Laplace transform input signals

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 CoutAA) term is the initial condition for the concentration within the tank. It is zero when the input is a delta function. Such a system is said to be initially relaxed. The term s[C (l)] is the Laplace transform of the input signal, a delta function in this case. The Laplace transform of S i) is 1. Substituting and solving for agutis) gives... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

Figure 5.1- Input signals and their Laplace transforms.

The 1/s is an operator or a transfer function showing what operation is performed on the input signal. This is a completely different idea than the simple Laplace transformation of a function. Remember, the Laplace transform of the unit step function was also equal to l/s. But this is the Laplace transformation of a function. The 1/s operator discussed above is a transfer function, not a function. 

Electron Drift in a Constant Electric Field. As an example, let us consider the system discussed in the time-of-flight section. In this system, charge carriers are generated close to the injecting contact and drift to the collecting contact under the force of a constant electric field. As discussed above, the current response on a laser pulse has a constant value of Jph = AQIr for 0 < t < t, and drops instantly to zero at t=r. The input signal is a delta function and the output response is a step function. Linear-response theory shows that the system function H s) is the Laplace transform of the impulse response function h(t). In our example ... 

Transfer fimction An s -domain function that determines the relationship between an exponential forcing function and the particular solution of a circuit s differential equation model. It also describes a relationship between the Laplace transform s-domain spectral representation of a circuit s input signal and the Laplace transform s -domain spectral description of its output signal when the circuit is initially at rest (no stored energy in its capacitors and inductors) and excited by the source. 

Theoretically, it is now possible to calculate the inverse problem. Experimentally, V t) can be determined as this is the recorded signal, usually an action potential. With some effort, VXt) can be specified as a function of time and its Laplace transform computed. The Laplace transform f (s) of the input signal (action potential, etc.) l (t) is computed from the relations... 

The actual signal which reaches the active circuitry of the preamplifier is modified from by the input circuitry and is given by (Laplace transform... 

For physically realistic systems the power of s of the denominator P should be higher or at least equal to the power of 5 of the numerator Q. The power in an exponent of a delay term is always negative, since physically predictive systems cannot be realized One of the mathematical advantages of Laplace transformation is, that it enables the input-output description of serial and parallel subsystems and description of signals easily. 

Laplace transformation can be used to analyze system behavior for control purposes and to solve a system analytically for different kinds input signals. 

Derive Laplace transforms of the input signals shown in Figs. E3.4a and E3.4b by summing component functions found in Table 3.1. 

In general (see Eq. (7.116)) Z is the transfer function between the Laplace transforms of excitation and response In the case of functions of type e the transfer function is also directly the quotient of the forced part of the output signal and the input signal [620], as we can readily convince ourselves using a capacitance as an example. In general this follows from the Laplace transform for cos(wt + p). For cos(wt + <,3) / coswt [623] we get cos — = sini this yields... 

The output of the simple moving average filter is the average of the M -F 1 most recent values of x(n). Intuitively, this corresponds to a smoothed version of the input, but its operation is more appropriately described by calculating the frequency response of the filter. First, however, the z-domain representation of the filter is introduced in analogy to the s- (or Laplace-) domain representation of analog filters. The z transform of a causal discrete-time signal x(n) is defined by... 

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