Stability in the z plane

6 Stability in the z-plane 7.6.1 Mapping from the s-plane into the z-plane 

Just as transient analysis of continuous systems may be undertaken in the. v-plane, stability and transient analysis on discrete systems may be conducted in the z-plane. It is possible to map from the. v to the z-plane using the relationship 

First we will look at the question of stability in the z plane. Then root locus and frequency response methods will be used to analyze sampled-data systems. Various types of processes and controllers will be studied. 

The stability of any system is determined by the location of the roots of its characteristic equation (or the poles of its transfer function). The characteristic equation of a continuous system is a polynomial in the complex variable s. If all the roots of this polynomial are in the left half of the s plane, the system is stable. For a continuous closedloop system, all the roots of 1 + must lie in the left 

The stability of a sampled-data system is determined by the location of the roots of a characteristic equation that is a polynomial in the complex variable z. This characteristic equation is the denominator of the system transfer function set equal to zero. The roots of this polynomial (the poles of the system transfer function) are plotted in the z plane. The ordinate is the imaginary part of z, and the abscissa is the real part of z. 

The region of stability in the z plane can be found directly from the region of stability in the s plane by using the basic relationship between the complex variables s and z. [Equation (18.23).] 

The stability region in the s plane is where a, the real part of s, is negative. Substituting Eq. (19.2) into Eq. (19.1) gives 

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The seript file examp75.m simulates the Jury stability test undertaken in Example 7.5. With the eontroller gain K in Example 7.5 (Figure 7.14) set to 9.58 for marginal stability see equation (7.75), the roots of the denominator of the elosed-loop pulse transfer funetion are ealeulated, and found to lie on the unit eirele in the z-plane. 

The bilinear transformation is another change of variables. We convert from the z variable into the lU variable. The transformation maps the unit circle in the z plane into the left half of the ID plane. This mapping converts the stability region back to the familiar LHP region. The Routh criterion can then be used. Root locus plots can be made in the 11 plane with the system going closedloop unstable when the loci cross over into the RHP. 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

Bilinear transformation to map the interior of the unit circle in the z-plane onto the left half of the complex variable -plane. (Application of the Routh-Hurwitz stability criterion). ... 

In Chapter 14 we define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary), develop transfer functions in the z domain, and discuss stability. Design of digital controllers is studied in Chapter 15 using root locus and frequency response methods in the z plane. We use practically all the stability analysis and controller design techniques that we introduced in the Laplace and frequency domains, now applying them in the z domain for sampled-data systems. 

The zeros of an FIR hlter may Ue anywhere in the z plane because they do not impact on the stability of the hlter however, if the weighting coefficients are real and symmetric, or anti-symmetric, about their center value M/2, any complex zeros of the hlter are constrained to Ue as conjugate pairs coincident with the unit circle or as quartets of roots off the unit circle with the form (l/p)e , (l/p)e ) where p and 9 are, respectively, the radius and angle of the hrst zero. Zeros that Ue within the unit circle are termed minimum phase, whereas those which Ue outside the unit circle are called maximum phase. This distinction describes the contribution made by a particular zero... 

This behaviour can result in severe problems due to cracking of plated through holes for laminates exposed to temperatures close to or higher than Tg. In addition, the copper foils in the inner layers of multilayer boards can undergo similar cracking. Thus excellent dimensional stability in the XY plane is achieved at the cost of low Z axis stability. This problem has precipitated a considerable interest in the development of higher Tg epoxy resins and the use of alternative polymers such as polyimides. Some of these materials will be reviewed in the following section. 

Thus the van der Waals stabilization should be maximized in the P3/2 3/2 state by allowing the rare gas atoms above and below the xy plane to approach the metal atom more closely than those lying in the xy plane. In terms of the substitutional site, this corresponds to an axial contraction along the z-axis, which... 

The examples below illustrate the use of the bilinear transformation to analyze the stability of sampled-data systems. We can use all the classical methods that we are used to employing in the s plane. The price that we pay is the additional algebra to convert to ID from z. 

It should be noted that the stability limits in the s plane, the ID plane, and the log-z plane are all the same the imaginary axis. However, lines of constant damping coefficients in the ID plane are not radial straight lines as they are in the s and log-z planes. 

Assuming that all combinations of neutrons and protons can exist, which atomic nuclei are stable enough to survive for as long as the Universe itself, that is, for around 10 billion years Estimates of nuclear stability are available to answer this query. The 270 or so nuclei found in nature in some lasting form all lie along what is known as the valley of stability in the (A, Z) plane (Eig. 4.2). 

Figure 9.4. Interaction diagram for the MOs of the trigonal bipyramid with a substituent in the equatorial plane a to the site of substitution (a) X (b) Z ( ) C . Only Z substituents are expected to have a strong stabilizing effect on the TS.

The intermediate cation has only a single bond and so rotation might be expected to lead to a. mixture of geometrical isomers of the product but this is not observed. The bonding interaction between the C-Si bond and the empty p orbital means that rotation is restricted. This stabilization weakens the C-Si bond and the silyl group is quickly removed before any further rotation can occur. The stabilization is effective only if the C-Si bond is correctly aligned with the vacant orbital, which means it must be in the same plane—rather like a it bond. Here is the result for both E- and Z-iso-mers of the vinyl silane. 

It is possible to study stability of equilibrium points for this Hamiltonian [62]. Because of symmetry, we know beforehand that the equilibrium point may exist only at z = 0 and either at x = 0,y = or at the two equivalent points turned 120° in the z = 0 plane. These three equilibrium configurations correspond to the three situations of Fig. 18. 

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