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title = "Advanced Motion Control Design"
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author = ["Thomas Dehaeze"]
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Reference
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: ([Levine 2011](#orgb7728f4)), chapter 27
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Author(s)
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: Levine, W. S.
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Year
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: 2011
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## Introduction {#introduction}
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The industrial state of the art control of motion systems can be summarized as follows.
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Most systems, by design, are either decoupled, or can be decoupled using static input-output transformations.
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Hence, most motion systems and their motion software architecture use SISO control design methods and solutions.
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Feedback design is mostly done in the frequency domain, using [Loop-Shaping](loop_shaping.md) techniques.
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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.
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In addition to the feedback controller, a feedforward controller is applied with acceleration, velocity from the reference signal.
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The setpoint itself is a result of a setpoint generator with jerk limitation profiles (see [Trajectory Generation](trajectory_generation.md)).
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If the requirements increase, the dynamic coupling between the various DOFs can no longer be neglected and more advanced MIMO control is required.
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<div class="definition">
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<div></div>
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Centralized control
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: the transfer function matrix of the controller is allowed to have any structure
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Decentralized control
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: diagonal controller transfer function, but constant decoupling manipulations of inputs and outputs are allowed
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Independent decentralized control
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: a single loop is designed without taking into account the effect of earlier or later designed loops
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Sequential decentralized control
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: a single loop is designed with taking into account the effect of all earlier closed loops
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</div>
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## Motion Systems {#motion-systems}
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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.
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The dynamics of such systems are often dominated by the mechanics, such that:
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\begin{equation}
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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}
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\end{equation}
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with \\(N\_{rb}\\) is the number of rigid body modes.
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The vectors \\(c\_i,b\_i\\) span the directions of the ith mode shapes.
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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:
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\begin{equation}
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G\_{yu} = T\_y G\_p(s) T\_u = \frac{1}{s^2} \begin{bmatrix}
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m & 0 & & \dots & & 0 \\\\\\
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0 & m & & & & \\\\\\
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& & m & \ddots & & \vdots \\\\\\
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\vdots & & \ddots & I\_x & & \\\\\\
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& & & & I\_y & 0 \\\\\\
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0 & & \dots & & 0 & I\_z
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\end{bmatrix} + G\_{\text{flex}}(s)
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\end{equation}
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## Feedback Control Design {#feedback-control-design}
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### [Loop-Shaping](loop_shaping.md) - The SISO case {#loop-shaping--loop-shaping-dot-md--the-siso-case}
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The key idea of loopshaping is the modification of the controller such that the open-loop is made according to specifications.
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The reason this works well is that the controller enters linearly into the open-loop transfer function \\(L(s) = G(s)K(s)\\).
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However, in practice all specifications are of course given in terms of the final system performance, that is, as _closed-loop_ specifications.
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So we should convert the closed-loop specifications into specifications on the open-loop.
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Take as an example the simple case of a disturbance being a sinusoid of known amplitude and frequency.
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If we know the specifications on the error amplitude, we can derive the requirement on the process sensitivity at that frequency.
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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.
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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.
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### Loop-Shaping - The MIMO case {#loop-shaping-the-mimo-case}
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In MIMO systems, it is much less trivial to apply loopshaping.
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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.
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A system with N inputs and N outputs has N characteristic loci.
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If each eigen value locus does not encircle the point (-1,0), the MIMO system is closed-loop stable.
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The shaping of these eigenvalue loci is not straightforward if the plant has large off-diagonal elements.
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In that case, a single element of the controller will affect more eigenvalue loci.
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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.
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<div class="important">
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<div></div>
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For that reason, the step-by-step approach is proposed:
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1. Interaction Analysis
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2. Decoupling Transformations
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3. Independent SISO design
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4. Sequential SISO design
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5. Norm-based MIMO design
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</div>
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#### Interaction Analysis {#interaction-analysis}
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The goal of the interaction analysis is to identify two-sided interactions in the plant dynamics.
