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@@ -3627,7 +3627,7 @@ However, the inclusion of parametric uncertainty may be more significant for MIM
|
||||
Unstructured perturbations are often used to get a simple uncertainty model.
|
||||
We here define unstructured uncertainty as the use of a "full" complex perturbation matrix \\(\Delta\\), usually with dimensions compatible with those of the plant, where at each frequency any \\(\Delta(j\w)\\) satisfying \\(\maxsv(\Delta(j\w)) < 1\\) is allowed.
|
||||
|
||||
Three common forms of **feedforward unstructured uncertainty** are shown [Table 4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
|
||||
Three common forms of **feedforward unstructured uncertainty** are shown [Figure 29](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
|
||||
|
||||
<div class="important">
|
||||
|
||||
@@ -3641,17 +3641,26 @@ Three common forms of **feedforward unstructured uncertainty** are shown [Table
|
||||
|
||||
</div>
|
||||
|
||||
<a id="table--fig:feedforward-uncertainty"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--fig:feedforward-uncertainty">Table 4</a>:</span>
|
||||
Common feedforward unstructured uncertainty
|
||||
|
||||
<figure class="subfigures" id="table--fig:feedforward-uncertainty">
|
||||
<div class="subfigure-row">
|
||||
<div class="subfigure" id="org-target--fig-additive-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_additive_uncertainty.png" alt="Additive uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(a)</span> Additive uncertainty</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-input-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_input_uncertainty.png" alt="Multiplicative input uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(b)</span> Multiplicative input uncertainty</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-output-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_output_uncertainty.png" alt="Multiplicative output uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(c)</span> Multiplicative output uncertainty</div>
|
||||
</div>
|
||||
</div>
|
||||
<figcaption><span class="figure-number">Figure 29: </span>Common feedforward unstructured uncertainty</figcaption>
|
||||
</figure>
|
||||
|
||||
|  |  |  |
|
||||
|-------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------|
|
||||
| <span class="org-target" id="org-target--fig-additive-uncertainty"></span> Additive uncertainty | <span class="org-target" id="org-target--fig-input-uncertainty"></span> Multiplicative input uncertainty | <span class="org-target" id="org-target--fig-output-uncertainty"></span> Multiplicative output uncertainty |
|
||||
|
||||
In [Table 5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
|
||||
In [Figure 30](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
|
||||
|
||||
<div class="important">
|
||||
|
||||
@@ -3665,15 +3674,24 @@ In [Table 5](#table--fig:feedback-uncertainty), three **feedback or inverse unst
|
||||
|
||||
</div>
|
||||
|
||||
<a id="table--fig:feedback-uncertainty"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--fig:feedback-uncertainty">Table 5</a>:</span>
|
||||
Common feedback unstructured uncertainty
|
||||
</div>
|
||||
|
||||
|  |  |  |
|
||||
|-------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------|
|
||||
| <span class="org-target" id="org-target--fig-inv-additive-uncertainty"></span> Inverse additive uncertainty | <span class="org-target" id="org-target--fig-inv-input-uncertainty"></span> Inverse multiplicative input uncertainty | <span class="org-target" id="org-target--fig-inv-output-uncertainty"></span> Inverse multiplicative output uncertainty |
|
||||
<figure class="subfigures" id="table--fig:feedback-uncertainty">
|
||||
<div class="subfigure-row">
|
||||
<div class="subfigure" id="org-target--fig-inv-additive-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_inv_additive_uncertainty.png" alt="Inverse additive uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(a)</span> Inverse additive uncertainty</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-inv-input-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_inv_input_uncertainty.png" alt="Inverse multiplicative input uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(b)</span> Inverse multiplicative input uncertainty</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-inv-output-uncertainty">
|
||||
<img src="/ox-hugo/skogestad07_inv_output_uncertainty.png" alt="Inverse multiplicative output uncertainty" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(c)</span> Inverse multiplicative output uncertainty</div>
|
||||
</div>
|
||||
</div>
|
||||
<figcaption><span class="figure-number">Figure 30: </span>Common feedback unstructured uncertainty</figcaption>
|
||||
</figure>
|
||||
|
||||
|
||||
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
|
||||
@@ -3768,12 +3786,12 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
|
||||
|
||||
### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining-p-n-and-m}
|
||||
|
||||
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 29](#figure--fig:input-uncertainty-set-feedback-weight).
|
||||
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 31](#figure--fig:input-uncertainty-set-feedback-weight).
|
||||
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
|
||||
|
||||
<a id="figure--fig:input-uncertainty-set-feedback-weight"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="<span class='figure-number'>Figure 29: </span>System with multiplicative input uncertainty and performance measured at the output" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="<span class='figure-number'>Figure 31: </span>System with multiplicative input uncertainty and performance measured at the output" >}}
|
||||
|
||||
We want to derive the generalized plant \\(P\\) which has inputs \\([u\_\Delta,\ w,\ u]^T\\) and outputs \\([y\_\Delta,\ z,\ v]^T\\).
