6.2. Generalized Model
The control system model was developed with the intent that a wide range of plant control systems can be simulated. For this purpose, two specific objectives were set. First, the model should be general enough to permit the user to assemble any set of control equations and specify how they interface to the plant solely through the input. And second, the model should employ a numerical method which is reliable in all foreseeable applications. Fulfilling these two goals led to the identification of a general equation form capable of representing all classes of plant control systems.
6.2.1. General Equations
The solution algorithms of the model are based on a general set of equations for the control system state variables and outputs. These equations are formulated under the assumption that the three components of a control system, the sensor, the controller, and the actuator, can all be modeled as ordinary differential equations. The general equation form is easy to deduce.
Since the sensor and actuator behavior are governed by physical laws and they are normally modeled in lumped parameter form, they are both described by
where
\(x \left( t \right)\) = n x 1 state vector;
\(u \left( t \right)\) = r x 1 input vector; and
\(y \left( t \right)\) = m x 1 output vector.
The controller also has the basic form of Eq. (6.2-1) as it consists of integrating and function elements. But in addition a derivative element is sometimes used in which case derivatives appear on the right hand side of Eq. (6.2-1). In practice the output signal from an integrator will be differentiated at most once so that the controller equation is
The general equation form results when the equations for the three components are coupled and the signals that link to the plant are explicitly labeled
where
\(u_{\text{mea}}\left( t \right) = 1 \times n_{\text{mea}}\) measured input vector;
\(u_{\text{dmd}}\left( t \right) = 1 \times n_{\text{dmd}}\) demand input vector; and
\(y_{\text{ctl}}\left( t \right) = 1 \times n_{\text{ctl}}\) control system output vector.
To guide the choice of initial conditions and their calculation for the above equations, we must consider the intended applications. Since the code is ultimately to be used for analysis of plant wide transients, the initial conditions must be compatible with the way in which these transients begin. Generally the user prescribes the plant steady state and therefore it should be reasonable to initialize the control system so that at time zero it preserves this steady state. In this case boundary conditions for the control system are taken from the plant, and control system time derivatives are set to zero. Writing the control equations explicitly in terms of the measured signals, control signals and the demand signals, Eq. (6.2-3) becomes
where
\({y^{*}}_{\text{ctl}}\left( 0 \right) = 1 \times n_{\text{ctl}}\) vector of plant values associated with \(y_{\text{ctl}}\left( 0 \right)\); and
\({u^{*}}_{\text{mea}}\left( 0 \right) = 1 \times n_{\text{mea}}\) vector of plant values associated with \(u_{\text{mea}}\left( 0 \right)\).
The asterisk denotes steady state conditions in the plant. The initial conditions then that place the control system in steady state equilibrium with the plant are the values of \(u_{\text{dmd}}\left( 0 \right)\) and \(x \left( 0 \right)\) that satisfy Eq. (6.2-4).
6.2.2. Block Diagram Approach
One might well ask what benefits can be obtained from a knowledge of this general equation. The principal benefit is a flexible modeling approach that permits the user to describe the plant control equations in a block diagram manner. The key to achieving this capability is the fact that the properties of the general equation form are well known and can be brought to bear on the development of a reliable numerical scheme.
The process by which the user describes their block diagram is analogous to the process of programming an analog computer. Basically, four types of information must be supplied. First, each mathematical block must be defined and the interconnections among them specified. Presently there are twenty-three blocks to choose from and these are shown in Table 6.8.2; additional blocks can be added if required. Each block can accept up to two signals at its input for processing and supply the result, termed a block signal, for further processing by other blocks. Second, each forcing function driving the collection of blocks must be defined. A demand signal is available for this purpose and references a user-supplied table of values that specifies the signal as a function of time. Third, plant measured quantities input to the collection of blocks must be defined. A measured signal is available for this purpose and permits access to a number of plant variables including temperature, flow, pressure and inventory in a number of reactor components. A complete list is given in Table 6.8.3. Finally, those block signals that are used to drive the plant must be defined. For that purpose, control signals can be defined by the user to represent, among other things, sources of external reactivity, feedwater mass flowrate and pump motor torque. A complete list is given in Table 6.8.3.