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Thomas Dehaeze
2024-12-17 16:38:55 +01:00
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content/book/skogestad07_multiv_feedb_contr.md
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@@ -4447,7 +4447,7 @@ The peak value is close to 6, meaning that even with 6 times less uncertainty, t
We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller.
We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity.
With \\(N\\) given by <eq:n_delta_structure_clasic>, we have:
With \\(N\\) given by \eqref{eq:n\_delta\_structure\_clasic}, we have:
\begin{equation\*}
\overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\overline{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\overline{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k})
@@ -5439,8 +5439,8 @@ for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
L &= (1-\gamma^2) I + XZ
\end{align}
The Matlab function `coprimeunc` can be used to generate the controller in <eq:control_coprime_factor>.
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from <eq:gamma_min_coprime> we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
The Matlab function `coprimeunc` can be used to generate the controller in \eqref{eq:control\_coprime\_factor}.
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \eqref{eq:gamma\_min\_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic-hinf-loop-shaping-design-procedure}
@@ -6273,7 +6273,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
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\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From <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}\\|\\).
From \eqref{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}\\|\\).
These arguments particularly apply at high frequencies.
More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\).
@@ -6289,7 +6289,7 @@ A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all
</div>
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in <eq:partial_control_partitioning> and <eq:partial_control>, and compute \\(P\_d\\) using <eq:tight_control_y2>.
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \eqref{eq:partial\_control\_partitioning} and \eqref{eq:partial\_control}, and compute \\(P\_d\\) using \eqref{eq:tight\_control\_y2}.
#### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control}
@@ -6459,7 +6459,7 @@ We then derive **necessary conditions for stability** which may be used to elimi
For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time.
Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable).
Using the **spectral radius condition** on the factorized \\(S\\) in <eq:S_factorization>, we have that the overall system is stable (\\(S\\) is stable) if
Using the **spectral radius condition** on the factorized \\(S\\) in \eqref{eq:S\_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
\begin{equation}
\rho(E\tilde{T}(j\omega)) < 1, \forall\omega
@@ -6684,7 +6684,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
|1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
\end{equation}
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by <eq:decent_contr_one_loop> is achievable.
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \eqref{eq:decent\_contr\_one\_loop} is achievable.
Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)
6. Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
@@ -6700,7 +6700,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4.
If one still cannot find any pairing which are controllable, then one should consider multivariable control.
7. If the chosen pairing is controllable, then <eq:decent_contr_one_loop> tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
7. If the chosen pairing is controllable, then \eqref{eq:decent\_contr\_one\_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)
@@ -6720,7 +6720,7 @@ Thus sequential design may involve many iterations.
#### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
A number of **conditions for the stability**, e.g. <eq:decent_contr_cond_stability> and <eq:decent_contr_necessary_cond_stability>, and **performance**, e.g. <eq:decent_contr_cond_perf_dist> and <eq:decent_contr_cond_perf_ref>, of decentralized control systems have been derived.
A number of **conditions for the stability**, e.g. \eqref{eq:decent\_contr\_cond\_stability} and \eqref{eq:decent\_contr\_necessary\_cond\_stability}, and **performance**, e.g. \eqref{eq:decent\_contr\_cond\_perf\_dist} and \eqref{eq:decent\_contr\_cond\_perf\_ref}, of decentralized control systems have been derived.
The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.