|
||||
|
||||
@@ -3906,7 +3924,7 @@ Then the \\(M\Delta\text{-system}\\) is stable for all perturbations \\(\Delta\\
|
||||
|
||||
#### Application of the Unstructured RS-condition {#application-of-the-unstructured-rs-condition}
|
||||
|
||||
We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Table 4](#table--fig:feedforward-uncertainty) and [Table 5](#table--fig:feedback-uncertainty) with
|
||||
We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Figure 29](#table--fig:feedforward-uncertainty) and [Figure 30](#table--fig:feedback-uncertainty) with
|
||||
|
||||
\begin{equation\*}
|
||||
E = W\_2 \Delta W\_1, \quad \hnorm{\Delta} \le 1
|
||||
@@ -3951,7 +3969,7 @@ In order to get tighter condition we must use a tighter uncertainty description
|
||||
Robust stability bound in terms of the \\(\hinf\\) norm (\\(\text{RS}\Leftrightarrow\hnorm{M}<1\\)) are in general only tight when there is a single full perturbation block.
|
||||
An "exception" to this is when the uncertainty blocks enter or exit from the same location in the block diagram, because they can then be stacked on top of each other or side-by-side, in an overall \\(\Delta\\) which is then full matrix.
|
||||
|
||||
One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 30](#figure--fig:coprime-uncertainty), for which the set of plants is
|
||||
One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 32](#figure--fig:coprime-uncertainty), for which the set of plants is
|
||||
|
||||
\begin{equation\*}
|
||||
G\_p = (M\_l + \Delta\_M)^{-1}(Nl + \Delta\_N), \quad \hnorm{[\Delta\_N, \ \Delta\_N]} \le \epsilon
|
||||
@@ -3963,7 +3981,7 @@ This uncertainty description is surprisingly **general**, it allows both zeros a
|
||||
|
||||
<a id="figure--fig:coprime-uncertainty"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="<span class='figure-number'>Figure 30: </span>Coprime Uncertainty" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="<span class='figure-number'>Figure 32: </span>Coprime Uncertainty" >}}
|
||||
|
||||
Since we have no weights on the perturbations, it is reasonable to use a normalized coprime factorization of the nominal plant.
|
||||
In any case, to test for RS we can rearrange the block diagram to match the \\(M\Delta\text{-structure}\\) with
|
||||
@@ -4007,12 +4025,12 @@ To this effect, introduce the block-diagonal scaling matrix
|
||||
|
||||
where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same dimension as the \\(i\\)'th perturbation block \\(\Delta\_i\\).
|
||||
|
||||
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 31](#figure--fig:block-diagonal-scalings).
|
||||
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 33](#figure--fig:block-diagonal-scalings).
|
||||
This clearly has no effect on stability.
|
||||
|
||||
<a id="figure--fig:block-diagonal-scalings"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="<span class='figure-number'>Figure 31: </span>Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="<span class='figure-number'>Figure 33: </span>Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
|
||||
|
||||
Note that with the chosen form for the scalings we have for each perturbation block \\(\Delta\_i = d\_i \Delta\_i d\_i^{-1}\\), that is we have \\(\Delta = D \Delta D^{-1}\\).
|
||||
|
||||
@@ -4302,7 +4320,7 @@ Note that \\(\mu\\) underestimate how bad or good the actual worst case performa
|
||||
|
||||
### Application: RP with Input Uncertainty {#application-rp-with-input-uncertainty}
|
||||
|
||||
We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 29](#figure--fig:input-uncertainty-set-feedback-weight)).
|
||||
We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 31](#figure--fig:input-uncertainty-set-feedback-weight)).
|
||||
|
||||
The performance requirement is then
|
||||
|
||||
@@ -4416,11 +4434,11 @@ with the decoupling controller we have:
|
||||
\overline{\sigma}(N\_{22}) = \overline{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right|
|
||||
\end{equation\*}
|
||||
|
||||
and we see from [Figure 32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
|
||||
and we see from [Figure 34](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
|
||||
|
||||
<a id="figure--fig:mu-plots-distillation"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="<span class='figure-number'>Figure 32: </span>\\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="<span class='figure-number'>Figure 34: </span>\\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
|
||||
|
||||
|
||||
##### RS {#rs}
|
||||
@@ -4431,7 +4449,7 @@ In this case \\(w\_I T\_I = w\_I T\\) is a scalar times the identity matrix:
|
||||
\mu\_{\Delta\_I}(w\_I T\_I) = |w\_I t| = \left|0.2 \frac{5s + 1}{(0.5s + 1)(1.43s + 1)}\right|
|
||||
\end{equation\*}
|
||||
|
||||
and we see from [Figure 32](#figure--fig:mu-plots-distillation) that RS is satisfied.
|
||||
and we see from [Figure 34](#figure--fig:mu-plots-distillation) that RS is satisfied.
|
||||
|
||||
The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may increase the uncertainty by a factor of \\(1/0.53 = 1.89\\) before the worst case uncertainty yields instability.
|
||||
|
||||
@@ -4439,7 +4457,7 @@ The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may in
|
||||
##### RP {#rp}
|
||||
|
||||
Although the system has good robustness margins and excellent nominal performance, the robust performance is poor.
|
||||
This is shown in [Figure 32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
|
||||
This is shown in [Figure 34](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
|
||||
The peak value is close to 6, meaning that even with 6 times less uncertainty, the weighted sensitivity will be about 6 times larger than what we require.
