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title = "An extended approach of inverted decoupling"
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author = ["Thomas Dehaeze"]
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draft = false
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title = "An extended approach of inverted decoupling"
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author = ["Thomas Dehaeze"]
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draft = false
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@ -8,7 +8,7 @@ Tags
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: [Decoupled Control](decoupled_control.md)
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Reference
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: ([Garrido, Vázquez, and Morilla 2012](#orgdf49544))
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: ([Garrido, Vázquez, and Morilla 2012](#org32a9ef5))
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Author(s)
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: Garrido, J., Francisco V\\'azquez, & Morilla, F.
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@ -19,7 +19,7 @@ Year
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## Introduction {#introduction}
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Most decoupling approaches use the conventional decoupling scheme in Figure [1](#orge2ae292) with:
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Most decoupling approaches use the conventional decoupling scheme in Figure [1](#org265d382) with:
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- \\(G(s)\\) the process matrix
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- \\(D(s)\\) the decoupler matrix
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@ -38,7 +38,7 @@ The main problem of this methodology is the fact that the complexity of the deco
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An alternative decoupling methods, called _inverted decoupling_, maintains very simple apparent processes and decoupler element independently of the system size.
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However, inverted decoupling cannot be applied to processes with multivariable [Right Half Plane Zeros](right_half_plane_zeros.md).
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<a id="orge2ae292"></a>
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<a id="org265d382"></a>
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{{< figure src="/ox-hugo/garrido12_decoupling_control_system.png" caption="Figure 1: Block diagram of a decoupling control system" >}}
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@ -115,4 +115,4 @@ It is usually necessary to approximate the expressions of \\(|G(s)|\\) and \\(\t
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## Bibliography {#bibliography}
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<a id="orgdf49544"></a>Garrido, Juan, Francisco Vázquez, and Fernando Morilla. 2012. “Centralized Multivariable Control by Simplified Decoupling.” _Journal of Process Control_ 22 (6):1044–62. <https://doi.org/10.1016/j.jprocont.2012.04.008>.
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<a id="org32a9ef5"></a>Garrido, Juan, Francisco Vázquez, and Fernando Morilla. 2012. “Centralized Multivariable Control by Simplified Decoupling.” _Journal of Process Control_ 22 (6):1044–62. <https://doi.org/10.1016/j.jprocont.2012.04.008>.
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@ -1,5 +0,0 @@
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+++
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title = "An extended approach of inverted decoupling"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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@ -1,5 +0,0 @@
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+++
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title = "An extended approach of inverted decoupling"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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@ -1,5 +0,0 @@
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+++
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title = "Dynamic decoupling of robot manipulators"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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@ -3,14 +3,14 @@ title = "Design, modeling and control of nanopositioning systems"
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author = ["Thomas Dehaeze"]
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description = "Talks about various topics related to nano-positioning systems."
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keywords = ["Control", "Metrology", "Flexible Joints"]
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draft = false
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draft = true
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Tags
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: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
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: [Piezoelectric Actuators](piezoelectric_actuators.md), [Flexible Joints](flexible_joints.md)
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Reference
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: ([Fleming and Leang 2014](#org9f0983e))
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: ([Fleming and Leang 2014](#orgc8028e0))
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Author(s)
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: Fleming, A. J., & Leang, K. K.
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@ -783,11 +783,11 @@ Year
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### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
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<a id="orgc11b95b"></a>
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<a id="orgded1d91"></a>
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{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
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Consider the electrical circuit shown in Figure [1](#orgc11b95b) where a voltage source is connected to a piezoelectric actuator.
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Consider the electrical circuit shown in Figure [1](#orgded1d91) where a voltage source is connected to a piezoelectric actuator.
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The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
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The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
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@ -911,4 +911,4 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
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## Bibliography {#bibliography}
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<a id="org9f0983e"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
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<a id="orgc8028e0"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
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@ -1,26 +0,0 @@
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+++
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title = "Mechatronics and robotics: new trends and challenges"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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Tags
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:
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Reference
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: ([Indri and Oboe 2020](#orge6b9fea))
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Author(s)
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: Indri, M., & Oboe, R.
