diff --git a/content/book/albertos04_multiv_contr_system.md b/content/book/albertos04_multiv_contr_system.md
index 1fb920d..33e4a84 100644
--- a/content/book/albertos04_multiv_contr_system.md
+++ b/content/book/albertos04_multiv_contr_system.md
@@ -5,10 +5,10 @@ draft = true
+++
Tags
-: [Multivariable Control]({{< relref "multivariable_control" >}})
+: [Multivariable Control](multivariable_control.md)
Reference
-: ([Albertos and Antonio 2004](#orga1617be))
+: ([Albertos and Antonio 2004](#org22a156f))
Author(s)
: Albertos, P., & Antonio, S.
@@ -17,19 +17,14 @@ Year
: 2004
-## Introduction to Multivariable Control {#introduction-to-multivariable-control}
+## [Decentralized Control](decentralized_control.md) and Decoupled Control {#decentralized-control--decentralized-control-dot-md--and-decoupled-control}
+Decentralized control is decomposed into two steps:
-## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
+1. decoupled the plant into several subsystems
+2. control the subsystems
-
-## Linear Systems Analysis {#linear-systems-analysis}
-
-
-## Solutions to the Control Problem {#solutions-to-the-control-problem}
-
-
-## Decentralised and Decoupled Control {#decentralised-and-decoupled-control}
+The initial effort of decoupling the system results in subsequent easier design, implementation and tuning.
### Decoupling {#decoupling}
@@ -37,7 +32,7 @@ Year
In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one.
This strategy is called **decoupling**.
-[Decoupled Control]({{< relref "decoupled_control" >}}) can be achieved in two ways:
+[Decoupled Control](decoupled_control.md) can be achieved in two ways:
- feedforward cancellation of the cross-coupling terms
- based on state measurements, via a feedback law
@@ -66,7 +61,7 @@ Although at first glance, decoupling seems an appealing idea, there are some dra
#### SVD Decoupling {#svd-decoupling}
-A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as:
+A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition](singular_value_decomposition.md) as:
\begin{equation}
M = U \Sigma V^T
@@ -77,10 +72,10 @@ 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](#orgd447864), 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 [1](#orgbba6502), 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.
-
+
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
@@ -104,29 +99,47 @@ If some of the vectors in \\(V\\) (input directions) have a significant componen
## Implementation and Other Issues {#implementation-and-other-issues}
+There are two main categories for the implementation of MIMO control:
-## Appendices {#appendices}
+- Decentralized, Decoupled, Cascade
+- Centralized, optimization based
+
+A fundamental reason to use cascade and decentralized control in most practical applications is because they require less modelling effort.
+Other advantages of cascade and decentralized control are:
+
+- its behaviour can be easily understood
+- standard equipment can be used (PID controllers, etc.)
+- their decoupled behavior enables easier tuning with model-free strategies
+- decentralized implementation tends to be more fault-tolerant, as individual loops will try to keep their set-points even in the case some other components have failed.
-### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
+### [Anti-Windup Control](anti_windup_control.md) {#anti-windup-control--anti-windup-control-dot-md}
+
+In practice, it is possible that an actuator saturate.
+In such case, the feedback path is broken, and this has several implications:
+
+- unstable processes: the process output might go out of control
+- multi-loop and centralized control: even with stable plants, opening a feedback path may cause the overall loop to become unstable
+
+The wind-up problem can appear with integral action regulators: during significative step changes in the set point, the integral of the error keeps accumulation and when reaching the desired set-point the accumulated integral action produces a significant overshoot increment.
+In SISO PID regulators, anti-windup schemes are implemented by either stopping integration if the actuator is saturated or by implementing the following control law:
+
+\begin{equation}
+u = K(r - y) - K T\_D \frac{dy}{dt} + \int K T\_i^{-1} (r - y) + T\_t^{-1} (u\_m - u) dt \label{eq:antiwindup\_pid}
+\end{equation}
+
+where \\(u\\) is the calculated control action and \\(u\_m\\) is the actual control action applied to the plant.
+In non-saturated behaviour, \\(u=u\_m\\) and the equation is the ordinary PID.
+In saturation, \\(u\_m\\) is a constant and the resulting equations drive \\(u\\) down towards \\(u\_m\\) dynamically, with time constant \\(T\_T\\).
-### Matrices {#matrices}
+### [Bumpless Transfer](bumpless_transfer.md) {#bumpless-transfer--bumpless-transfer-dot-md}
-
-### Signal and System Norms {#signal-and-system-norms}
-
-
-### Optimisation {#optimisation}
-
-
-### Multivariable Statistics {#multivariable-statistics}
-
-
-### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
+When switching on the regulator, significant transient behavior can be seen and the controller may saturate the actuators.
+The solution is similar to that of the wind-up phenomenon: the regulator should be always on, carrying out calculations by using \eqref{eq:antiwindup_pid}.
## Bibliography {#bibliography}
-Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. .
+Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. .
diff --git a/content/zettels/decentralized_control.md b/content/zettels/decentralized_control.md
new file mode 100644
index 0000000..c95ec22
--- /dev/null
+++ b/content/zettels/decentralized_control.md
@@ -0,0 +1,10 @@
++++
+title = "Decentralized Control"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+Decentralized control is a way to control a MIMO system by individually controlling the inputs and outputs using SISO controllers.
diff --git a/content/zettels/stewart_platforms.md b/content/zettels/stewart_platforms.md
index f65509f..1a1ae69 100644
--- a/content/zettels/stewart_platforms.md
+++ b/content/zettels/stewart_platforms.md
@@ -36,37 +36,37 @@ Tags
Papers by J.E. McInroy:
-- ([O’Brien et al. 1998](#orgf349082))
-- ([McInroy, O’Brien, and Neat 1999](#org5767dd1))
-- ([McInroy 1999](#org2c21b8d))
-- ([McInroy and Hamann 2000](#org4297441))
-- ([Chen and McInroy 2000](#org8bb7a6a))
-- ([McInroy 2002](#org9b28444))
-- ([Li, Hamann, and McInroy 2001](#orgf3f89be))
-- ([Lin and McInroy 2003](#org950f17b))
-- ([Jafari and McInroy 2003](#orge2968a9))
-- ([Chen and McInroy 2004](#org5ac50ad))
+- ([O’Brien et al. 1998](#org320ae46))
+- ([McInroy, O’Brien, and Neat 1999](#orgea92441))
+- ([McInroy 1999](#orgfb23e22))
+- ([McInroy and Hamann 2000](#org90e178e))
+- ([Chen and McInroy 2000](#org418808c))
+- ([McInroy 2002](#orgaa607f0))
+- ([Li, Hamann, and McInroy 2001](#org7176257))
+- ([Lin and McInroy 2003](#org6427256))
+- ([Jafari and McInroy 2003](#orga8add64))
+- ([Chen and McInroy 2004](#org3f36881))
## Bibliography {#bibliography}
-Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. .
+Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. .
-Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. .
+Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. .
-Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. .
+Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. .
-Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. .
+Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. .
-Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. .
+Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. .
-McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. .
+McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. .
-———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
+———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. .
-McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. .
+McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. .
-McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. .
+McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. .
-O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. .
+O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. .