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content/book/levine11_contr_system_applic.md

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+title = "Advanced Motion Control Design"
+author = ["Thomas Dehaeze"]
+draft = false
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+
+Tags
+:
+
+
+Reference
+: ([Levine 2011](#org5f2e773)), chapter 27
+
+Author(s)
+: Levine, W. S.
+
+Year
+: 2011
+
+
+## Introduction {#introduction}
+
+The industrial state of the art control of motion systems can be summarized as follows.
+Most systems, by design, are either decoupled, or can be decoupled using static input-output transformations.
+Hence, most motion systems and their motion software architecture use SISO control design methods and solutions.
+Feedback design is mostly done in the frequency domain, using [Loop-Shaping](loop_shaping.md) techniques.
+A typical motion controller has a PID structure, with a low pass at high frequencies and one or two notch filters to compensate flexible dynamics.
+In addition to the feedback controller, a feedforward controller is applied with acceleration, velocity from the reference signal.
+The setpoint itself is a result of a setpoint generator with jerk limitation profiles (see [Trajectory Generation](trajectory_generation.md)).
+If the requirements increase, the dynamic coupling between the various DOFs can no longer be neglected and more advanced MIMO control is required.
+
+<div class="important">
+  <div></div>
+
+Step by step procedure:
+
+1.  Interaction Analysis
+2.  Decoupling
+3.  Independent SISO design
+4.  Sequential SISO design
+5.  Norm-based MIMO design
+
+</div>
+
+<div class="definition">
+  <div></div>
+
+Centralized control
+: the transfer function matrix of the controller is allowed to have any structure
+
+Decentralized control
+: diagonal controller transfer function, but constant decoupling manipulations of inputs and outputs are allowed
+
+Independent decentralized control
+: a single loop is designed without taking into account the effect of earlier or later designed loops
+
+Sequential decentralized control
+: a single loop is designed with taking into account the effect of all earlier closed loops
+
+</div>
+
+
+## Motion Systems {#motion-systems}
+
+Here, we focus on the control of linear time invariant electromechanical motion systems that have the same number of actuators and sensors as Rigid Body modes.
+The dynamics of such systems are often dominated by the mechanics, such that:
+
+\begin{equation}
+G\_p(s) = \sum\_{i=1}^{N\_{rb}} \frac{c\_i b\_i^T}{s^2} + \sum\_{i=N\_{rb} + 1}^{N} \frac{c\_ib\_i^T}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
+\end{equation}
+
+with \\(N\_{rb}\\) is the number of rigid body modes.
+The vectors \\(c\_i,b\_i\\) span the directions of the ith mode shapes.
+
+If the resonance frequencies \\(\omega\_i\\) are high enough, the plant can be approximately decoupled using static input/output transformations \\(T\_u,T\_y\\) so that:
+
+\begin{equation}
+G\_{yu} = T\_y G\_p(s) T\_u = \frac{1}{s^2} \begin{bmatrix}
+m      & 0       & & \dots  &     & 0 \\\\\\
+0      & m       & &        &     & \\\\\\
+       & &       m & \ddots &     & \vdots \\\\\\
+\vdots & & \ddots & I\_x      &     & \\\\\\
+       & &       &          & I\_y & 0 \\\\\\
+0      & & \dots &          & 0   & I\_z
+\end{bmatrix} + G\_{\text{flex}}(s)
+\end{equation}
+
+
+## Feedback Control Design {#feedback-control-design}
+
+
+### [Loop-Shaping](loop_shaping.md) - The SISO case {#loop-shaping--loop-shaping-dot-md--the-siso-case}
+
+The key idea of loopshaping is the modification of the controller such that the open-loop is made according to specifications.
+The reason this works well is that the controller inters linearly into the open-loop transfer function \\(L(s) = G(s)K(s)\\).
+However, in practice all specifications are of course given in terms of the final system performance, that is, as closed-loop specifications.
+So we should convert the closed-loop specifications into specifications on the open-loop.
+
+Let us assume we know the spectral contents of the disturbance.
+Take as an example the simple case of a disturbance being a sinusoid of known amplitude and frequency.
+If we know the specifications on the error amplitude, we can derive the requirement on the process sensitivity at that frequency.
+Since at low frequency the sensitivity can be approximated as the inverse of the open-loop, we can translate this into a specification of the open-loop at that frequency.
+Because we know that the slope of the open-loop of a well tuned motion system will be between -2 and -1, we can estimate the required crossover frequency.
+
+
+### Loop-Shaping - The MIMO case {#loop-shaping-the-mimo-case}
+
+
+## Bibliography {#bibliography}
+
+<a id="org5f2e773"></a>Levine, W. S. 2011. _Control System Applications_. The Control Handbook. Boca Raton: CRC Press.