diff --git a/content/article/oomen18_advan_motion_contr_precis_mechat.md b/content/article/oomen18_advan_motion_contr_precis_mechat.md index 53e6e2d..a8ed478 100644 --- a/content/article/oomen18_advan_motion_contr_precis_mechat.md +++ b/content/article/oomen18_advan_motion_contr_precis_mechat.md @@ -1,7 +1,7 @@ +++ title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems" author = ["Dehaeze Thomas"] -draft = true +draft = false +++ Tags @@ -16,12 +16,171 @@ Author(s) Year : 2018 + +## Introduction {#introduction} + +Control of positioning systems is traditionally simplified by an excellent mechanical design. +In particular, the mechanical design is such that the system is stiff and highly reproducible. +In conjunction with moderate performance requirements, the control bandwidth is well-below the resonance frequency of the flexible mechanics as is shown in Figure [1](#figure--fig:oomen18-next-gen-loop-gain) (a). +As a result, the system can often be completely **decoupled** in the frequency range relevant for control. +Consequently, the control design is divided into well-manageable SISO control loops. + +Although motion control design is well developed, presently available techniques mainly apply to positioning systems that behave as a rigid body in the relevant frequency range. +On one hand, increasing performance requirements hamper the validity of this assumption, since the bandwidth has to increase, leading to flexible dynamics in the cross-over region, see Figure [1](#figure--fig:oomen18-next-gen-loop-gain) (b). + {{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}} -## Bibliography {#bibliography} +## Traditional motion control {#traditional-motion-control} + +In the frequency range that is relevant for control, the dynamical behavior is mainly determined by the mechanics. +In particular, the mechanics can typically be described as: + +\begin{equation} +G\_m = \sum\_{i=1}^{n\_{RB}} \frac{c\_i b\_i^T}{s^2} + \sum\_{n\_{RB} + 1}^{n\_s} \frac{c\_i b\_i^T}{s^2 + 2\xi \omega\_i s + \omega\_i^2} +\end{equation} + +where the first term refers to rigid body modes and the second term to flexible modes. + +- \\(n\_{RB}\\) is the number of rigid body modes +- \\(c\_i \in \mathbb{R}^{n\_y}\\) and \\(b\_i \in \mathbb{R}^{n\_u}\\) are associated with the mode shapes +- \\(\xi\_i, \omega\_i \in \mathbb{R}\_+\\) + +In traditional positioning systems, the number of actuators \\(n\_u\\) and sensors \\(n\_y\\) equals the number of rigid body modes \\(n\_{RB}\\) and are positioned such that the matrix \\(\sum\_{i=1}^{n\_{RB}} c\_i b\_i^T\\) is invertible. +In this case, matrices \\(T\_u\\) and \\(T\_y\\) can be selected such that: + +\begin{equation} +G = T\_y G\_m T\_u = \frac{1}{s^2} I\_{n\_{RB}} + G\_{\text{flex}} +\end{equation} + +A tradition motion control architecture is shown in Figure [2](#figure--fig:oomen18-control-architecture). + + + +{{< figure src="/ox-hugo/oomen18_control_architecture.png" caption="Figure 2: Traditional motion control architecture" >}} + + +### Traditional feedforward design {#traditional-feedforward-design} + +[Feedforward Control]({{< relref "feedforward_control.md" >}}) can effectively compensate for reference induced error signals. +In particular, \\(f\\) should be selected such that \\(r - G f\\) is minimized. +In the low frequency range, the system is decoupled and \\(G\_{\text{flex}}\\) can be ignored, in which case \\(f = G^{-1} r\\). +In practice, the feedforward signal is selected as \\(f = ms^2 r\\). + + +### Traditional feedback design {#traditional-feedback-design} + +The [Feedback Controller]({{< relref "feedback_control.md" >}}) has to minimize \\((1 + GK)^{-1}(\delta - v)\\). +The main idea is that rigid body decoupling of \\(G\\) enables the shaping of the diagonal elements of \\(K\\) through a decentralized feedback controller. +As a result, each diagonal element of \\(K\\) may be tuned independently. +Typically, a PID controller is tuned through manual loop-shaping, followed by notch filters to account the the flexible modes that hamper stability and/or performance. + + +### Traditional design procedure {#traditional-design-procedure} + +Traditional motion control design divides the multi-variable control design problems into sub-problems that are manageable by manual control design. +The traditional procedure consists of the following steps: + +- identify an FRF of \\(G\_m\\) +- decouple the plant to obtain an FRF of \\(G\\) +- design \\(K\\) using manual loop-shaping, consisting of PID with notches +- tune a feedforward controller, e.g. \\(f = m s^2 r\\) + + +## Precision motion control developments {#precision-motion-control-developments} + + +### Challenges {#challenges} + +High performance mechatronic systems are becoming lighter and lighter. +Such lightweight systems exhibit predominant flexible dynamical behavior, as well as an increased susceptibility to disturbances. + +This leads to several