Characteristic equation closedloop

Closedloop Characteristic Equation and Closedloop Transfer Functions... 

Since the characteristic equation of any system (openloop or closedloop) is the denominator of the transfer function describing it, the closedloop characteristic equation for this system is... 

This equation shows that closedloop dynamics depend on the process openloop transfer functions (G, Gv, and Gj) and on the feedback controller transfer function (fl). Equation (10.10) applies for simple single-input-single-output systems. We will derive closedloop characteristic equations for other systems in later chapters. 

The Routh method can be used to find out if there are any roots of a polynomial in the RHP. It can be applied to either closedloop or openloop systems by using the appropnate characteristic equation. 

Example 10. Suppose the closedloop characteristic equation for a system is... 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

Derive the closedloop characteristic equation for the system when both jacket and condenser water are used. 

Next we look at the controlled output variable X2. Figure 11.Id shows the reduced block diagram of the system in the conventional form. We can deduce the closedloop characteristic equation of this system by inspection. 

So Eq. (11.4) gives the closedloop characteristic equation of this series cascade system. A little additional rearrangement leads to a completely equivalent form of Eq. (11.8). 

If only a single controller flj ed to control X by manipulating M, the closedloop characteristic equation is the conventional... 

If, however, a cascade control system is used, as sketched in Fig. ll.3b, the closedloop characteristic equation is not that given in Eq, (11.21). To derive it, let us start with the secondary loop. 

Now we solve for the closedloop transfer function for the primary loop with the secondary loop on automatic. Figure 11.3c shows the simplified block diagrana. By inspection we can see that the closedloop characteristic equation is... 

Can we malte this system closedloop stable A pioportional feedback controller gives a closedloop characteristic equation ... 

C. THIRD-ORDER OPENLOOP UNSTABLE PROCESS. If an additional lag is added to the system and a proportional controller is used, the closedloop characteristic equation becomes... 

With a proportional feedback controller the closedloop characteristic equation is... 

However, since Mi is more expensive than Mj, we wish to minimize the long-term steadystate use of Mi- Therefore, a valve position controller" is used to control Mi at M . What is the closedloop characteristic equation of the system ... 

The Nyquist stability criterion is, on the surface, quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical, magical (— 1, 0) point somehow tells us that the system is closedloop unstable. This all looks like blue smoke and mirrors However, as we will prove below, it all goes back to finding out if there are any roots of the closedloop characteristic equation in the RHP. 

B, APPLICATION OF THEOREM TO CLOSEDLOOP STABILITY. To check the stability of a system, we are interested in the roots or zeros of the characteristic equation. If any of them lie in the right half of the s plane, the system is unstable. For a closedloop system, the characteristic equation is... 

If the B plot crosses the negative real axis between the origin and the ( — 1,0) point, the system is closedloop stable. Now N = 0 and therefore Z = JV = 0. There are no zeros of the closedloop characteristic equation in the RHP. 

Remember also that for gains greater than the ultimate gain, the root locus plot showed two roots of the closedloop characteristic equation in the... 

The Nyquist stabihty criterion can be used for openloop unstable processes, but we have to use the complete, rigorous version with P (the number of poles of the closedloop characteristic equation in the RHP) no longer equal to zero. 

Thus the characteristic matrix for this closedloop system is the 4cl niatrix. Its eigenvalues will be the close oop eigenvalues, and they will be the roots of the closedloop characteristic equation. 

For closedloop systems, the denominator of the transfer functions in the closedloop servo and load transfer function matrices gives the closed-loop characteristic equation. This denominator was shown in Chap. 15 to be [I + which is a scalar Nth-order polynomial in s. Therefore, the... 

Notice that the closedloop characteristic equation depends on the tuning of both feedback controllers. 

We can use this theorem to find out if the closedloop characteristic eqga-tion has any roots or zeros in the right half of the s plane. The s variable follows a closed contour that completely surrounds the entire right half of the s plane. Since the closedloop characteristic equation is given in Eq. (16.1), the function that we are interested in is... 

Then the number of encirclements of the (-1, 0) point made by (, ) as co varies from 0 to oo gives the number of zeros of the closedloop characteristic equation in the right half of the s plane. 

Also shown in Fig. 16.1 is the W plot when only proportional controllers are used. Note that the curves with P controllers start on the positive real axis. However, with PI controllers the curves start on the negative real axis. This is due to the two integrators, one in each controller, which give 180 degrees of phase angle lag at low frequencies. As shown in Eq. (16.3), the product of the and B2 controllers appears in the closedloop characteristic equation. 

A brief justification for the characteristic loci method (thanks to C. C. Yu) is sketched below. For a more rigorous treatment see McFarland and Belletrutti (Automatica 1973, Vol. 8, p. 455). We assume an openloop stable system so the closedloop characteristic equation has no poles in the right half of the s plane. 

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