Time invariance

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

For a linear, time invariant plant, equation (9.1) beeomes... 

As it stands, of course, the dynamical system defined by equation 8.3, is reversible- -but is clearly not a CA, since the system is, in general, neither space nor time invariant. We can restore this invariance, however, by choosing TTi t) in such a manner that its dependence on H and C is only an indirect one - specifically, so that it depends only on the particular state of the neighborhood (or landscape) of site i (excluding i itself) at time C. 

The function Y(t) also can be visualized as the result of passing the impulse train N (t) through a linear, time-invariant filter whose impulse response is This observation coupled with the fact that the... 

The reader unfamiliar with the notion of a linear, time-invariant filter can profit from W. M. Siebert s article in E. J. Baghdady, Lectures on Communication System Theory, McGraw-Bftll Book Co., New York, 1901. 

Now, our previous result shows that F0(f), being the result of passing X(t) through the linear, time-invariant filter h0(t )> must have a gaussian first-order distribution therefore,... 

In this connection, it should be carefully noted that, even if X(t) is not a gaussian process, the mean and the autocorrelation function of the output of a linear, time-invariant filter are related to the mean and autocorrelation function of the input process according to Eqs. (3-293) and (3-294).64 This is an important fact of which use will be made in the next section. 

Harmonic Analysis of Random Processes.—The response Y(t) of a linear, time-invariant electrical filter to an input X(t) can be expressed in the familiar form 66... 

With the aid of the power density spectrum, we can now give a complete description of how a linear, time-invariant filter affects the frequency distribution of power of the input time function X(t). To accomplish this, we must find the relationship between the power... 

The resemblances (and differences ) between Eq. (3-321), which relates the input and output power density spectra of a linear, time-invariant... 

Since the energy required to strip the electrons from plutonium metal at the anode is exactly matched by the energy returned at the cathode, the potential required by the process is only that required to overcome time invariant (i2r) losses in the cell circuit, and time dependent resistance (electrode polarization). 

In this relation a(r, t) is the experimentally observed signal, s represents random noise, axi r) represents the time invariant systematic noise and aRi(f) the radial invariant systematic noise Schuck [42] and Dam and Schuck [43] describe how this systematic noise is ehminated. x is the normahsed concentration at r and t for a given sedimenting species of sedimentation coefficient 5 and translational diffusion coefficient D it is normalised to the initial loading concentration so it is dimensionless. 

Bhawe (14) has simulated the periodic operation of a photo-chemically induced free-radical polymerization which has both monomer and solvent transfer steps and a recombination termination reaction. An increase of 50% in the value of Dp was observed over and above the expected value of 2.0. An interesting feature of this work is that when very short period oscillations were employed, virtually time-invariant products were predicted. 

Similarly, if a process is described as. statistically stationary, then its statistical moments are time invariant. 

Chapter 8 combined transport with kinetics in the purest and most fundamental way. The flow fields were deterministic, time-invariant, and calculable. The reactor design equations were applied to simple geometries, such as circular tubes, and were based on intrinsic properties of the fluid, such as molecular dif-fusivity and viscosity. Such reactors do exist, particularly in polymerizations as discussed in Chapter 13, but they are less typical of industrial practice than the more complex reactors considered in this chapter. 

The models of Chapter 9 contain at least one empirical parameter. This parameter is used to account for complex flow fields that are not deterministic, time-invariant, and calculable. We are specifically concerned with packed-bed reactors, turbulent-flow reactors, and static mixers (also known as motionless mixers). We begin with packed-bed reactors because they are ubiquitous within the petrochemical industry and because their mathematical treatment closely parallels that of the laminar flow reactors in Chapter 8. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

In the steady state, the total flux is constant along the entire path. This condition (i.e., that of flux continuity) is a reflection of mass balance nowhere in a steady flux will the ions accumulate or vanish (i.e., their local concentrations are time invariant). The condition of continuity of the steady flux is disturbed in those places where ions are consumed (sinks) or produced (sources) by chemical reactions. It is necessary to preserve the balance that any excess of ions supplied correspond to the amount of ions reacting, and that any excess of ions eliminated correspond to the amount of ions formed in the reaction. 

In electrochemical systems, a steady state during current flow implies that a time-invariant distribution of the concentrations of ions and neutral species, of potential, and of other parameters is maintained in any section of the cell. The distribution may be nonequilibrium, and it may be a function of current, but at a given current it is time invariant. 

We see that the expression for the current consists of two terms. The first term depends on time and coincides completely with Eq. (11.14) for transient diffusion to a flat electrode. The second term is time invariant. The first term is predominant initially, at short times t, where diffusion follows the same laws as for a flat electrode. During this period the diffusion-layer thickness is still small compared to radius a. At longer times t the first term decreases and the relative importance of the current given by the second term increases. At very long times t, the current tends not to zero as in the case of linear diffusion without stirring (when is large) but to a constant value. For the characteristic time required to attain this steady state (i.e., the time when the second term becomes equal to the first), we can write... 

When alternating current is used for the measurements, a transient state arises at the electrode during each half-period, and the state attained in any half-period changes to the opposite state during the next half-period. These changes are repeated according to the ac frequency, and the system will be quasisteady on the whole (i.e., its average state is time invariant). 

From a green perspective, decreased yields, by-product formation and inability to reproduce key product properties will invariably increase waste and require greater materials and energy use. Longer cycle times invariably will lead to increased energy use and in some cases, increased materials use. 

Kalman filter algorithm equations for time-invariant system states... 

For the time invariant calibration model discussed in Section 41.2, eq. (41.14) reduces to ... 

In Sections 41.2 and 41.3 we applied a recursive procedure to estimate the model parameters of time-invariant systems. After each new measurement, the model parameters were updated. The updating procedure for time-variant systems consists of two steps. In the first step the system state j - 1) at time /), is extrapolated to the state x(y) at time by applying the system equation (eq. (41.15)) in Table 41.10). At time tj a new measurement is carried out and the result is used to... 

The algorithm is initialized in the same way as for a time-invariant system. The sequence of the estimations is as follows ... 

Time-invariant systems can also be solved by the equations given in Table 41.10. In that case, F in eq. (41.15) is substituted by the identity matrix. The system state, x(j), of time-invariant systems converges to a constant value after a few cycles of the filter, as was observed in the calibration example. The system state. 

The measurement model of the time-invariant calibration system (eq. (41.5)) should now be expanded in the following way ... 

For a time-invariant system, the expected standard deviation of the innovation consists of two parts the measurement variance (r(/)), and the variance due to the uncertainty in the parameters (P(y)), given by [4] ... 

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