|
||||
|
||||
|
||||
@@ -4576,11 +4594,11 @@ The latter is an attempt to "flatten out" \\(\mu\\).
|
||||
#### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example-mu-text-synthesis-with-dk-iteration}
|
||||
|
||||
For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
|
||||
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 33](#figure--fig:weights-distillation).
|
||||
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 35](#figure--fig:weights-distillation).
|
||||
|
||||
<a id="figure--fig:weights-distillation"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="<span class='figure-number'>Figure 33: </span>Uncertainty and performance weights" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="<span class='figure-number'>Figure 35: </span>Uncertainty and performance weights" >}}
|
||||
|
||||
The objective is to minimize the peak value of \\(\mu\_{\tilde{\Delta}}(N)\\), \\(\tilde{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\).
|
||||
\\(\Delta\_I\\) is a diagonal \\(2 \times 2\\) matrix representing the diagonal input uncertainty and \\(\Delta\_P\\) is a full \\(2 \times 2\\) matrix representing the performance specifications.
|
||||
@@ -4592,8 +4610,8 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
|
||||
|
||||
- Iteration No. 1.
|
||||
Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights).
|
||||
Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 34](#figure--fig:dk-iter-mu).
|
||||
Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 35](#figure--fig:dk-iter-d-scale)
|
||||
Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 36](#figure--fig:dk-iter-mu).
|
||||
Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 37](#figure--fig:dk-iter-d-scale)
|
||||
- Iteration No. 2.
|
||||
Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller.
|
||||
Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\).
|
||||
@@ -4603,25 +4621,25 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
|
||||
|
||||
<a id="figure--fig:dk-iter-mu"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="<span class='figure-number'>Figure 34: </span>Change in \\(\mu\\) during DK-iteration" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="<span class='figure-number'>Figure 36: </span>Change in \\(\mu\\) during DK-iteration" >}}
|
||||
|
||||
<a id="figure--fig:dk-iter-d-scale"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="<span class='figure-number'>Figure 35: </span>Change in D-scale \\(d\_1\\) during DK-iteration" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="<span class='figure-number'>Figure 37: </span>Change in D-scale \\(d\_1\\) during DK-iteration" >}}
|
||||
|
||||
The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 36](#figure--fig:mu-plot-optimal-k3).
|
||||
The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 38](#figure--fig:mu-plot-optimal-k3).
|
||||
The objectives of RS and NP are easily satisfied.
|
||||
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\overline{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
|
||||
|
||||
<a id="figure--fig:mu-plot-optimal-k3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="<span class='figure-number'>Figure 36: </span>\\(mu\text{-plots}\\) with \\(\mu\\) "optimal" controller \\(K\_3\\)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="<span class='figure-number'>Figure 38: </span>\\(mu\text{-plots}\\) with \\(\mu\\) "optimal" controller \\(K\_3\\)" >}}
|
||||
|
||||
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [Figure 37](#figure--fig:perturb-s-k3).
|
||||
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [Figure 39](#figure--fig:perturb-s-k3).
|
||||
|
||||
<a id="figure--fig:perturb-s-k3"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="<span class='figure-number'>Figure 37: </span>Perturbed sensitivity functions \\(\overline{\sigma}(S^\prime)\\) using \\(\mu\\) "optimal" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="<span class='figure-number'>Figure 39: </span>Perturbed sensitivity functions \\(\overline{\sigma}(S^\prime)\\) using \\(\mu\\) "optimal" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
|
||||
|
||||
|
||||
### Further Remarks on \\(\mu\\) {#further-remarks-on-mu}
|
||||
@@ -4696,7 +4714,7 @@ By multivariable transfer function shaping, therefore, we mean the shaping of th
|
||||
|
||||
The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values.
|
||||
|
||||
Consider the one degree-of-freedom system as shown in [Figure 38](#figure--fig:classical-feedback-small).
|
||||
Consider the one degree-of-freedom system as shown in [Figure 40](#figure--fig:classical-feedback-small).
|
||||
We have the following important relationships:
|
||||
|
||||
\begin{align}
|
||||
@@ -4706,7 +4724,7 @@ We have the following important relationships:
|
||||
|
||||
<a id="figure--fig:classical-feedback-small"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="<span class='figure-number'>Figure 38: </span>One degree-of-freedom feedback configuration" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="<span class='figure-number'>Figure 40: </span>One degree-of-freedom feedback configuration" >}}
|
||||
|
||||
<div class="important">
|
||||
|
||||
@@ -4750,11 +4768,11 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
|
||||
|
||||
</div>
|
||||
|
||||
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [Figure 39](#figure--fig:design-trade-off-mimo-gk).
|
||||
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [Figure 41](#figure--fig:design-trade-off-mimo-gk).
|
||||
|
||||
<a id="figure--fig:design-trade-off-mimo-gk"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="<span class='figure-number'>Figure 39: </span>Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="<span class='figure-number'>Figure 41: </span>Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
|
||||
|
||||
The control engineer must design \\(K\\) such that \\(\minsv(GK)\\) lies above a performance boundary for all \\(\omega\\) up to \\(\omega\_l\\), and such that \\(\maxsv(GK)\\) lies below a robustness boundary for all \\(\omega\\) above \\(\omega\_h\\).<br />
|
||||
|
||||
@@ -4810,11 +4828,11 @@ The optimal state estimate is given by a **Kalman filter**.
