From 2829dc2362274bb851492fd4f088dfba8f4e6f0f Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 3 Sep 2021 10:20:20 +0200 Subject: [PATCH] Update Content - 2021-09-03 --- .../inbook/levine11_contr_system_applic.md | 236 ------------------ 1 file changed, 236 deletions(-) delete mode 100644 content/inbook/levine11_contr_system_applic.md diff --git a/content/inbook/levine11_contr_system_applic.md b/content/inbook/levine11_contr_system_applic.md deleted file mode 100644 index 8358e78..0000000 --- a/content/inbook/levine11_contr_system_applic.md +++ /dev/null @@ -1,236 +0,0 @@ -+++ -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. - -
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- -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. - -
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- -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 - -
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- - 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 - -
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- - 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)\\). - -
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- -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.