Control-to-output transfer function

The transfer function of the plant is also called the control-to-output transfer function ... 

The control-to-output transfer function is a product of the transfer functions of the PWM, the switch and the LC filter (since these are cascaded stages). Alternatively, the control-to-output transfer function is a product of the transfer functions of the PWM stage and the transfer function of the power stage . 

Finally, the control-to-output transfer function is the product of three (cascaded) transfer functions, that is, it becomes... 

Control-to-output Transfer Function Here are the steps for this topology ... 

We ignored the ESR of the output capacitor in Figure 7-14, and also in the list of transfer functions provided earlier. For example, earlier we had provided the following control-to-output transfer function for a buck... 

For example, looking at the earlier equation, we see that the line-to-output transfer function for the buck is the same as its control-to-output transfer function, except that the Vin/Vramp factor is replaced by D. So for example, if Vramp = 2.14 V, and D = 0.067 (as for 1 V output from a 15 V input), then the control-to-output gain at low frequencies is... 

So when feedback is present ( loop closed ), it can be shown by control loop theory that the line-to-output transfer function changes to... 

The use of block diagrams to illustrate cause and effect relationship is prevalent in control. We use operational blocks to represent transfer functions and lines for unidirectional information transmission. It is a nice way to visualize the interrelationships of various components. Later, they are crucial to help us identify manipulated and controlled variables, and input(s) and output(s) of a system. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

The Ridge controller (del Castillo, 2002) based on a niinimrim variance criterion was found to have good characteristics to be adapted to the transfer function defined in (1). This controller has a tuning parameter that balances the variances of the output with the inputs. 

There are two sets of transfer functions one should be concerned with target settings and effects of variability.These are illustrated in Fig. 8, which shows the relationship between controllable inputs and their variability with the desired output and its variability. Thus, QFD defines how one flows the customer CTQs downwards, while the transfer functions define how one predicts process capability and defines the critical control points.The transfer functions allow the team to establish this linkage early in the development, rather than try to do it once products are in production. In addition when customer requirements change, the team does not have to begin a new project. With the transfer functions in hand, they can quickly evaluate capability and predicted reliability for the required process changes. 

We are now in a position to start tying all the loose ends together For each of the three topologies, we now know both the forward transfer function G(s) (control-to-output) and also the feedback transfer function H(s). Going back to the basic equation for the closed-loop transfer function,... 

After evaluating feasible pairing, we can estimate their performance by using Closed Loop Disturbances Gain (CLDG) index. To keep the control error between acceptable bounds, the closed loop disturbance gain should be smaller than the loop transfer function (l+g (s)A,(s)), for each disturbance, where g,j(s) is the open loop input-output transfer function and Aj(s) is the controller model. 

The classical approach to process sensitivity, which is the approach used in courses on process dynamics and control, is through transfer function analysis. The transfer function contains the response of the output to small input disturbances at all... 

In Fig. 32.1 it is assumed that measurement and control are ideal, i.e. they do not have any dynamics and the gain is equal to one. The controller has a transfer function Gc, the process transfer function is Gp and the disturbance transfer function G. The process output can now be written as the sum of all dependencies ... 

Figure 8-41 includes two conventional feedback controllers G i controls Cl by manipulating Mi, and G o controls C9 by manipidating Mo. The output sign s from the feedback controllers serve as input signals to the two decouplers D o and D91. The block diagram is in a simplified form because the load variables and transfer functions for the final control elements and sensors have been omitted. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

The temperature of a gas leaving an electric furnace is measured at X by means of a thermocouple. The output of the thermocouple is sent, via a transmitter, to a two-level solenoid switch which controls the power input to the furnace. When the outlet temperature of the gas falls below 673 K (400°C) the solenoid switch closes and the power input to the furnace is raised to 20 kW. When the temperature of the gas falls below 673 K (400°C) the switch opens and only 16 kW is supplied to the furnace. It is known that the power input to the furnace is related to the gas temperature at X by the transfer function ... 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

Rearranging to get the output over the input gives the transfer function for the controller. 

Figure 11.11 gives the IMC structure. The model of the process is run in parallel with the actual process. The output of the model is subtracted from the actual output of the process X and this signal is fed hack into the controller. This signal is, in a sense, the effect of load disturbances on the output (since we have subtracted the effect of the manipulated variable M). Thus we are inferring the load disturbance without having to measure it. This signal is Xi, the output of the process load transfer function, and is equal to We... 

Draw a block diagram of a process that has two manipulated variable inputs (Mi and M]) that each affect the output (2T). A feedback controller Si is used to control X by manipulating Mi since the transfer function between Mj and X has a small time constant and smaU deadtime. 

In a digital computer-control system, the feedback controller has a pulse transfer function. What we need is an equation or algorithm that can be programmed into the digital computer. At the sampling time for a given loop, the computer looks at the current process output x, compares it to a setpoint, and calculates a current value of the error. This error, plus some old values of the error and old values of the controller output or manipulated variable that have been stored in computer memory, are then used to calculate a new value of the controller output m,. 

If we specify the form of the input AT and the desired form of the output (Z, and if the process and hold transfer functions are known, we can rearrange Eq. (20.14) to give the required controller. We will define this controller that is designed for setpoint changes. ... 

Whereas the microprocessor controls an individual basic operation, the central computer, which has all the analytical procedures held in its memory, controls the particular analytical procedure required. At the appropriate time, the central computer transmits the relevant set of parameters to the corresponding units and provides the schedule for the sample-transport operation. All units are monitored to ensure proper functioning. If one of the units signals an error, a predetermined action, such as disposing of the sample, is taken. The basic results from the units are transferred to the central computer, the final results are calculated, and the report is passed to the output terminal. These results can also be transmitted to other data processing equipment for administrative or management purposes. The central control is, therefore, the leading element in a hierarchy of... 

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