|
||||
|
||||
The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\).
|
||||
|
||||
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
|
||||
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 42](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
|
||||
|
||||
<a id="figure--fig:lqg-separation"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="<span class='figure-number'>Figure 40: </span>The separation theorem" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="<span class='figure-number'>Figure 42: </span>The separation theorem" >}}
|
||||
|
||||
<div class="important">
|
||||
|
||||
@@ -4842,7 +4860,7 @@ and \\(X\\) is the unique positive-semi definite solution of the algebraic Ricca
|
||||
|
||||
<div class="important">
|
||||
|
||||
The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), with:
|
||||
The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 43](#figure--fig:lqg-kalman-filter), with:
|
||||
|
||||
\begin{equation} \label{eq:kalman\_filter\_structure}
|
||||
\dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x})
|
||||
@@ -4864,9 +4882,9 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
|
||||
|
||||
<a id="figure--fig:lqg-kalman-filter"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="<span class='figure-number'>Figure 41: </span>The LQG controller and noisy plant" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="<span class='figure-number'>Figure 43: </span>The LQG controller and noisy plant" >}}
|
||||
|
||||
The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
|
||||
The structure of the LQG controller is illustrated in [Figure 43](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
|
||||
|
||||
\begin{align\*}
|
||||
L\_{\text{LQG}}(s) &= \left[ \begin{array}{c|c}
|
||||
@@ -4881,11 +4899,11 @@ The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:l
|
||||
|
||||
It has the same degree (number of poles) as the plant.<br />
|
||||
|
||||
For the LQG-controller, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
|
||||
For the LQG-controller, as shown on [Figure 43](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 44](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
|
||||
|
||||
<a id="figure--fig:lqg-integral"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="<span class='figure-number'>Figure 42: </span>LQG controller with integral action and reference input" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="<span class='figure-number'>Figure 44: </span>LQG controller with integral action and reference input" >}}
|
||||
|
||||
For an LQG-controller system with a combined Kalman filter and LQR control law, there are **no guaranteed stability margins**, and there exist LQG combinations with arbitrary small gain margins.
|
||||
However, there are procedures for improving robustness properties of LQG control such as **Loop Transfer Recovery** (LTR).
|
||||
@@ -4905,11 +4923,11 @@ Their main limitation is that they can only be applied to minimum phase plants.
|
||||
There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
|
||||
It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
|
||||
|
||||
Such a general formulation is afforded by the general configuration shown in [Figure 43](#figure--fig:general-control).
|
||||
Such a general formulation is afforded by the general configuration shown in [Figure 45](#figure--fig:general-control).
|
||||
|
||||
<a id="figure--fig:general-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="<span class='figure-number'>Figure 43: </span>General control configuration" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="<span class='figure-number'>Figure 45: </span>General control configuration" >}}
|
||||
|
||||
The system is described by
|
||||
|
||||
@@ -5085,7 +5103,7 @@ Then the LQG cost function is
|
||||
|
||||
#### \\(\hinf\\) Optimal Control {#hinf-optimal-control}
|
||||
|
||||
With reference to the general control configuration on [Figure 43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
|
||||
With reference to the general control configuration on [Figure 45](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
|
||||
|
||||
\begin{equation\*}
|
||||
\hnorm{F\_l(P, K)} = \max\_{\omega} \maxsv\big(F\_l(P, K)(j\omega)\big)
|
||||
@@ -5196,7 +5214,7 @@ In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can b
|
||||
This can be useful for **systems with channels of quite different bandwidths**.
|
||||
In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.<br />
|
||||
|
||||
To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection).
|
||||
To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 46](#figure--fig:mixed-sensitivity-dist-rejection).
|
||||
We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required.
|
||||
The elements of the generalized plant are
|
||||
|
||||
@@ -5215,16 +5233,16 @@ The elements of the generalized plant are
|
||||
|
||||
<a id="figure--fig:mixed-sensitivity-dist-rejection"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="<span class='figure-number'>Figure 44: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="<span class='figure-number'>Figure 46: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
|
||||
|
||||
Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 45](#figure--fig:mixed-sensitivity-ref-tracking).
|
||||
Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 47](#figure--fig:mixed-sensitivity-ref-tracking).
|
||||
Here we consider a tracking problem.
|
||||
The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
|
||||
As the regulation problem of [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
|
||||
As the regulation problem of [Figure 46](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
|
||||
|
||||
<a id="figure--fig:mixed-sensitivity-ref-tracking"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="<span class='figure-number'>Figure 45: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="<span class='figure-number'>Figure 47: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
|
||||
|
||||
Another useful mixed sensitivity optimization problem, is to find a stabilizing controller which minimizes
|
||||
|
||||
@@ -5235,7 +5253,7 @@ Another useful mixed sensitivity optimization problem, is to find a stabilizing
|
||||
The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation.
|
||||
It is also important for robust stability with respect to multiplicative perturbations at the plant output.
|
||||
|
||||
The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 46](#figure--fig:mixed-sensitivity-s-t).
