Update Content - 2021-08-24

content/book/albertos04_multiv_contr_system.md

@@ -8,7 +8,7 @@ Tags
 : [Multivariable Control](multivariable_control.md)
 
 Reference
-: ([Albertos and Antonio 2004](#org22a156f))
+: ([Albertos and Antonio 2004](#orga6ef935))
 
 Author(s)
 : Albertos, P., & Antonio, S.
@@ -19,6 +19,9 @@ Year
 
 ## [Decentralized Control](decentralized_control.md) and Decoupled Control {#decentralized-control--decentralized-control-dot-md--and-decoupled-control}
 
+
+### Introduction {#introduction}
+
 Decentralized control is decomposed into two steps:
 
 1.  decoupled the plant into several subsystems
@@ -26,6 +29,42 @@ Decentralized control is decomposed into two steps:
 
 The initial effort of decoupling the system results in subsequent easier design, implementation and tuning.
 
+Decentralized control tries to control multivariable plants by a suitable decomposition into SISO control loops.
+If the process has strong coupling or conditioning problems, centralized control may be required.
+It however requires the availability of a precise model.
+
+Two approaches can be used to control a coupled system with SISO techniques:
+
+-   **decentralized control** tries to divide the plant and design _independent_ controllers for each subsystems.
+    Two alternative arise:
+    -   neglect the coupling
+    -   carry out a _decoupling_ operation by "canceling" coupling by transforming the system into a diagonal or triangular structure bia a transformation matrix
+-   **cascade control**
+
+
+### Mutli-Loop Control, Pairing Selection {#mutli-loop-control-pairing-selection}
+
+The strategy called _multi-loop control_ consists of first proper input/output pairing, and then design of several SISO controllers.
+In this way, a complex control problem is divided into several simpler ones.
+
+The multi-loop control may not work in strongly coupled systems.
+Therefore, a methodology the access the degree of interaction between the loops is needed.
+
+
+#### [Relative Gain Array](relative_gain_array.md) {#relative-gain-array--relative-gain-array-dot-md}
+
+The Relative Gain Array (RGA) \\(\Lambda(s)\\) is defined as:
+
+\begin{equation}
+\Lambda(s) = G(s) \times (G(s)^T)^{-1}
+\end{equation}
+
+The RGA is scaling-independent and controller-independent.
+These coefficients can be interpreted as the ratio between the open-loop SISO static gain and the gain with "perfect" control on the rest of the loops.
+
+For demanding control specifications, the values of \\(\Lambda\\) car be drawn as a function of frequency.
+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.
+
 
 ### Decoupling {#decoupling}
 
@@ -40,16 +79,17 @@ This strategy is called **decoupling**.
 
 #### Feedforward Decoupling {#feedforward-decoupling}
 
-A pre-compensator can be added to transform the open-loop characteristics into a new one as chosen by the designer.
+A pre-compensator (Figure [1](#org7023330)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
 This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
 
+<a id="org7023330"></a>
+
+{{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="Figure 1: Decoupler pre-compensator" >}}
+
 **Approximate decoupling**:
 To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
 If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
 
-
-#### Feedback Decoupling {#feedback-decoupling}
-
 Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
 
 -   as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops.
@@ -72,12 +112,12 @@ where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
 The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
 In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
 
-If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#orgbba6502), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
+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.
 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.
 
-<a id="orgbba6502"></a>
+<a id="org2de6de7"></a>
 
-{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
+{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 2: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
 
 The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control".
 
@@ -88,13 +128,22 @@ SVD decoupling produces the most suitable combinations for independent "multi-lo
 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.
 
 
-## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
+### Conclusions {#conclusions}
 
+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.
 
-## Optimisation-based Control {#optimisation-based-control}
+Multi-loop strategies, if suitable, may present th advantages of fault tolerance, as well as simplicity.
+However, in some cases, tuning may be difficult and coupling may severely limit their performance.
 
+Decoupling is based on mathematical transformations of the system models into diagonal form.
+Feedforward decoupling can be used in many cases.
+Feedback decoupling achieves its objective if state is measurable and system is minimum-phase.
+However, decoupling may be very sensitive to modelling errors and it is not the optimal strategy for disturbance rejection.
 
-## Designing for Robustness {#designing-for-robustness}
+Cascade control is widely used in industry to improve the behaviour of basic SISO loops via the addition of extra sensors and actuators.
+However, ease of tuning requires that different time constants are involved in different subsystems.
+In general, addition of extra sensors and actuators in a SISO or MIMO loop, will improve achievable performance and/or tolerance to modelling errors.
+The level of improvement must be traded off against the cost of additional instrumentation.
 
 
 ## Implementation and Other Issues {#implementation-and-other-issues}
@@ -142,4 +191,4 @@ The solution is similar to that of the wind-up phenomenon: the regulator should
 
 ## Bibliography {#bibliography}
 
-<a id="org22a156f"></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>.
+<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>.