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-title = "Advanced Motion Control Design"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-:
-
-
-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.
-
-
-
-
-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
-
-
-
-
-## 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.
-
-
-
-
-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
-
-
-
-
-#### 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
-
-
-
-
- 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.
-
-
-- Structure Singular Value (SSV) of interaction as multiplicative output uncertainty
-
-
-
-
- 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.
-
-
-
-
-#### 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)\\).
-
-
-
-
-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))\\)
-
-
-
-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}
-
-Levine, W. S. 2011. _Control System Applications_. The Control Handbook. Boca Raton: CRC Press.