|
||||
The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 48](#figure--fig:mixed-sensitivity-s-t).
|
||||
|
||||
The elements of the generalized plant are
|
||||
|
||||
@@ -5254,7 +5272,7 @@ The elements of the generalized plant are
|
||||
|
||||
<a id="figure--fig:mixed-sensitivity-s-t"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="<span class='figure-number'>Figure 46: </span>\\(S/T\\) mixed-sensitivity optimization in standard form" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="<span class='figure-number'>Figure 48: </span>\\(S/T\\) mixed-sensitivity optimization in standard form" >}}
|
||||
|
||||
The shaping of closed-loop transfer functions as described above with the stacked cost functions becomes difficult with more than two functions whereas with two, the process is relatively easy.
|
||||
The bandwidth requirements on each are usually complementary and simple, stable low-pass and high-pass filters are sufficient to carry out the required shaping and trade-offs.<br />
|
||||
@@ -5277,20 +5295,20 @@ The focus of attention has moved to the size of signals and away from the size a
|
||||
</div>
|
||||
|
||||
Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals.
|
||||
Weights are also used if a perturbation is used to model uncertainty, as in [Figure 47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
|
||||
Weights are also used if a perturbation is used to model uncertainty, as in [Figure 49](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
|
||||
|
||||
<a id="figure--fig:input-uncertainty-hinf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="<span class='figure-number'>Figure 47: </span>Multiplicative dynamic uncertainty model" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="<span class='figure-number'>Figure 49: </span>Multiplicative dynamic uncertainty model" >}}
|
||||
|
||||
LQG control is a simple example of the signal based approach, in which the exogenous signals are assumed to be stochastic and the error signals are measured in terms of the 2-norm.
|
||||
As we have seen, the weights \\(Q\\) and \\(R\\) are constant, but LQG can be generalized to include frequency dependent weights on the signals leading to what is called Wiener-Hopf design or \\(\htwo\\) control.<br />
|
||||
|
||||
When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm.
|
||||
In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm.
|
||||
However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 47](#figure--fig:input-uncertainty-hinf).<br />
|
||||
However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 49](#figure--fig:input-uncertainty-hinf).<br />
|
||||
|
||||
A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 48](#figure--fig:hinf-signal-based).
|
||||
A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 50](#figure--fig:hinf-signal-based).
|
||||
\\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed.
|
||||
The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals.
|
||||
The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\).
|
||||
@@ -5313,9 +5331,9 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
|
||||
|
||||
<a id="figure--fig:hinf-signal-based"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="<span class='figure-number'>Figure 48: </span>A signal-based \\(\hinf\\) control problem" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="<span class='figure-number'>Figure 50: </span>A signal-based \\(\hinf\\) control problem" >}}
|
||||
|
||||
Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 49](#figure--fig:hinf-signal-based-uncertainty).
|
||||
Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 51](#figure--fig:hinf-signal-based-uncertainty).
|
||||
The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\).
|
||||
We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**.
|
||||
This problem is a non-standard \\(\hinf\\) optimization.
|
||||
@@ -5327,7 +5345,7 @@ It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) proc
|
||||
|
||||
<a id="figure--fig:hinf-signal-based-uncertainty"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="<span class='figure-number'>Figure 49: </span>A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="<span class='figure-number'>Figure 51: </span>A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
|
||||
|
||||
However, whilst the structured singular value is a useful analysis tool for assessing designs, \\(\mu\text{-synthesis}\\) is sometimes difficult to use and often too complex for the practical problems.
|
||||
|
||||
@@ -5378,7 +5396,7 @@ The objective of robust stabilization is to stabilize not only the nominal model
|
||||
|
||||
where \\(\epsilon > 0\\) is then the **stability margin**.<br />
|
||||
|
||||
For the perturbed feedback system of [Figure 50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
|
||||
For the perturbed feedback system of [Figure 52](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
|
||||
|
||||
\begin{equation\*}
|
||||
\gamma \triangleq \hnorm{\begin{bmatrix}
|
||||
@@ -5391,7 +5409,7 @@ Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bm
|
||||
|
||||
<a id="figure--fig:coprime-uncertainty-bis"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="<span class='figure-number'>Figure 50: </span>\\(\hinf\\) robust stabilization problem" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="<span class='figure-number'>Figure 52: </span>\\(\hinf\\) robust stabilization problem" >}}
|
||||
|
||||
The lowest achievable value of \\(\gamma\\) and the corresponding maximum stability margin \\(\epsilon\\) are given as
|
||||
|
||||
@@ -5456,11 +5474,11 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
|
||||
G\_s = W\_2 G W\_1
|
||||
\end{equation}
|
||||
|
||||
as shown in [Figure 51](#figure--fig:shaped-plant).
|
||||
as shown in [Figure 53](#figure--fig:shaped-plant).
|
||||
|
||||
<a id="figure--fig:shaped-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="<span class='figure-number'>Figure 51: </span>The shaped plant and controller" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="<span class='figure-number'>Figure 53: </span>The shaped plant and controller" >}}
|
||||
|
||||
The controller \\(K\_s\\) is synthesized by solving the robust stabilization problem for the shaped plant \\(G\_s\\) with a normalized left coprime factorization \\(G\_s = M\_s^{-1}N\_s\\).
