Non-minimum-phase

However, the stiffness/ill conditioning of the model (3.31) will strongly impact on the implementation of optimization controllers (e.g., a model predictive controller) (Baldea et al. 2010). On the other hand, for any choice of four flow rates as manipulated inputs (keeping the remaining one constant at its nominal value), the system is non-minimum phase (Kumar and Daoutidis 2002) and thus potentially closed-loop unstable with an inversion-based controller.3 As discussed in the previous section, a more systematic controller-design approach would... 

A linear system is referred to as minimum phase if all the zeros of its transfer function lie in the left-hand plane else, the system is non-minimum phase. Naturally, the inverse of the transfer function of a non-minimum phase system will be unstable. In the case of nonlinear systems, the concept of transfer function zeros is replaced by the zero dynamics (Isidori 1995) a nonlinear system is minimum phase if its zero dynamics are stable. 

On the basis of the arguments regarding the cause of the non-minimum-phase behavior of the reactor-external-heat-exchanger process, the term... 

Jacobsen, E. W. (1999). On the dynamics of integrated plants - non-minimum phase behavior. J. Proc. Contr., 9, 439-451. 

As shown by Bristol (72) for controllers with heavy reset action, this measure has very interesting properties. Input/output pairs are selected for those Pij approaching 1. A negative element in Pij indicates instability or non-minimum phase behavior, so a glimpse of the dynamics can be obtained from Bristol s method. 

Figure 9 shows the steady state convergent results for reducing the exit gas pressure from 2.84 MPa (28 atm) to 2.53 MPa (25 atm). The pressure decrease lowers the gas density but increases the gas velocity since the molar feed rate of the inlet gas remains constant. This puts less heat into the reactor which lowers the exit gas temperature and causes the burning zone to shift down the reactor a short distance. This is reflected in the Figure 9 solids temperature response which starts out with a short false temperature rise but then exhibits a rapid temperature drop to a final steady state condition (non-minimum phase response). 

Both formulations EKF and CEKF were implemented in MatLab 7.3.0.267 (R2006b) and applied in the process dynamic model, previously presented. The system initial condition is an operating point that presents a minimum-phase behavior (lstep changes in the valve distribution flow factors during the process simulation the system moves to an operating region presenting non-minimum phase behavior (l[Pg.523]

Engell and co-workers in Chapter C4 deal with the control structure selection based on input/output controllability measures. The limitations imposed by non-minimum phase characteristics on the attainable closed-loop performance are considered in the evaluation of the candidate set of control structure configurations. The optimisation of the attainable performance over the set of all linear stabilizing controllers can refine the controller structure with input constraints and coupling properties directly accounted for. 

During the last decades, plants in the process industries have been steadily tighter optimized, both with respect to economic and environmental factors. One consequence of this optimization has been more complex plant structures, involving recycle flows of material and energy. For such plants, the dynamic properties, and hence the system controllability, is to a large extent determined by interactions between the process units. Previous studies have shown that the interactions due to recycling can affect stability [1], disturbance sensitivity and response time [2], oscillatory modes [3] and non-minimum phase behavior [4]. Recycling may also introduce... 

From the above results it is clear that we need to modify the process design in order to remove the non-minimum phase behavior and simultaneously increase the effect of the control input on the product composition. However, since the disturbance sensitivity requires a relatively high bandwidth of the control system, i.e., as > 0.2, it may be relevant to also modify the design with the aim of reducing the disturbance sensitivity at higher frequencies. In order to achieve these goals it is necessary to understand the source of the relevant behaviors, and for this purpose we shall in the next section consider decomposition of models for integrated process systems by means of tools from linear systems theory. 

As stated above, it is usually difficult to predict whether a specific behavior in an integrated plant can be attributed to a single unit, or is caused by process unit interactions. For instance, for the reactor-separator example it is not obvious whether the non-minimum phase behavior is a property of the distillation column alone, or caused by interactions between the column and the reactor. Nor is it clear to what extent the relatively large disturbance sensitivity is due to unit interactions. 