challenges for motion control design: + +- **Unmeasured performance variables** due to spatio-temporal deformations. + In particular, the location where the performance is desired may not be directly measured. +- **Many additional inputs and outputs** can be exploited to actively control the flexible dynamical behavior. + Spatially distributed actuators can actively provide stiffness and damping to the mechanical deformations. +- **Position dependent behavior** is almost unavoidable. + For instance in gantry stage designs, mass distribution change due to motion, leading to additional position-dependent behavior. + A key challenge lies in handling the position dependence of future systems +- A **system-of-systems perspective** on motion control design provides a strong potential for performance enhancement of the overall system. + In particular, typical manufacturing machines and scientific instruments involves multiple controlled subsystems where the two subsystems have to move relative to each other. + Performance limitations in each subsystem will negatively impact the overall performance. + A joint design enables that individual subsystems will be able to compensate each other's limitations. + A main challenge lies in an increase of the complexity of the control problem. +- **Thermal dynamics**, in addition to mechanical deformations are expected to become substantially more important due to increasing performance specifications. +- **Vibrations**, such as flow induced vibrations of cooling liquids and floor vibrations, have to be attenuated. + + +### Generalized plant approach {#generalized-plant-approach} + +A generalized plant framework allows for a systematic way to address the future challenges in advanced motion control. + +The generalized plant is depicted in Figure [3](#figure--fig:oomen18-generalized-plant): + +- \\(z\\) are the performance variables +- \\(y\\) and \\(u\\) are the measured variables and measured variables, respectively +- \\(w\\) contains the exogenous inputs, typically including both reference signals and disturbances. + + + +{{< figure src="/ox-hugo/oomen18_generalized_plant.png" caption="Figure 3: Generalized plant setup" >}} + + +## Feedback and Identification for Control {#feedback-and-identification-for-control} + +Feedback control is essential to deal with uncertainty in the system dynamics \\(G\\) and disturbances \\(v\\). +Indeed, the main goal of feedback si to render the system insensitive to such uncertainties. + + +### Norm-based control {#norm-based-control} + +A model-based design is foreseen to be able to systematically address the above mentioned challenges. + +To specify the control goal, the criterion: + +\begin{equation} +J(G, K) = \\| \mathcal{F}\_l(P(G), K) \\| +\end{equation} + +is posed, where the goal is to compute: + +\begin{equation} +K\_{\text{opt}} = \text{arg} \text{min}\_{K} J(G\_0, K) +\end{equation} + +Where \\(\\| \cdot \\|\\) denotes a suitable norm, e.g. \\(\mathcal{H}\_2\\) or \\(\mathcal{H}\_\infty\\), and \\(\mathcal{F}\_l\\) denotes a lower linear fractional transformation. + +\\(G\_0\\) denotes the true system, which is generally unknown and represented by a model \\(\hat{G}\\). + + +### Nominal modeling for control {#nominal-modeling-for-control} + +To arrive at a mathematically tractable optimization problem, knowledge of the true system is represented through a model \\(\hat{G}\\). +The central question is how to obtain such a model that is suitable for controller design. +[System Identification]({{< relref "system_identification.md" >}}) as opposed to first principles modeling, is an inexpensive, fast and accurate approach to obtain such a model. +Indeed, the machine is often already built, enabling direct experimentation. + +The model \\(\hat{G}\\) that results from system identification is an approximation of the true system \\(G\_0\\) for several reasons: + +- motion systems often contains an infinite number of modes \\(n\_s\\), while a model of limited complexity may be desirable from a control perspective +- parasitic non-linearities are present, including nonlinear damping +- identification experiments are based on finite time disturbed observations, leading to uncertainties on estimated parameters + + +### Toward robust motion control {#toward-robust-motion-control} + +Doing a model based control design using an identified model may not work well due to a lack of robustness. +Indeed, if \\(K(\hat{G})\\) is designed solely based on \\(\hat{G}\\), there is no reason to assume that it achieves a suitable level of performance on \\(G\_0\\). +This motivates a robust control design, where the **model quality is explicitly addressed during controller synthesis**. + + +## Feedforward and learning {#feedforward-and-learning} + +## References
Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” Ieej Journal of Industry Applications 7 (2): 127–40. doi:10.1541/ieejjia.7.127.