|
||||
The feedback controller for the plant \\(G\\) is then \\(K = W\_1 K\_s W\_2\\).<br />
|
||||
@@ -5491,13 +5509,13 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
|
||||
- A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements
|
||||
7. **Analyze the design** and if not all the specification are met, make further modifications to the weights
|
||||
8. **Implement the controller**.
|
||||
The configuration shown in [Figure 52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 38](#figure--fig:classical-feedback-small).
|
||||
The configuration shown in [Figure 54](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 40](#figure--fig:classical-feedback-small).
|
||||
This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
|
||||
The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
|
||||
|
||||
<a id="figure--fig:shapping-practical-implementation"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="<span class='figure-number'>Figure 52: </span>A practical implementation of the loop-shaping controller" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="<span class='figure-number'>Figure 54: </span>A practical implementation of the loop-shaping controller" >}}
|
||||
|
||||
We will conclude this section with a summary of the **advantages** offered by the above \\(\hinf\\) loop-shaping design procedure:
|
||||
|
||||
@@ -5518,25 +5536,25 @@ Many control design problems possess two degrees-of-freedom:
|
||||
Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output.
|
||||
But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.<br />
|
||||
|
||||
A general two degrees-of-freedom feedback control scheme is depicted in [Figure 53](#figure--fig:classical-feedback-2dof-simple).
|
||||
A general two degrees-of-freedom feedback control scheme is depicted in [Figure 55](#figure--fig:classical-feedback-2dof-simple).
|
||||
The commands and feedbacks enter the controller separately and are independently processed.
|
||||
|
||||
<a id="figure--fig:classical-feedback-2dof-simple"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="<span class='figure-number'>Figure 53: </span>General two degrees-of-freedom feedback control scheme" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="<span class='figure-number'>Figure 55: </span>General two degrees-of-freedom feedback control scheme" >}}
|
||||
|
||||
The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
|
||||
However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.<br />
|
||||
|
||||
The design problem is illustrated in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
|
||||
The design problem is illustrated in [Figure 56](#figure--fig:coprime-uncertainty-hinf).
|
||||
The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
|
||||
A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
|
||||
|
||||
<a id="figure--fig:coprime-uncertainty-hinf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="<span class='figure-number'>Figure 54: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="<span class='figure-number'>Figure 56: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
|
||||
|
||||
The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
|
||||
The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 56](#figure--fig:coprime-uncertainty-hinf).
|
||||
This problem is easily cast into the general configuration.
|
||||
|
||||
The control signal to the shaped plant \\(u\_s\\) is given by:
|
||||
@@ -5566,11 +5584,11 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
|
||||
5. Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state.
|
||||
6. Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\)
|
||||
|
||||
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 55](#figure--fig:hinf-synthesis-2dof).
|
||||
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 57](#figure--fig:hinf-synthesis-2dof).
|
||||
|
||||
<a id="figure--fig:hinf-synthesis-2dof"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="<span class='figure-number'>Figure 55: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="<span class='figure-number'>Figure 57: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
|
||||
|
||||
|
||||
#### Observer-Based Structure for \\(\hinf\\) Loop-Shaping Controllers {#observer-based-structure-for-hinf-loop-shaping-controllers}
|
||||
@@ -5650,11 +5668,11 @@ When implemented in Hanus form, the expression for \\(u\\) becomes
|
||||
|
||||
where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation.
|
||||
|
||||
The situation is illustrated in [Figure 56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
|
||||
The situation is illustrated in [Figure 58](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
|
||||
|
||||
<a id="figure--fig:weight-anti-windup"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="<span class='figure-number'>Figure 56: </span>Self-conditioned weight \\(W\_1\\)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="<span class='figure-number'>Figure 58: </span>Self-conditioned weight \\(W\_1\\)" >}}
|
||||
|
||||
The Hanus form prevents windup by keeping the states of \\(W\_1\\) consistent with the actual plant input at all times.
|
||||
When there is no saturation, \\(u\_a=u\\), the dynamics of \\(W\_1\\) remains unaffected.
|
||||
@@ -5713,11 +5731,11 @@ Moreover, one should be careful about combining controller synthesis and analysi
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
||||
In previous sections, we considered the general problem formulation in [Figure 57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
|
||||
In previous sections, we considered the general problem formulation in [Figure 59](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
|
||||
|
||||
<a id="figure--fig:general-control-names-bis"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="<span class='figure-number'>Figure 57: </span>General Control Configuration" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="<span class='figure-number'>Figure 59: </span>General Control Configuration" >}}
|
||||
|
||||
In this chapter we are concerned with the **structural decisions** associated with the following selection tasks of control structure design:
|
||||
|
||||
@@ -5748,31 +5766,40 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
|
||||
- **Optimization layer**: computes the desired reference commands \\(r\\)
|
||||
- **Control layer**: implements these commands to achieve \\(y \approx r\\)
|
||||
|
||||
Additional layers are possible, as is illustrated in [Figure 58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
|
||||
Additional layers are possible, as is illustrated in [Figure 60](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
|
||||
|
||||
<a id="figure--fig:control-system-hierarchy"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="<span class='figure-number'>Figure 58: </span>Typical control system hierarchy in a chemical plant" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="<span class='figure-number'>Figure 60: </span>Typical control system hierarchy in a chemical plant" >}}
|
||||
|
||||
In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [ 6](#org-target--fig-optimize-control-b)).
