false
 +++
 
 Tags
-: [Active Damping]({{< relref "active_damping" >}})
+: [Active Damping](active_damping.md)
 
 Reference
-: ([Souleille et al. 2018](#org34cc88f))
+: ([Souleille et al. 2018](#orgdd47abc))
 
 Author(s)
 : Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
@@ -23,10 +23,10 @@ This article discusses the use of Integral Force Feedback with amplified piezoel
 
 ## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
 
-Figure [1](#org5ad28c2) shows a picture of the amplified piezoelectric stack.
+Figure [1](#org4d65c6e) shows a picture of the amplified piezoelectric stack.
 The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
 
-<a id="org5ad28c2"></a>
+<a id="org4d65c6e"></a>
 
 {{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
 
@@ -61,34 +61,34 @@ and the control force is given by:
   f = F\_s G(s) = F\_s \frac{g}{s}
 \end{equation}
 
-The effect of the controller are shown in Figure [2](#org985b671):
+The effect of the controller are shown in Figure [2](#org3336e8f):
 
 -   the resonance peak is almost critically damped
 -   the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
 -   the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\)
 -   the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
 
-<a id="org985b671"></a>
+<a id="org3336e8f"></a>
 
 {{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
 
-<a id="orgd326cea"></a>
+<a id="org20a69be"></a>
 
 {{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}}
 
 
 ## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators}
 
-A heavy payload is mounted on a set of three isolators (Figure [4](#org62f47a3)).
+A heavy payload is mounted on a set of three isolators (Figure [4](#orga310d92)).
 The payload consists of two  masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
 
-<a id="org62f47a3"></a>
+<a id="orga310d92"></a>
 
 {{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
 
-As shown in Figure [5](#org5b0f55b), both the suspension modes and the flexible modes of the payload can be critically damped.
+As shown in Figure [5](#org3c2e029), both the suspension modes and the flexible modes of the payload can be critically damped.
 
-<a id="org5b0f55b"></a>
+<a id="org3c2e029"></a>
 
 {{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
 
@@ -96,4 +96,4 @@ As shown in Figure [5](#org5b0f55b), both the suspension modes and the flexible
 
 ## Bibliography {#bibliography}
 
-<a id="org34cc88f"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.
+<a id="orgdd47abc"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.

content/article/stein03_respec_unstab.md

@@ -9,7 +9,7 @@ Tags
 
 
 Reference
-: ([Stein 2003](#org50b7ac8))
+: ([Stein 2003](#orge299f80))
 
 Author(s)
 : Stein, G.
@@ -73,15 +73,15 @@ It is here proposed to call it **dirt**
 
 </div>
 
-The job of a serious control designer is then to more dirt from one place to another, using appropriate tools, without being able to get rid of any of it (illustrated in Figure [1](#org956e0bf)).
+The job of a serious control designer is then to more dirt from one place to another, using appropriate tools, without being able to get rid of any of it (illustrated in Figure [1](#orgf166bde)).
 
-<a id="org956e0bf"></a>
+<a id="orgf166bde"></a>
 
 {{< figure src="/ox-hugo/stein03_serious_design.png" caption="Figure 1: Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies" >}}
 
-In the same spirit, the job of a more academic control designer with more abstract tools such as LQG, \\(\mathcal{H}\_\infty\\), is to set parameters (weights) of a synthesis machine to adjust the contours of the machine's digging blades to get just the right shape for the sensitivity function (Figure [2](#org73c6dd3)).
+In the same spirit, the job of a more academic control designer with more abstract tools such as LQG, \\(\mathcal{H}\_\infty\\), is to set parameters (weights) of a synthesis machine to adjust the contours of the machine's digging blades to get just the right shape for the sensitivity function (Figure [2](#org29aa88f)).
 
-<a id="org73c6dd3"></a>
+<a id="org29aa88f"></a>
 
 {{< figure src="/ox-hugo/stein03_formal_design.png" caption="Figure 2: Sensitivity shaping automated by modern control tools" >}}
 
@@ -89,9 +89,9 @@ In the same spirit, the job of a more academic control designer with more abstra
 ## Available bandwidth {#available-bandwidth}
 
 An argument is sometimes made that the Bode integrals are not really restrictive because we only seek to dig holes over finite frequency bands.
-We then have an infinite frequency range left over into which to dump the dirt, so we can make the layer arbitrarily thin (Figure [3](#orgb2839fe)).
+We then have an infinite frequency range left over into which to dump the dirt, so we can make the layer arbitrarily thin (Figure [3](#orgf23d7a5)).
 
-<a id="orgb2839fe"></a>
+<a id="orgf23d7a5"></a>
 
 {{< figure src="/ox-hugo/stein03_spreading_it_thin.png" caption="Figure 3: It is possible to spead the increase of the sensitivity function over a larger frequency band" >}}
 
@@ -121,6 +121,7 @@ All the action of the feedback design, the sensitivity improvements as well as t
 Only a small error \\(\delta\\) occurs outside that range, associated with the tail of the complete integrals.
 