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Year
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: 2020
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- [ ] Read Chapter 4
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## Bibliography {#bibliography}
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<a id="orge6b9fea"></a>Indri, Marina, and Roberto Oboe. 2020. _Mechatronics and Robotics: New Trends and Challenges_. CRC Press.
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@ -1,111 +0,0 @@
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+++
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title = "Advanced Motion Control Design"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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Tags
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:
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Reference
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: ([Levine 2011](#org5f2e773)), 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="important">
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<div></div>
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Step by step procedure:
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1. Interaction Analysis
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2. Decoupling
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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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<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 inters 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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Let us assume we know the spectral contents of the disturbance.
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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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## Bibliography {#bibliography}
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<a id="org5f2e773"></a>Levine, W. S. 2011. _Control System Applications_. The Control Handbook. Boca Raton: CRC Press.
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@ -1,23 +0,0 @@
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+++
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title = "System identification : a frequency domain approach"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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Tags
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: [System Identification](system_identification.md)
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Reference
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: ([Pintelon and Schoukens 2012](#org88596ad))
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Author(s)
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: Pintelon, R., & Schoukens, J.
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Year
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: 2012
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## Bibliography {#bibliography}
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<a id="org88596ad"></a>Pintelon, Rik, and Johan Schoukens. 2012. _System Identification : a Frequency Domain Approach_. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. <https://doi.org/10.1002/9781118287422>.
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@ -1,7 +1,7 @@
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+++
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title = "Mastering system identification in 100 exercises"
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author = ["Thomas Dehaeze"]
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draft = false
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draft = true
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+++
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Tags
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@ -9,7 +9,7 @@ Tags
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Reference
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: ([Schoukens, Pintelon, and Rolain 2012](#org4021f9f))
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: ([Schoukens, Pintelon, and Rolain 2012](#org1193e9b))
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Author(s)
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: Schoukens, J., Pintelon, R., & Rolain, Y.
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@ -20,4 +20,4 @@ Year
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## Bibliography {#bibliography}
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<a id="org4021f9f"></a>Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012. _Mastering System Identification in 100 Exercises_. John Wiley & Sons.
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<a id="org1193e9b"></a>Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012. _Mastering System Identification in 100 Exercises_. John Wiley & Sons.
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@ -1,14 +1,14 @@
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+++
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title = "Multivariable control systems: an engineering approach"
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title = "Decentralized and decoupled control"
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author = ["Thomas Dehaeze"]
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draft = true
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draft = false
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+++
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Tags
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: [Multivariable Control](multivariable_control.md)
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: [Decentralized Control](decentralized_control.md), [Multivariable Control](multivariable_control.md)
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Reference
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: ([Albertos and Antonio 2004](#orga6ef935))
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: ([Albertos and Antonio 2004](#org86b8145))
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Author(s)
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: Albertos, P., & Antonio, S.
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@ -17,10 +17,7 @@ Year
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: 2004
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## [Decentralized Control](decentralized_control.md) and Decoupled Control {#decentralized-control--decentralized-control-dot-md--and-decoupled-control}
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### Introduction {#introduction}
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## Introduction {#introduction}
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Decentralized control is decomposed into two steps:
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@ -42,7 +39,7 @@ Two approaches can be used to control a coupled system with SISO techniques:
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- **cascade control**
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### Mutli-Loop Control, Pairing Selection {#mutli-loop-control-pairing-selection}
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## Mutli-Loop Control, Pairing Selection {#mutli-loop-control-pairing-selection}
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The strategy called _multi-loop control_ consists of first proper input/output pairing, and then design of several SISO controllers.
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In this way, a complex control problem is divided into several simpler ones.