diff --git a/content/zettels/feedback_control.md b/content/zettels/feedback_control.md new file mode 100644 index 0000000..d552db6 --- /dev/null +++ b/content/zettels/feedback_control.md @@ -0,0 +1,13 @@ ++++ +title = "Feedback Control" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: + +## References + +
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diff --git a/content/zettels/feedforward_control.md b/content/zettels/feedforward_control.md new file mode 100644 index 0000000..403c2d5 --- /dev/null +++ b/content/zettels/feedforward_control.md @@ -0,0 +1,13 @@ ++++ +title = "Feedforward Control" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: + +## References + +
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diff --git a/content/zettels/passive_damping.md b/content/zettels/passive_damping.md new file mode 100644 index 0000000..b6029b0 --- /dev/null +++ b/content/zettels/passive_damping.md @@ -0,0 +1,16 @@ ++++ +title = "Passive Damping" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: + + +## Bibliography {#bibliography} + +## References + +
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diff --git a/content/zettels/tuned_mass_damper.md b/content/zettels/tuned_mass_damper.md index 4c299da..75a3897 100644 --- a/content/zettels/tuned_mass_damper.md +++ b/content/zettels/tuned_mass_damper.md @@ -23,16 +23,16 @@ The TMD then has large internal damping such that the energy is dissipated (i.e. The optimal parameters of the tuned mass damper can be roughly estimated as follows: -- Choose the maximum mass of the TMD \\(m\\) and note: - \\[ \mu = m/M \\] - where \\(M\\) is the mass of the system to damp +- Choose the maximum acceptable mass of the TMD \\(m\_2\\) and note: + \\[ \mu = m\_2/m\_1 \\] + where \\(m\_1\\) is the mass of the system to damp - The resonance frequency of the tuned mass damper should be chosen to be \\[ \nu = \frac{1}{1 + \mu} \approx 1 \\] As usually we have \\(\mu \ll 1\\) (i.e. TMD mass small compared to the structure mass, for instance few percent) - This allows to compute the stiffness of the TMD: - \\[ k = \nu^2 K \mu = K \frac{\mu}{(1 + \mu)^2} \\] + \\[ k\_2 = \nu^2 k\_1 \mu = k\_1 \frac{\mu}{(1 + \mu)^2} \\] - Finally, the optimal damping of the TMD is: - \\[ \xi = \sqrt{\frac{3\mu}{8 (1 + \mu)}} \Longrightarrow c = 2 \xi \sqrt{k m} \\] + \\[ \xi\_2 = \sqrt{\frac{3 \mu}{8 (1 + \mu)}} \Longrightarrow c\_2 = 2 \xi\_2 \sqrt{k\_2 m\_2} \\] ## Simple TMD model {#simple-tmd-model} diff --git a/static/ox-hugo/oomen18_control_architecture.png b/static/ox-hugo/oomen18_control_architecture.png new file mode 100644 index 0000000..de9f86d Binary files /dev/null and b/static/ox-hugo/oomen18_control_architecture.png differ diff --git a/static/ox-hugo/oomen18_generalized_plant.png b/static/ox-hugo/oomen18_generalized_plant.png new file mode 100644 index 0000000..cc85e28 Binary files /dev/null and b/static/ox-hugo/oomen18_generalized_plant.png differ