 
+
 ## Bibliography {#bibliography}
 
-<a id="org50b7ac8"></a>Stein, Gunter. 2003. “Respect the Unstable.” _IEEE Control Systems Magazine_ 23 (4). IEEE:12–25.
+<a id="orge299f80"></a>Stein, Gunter. 2003. “Respect the Unstable.” _IEEE Control Systems Magazine_ 23 (4). IEEE:12–25.

content/article/sun16_pract_multiv_contr_approac_based.md

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++++
+title = "A practical multivariable control approach based on inverted decoupling and decentralized active disturbance rejection control"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Decoupled Control](decoupled_control.md)
+
+Reference
+: ([Sun et al. 2016](#org2268976))
+
+Author(s)
+: Sun, L., Dong, J., Li, D., & Lee, K. Y.
+
+Year
+: 2016
+
+
+
+## Bibliography {#bibliography}
+
+<a id="org2268976"></a>Sun, Li, Junyi Dong, Donghai Li, and Kwang Y. Lee. 2016. “A Practical Multivariable Control Approach Based on Inverted Decoupling and Decentralized Active Disturbance Rejection Control.” _Industrial & Engineering Chemistry Research_ 55 (7):2008–19. <https://doi.org/10.1021/acs.iecr.5b03738>.

content/article/tlkj.md

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++++
+title = "An extended approach of inverted decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
++++

content/article/voorhoeve15_ident_high_tech_motion_system.md

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++++
+title = "Identification of high-tech motion systems: an active vibration isolation benchmark"
+author = ["Thomas Dehaeze"]
+draft = false
++++

content/book/albertos04_multiv_contr_system.md

@@ -8,7 +8,7 @@ Tags
 : [Multivariable Control](multivariable_control.md)
 
 Reference
-: ([Albertos and Antonio 2004](#org22a156f))
+: ([Albertos and Antonio 2004](#orga6ef935))
 
 Author(s)
 : Albertos, P., & Antonio, S.
@@ -19,6 +19,9 @@ Year
 
 ## [Decentralized Control](decentralized_control.md) and Decoupled Control {#decentralized-control--decentralized-control-dot-md--and-decoupled-control}
 
+
+### Introduction {#introduction}
+
 Decentralized control is decomposed into two steps:
 
 1.  decoupled the plant into several subsystems
@@ -26,6 +29,42 @@ Decentralized control is decomposed into two steps:
 
 The initial effort of decoupling the system results in subsequent easier design, implementation and tuning.
 
+Decentralized control tries to control multivariable plants by a suitable decomposition into SISO control loops.
+If the process has strong coupling or conditioning problems, centralized control may be required.
+It however requires the availability of a precise model.
+
+Two approaches can be used to control a coupled system with SISO techniques:
+
+-   **decentralized control** tries to divide the plant and design _independent_ controllers for each subsystems.
+    Two alternative arise:
+    -   neglect the coupling
+    -   carry out a _decoupling_ operation by "canceling" coupling by transforming the system into a diagonal or triangular structure bia a transformation matrix
+-   **cascade control**
+
+
+### Mutli-Loop Control, Pairing Selection {#mutli-loop-control-pairing-selection}
+
+The strategy called _multi-loop control_ consists of first proper input/output pairing, and then design of several SISO controllers.
+In this way, a complex control problem is divided into several simpler ones.
+
+The multi-loop control may not work in strongly coupled systems.
+Therefore, a methodology the access the degree of interaction between the loops is needed.
+
+
+#### [Relative Gain Array](relative_gain_array.md) {#relative-gain-array--relative-gain-array-dot-md}
+
+The Relative Gain Array (RGA) \\(\Lambda(s)\\) is defined as:
+
+\begin{equation}
+\Lambda(s) = G(s) \times (G(s)^T)^{-1}
+\end{equation}
+
+The RGA is scaling-independent and controller-independent.
+These coefficients can be interpreted as the ratio between the open-loop SISO static gain and the gain with "perfect" control on the rest of the loops.
+
+For demanding control specifications, the values of \\(\Lambda\\) car be drawn as a function of frequency.
+In this case, at frequencies important for control stability robustness (around the peak of the sensitivity transfer function), if \\(\Lambda(j\omega)\\) approaches the identity matrix, stability problems are avoided in multi-loop control.
+
 
 ### Decoupling {#decoupling}
 
@@ -40,16 +79,17 @@ This strategy is called **decoupling**.
 