Reactor-Separator problem revisited. In the previous section we modified the process design with the aim of removing the non-minimum phase behavior. This was achieved by a modification of the reactor design which reduced the individual disturbance sensitivity gR s) of the reactor. Since this disturbance sensitivity also affects the recycle loop gaing22( S)g/f( S)) this also resulted in a significant reduction in the disturbance sensitivity of the overall plant, i.e., from to yo- This can be seen from the lower plot in Fig. 5. At steady-state, a 20%... 

We stress that design for controllability can either aim at reducing control bandwidth limitations, imposed by fundamental process properties, or at reducing the control requirements imposed by disturbance sensitivities. Based on results from linear systems theory we have presented simple model based tools, based on the decomposed models above, which can be used to improve stability, non-minimum phase behavior and disturbance sensitivities in plants with recycle. One important conclusion of the presented results is that the phase-lag properties of the individual process units play a crucial role for the disturbance sensitivity of an integrated plant. In particular, by a careful design of the recycle loop phase lag, it is possible to tailor the effect of process interactions such that they serve to effectively dampen the effect of disturbances in the most critical frequency region, that is, around the bandwidth of the control system. 

As mentioned above, non-minimum-phase dynamics of the plant, in particular, RHP zeros close to the origin or large time delays, impose strong limitations on the attainable control performance, and this should be reflected in the specified attainable desired closed-loop dynamics that are used to compute the RPN. However, especially if controllability indices are used by non-experts in control, this cannot be guaranteed, and therefore it is desirable to have a means to alert the user to such restrictions and to indicate the best attainable performance in a simple maimer. Therefore the computed RPN for a particular choice of inputs and outputs should be compared to the minimum RPN for a given desired performance. It is defined as follows ... 

In a first control structure selection step, the plant dynamics is checked for non-minimum phase elements for all possible structures. For the air separation plant, the possible closed loop bandwidth of several possible control structures is limited by RHP zeros. 6 control structures have very small RHP zeros which make good control of the plant impossible. 6 other structures have no or relatively large RHP zeros and thus are prime candidates The selection... 

In the first selection step, the non-minimum phase behavior of the various combinations was analyzed. For this distillation column, the closed-loop bandwidth is mainly limited by the RHP zeros. Therefore, all combinations with RHP zeros smaller than 0.01 rad/s were rejected and the number of control structures was reduced to seven. 

As illustrated by the examples, the I/O-controllability indices RPN and rRPN are very useful to quantify the achievable closed-loop performance of a given control structure. Because of its dependence on the attainable closed-loop performance, the RPN takes the effect of non-minimum-phase behaviour and the desired performance of the closed loop into account. In addition, the frequency-dependent directionality of the system is quantified correctly. Based solely on RPN, it is not possible to evaluate the effect of the desired performance on the closed-loop response, but using the relative RPN, it can be concluded whether a given desired performance is realizable. The relative RPN (rRPN) helps to compare how far the desired performance deviates from the attainable performance. rRPN values larger than 1 indicate that the desired performance is unrealizable, values smaller than 1 indicate that it can be achieved. 

This chapter will analyze the mixing process in more detail. The process was already introduced in chapter 4, but some special properties of the Laplace transform and some special cases of the mixing process will be reviewed. In subsequent chapters other types of processes will be analyzed for their dynamic behavior. The purpose of the Laplace transform is to analyze how the process output of interest changes if the process input is changed. This will result in knowledge about the behavioral properties of the system, such as order, stability, integrating or non-minimum phase response behavior. 

The reactor temperature response is shown in Fig. 12.13, as can be expected this response also has a non-minimum phase character. 

Fig. 4 is a simplified schematic of a boost converter. The circuit consists of a control block, a switch Q, a diode D, inductor L and output capacitor C. The key problems with the boost converter are two-fold because D carries high current due to lower fuel cell stack voltage, its reverse-recovery when Q turns on could be a problem and non-minimum phase and nonlinear behavior of the converter limiting the system... 

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