6.4. Guide to User Application¶
This section provides some guidelines that should help tie together the model equations and solution techniques just described and the card input description given in Section 6.8. In this section signal definition rules that must be observed are stated, model capabilities are highlighted, and rules of thumb for choosing the values of solution control parameters are given.
6.4.1. Signal Definition Rules¶
Signals are defined through the input deck and the definitions must conform to certain rules. Any of the four signal kinds can appear anywhere in the signal card region of the input deck, subject to the following rules.
Rule 1 |
A block signal that is output from other than an integrator, lag compensator or lead-lag compensator must have been previously defined in the input stream before it can be used as an input to another block. This rule is intended to avoid circular references and to maintain proper sequencing of signals during numerical integration. |
Rule 2 |
A demand signal or measured signal must pass through at least one block before it can be used as a control signal. |
Rule 3 |
Each signal must be assigned a unique signal number between 1 and 998. |
The card format for defining a signal is given in Section 6.8.
6.4.2. Units¶
Generally all measured signals are in MKS units while all control signals should be calculated in these same units. The exceptions are those signals that are normalized to a steady state value; these are appropriately noted in Table 6.8.3.
The convention for demand signals is that demand tables are always entered by the user, normalized to a time zero value of unity. The actual value for a demand signal is calculated in the code by multiplying the current time entry in the demand table by the initial condition value. The next subsection describes how the initial condition value is obtained.
The units of a block output signal are determined solely by the units of the input signals and any conversion factors that are entered by the user as constants on the block definition card.
6.4.3. Initial Conditions¶
In order to begin a transient calculation, initial condition values are required for demand signals and for the integrator, lag compensator and lead-lag compensator blocks. There are basically three options available for setting these values. In the first option, all values are supplied by the user through input cards: in this case the steady state solution finder is bypassed. If the user is seeking the steady state solution, then a null transient may have to be run. In the second option, the steady state solution finder is used to solve for the steady state values. In the final option, a mixed set of initial conditions are used with some values read directly from the signal cards while the remaining values are solved for such that Eq. 6.2-4 is satisfied. The card input data required for each of these options is described below.
If the initial condition values are to be read from the input cards then the steady state solution finder should be bypassed by setting the J1SIG field on the 999 card to ‘0’. Then the value for a demand signal and for the block signal of each integrator, lag compensator and lead-lag compensator is taken form the F4SIG field on the associated signal definition card.
If the initial condition values are to be calculated by the steady state solution finder then the J1SIG field on the 999 card is set to ‘1’. An initial guess for each demand signal and integrator initial condition variable must be supplied on the F4SIG field of the signal definition card. In addition the F3SIG field must be set to ‘0’. As a rule of thumb, the initial guess should be within 15% of the actual steady state value to ensure convergence. The lag compensator and lead-lag compensator are special cases and do not require initial condition information from the user.
Finally, if a mixed set of initial conditions is to be used, the card data is identical to the case directly above, except for those demand and integrator signals whose initial conditions are to be read from cards. For these signals the F3SIG field is set to ‘1.0’ and the F4SIG field is set to the initial condition value desired.
6.4.4. Solution Accuracy¶
The control system modeling capability attempts to limit the solution error that is introduced during the numerical integration of the control equations over a subinterval. Recall the error is controlled by repeatedly bisecting the subinterval time step into substeps until integrating across the subinterval gives a relative error between successive iterations that is less than the user-supplied value for EPSCS. [The method was described in Section 6.3]. The value of EPSCS is input on a table card and occupies location 8001. A value of 0.01 is suggested for most applications.
6.4.5. Solution Stability¶
The modeling capability also attempts to maintain a stable solution to the coupled control system and plant equations. The basic idea is that stability is enhanced if the relative change in a control signal across a subinterval is maintained less than the user supplied value for EPSCPL. [The method was described in Section 6.3]. The value of EPSCPL is input on a table card and occupies location 8002. A value of 0.1 is suggested for most applications.
6.4.6. Zero Crossing Parameter¶
The zero crossing parameter in Eq. 6.3-2 is intended to prevent unnecessarily small time step size when a signal passes close to zero. The situation we seek to avoid occurs when the zero crossing parameter F5SIG is zero. Then the denominator in Eq. 6.3-2 is very small so that the relative error is very large. Time step size is severely reduced even though the absolute error in the signal may well be acceptably small. The solution is to control absolute error at the zero crossing and we do it through the relative error control mechanisms associated with Eq. 6.3-2 by proper choice of a value for F5SIG.
The appropriate value of F5SIG is problem dependent and is selected by the user for input to the code. The goal is to select a value that gives a desired level of absolute error near the zero crossing yet does not significantly impact the calculation of relative error away from the zero crossing. To do so we note that the code controls integration error using Eq. 6.3-2 so that on convergence the solution satisfies
(6.4‑1)
where the value of \(m\) is restricted to those signals that are output by dynamic blocks. Near the zero crossing \(S_{\text{m}}^{j}\) will be insignificant so that Eq. 6.4-1 is equivalently
(6.4‑2)
Note that the numerator is the absolute error in the solution at convergence. We can arrange for the numerator to take on a specific value by appropriately choosing the value of \(\text{F5SIG} \left( m \right)\) once the value of EPSCS has been selected. For example, suppose we want the absolute error on convergence near the zero crossing to be \(S_{\text{m}}^{o} \times 10^{-4}\) where \(S_{\text{m}}^{o}\) is the maximum magnitude signal \(m\) is to take on over all time. If, the sake of illustration, a value of \(10^{-2}\) was input for EPSCS, then we can achieve our absolute error objective by calculating the value of \(\text{F5SIG} \left( m \right)\) from Eq. 6.4-2,
Away from the zero crossing, the impact of \(\text{F5SIG} \left( m \right)\) is insignificant.
Similarly, the value of \(\text{F5SIG} \left( m \right)\) associated with a control signal should be selected as follows. The time step is adjusted down if necessary so that the largest relative change in a control signal is limited by Eq. 6.3-3 to
(6.4‑3)
where the value of m is restricted to those signals that are control signals. Near the zero crossing \(S_{m}^{j}\)will be insignificant so that Eq. 6.4-3 is equivalently
(6.4‑4)
Note that the numerator is the absolute change in the solution across the time step. We can arrange for the numerator to take on a specific value by appropriately choosing the value of \(\text{F5SIG} \left( m \right)\) once the value of EPSCPL has been selected. For example, suppose we want the absolute change in the control signal near the zero crossing to be as large as \(S_{\text{m}}^{o} \times 10^{- 3}\) before time step size is reduced. If, for the sake of illustration, a value of \(10^{-1}\) was input for EPSCPL, then the absolute change objective will be met if \(\text{F5SIG} \left( m \right)\) is calculated from Eq. 6.4-4,
Away from the zero crossing, the impact of \(\text{F5SIG} \left( m \right)\) on the control of fractional change is insignificant.