Tracking error

The idea of selecting waveforms adaptively based on tracking considerations was introduced in the papers of Kershaw and Evans [3, 4], There they used a cost function based on the predicted track error covariance matrix. 

We have compared one-step and two-step ahead scheduling using two performance measures. The first is the root mean square error of the track estimation this is a fairly obvious measure of the performance of the tracker. The second measure was the number of track updates. Since the sensor is managed in such a way that track updating is done only when the predicted track error exceeds a threshold, this also gives a measure of how far the estimation process is diverging from the actual target state. 

We have, on the other hand done simple simulations for the case of one-step ahead and two-step ahead scheduling. In the latter case, the revisit times and waveforms are calculated while the target states are propagated forward over two measurements, with the cost function being the absolute value of the determinant of the track error covariance after the second measurement. Only the first of these measurements is done before the revisit calculation is done again for that target, so that the second may never be implemented. 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

In other words, the regulation problem is equivalent to the problem of finding a subset Z of the Cartesian product R x R on which the output tracking error e t) is zeroed, and an input signal which makes attractive and invariant this subset (Figure 3). The trajectories described by the state and the input on the invariant subset Z, are thereafter referred as the steady state... 

In this setting, the exosystem may be also nonlinear and it is supposed that it does not depend on p, which is often verified in practical problems since the exosystem models the reference and disturbance signals affecting the plant. Moreover, the third nonlinear function in (31) describes the output tracking error e G R, which, in many cases, is given as a difference between the system output /i(x,p) and the reference signal, described by r uj), namely... 

Remark 3. Equation (32) is known as the Francis-Isidori-Byrnes equation (FIB) [8] and is the nonlinear version of equation (10) used to find the subset Z on the Cartesian product R x called, so far, the zero tracking error submanifold. 

To find the steady state mappings, we proceed as follows. The tracking errors are zero when... 

B. Castillo-Toledo and G. Obregon-Puhdo. Guaranteeing asymptotic zero intersampling tracking error via a discretized regulator and exponential holder for nonlinear systems. J. App. Reserch and Tech., 1 203-214, 2003. 

The prediction of the control input is computed via an optimization method that minimizes a suitably defined objective function, usually composed by two terms the first one is related to the deviation of the predicted output from the reference trajectory (i.e., the tracking error), while the second term takes into account control input changes. Hence, the optimization problem has the form... 

In the following, the model-based controller-observer adaptive scheme in [15] is presented. Namely, an observer is designed to estimate the effect of the heat released by the reaction on the reactor temperature dynamics then, this estimate is used by a cascade temperature control scheme, based on the closure of two temperature feedback loops, where the output of the reactor temperature controller becomes the setpoint of the cooling jacket temperature controller. Model-free variants of this control scheme are developed as well. The convergence of the overall controller-observer scheme, in terms of observer estimation errors and controller tracking errors, is proven via a Lyapunov-like argument. Noticeably, the scheme is developed for the general class of irreversible nonchain reactions presented in Sect. 2.5. 

The whole control scheme is represented in Fig. 5.2. The first control loop (inner loop) is closed around the jacket temperature in such a way to track a desired temperature, 7j,des(0 = J2,des(0> to be determined then, an outer loop is closed around the reactor temperature so as to track the desired reactor temperature profile, 7r,des(0 = yi,des(0- The outer controller computes the desired jacket temperature on the basis of the reactor tracking error e = ypdes - yi and of the estimate of aq, while the inner controller receives y2,des as input and computes the temperature of the fluid entering the jacket, i.e., the manipulated input u. 

Theorem 5.3 If the rate constants are bounded as in (2.32) and (2.33), then, there exists a set of observer gains such that the state estimation error x and the tracking error e globally uniformly converge to 0 as t oo, for any positive set of control... 

Figure 5.8 is referred to the performance of the PID controller. It can be recognized that the temperature tracking error is always below 0.5 K and is characterized by a very similar time history of the control input with respect to the previously tested schemes (Fig. 5.5). By comparing the results in Fig. 5.8 with those in Fig. 5.4, the adaptive model-based scheme presents better performance than the linear PID control, especially during the heating and cooling phases, i.e., when the reference temperature is not constant.

Fig. 5.8 Reactor temperature tracking error left) and commanded temperature of the fluid in the jacket right) obtained by using PID controller...

A double monochromator has two separate gratings which are separated by an adjustable slit. Both of these gratings are mechanically coupled onto the same shaft so that tracking errors are virtually eliminated. The gratings are rotated concurrently and the... 

The tests were also performed on computer-generated data in which additional uniform or non-uniform motion was added, to study how far the CG algorithm could be pushed beyond its original design parameters. For uniform motion, CG tracking was as successful as in the quiescent case for small drifts but failed for drifts of the order of half the particle-particle separation. For non-uniform (linear shear) flows with small strains between frames the identification worked correctly, but large non-uniform displacements caused major tracking errors. 

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