digital-brain/content/inbook/levine11_contr_system_applic.md

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title = "Advanced Motion Control Design"
author = ["Thomas Dehaeze"]
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Reference
: ([Levine 2011](#orgb7728f4)), 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="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 enters 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.

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}

In MIMO systems, it is much less trivial to apply loopshaping.
The stability is determined by the closed-loop polynomial, \\(\det(I + L(s))\\), and the characteristic loci (eigenvalues of the FRF \\(L(j\omega)\\) in the complex plane) can be used for this graphically.
A system with N inputs and N outputs has N characteristic loci.

If each eigen value locus does not encircle the point (-1,0), the MIMO system is closed-loop stable.
The shaping of these eigenvalue loci is not straightforward if the plant has large off-diagonal elements.
In that case, a single element of the controller will affect more eigenvalue loci.

The strong non-intuitive aspect of MIMO loopshaping and the fact that SISO loopshaping is used often, are major obstacles in application of modern design tools in industrial motion systems.

<div class="important">
  <div></div>

For that reason, the step-by-step approach is proposed:

1.  Interaction Analysis
2.  Decoupling Transformations
3.  Independent SISO design
4.  Sequential SISO design
5.  Norm-based MIMO design

</div>


#### Interaction Analysis {#interaction-analysis}

The goal of the interaction analysis is to identify two-sided interactions in the plant dynamics.
Two measured for plant interactions can be used:

-   Relative Gain Array (RGA) per frequency

    <div class="definition">
      <div></div>

    The frequency dependent relative gain array is calculated as:

    \begin{equation}
    \text{RGA}(G(j\omega)) = G(j\omega) \times (G(j\omega)^{-1})^{T}
    \end{equation}

    where \\(\times\\) denotes element wise multiplication.

    </div>
-   Structure Singular Value (SSV) of interaction as multiplicative output uncertainty

    <div class="definition">
      <div></div>

    The structured singular value interaction measure is the following condition:

    \begin{equation}
    \mu\_D(E\_T(j\omega)) < \frac{1}{2}, \forall \omega
    \end{equation}

    with \\(E\_T(j\omega) = G\_{nd}(j\omega) G\_d^{-1}(j\omega)\\), \\(\mu\_D\\) is the structured singular value, with respect to the diagonal structure of the feedback controller.
    \\(G\_d(s)\\) are the diagonal terms of the transfer function matrix, and \\(G\_{nd}(s) = G(s) - G\_d(s)\\).

    If a diagonal transfer function matrix is used, controllers gains must be small at frequencies where this condition is not met.

    </div>


#### Decoupling Transformations {#decoupling-transformations}

A common method to reduce plant interaction is to redefine the input and output of the plant.
One can combine several inputs or outputs to control the system in more decoupled coordinates.
For motion systems most of these transformations are found on the basis of _kinematic models_.
Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
Likewise, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
This is basically the inversion of a kinematic model of the plant.

As motion systems are often designed to be light and stiff, kinematic decoupling is often sufficient to achieve acceptable decoupling at the crossover frequency.


#### Independent SISO design {#independent-siso-design}

For systems where interaction is low, or the decoupling is almost successful, one can design a _diagonal_ controller by closing each control loop independently.
The residual interaction can be accounted for in the analysis.

For this, we make use of the following decomposition:

\begin{equation}
\det(I + GK) = \det(I + E\_T T\_d) \det(I + G\_d K)
\end{equation}

with \\(T\_d = G\_d K (I + G\_d K)^{-1}\\).
\\(G\_d(s)\\) is defined to be only the diagonal terms of the plant transfer function matrix.
The effect of the non-diagonal terms of the plant \\(G\_{nd}(s) = G(s) - G\_d(s)\\) is accounted for in \\(E\_T(s)\\).

<div class="important">
  <div></div>

Then the MIMO closed-loop stability assessment can be slit up in two assessments:

-   the first for stability of N non-interacting loops, namely \\(\det(I + G\_d(s)K(s))\\)
-   the second for stability of \\(\det(I + E\_T(s)T\_d(s))\\)

</div>

If \\(G(s)\\) and \\(T\_d(s)\\) are stable, one can use the _small gain theorem_ to find a sufficient condition of stability of \\(\det(I + E\_TT\_d)\\) as

\begin{equation}
\rho(E\_T(j\omega) T\_d(j\omega)) < 1, \forall \omega
\end{equation}

where \\(\rho\\) is the spectral radius.

Due to the fact that a sufficient condition is used, independent loop closing usually leads to conservative designs.


#### Sequential SISO design {#sequential-siso-design}

If the interaction is larger, the sequential loop closing method is appropriate.
The controller is still a diagonal transfer function matrix, but each control designs are now dependent.
In principle, one starts with the open-loop FRF of the MIMO Plant.
Then one loop is closed using SISO loopshaping.
The controller is taken into the plant description, and a new FRF is obtained with one input and output less.
Then, the next loop is designed and so on.

The multivariable system is nominally closed-loop stable if in each design step the system is closed-loop stable.
However, the robustness margins in each design step do not guarantee robust stability of the final multivariable system.

Drawbacks of sequential design are:

-   the ordering of the design steps may have great impact on the achievable performance.
    There is no general approach to determine the best sequence.
-   there are no guarantees that robustness margins in earlier loops are preserved.
-   as each design step usually considers only a single output, the responses in earlier designed loops may degrade.


#### Norm-based MIMO design {#norm-based-mimo-design}

If sequential SISO design is not successful, the next step is to start norm-based control design.
This method requires a parametric model and weighting filters to express the control problem in terms of an operator norm like \\(H\_2\\) or \\(H\_\infty\\).

Parametric models are usually build up step-by-step, first considering the unmodeled dynamics as (unstructured) uncertainty.


## Bibliography {#bibliography}

<a id="orgb7728f4"></a>Levine, W. S. 2011. _Control System Applications_. The Control Handbook. Boca Raton: CRC Press.