Robust stability

8 Robust stability and robust performance 9.8.1 Robust stability 

Robust stability can be investigated in the frequency domain, using the Nyquist stability criterion, defined in section 6.4.2. 

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as 

From equation (9.135) the disk radius (bound of uncertainty) is 

From the Nyquist stability criterion, let N k, G(iuj)) be the net number of clockwise encirclements of a point (k, 0) of the Nyquist contour. Assume that all plants in the family tt, expressed in equation (9.132) have the same number ( ) of right-hand plane (RHP) poles. 

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There will be robust stability of a speeifie eontroller C(juj) if and only if... 

From Figure 9.23 robust stability oeeurs when the veetor magnitude 1 + Gm(jtu)C(juj) (see also Figure 6.25) exeeeds the disk radius Gm(jtu)C(juj) ( m(uj)... 

Robust stability ean therefore be stated as If all plants G(.v) in the family tt have the same number of RHP poles and that a partieular eontroller C(.v) stabilizes the nominal plant Gmfv), then the system is robustly stable with the eontroller C(.v) if and only if the eomplementary sensitivity funetion T s) for the nominal plant Gmfv) satisfies the following bound... 

Robust stability provides a minimum requirement in an environment where there is plant model uneertainty. For a eontrol system to have robust performanee it should be eapable of minimizing the error for the worst plant (i.e. the one giving the largest error) in the family G(jtu) [Pg.308]

From Figure 9.23 representing robust stability, the aetual frequeney response G juj)C juj) will always lie inside the region of uneertainty denoted by the disk, or... 

What is the maximum value that K ean have for robust stability ... 

At frequeneies below 1 rad/s, fmCtu) 0.5 and at frequeneies above 4rad/s ftnfa ) 2.0. From equation (9.141), for robust stability... 

In Example 6.4, when there was no model uneertainty, K for marginal stability was 8, and for a gain margin of 6dB, K was 4. In this example with model uneertainty, from equation (9.154) marginal stability oeeurs with K = 3.5, so this is the maximum value for robust stability. For robust performanee, equation (9.150) applies. For a speeifie step input let lV(s) = 1 /s now... 

As seen in the retro-synthetic Scheme 5.3, intermediate 15 is useful for both routes. The choice of benzyl protection group was made based on the robust stability of benzyl phenol ethers toward most reactions and several possible avenues to remove it, although it was reported from Medicinal Chemistry that benzyl group removal via hydrogenolysis posed challenges in this compound. The choice of iodide substitution was born out of the known high reactivity of iodides in the Ullmann-type coupling reaction with alcohols and the robust stability of aryl iodides in many other common reactions. 

Although adequate materials and devices are essential, successful manufacturing will require other capabilities as well. First, the process must have high yield, which implies low variability, and provide robust stability to environmental factors. To produce the envisioned products, there must be readily available electronic design tools that can adequately simulate both device and circuit performance. Although some of these computer-aided design tools are available from microelectronics technology, others must either be modified, because of the differences in the thin-hlm devices, or created anew because the devices have no equivalent (nanowires and nanotubes). 

J. Alvarez-Ramirez, J. Snarez, and R. Femat. Robust stabilization of temperature in continuous-stirred tank reactors. Chem. Eng. Sci., 52(14) 2223-2230, 1997. 

W. Zhou and P.L. Lee. Robust stability analysis of generic model control. Chemical Engineering Communications, 117(1) 41—72, 1992. 

Robust stability over long periods of time and at high volume fractions, and... 

Example 10—Ensuring robust stability of a heavy oil fractionator. 

For this MFC system, Eqs. (118) through (125), Vuthandam et al. (1995) developed sufficient conditions for robust stability with zero offset. These conditions can be used directly for calculation of minimum values for the prediction and control horizon lengths, p and m, respectively, as well as for the move suppression coefficients rji, which are not equal over the... 

A third proposition has been discussed in Section V,B,l,b. The idea is that if a robust stability constraint [Eqs. (147) or (158)], is used, MFC will be stabilizing therefore, true performance objectives may translated into values for the tuning parameters of MFC, with no need to worry about potential instabilities resulting from poor tuning. However, that translation of performance objectives to values for MFC tuning parameters is not always straightforward. 

A fourth proposition was discussed by Vuthandam et al. (1995). Their idea is that the values of the MFC tuning parameters must satisfy robust stability requirements. It turns out that for the robust stability requirements developed by these authors, performance improves as the prediction horizon length, p, increases from its minimum value to larger values, but after a certain point performance deteriorates as p increases further. This happens because for very large p the input move terms in the on-line objective function must be penalized so much that the controller becomes very sluggish and performance suffers. Results such as these depend on the form of the robust stability conditions. If such conditions are only sufficient, as is the case with Vuthandam et al (1995), then performance-related results may be conservative. 

Genceli, H., Robust Stability Analysis of Constrained /j-Norm Model Predictive Control. PhD Thesis, Texas A M University (1993). 

Genceli, H., and Nikolaou, M., Robust stability analysis of constrained /i-norm model predictive control, AIChEJ. 39(12), 1954-1965 (1993). 

Zheng, Z. Q., and Morari, M., Robust stability of constrained model predictive control. Proceedings of the American Control Conference, session WM7, 379-383, San Francisco (1993). 

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