|
||||
In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [Figure 61b](#org-target--fig-optimize-control-b)).
|
||||
There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.<br />
|
||||
|
||||
The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information.
|
||||
The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.<br />
|
||||
|
||||
From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [ 6](#org-target--fig-optimize-control-c)).
|
||||
From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [Figure 61c](#org-target--fig-optimize-control-c)).
|
||||
All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant.
|
||||
However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power.
|
||||
|
||||
<a id="table--fig:optimize-control"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--fig:optimize-control">Table 6</a>:</span>
|
||||
Alternative structures for optimization and control
|
||||
</div>
|
||||
|
||||
|  |  |  |
|
||||
|-------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------|
|
||||
| <span class="org-target" id="org-target--fig-optimize-control-a"></span> Open loop optimization | <span class="org-target" id="org-target--fig-optimize-control-b"></span> Closed-loop implementation with separate control layer | <span class="org-target" id="org-target--fig-optimize-control-c"></span> Integrated optimization and control |
|
||||
<figure class="subfigures" id="table--fig:optimize-control">
|
||||
<div class="subfigure-row">
|
||||
<div class="subfigure" id="org-target--fig-optimize-control-a" style="flex-basis: 32%">
|
||||
<img src="/ox-hugo/skogestad07_optimize_control_a.png" alt="Open loop optimization" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(a)</span> Open loop optimization</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-optimize-control-b" style="flex-basis: 32%">
|
||||
<img src="/ox-hugo/skogestad07_optimize_control_b.png" alt="Closed-loop implementation with separate control layer" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(b)</span> Closed-loop implementation with separate control layer</div>
|
||||
</div>
|
||||
<div class="subfigure" id="org-target--fig-optimize-control-c" style="flex-basis: 32%">
|
||||
<img src="/ox-hugo/skogestad07_optimize_control_c.png" alt="Integrated optimization and control" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(c)</span> Integrated optimization and control</div>
|
||||
</div>
|
||||
</div>
|
||||
<figcaption><span class="figure-number">Figure 61: </span>Alternative structures for optimization and control</figcaption>
|
||||
</figure>
|
||||
|
||||
|
||||
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
|
||||
@@ -5885,7 +5912,7 @@ Thus, the selection of controlled and measured outputs are two separate issues.
|
||||
|
||||
### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements}
|
||||
|
||||
We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 57](#figure--fig:general-control-names-bis).
|
||||
We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 59](#figure--fig:general-control-names-bis).
|
||||
Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to:
|
||||
|
||||
- Stabilize the plant, or more generally change its dynamics
|
||||
@@ -5977,19 +6004,24 @@ Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degre
|
||||
A cascade control structure results when either of the following two situations arise:
|
||||
|
||||
- The reference \\(r\_i\\) is an output from another controller.
|
||||
This is the **conventional cascade control** ([ 7](#org-target--fig-cascade-extra-meas))
|
||||
This is the **conventional cascade control** ([Figure 62a](#org-target--fig-cascade-extra-meas))
|
||||
- The "measurement" \\(y\_i\\) is an output from another controller.
|
||||
This is referred to as **input resetting** ([ 7](#org-target--fig-cascade-extra-input))
|
||||
This is referred to as **input resetting** ([Figure 62b](#org-target--fig-cascade-extra-input))
|
||||
|
||||
<a id="table--fig:cascade-implementation"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--fig:cascade-implementation">Table 7</a>:</span>
|
||||
Cascade Implementations
|
||||
|
||||
<figure class="subfigures" id="table--fig:cascade-implementation">
|
||||
<div class="subfigure-row">
|
||||
<div class="subfigure" id="org-target--fig-cascade-extra-meas">
|
||||
<img src="/ox-hugo/skogestad07_cascade_extra_meas.png" alt="Extra measurements \(y_2\)" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(a)</span> Extra measurements \(y_2\)</div>
|
||||
</div>
|
||||
|
||||
|  |  |
|
||||
|--------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
|
||||
| <span class="org-target" id="org-target--fig-cascade-extra-meas"></span> Extra measurements \\(y\_2\\) | <span class="org-target" id="org-target--fig-cascade-extra-input"></span> Extra inputs \\(u\_2\\) |
|
||||
<div class="subfigure" id="org-target--fig-cascade-extra-input">
|
||||
<img src="/ox-hugo/skogestad07_cascade_extra_input.png" alt="Extra inputs \(u_2\)" loading="lazy">
|
||||
<div class="subfigure-caption"><span class="subfigure-label">(b)</span> Extra inputs \(u_2\)</div>
|
||||
</div>
|
||||
</div>
|
||||
<figcaption><span class="figure-number">Figure 62: </span>Cascade Implementations</figcaption>
|
||||
</figure>
|
||||
|
||||
|
||||
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
|
||||
@@ -6013,7 +6045,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c
|
||||
|
||||
##### Cascade implementation {#cascade-implementation}
|
||||
|
||||
To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [ 7](#org-target--fig-cascade-extra-meas):
|
||||
To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [Figure 62a](#org-target--fig-cascade-extra-meas):
|
||||
|
||||
\begin{align\*}
|
||||
r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\
|
||||
@@ -6023,12 +6055,12 @@ To obtain an implementation with two SISO controllers, we may cascade the contro
|
||||
Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\).