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@ -51,7 +48,7 @@ The multi-loop control may not work in strongly coupled systems.
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Therefore, a methodology the access the degree of interaction between the loops is needed.
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#### [Relative Gain Array](relative_gain_array.md) {#relative-gain-array--relative-gain-array-dot-md}
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### [Relative Gain Array](relative_gain_array.md) {#relative-gain-array--relative-gain-array-dot-md}
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The Relative Gain Array (RGA) \\(\Lambda(s)\\) is defined as:
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@ -66,7 +63,7 @@ For demanding control specifications, the values of \\(\Lambda\\) car be drawn a
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In this case, at frequencies important for control stability robustness (around the peak of the sensitivity transfer function), if \\(\Lambda(j\omega)\\) approaches the identity matrix, stability problems are avoided in multi-loop control.
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### Decoupling {#decoupling}
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## Decoupling {#decoupling}
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In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one.
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This strategy is called **decoupling**.
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@ -77,12 +74,12 @@ This strategy is called **decoupling**.
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- based on state measurements, via a feedback law
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#### Feedforward Decoupling {#feedforward-decoupling}
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### Feedforward Decoupling {#feedforward-decoupling}
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A pre-compensator (Figure [1](#org7023330)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
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A pre-compensator (Figure [1](#org860681c)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
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This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
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<a id="org7023330"></a>
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<a id="org860681c"></a>
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{{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="Figure 1: Decoupler pre-compensator" >}}
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@ -99,7 +96,7 @@ Although at first glance, decoupling seems an appealing idea, there are some dra
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- feedback decoupling needs full state measurements
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#### SVD Decoupling {#svd-decoupling}
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### SVD Decoupling {#svd-decoupling}
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A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition](singular_value_decomposition.md) as:
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@ -112,10 +109,10 @@ where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
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The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
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In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
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If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [2](#org2de6de7), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
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If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [2](#orgbb644a3), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
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Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
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<a id="org2de6de7"></a>
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<a id="orgbb644a3"></a>
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{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 2: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
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@ -128,7 +125,7 @@ SVD decoupling produces the most suitable combinations for independent "multi-lo
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If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control.
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### Conclusions {#conclusions}
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## Conclusions {#conclusions}
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In this chapter, the control of systems with multiple inputs and outputs is discussed using SISO-based tools, either directly or after some multivariable decoupling transformations.
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@ -188,7 +185,6 @@ When switching on the regulator, significant transient behavior can be seen and
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The solution is similar to that of the wind-up phenomenon: the regulator should be always on, carrying out calculations by using \eqref{eq:antiwindup_pid}.
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## Bibliography {#bibliography}
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<a id="orga6ef935"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
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<a id="org86b8145"></a>Albertos, P., and S. Antonio. 2004. “Decentralized and Decoupled Control.” In _Multivariable Control Systems: An Engineering Approach_, 125–62. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
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236
content/inbook/levine11_contr_system_applic.md
Normal file
236
content/inbook/levine11_contr_system_applic.md
Normal file
@ -0,0 +1,236 @@
|
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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.
|
||||
|
||||
<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.
|
||||
236
content/inbook/steinbuch11_advan_motion_contr_desig.md
Normal file
236
content/inbook/steinbuch11_advan_motion_contr_desig.md
Normal file
@ -0,0 +1,236 @@
|
||||
+++
|
||||
title = "Advanced Motion Control Design"
|
||||
author = ["Thomas Dehaeze"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: ([Steinbuch et al. 2011](#org16dd9d1))
|
||||
|
||||
Author(s)
|
||||
: Steinbuch, M., Merry, R., Boerlage, M., Ronde, M., & Molengraft, M.
|
||||
|
||||
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="org16dd9d1"></a>Steinbuch, Maarten, Roel Merry, Matthijs Boerlage, Michael Ronde, and Marinus Molengraft. 2011. “Advanced Motion Control Design.” In _Control System Applications_, 651–76. CRC Press.
|
||||
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