Imaginaries. Axis

In a quadrupole mass analyzer, four parallel rods are arranged equidistantly from a central (imaginary) axis. 

In the semiclassical evaluation of the barrier partition function the integration goes along the whole imaginary axis in the c, plane (see fig. 21). 

When a iocus crosses the imaginary axis, cr = 0. This is the condition of marginai stabiiity, i.e. the controi system is on the verge of instabiiity, where transient osciiia-tions neither increase, nor decay, but remain at a constant vaiue. 

Imaginary axis crossover The loeation on the imaginary axis of the loei (marginal stability) ean be ealeulated using either ... 

Imaginary axis crossover (Rule 9) From characteristic equation (5.ii2)... 

This script file produces the root locus shown in Figure 5.24 and allows the user to select the value of K furthest from the imaginary axis that corresponds to ( = 0.7. The command window text is... 

In a transient or an AC circuit we term the sum of resistance, inductance, and capacitance as impedance. Using complex notation, the energy storage properties of inductance and capacitance are represented as purely imaginary quantities, while the resistance is represented as a (+) real quantity. Capacitance is represented as the negative imaginary axis, and current through a pure capacitance is said to lead... 

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

We may observe that if a, is zero, both roots, jVa0, are on the imaginary axis. If a, is zero, one of the two roots is at the origin. We can expand the pole form to give... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

If any one of the coefficients is negative, at least one root has a positive real part (i.e., in the right hand plane). If any of the coefficients is zero, not all of the roots are in the left hand plane it is likely that some of them are on the imaginary axis. Either way, stop. This test is a necessary condition for BIBO stability. There is no point in doing more other than to redesign the controller. 

The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jco in the closed-loop characteristic equation to find the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this specific proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. 

After entering the riocf ind () command, MATLAB will prompt us to click a point on the root locus plot. In this problem, we select the intersection between the root locus and the imaginary axis for the ultimate gain. 

The points at which the loci cross the imaginary axis can be found by the Routh-Hurwitz criterion or by substituting s = jco in the characteristic equation. (Of course, we can also use MATLAB to do that.)... 

All comments on Nyquist plots are made without the need of formal hand sketching techniques. Strictly speaking, the polar plot is a mapping of the imaginary axis from co = 0+ to You ll see... 

The magnitude log-log plot is a line with slope -1. The phase angle plot is a line at -90°. The polar plot is the negative imaginary axis, approaching from negative infinity with co = 0 to the origin with to —> °°. 

When we make a Nyquist plot, we usually just map the positive imaginary axis from co = 0 to infinity, as opposed to the entire axis starting from negative infinity. If a system is unstable, the resulting plot will only contribute jt to the (-1,0) point as opposed to 2k—what encirclement really means. However, just mapping the positive imaginary axis is sufficient to observe if the plot may encircle the (-1,0) point. 

From the characteristic polynomial, it is probable that we ll get either overdamped or underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that no matter what, both Zq and p0 should be larger than the value of l/xp in Gp. That s how we may "steer" the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.)... 

We should find a gain margin of 1.47 (3.34 dB) and a phase margin of 12.3°. Both margins are a bit small. If we do a root locus plot on each case and with the help of riocf ind () in MATLAB, we should find that the corresponding closed-loop poles of these results are indeed quite close to the imaginary axis. 

We may want to find the ultimate gain when the loci cross the imaginary axis. Again there are many ways to do it. The easiest method is to estimate with the MATLAB function rlocf ind (), which we will introduce next. 

Since the real and imaginary parts of a complex number are independent of each other, a complex number is always specified in terms of two real numbers, like the coordinates of a point in a plane, or the two components of a two-dimensional vector. In an Argand diagram a complex number is represented as a point in the complex plane by a real and an imaginary axis. 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

We know the system is stable if all the roots of the characteristic equation are in the LHP and unstable if any of the roots are in the RHP. Therefore the imaginary axis represents the stability boundary. On the imaginary axis s is equal to some pure imaginary number s = ioi. 

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