Poles closed-loop

The position of the closed-loop poles in the. v-plane determine the nature of the transient behaviour of the system as can be seen in Figure 5.5. Also, the open-loop transfer function may be expressed as... 

Fig. 5.5 Effect of closed-loop pole position in the s-plane on system transient response.

Closed-loop poles (For K = 11.35) Since the closed-loop system is third-order, there are three closed-loop poles. Two of them are given in equation (5.81). The third lies on the real locus that extends from —5 to —oo. Its value is calculated using the magnitude criterion as shown in Figure 5.15. 

The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

Converting from polar to cartesian co-ordinates gives the closed-loop poles in the z-plane... 

The roots of equation (8.96) are the closed-loop poles or eigenvalues. 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

Calculate desired closed-loop poles desiredpoles=rcots(chareqn)... 

Actual closed-loop poles = desired closed-loop poles... 

This transfer function has closed-loop poles at -0.29, -0.69, and -10.02. (Of course, we computed them using MATLAB.)... 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

Recall Eq. (5-11), the closed-loop characteristic equation is the denominator of the closed-loop transfer function, and the probable locations of the closed-loop pole are given by... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

The controller setting is different depending on which error integral we minimize. Set point and disturbance inputs have different differential equations, and since the optimization calculation depends on the time-domain solution, the result will depend on the type of input. The closed-loop poles are the same, but the zeros, which affect the time-independent coefficients, are not. 

The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6-20) that Gc is sort of the reciprocal of Gp. The zeros of Gc are by choice the poles of Gp. The product of GcGp cancels everything out—hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as... 

Since the system characteristic equation is 1 + GcGp = 0, our closed-loop poles are only dependent on our design parameter xc. A closed-loop system designed on the basis of pole-zero cancellation has drastically different behavior than a system without such cancellation. 

There is now only one real and negative closed-loop pole (presuming Kc > 0). This situation is exactly what direct synthesis leads us to. 

Stability analysis methods Routh-Hurwitz criterion Apply the Routh test on the closed-loop characteristic polynomial to find if there are closed-loop poles on the right-hand-plane. 

Direct substitution Substitute s = jto in characteristic polynomial and solve for closed-loop poles on /m-axis. The Im and Re parts of the equation allow the ultimate gain and ultimate frequency to be solved. 

Root-locus With each chosen value of proportional gain, plot the closed-loop poles. Generate the loci with either hand-sketching or computer. 

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... 

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. 

If we have chosen the other possibility of u = 0, meaning that the closed-loop poles are on the real axis, the ultimate gain is Kc u = -6, which is consistent with the other limit obtained with the Routh criterion. 

Root locus is a graphical representation of the roots of the closed-loop characteristic polynomial (i.e., the closed-loop poles) as a chosen parameter is varied. Only the roots are plotted. The values of the parameter are not shown explicitly. The analysis most commonly uses the proportional gain as the parameter. The value of the proportional gain is varied from 0 to infinity, or in practice, just "large enough." Now, we need a simple example to get this idea across. 

The solution, meaning the closed-loop poles of the system, is... 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

The closed-loop pole "runs" from the point s = — l/xp at the mathematical limit of Kc = 0 to the point s = -l/xD as Kc approaches infinity. 

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