 #### Feedforward Decoupling {#feedforward-decoupling}
 
-A pre-compensator can be added to transform the open-loop characteristics into a new one as chosen by the designer.
+A pre-compensator (Figure [1](#org7023330)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
 This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
 
+<a id="org7023330"></a>
+
+{{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="Figure 1: Decoupler pre-compensator" >}}
+
 **Approximate decoupling**:
 To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
 If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
 
-
-#### Feedback Decoupling {#feedback-decoupling}
-
 Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
 
 -   as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops.
@@ -72,12 +112,12 @@ where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
 The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
 In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
 
-If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#orgbba6502), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
+If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [2](#org2de6de7), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
 Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
 
-<a id="orgbba6502"></a>
+<a id="org2de6de7"></a>
 
-{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
+{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 2: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
 
 The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control".
 
@@ -88,13 +128,22 @@ SVD decoupling produces the most suitable combinations for independent "multi-lo
 If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control.
 
 
-## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
+### Conclusions {#conclusions}
 
+In this chapter, the control of systems with multiple inputs and outputs is discussed using SISO-based tools, either directly or after some multivariable decoupling transformations.
 
-## Optimisation-based Control {#optimisation-based-control}
+Multi-loop strategies, if suitable, may present th advantages of fault tolerance, as well as simplicity.
+However, in some cases, tuning may be difficult and coupling may severely limit their performance.
 
+Decoupling is based on mathematical transformations of the system models into diagonal form.
+Feedforward decoupling can be used in many cases.
+Feedback decoupling achieves its objective if state is measurable and system is minimum-phase.
+However, decoupling may be very sensitive to modelling errors and it is not the optimal strategy for disturbance rejection.
 
-## Designing for Robustness {#designing-for-robustness}
+Cascade control is widely used in industry to improve the behaviour of basic SISO loops via the addition of extra sensors and actuators.
+However, ease of tuning requires that different time constants are involved in different subsystems.
+In general, addition of extra sensors and actuators in a SISO or MIMO loop, will improve achievable performance and/or tolerance to modelling errors.
+The level of improvement must be traded off against the cost of additional instrumentation.
 
 
 ## Implementation and Other Issues {#implementation-and-other-issues}
@@ -142,4 +191,4 @@ The solution is similar to that of the wind-up phenomenon: the regulator should
 
 ## Bibliography {#bibliography}
 
-<a id="org22a156f"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
+<a id="orga6ef935"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.

content/inproceedings/vaes04_optim_mimo.md

@@ -0,0 +1,23 @@
++++
+title = "Optimal decoupling for mimo-controller design with robust performance"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: ([Vaes, Swevers, and Sas 2004](#org5d17a4b))
+
+Author(s)
+: Vaes, D., Swevers, J., & Sas, P.
+
+Year
+: 2004
+
+
+## Bibliography {#bibliography}
+
+<a id="org5d17a4b"></a>Vaes, D., J. Swevers, and P. Sas. 2004. “Optimal Decoupling for MIMO-Controller Design with Robust Performance.” In _Proceedings of the 2004 American Control Conference_, nil. <https://doi.org/10.23919/acc.2004.1384036>.

content/sldkfj/garrido11_exten_approac_inver_decoup.md

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+title = "An extended approach of inverted decoupling"
+author = ["Thomas Dehaeze"]
+draft = false
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content/techreport/merlet87_paral_manip.md

@@ -5,10 +5,10 @@ draft = false
 +++
 
 Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}})
+: [Stewart Platforms](stewart_platforms.md)
 
 Reference
-: ([Merlet 1987](#org07bdf3f))
+: ([Merlet 1987](#org6a131ba))
 
 Author(s)
 : Merlet, J.
@@ -17,6 +17,7 @@ Year
 : 1987
 
 
+
 ## Bibliography {#bibliography}
 
-<a id="org07bdf3f"></a>Merlet, Jean-Pierre. 1987. “Parallel Manipulators. Part I: Theory Design, Kinematics, Dynamics and Control.” INRIA.
+<a id="org6a131ba"></a>Merlet, Jean-Pierre. 1987. “Parallel Manipulators. Part I: Theory Design, Kinematics, Dynamics and Control.” INRIA.

content/zettels/right_half_plane_zeros.md

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+title = "Right Half Plane Zeros"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+Right Half Plane (RHP) Zeros are present in a broad range of systems.
+
+-   SISO RHP zeros:
+    -   \\(G(z) = 0\\). The real part of \\(z\\) is positive
+-   MIMO RHP zeros:
+    -   RHP zeros of the determinant of the transfer function matrix
+    -   For MIMO plants, the zero is associated with an output direction
+-   Consequences:
+    -   Non minimum phase
+    -   Usually, if looking at the step response, at first the response goes in the wrong direction
+    -   The frequency of the RHP zero should be outside the controller bandwidth.
+        It therefore impose a fundamental limitation to the controller bandwidth.
+        The reason is that inside the control bandwidth, the controller basically inverts the plant, and the inverse of a RHP zero is an unstable pole.

content/zettels/test_page_ok.md

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+title = "test page ok"
+author = ["Thomas Dehaeze"]
+draft = false
++++