Openloop Characteristic Equation

The dynamics of this openloop system depend on the roots of the openloop characteristic equation, i.e., on the roots of the polynomials in the denominators of the openloop transfer functions. These are the poles of the openloop transfer functions. If all the roots lie in the left half of the s plane, the system is openloop stable. For the two-heated-tank example shown in Fig. 10.16, the poles of the openloop transfer function are 5 = 1 and s = — j, so the system is openloop stable. 

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. 

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... 

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. 

If the process is openloop unstMe, Gm<.) will have one or more poles in the RHP, so F(,y = 1 + Gm( , B( will also have one or more poles in the RHP. We can find out how many poles there are by solving for the roots of the openloop characteristic equation or by using the Routh stability criterion on the openloop characteristic equation (the denominator of Gj, ). Once the number of poles P is known, the number of zeros can be found from Eq. (13.6). 

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. 

The system considered in the example above has a characteristic equation that is the denominator of the transfer function set equal to zero. This true, of course, for any system. Since the system is uncontrolled, the openloop characteristic equation is [using Eq. (15.60)]... 

The roots of the openloop characteristic equation are s = — 2 and s = 4, These are exactly the values we calculated for the eigenvalues of the matrix of this system ... 

Exarngde 15.15. Determine the dosedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 15.14. Use a diagonal controller structure (two SI SO.controllers) that are proportional only. 

The eigenvalues of the 4 matrix, will be the openloop eigenvalues and will be equal to the roots of the openloop characteristic equation. In order to help us... 

For openloop systems, the denominator of the transfer functions in the matrix gives the openloop characteristic equation. In Example 15.14 the denominator of the elements in was (s + 2X + 4). Therefore the openloop characteristic equation was... 

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. 

So the root of the openloop characteristic equation is b. Since b is less than 1, this root lies inside the unit circle and the system is openloop stable. 

Example 19.6. The chromatographic system studied in Example 18.9 had a first-order lag openloop process transfer function and a deadtime of one sampling period. The closedloop characteristic equation was [see Eq. (18.100)]... 

Figure 3.2 gives root locus plots for the two designs at reactor temperature of 350 K with the measurement lags included. You may remember that a root locus plot is a plot of the roots of the closedloop characteristic equation as a function of the controller gain Kc. The plots start (Kc = 0) at the poles of the openloop transfer function and end (Kc —> oo) at its zeros. 

OPENLOOP AND CLOSEDLOOP SYSTEMS 8.1.1 Openloop Characteristic Equation... 

We want to look at the stability of the closedloop system with a proportional controller Gc(j) = K- First, however, let us check the openloop stability of this system. The open-loop characteristic equation is the denominator of the openloop transfer function set equal to zero. 

Can we make the system stable by using feedback controk That is, can an openloop-unstable process be made closedloop stable by appropriate design of the feedback controller Let us try a proportional controller Gc(.o = Kc- The closedloop characteristic equation is... 

First of all, we know immediately that the openloop system transfer function Gm s)Gc s) has one pole (at = + I/t ) in the RHP. Therefore, the closedloop characteristic equation... 

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