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title = "Dynamic error budgeting, a design approach"
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author = ["Thomas Dehaeze"]
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Tags
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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Reference
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: <sup id="651e626e040250ee71a0847aec41b60c"><a class="reference-link" href="#monkhorst04_dynam_error_budget" title="@phdthesis{monkhorst04_dynam_error_budget,
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author = {Wouter Monkhorst},
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school = {Delft University},
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title = {Dynamic Error Budgeting, a design approach},
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year = 2004,
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}">@phdthesis{monkhorst04_dynam_error_budget,
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author = {Wouter Monkhorst},
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school = {Delft University},
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title = {Dynamic Error Budgeting, a design approach},
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year = 2004,
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}</a></sup>
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Author(s)
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: Monkhorst, W.
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Year
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: 2004
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## Introduction {#introduction}
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Challenge definition of this thesis:
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> Develop a tool which enables the designer to account for stochastic disturbances during the design of a mechatronics system.
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Develop tools should enable the designer to:
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- Predict the final performance level of a system which is subject to stochastic disturbances
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- Gain insight in performance limiting factors of the system.
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This insight should enable the designer to point out critical system components/properties and to improve the performance of the system
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- Objectively compare the performance of different system designs
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## Dynamic Error Budgeting {#dynamic-error-budgeting}
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### Motivations {#motivations}
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Main motivations are:
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- Cutting costs in the design phase
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- Speeding up the design process
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- Enhancing design insight
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### DEB design process {#deb-design-process}
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Step by step, the process is as follows:
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- design a concept system
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- model the concept system, such that the closed loop transfer functions can be determined
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- Identify all significant disturbances.
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Model them with their _Power Spectral Density_
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- Define the performance outputs of the system and simulate the output error.
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Using the theory of _propagation_, the contribution of each disturbance to the output error can be analyzed and the critical disturbance can be pointed out
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- Make changes to the system that are expected to improve the performance level, and simulate the output error again.
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Iterate until the error budget is meet.
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### Assumptions {#assumptions}
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The assumptions when applying DEB are:
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- the system can be accurately described with a **linear time invariant model**.
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This is usually the case as much effort is put in to make systems have a linear behavior and because feedback loops have a " linearizing" effect on the closed loop behavior.
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- the disturbances action on the system must be **stationary** (their statistical properties are not allowed to change over time).
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- the disturbances are **uncorrelated** with each other.
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This is more difficult to satisfy for MIMO systems and the designer must make sure that the separate disturbances all originate from separate independent sources.
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- the disturbance signals are modeled by their **Power Spectral Density**.
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This implies that only stochastic disturbances are allowed.
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Deterministic components like sinusoidal and DC signals are infinite peaks in their PSD and should not be used.
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For the deterministic part, other techniques can be used to determine their influence to the error.
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- the calculation method makes no assumption on the distribution of the distribution functions of the disturbances.
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In practice, many disturbances will have a normal like distribution.
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### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2--control-maximizing-performance}
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#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the--mathcal-h-2--norm-and-variance-of-the-output}
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The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
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\\[ \\|H\\|\_2^2 = \int\_{-\infty}^\infty |H(j2\pi f)|^2 df \\]
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Stochastic interpretation of the \\(\mathcal{H}\_2\\) norm: the squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
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#### The \\(\mathcal{H}\_2\\) control problem {#the--mathcal-h-2--control-problem}
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Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_2\\) norm of the closed loop system \\(H\\):
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\\[ C\_{\mathcal{H}\_2} \in \arg \min\_C \\|H\\|\_2 \\]
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#### Using weighting filters to model disturbances {#using-weighting-filters-to-model-disturbances}
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In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
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This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org7f8d04e)).
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<a id="org7f8d04e"></a>
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{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
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The white noise input \\(w(t)\\) is dimensionless, and when the weighting filter has units [SI], the resulting weighted signal \\(\bar{w}(t)\\) has units [SI].
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The PSD \\(S\_w(f)\\) of the weighted signal is:
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\\[ |S\_w(f)| = V\_w(j 2 \pi f) V\_w^T(-j 2 \pi f) \\]
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Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
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However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org4f416df)).
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<a id="org4f416df"></a>
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{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
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#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
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In this research, the outputs of the closed loop system (Figure [3](#orgc347ae6)) are:
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- the performance (error) signal \\(e\\)
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- the controller output \\(u\\)
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In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
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<a id="orgc347ae6"></a>
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{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
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The resulting problem is a multi-objective control problem: while constraining the variance of the controller output \\(u\\), the variance of the performance channel should be minimized.
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This problem can be solved by scaling the controller output \\(u\\) with a factor \\(\alpha\\) during the \\(\mathcal{H}\_2\\) synthesis.
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When varying \\(\alpha\\), one can plot the amount of control effort at one axis and the achieve performance on the other axis.
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The resulting points lie on the so-called **Pareto curve**.
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## Conclusions {#conclusions}
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\\(\mathcal{H}\_2\\) control strategy is an extension of the DEB approach.
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It offers the designer the opportunity to optimize over the degree of freedom given by the controller, enabling the designer to predict the maximum achievable performance level of a system concept.
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Using this technique, the designer is able to objectively compare the performance potential of different system concepts.
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The accuracy of the predicted performance by DEB with respect to the measured results can be improved by using higher order models of the disturbances.
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Increasing of order of the disturbance model might even allow modelling of harmonic disturbances by using inverse notches.
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To achieve the highest degree of prediction accuracy, it is recommended to use to actual measured disturbance spectra in the simulations.
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When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, it can give the control designer useful hints about how to control the system best for optimal performance.
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Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
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# Bibliography
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<a class="bibtex-entry" id="monkhorst04_dynam_error_budget">Monkhorst, W., *Dynamic error budgeting, a design approach* (2004). Delft University.</a> [↩](#651e626e040250ee71a0847aec41b60c)
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## Backlinks {#backlinks}
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- [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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