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Two measured for plant interactions can be used:
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- Relative Gain Array (RGA) per frequency
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<div class="definition">
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<div></div>
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The frequency dependent relative gain array is calculated as:
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\begin{equation}
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\text{RGA}(G(j\omega)) = G(j\omega) \times (G(j\omega)^{-1})^{T}
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\end{equation}
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where \\(\times\\) denotes element wise multiplication.
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</div>
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- Structure Singular Value (SSV) of interaction as multiplicative output uncertainty
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<div class="definition">
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<div></div>
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The structured singular value interaction measure is the following condition:
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\begin{equation}
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\mu\_D(E\_T(j\omega)) < \frac{1}{2}, \forall \omega
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\end{equation}
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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.
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\\(G\_d(s)\\) are the diagonal terms of the transfer function matrix, and \\(G\_{nd}(s) = G(s) - G\_d(s)\\).
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If a diagonal transfer function matrix is used, controllers gains must be small at frequencies where this condition is not met.
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</div>
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#### Decoupling Transformations {#decoupling-transformations}
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A common method to reduce plant interaction is to redefine the input and output of the plant.
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One can combine several inputs or outputs to control the system in more decoupled coordinates.
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For motion systems most of these transformations are found on the basis of _kinematic models_.
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Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
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Likewise, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
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This is basically the inversion of a kinematic model of the plant.
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As motion systems are often designed to be light and stiff, kinematic decoupling is often sufficient to achieve acceptable decoupling at the crossover frequency.
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#### Independent SISO design {#independent-siso-design}
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For systems where interaction is low, or the decoupling is almost successful, one can design a _diagonal_ controller by closing each control loop independently.
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The residual interaction can be accounted for in the analysis.
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For this, we make use of the following decomposition:
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\begin{equation}
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\det(I + GK) = \det(I + E\_T T\_d) \det(I + G\_d K)
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\end{equation}
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with \\(T\_d = G\_d K (I + G\_d K)^{-1}\\).
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\\(G\_d(s)\\) is defined to be only the diagonal terms of the plant transfer function matrix.
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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)\\).
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<div class="important">
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<div></div>
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Then the MIMO closed-loop stability assessment can be slit up in two assessments:
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- the first for stability of N non-interacting loops, namely \\(\det(I + G\_d(s)K(s))\\)
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- the second for stability of \\(\det(I + E\_T(s)T\_d(s))\\)
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</div>
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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
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\begin{equation}
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\rho(E\_T(j\omega) T\_d(j\omega)) < 1, \forall \omega
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\end{equation}
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where \\(\rho\\) is the spectral radius.
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Due to the fact that a sufficient condition is used, independent loop closing usually leads to conservative designs.
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#### Sequential SISO design {#sequential-siso-design}
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If the interaction is larger, the sequential loop closing method is appropriate.
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The controller is still a diagonal transfer function matrix, but each control designs are now dependent.
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In principle, one starts with the open-loop FRF of the MIMO Plant.
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Then one loop is closed using SISO loopshaping.
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The controller is taken into the plant description, and a new FRF is obtained with one input and output less.
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Then, the next loop is designed and so on.
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The multivariable system is nominally closed-loop stable if in each design step the system is closed-loop stable.
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However, the robustness margins in each design step do not guarantee robust stability of the final multivariable system.
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Drawbacks of sequential design are:
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- the ordering of the design steps may have great impact on the achievable performance.
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There is no general approach to determine the best sequence.
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- there are no guarantees that robustness margins in earlier loops are preserved.
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- as each design step usually considers only a single output, the responses in earlier designed loops may degrade.
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#### Norm-based MIMO design {#norm-based-mimo-design}
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If sequential SISO design is not successful, the next step is to start norm-based control design.
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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\\).
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Parametric models are usually build up step-by-step, first considering the unmodeled dynamics as (unstructured) uncertainty.
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## Bibliography {#bibliography}
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<a id="orgb7728f4"></a>Levine, W. S. 2011. _Control System Applications_. The Control Handbook. Boca Raton: CRC Press.
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