Openloop Systems

Now let us make life a little more interesting. The system considered above is an openloop system, i.e no feedback control is used. If we add a feedback controller, we have a closedloop system. The controller looks at the product concentration leaving the third tank and makes adjustments in the inlet concentration to the first reactor C>o in order to keep near its desired setpoint value C 3. The variable is a disturbance concentration and the variable is a manipulated concentration that is changed by the controller. We... 

Consider the general openloop system sketched in Fig, 10.1a. The load variable L(,) enters through the openloop process transfer function The manipulated... 

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. 

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. 

There are no sign changes in the first column, so the system is openloop stable. This finding should be no great shock since, from our simulations, we know the openloop system is stable. We also can see by inspection of Eq. (10.19) that the three poles of the openloop transfer function are located at —1 in the LHP, which is the stable region. 

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. 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

First of all, we know immediately that the openloop system transfer function has one pole (at s = -I- 1/ip) in the RHP, Therefore the closedloop... 

Derive an analytical relationship between openloop oiajumum log modulus and damping coeflident for a second-order underdamped openloop system with a gain of unity. Show that a damping coeflident of 0.4 corresponds to a maximum log modulus of +2J dedbels. 

A. OPENLOOP SYSTEM. Let us first consider an openloop process with N controlled variables, N manipulated variables, and one load disturbance. The system can be described in the Laplace domain by N equations that show how all of the manipulated variables and the load disturbance affect each of the controlled variables through their appropriate transfer functions. 

We have considered openloop systems up to this point, but the mathematics applies to any system, openloop or closedloop, as we will see in the next section. 

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... 

First let s consider an openloop system with the openloop transfer function... 

The stability of this openloop system will depend on the values of the poles of the openloop transfer function. If all the p, lie inside the unit circle, the system is openloop stable. 

There is a single root. It lies on the real axis in the z plane and its location depends on the value of the feedback controller gain. When the feedback controller gain is zero (the openloop system), the root lies at z = b. As is increased, the closed-loop root moves to the left along the real axis in the z plane. We will return to this example in the next section. 

Example 19.7. The first-order lag process, zero-order hold, and proportional sampled-data controller from Example 19.1 gave an openloop system transfer function... 

Example 20.10. Suppose we add a one-sampling period deadtime to the first-order system. The openloop system transfer function becomes... 

The linear model permits the use of all the linear analysis tools available to the process control engineer. For example, the poles and zeros of the openloop transfer function reveal the dynamics of the openloop system. A root locus plot shows the range of controller gains over which the system will be closedloop-stable. 

The first reactor in the 3-CSTR process has a conversion rate of 72.8%, and the reactant concentration in this first reactor is 2.18 kmol/m3. The reactor volume is low (14.3 m3), and the jacket heat transfer area is only 24.5 m2. The resulting jacket temperature (300 K) is almost down to the inlet cooling water temperature of 294 K. Linear analysis gives a Nyquist plot that never drops into the third quadrant, so the critical (—1,0) point cannot be encircled in a counterclockwise direction. This is required for closedloop stability because the openloop system is unstable and has a positive pole. Thus a proportional controller cannot stabilize this first reactor. 

For a first-order system, the closedloop root is always real, so the system can never be underdamped or oscillatory. The closedloop damping coefficient of this system is always greater than 1. The larger the value of controller gain, the smaller is the closedloop time constant because the root moves farther away from the origin (remember, the time constant is the reciprocal of the distance from the root to the origin). If we wanted a closedloop time constant of (i.e., the closedloop system is 10 times faster than the openloop system), we would set Kc equal to 9/K,. Equation (8.31) shows that at this value of gain the closedloop root is equal to - 10/to. 

On the surface, the Nyquist stability criterion is 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. 

C+ contour. On the contour the variable s is a pure imaginary number. Thus, s = io) as (o goes from 0 to +oc. Substituting io) for in the total openloop system transfer function gives... 

Keep in mind that we are talking about closedloop stability and that we are studying it by making frequency response plots of the total openloop system transfer function. These log modulus and phase angle plots are for the openloop system. So we could use the terminology Lq and 6q for our Bode and Nichols plots of the openloop Gm Gq frequency response plots. 

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