|
||||
Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**.
|
||||
|
||||
In the general case ([ 7](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
|
||||
However, it is common to encounter the situation in [Figure 59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [ 7](#org-target--fig-cascade-extra-meas).
|
||||
In the general case ([Figure 62a](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
|
||||
However, it is common to encounter the situation in [Figure 63](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [Figure 62a](#org-target--fig-cascade-extra-meas).
|
||||
|
||||
<div class="important">
|
||||
|
||||
With reference to the special (but common) case of cascade control shown in [Figure 59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
|
||||
With reference to the special (but common) case of cascade control shown in [Figure 63](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
|
||||
|
||||
- The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
|
||||
If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
|
||||
@@ -6039,7 +6071,7 @@ With reference to the special (but common) case of cascade control shown in [Fig
|
||||
|
||||
<a id="figure--fig:cascade-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="<span class='figure-number'>Figure 59: </span>Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="<span class='figure-number'>Figure 63: </span>Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
|
||||
|
||||
In terms of design, it is recommended to first design \\(K\_2\\) to minimize the effect of \\(d\_2\\) on \\(y\_1\\) and then to design \\(K\_1\\) to minimize the effect of \\(d\_1\\) on \\(y\_1\\).
|
||||
|
||||
@@ -6065,7 +6097,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t
|
||||
|
||||
##### Cascade implementation {#cascade-implementation}
|
||||
|
||||
To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [ 7](#org-target--fig-cascade-extra-input).
|
||||
To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [Figure 62b](#org-target--fig-cascade-extra-input).
|
||||
We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**.
|
||||
The fast control loop is then
|
||||
|
||||
@@ -6086,7 +6118,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
|
||||
|
||||
<div class="exampl">
|
||||
|
||||
Consider the system in [Figure 60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
|
||||
Consider the system in [Figure 64](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
|
||||
Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
|
||||
Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
|
||||
The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
|
||||
@@ -6100,7 +6132,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
|
||||
|
||||
<a id="figure--fig:cascade-control-two-layers"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="<span class='figure-number'>Figure 60: </span>Control configuration with two layers of cascade control" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="<span class='figure-number'>Figure 64: </span>Control configuration with two layers of cascade control" >}}
|
||||
|
||||
|
||||
#### Selectors {#selectors}
|
||||
@@ -6173,7 +6205,7 @@ Four applications of partial control are:
|
||||
The outputs \\(y\_1\\) have an associated control objective but are not measured.
|
||||
Instead, we aim at indirectly controlling \\(y\_1\\) by controlling the secondary measured variables \\(y\_2\\).
|
||||
|
||||
The table [Table 8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
|
||||
The table [Table 4](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
|
||||
In all cases, there is a control objective associated with \\(y\_1\\) and a feedback involving measurement and control of \\(y\_2\\) and we want:
|
||||
|
||||
- The effect of disturbances on \\(y\_1\\) to be small (when \\(y\_2\\) is controlled)
|
||||
@@ -6181,7 +6213,7 @@ In all cases, there is a control objective associated with \\(y\_1\\) and a feed
|
||||
|
||||
<a id="table--tab:partial-control"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:partial-control">Table 8</a>:</span>
|
||||
<span class="table-number"><a href="#table--tab:partial-control">Table 4</a>:</span>
|
||||
Applications of partial control
|
||||
</div>
|
||||
|
||||
@@ -6201,7 +6233,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 61](#figure--fig:partial-control)).
|
||||
Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 65](#figure--fig:partial-control)).
|
||||
We get:
|
||||
|
||||
\begin{equation} \label{eq:partial\_control}
|
||||
@@ -6214,7 +6246,7 @@ We get:
|
||||
|
||||
<a id="figure--fig:partial-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="<span class='figure-number'>Figure 61: </span>Partial Control" >}}
|
||||
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="<span class='figure-number'>Figure 65: </span>Partial Control" >}}
|
||||
|
||||
|
||||
##### Tight control of \\(y\_2\\) {#tight-control-of-y-2}
|
||||
@@ -6270,7 +6302,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
|
||||
|
||||
##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems}
|
||||
|
||||
Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
|
||||
Consider the conventional cascade control system in [Figure 62a](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
|
||||
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
|
||||
|
||||
From \ref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
|
||||
@@ -6338,11 +6370,11 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
|
||||
|
||||
### Decentralized Feedback Control {#decentralized-feedback-control}
|
||||
|
||||
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 62](#figure--fig:decentralized-diagonal-control)).
|
||||
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 66](#figure--fig:decentralized-diagonal-control)).
|
||||
|
||||
<a id="figure--fig:decentralized-diagonal-control"></a>
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{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="<span class='figure-number'>Figure 62: </span>Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
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{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="<span class='figure-number'>Figure 66: </span>Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
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The design of **decentralized diagonal control systems** involves two steps:
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Reference in New Issue
Block a user