Open-loop unstable

Addition of a feedback control loop can stabilize or destabilize a process. We will see plenty examples of the latter. For now, we use the classic example of trying to stabilize an open-loop unstable process. 

Nyquist stability ciiteiion applied to an open-loop unstable process. 

Example 1. Periodic Control of an Open Loop Unstable CSTR... 

Fig. 6. Load response of the open-loop unstable system (proposed continuous line, Lee et al. [18] dotted line).

H.P. Huang, C.C. Chen, Control system synthesis for open-loop unstable processes with time delay, IEE Control Theory Appl. 144 (1997) 334-338. 

V. Saraf, F. Zhao, B.W. Bequette, Relay autotuning of cascaded-controlled open-loop unstable reactors, Ind. Eng. Chem. Res. 42 (2003) 4488 1494. 

Inspecting Equation (5.29), we notice that three of the state variables (namely, Mr, My, and Ml) are material holdups, which act as integrators and render the system open-loop unstable. Our initial focus will therefore be a pseudo-open loop analysis consisting of simulating the model in Equation (5.29) after the holdup of the reactor, and the vapor and liquid holdup in the condenser, have been stabilized. This task is accomplished by defining the reactor effluent, recycle, and liquid-product flow rates as functions of Mr, My, and Ml via appropriate control laws (specifically, via the proportional controllers (5.42) and (5.48), as discussed later in this section). With this primary control structure in place, we carried out a simulation using initial conditions that were slightly perturbed from the steady-state values in Table 5.1. 

Subsequently, we used Aspen Dynamics for time-domain simulations. A basic control system was implemented with the sole purpose of stabilizing the (open-loop unstable) column dynamics. Specifically, the liquid levels in the reboiler and condenser are controlled using, respectively, the bottoms product flow rate and the distillate flow rate and two proportional controllers, while the total pressure in the column is controlled with the condenser heat duty and a PI controller (Figure 7.4). A controller for product purity was not implemented. 

The completely mixed model succeeds in representing part of the experimental data and predicts that at industrial conditions the reactor is open-loop unstable. Initiator productivity decreases are accounted quite accurately only by the second reactor model which details the mixing conditions at the initiator feed point. Independent estimates of the model parameters result in an excellent match with experimental data for several initiator types. Imperfect mixing is shown to have a tendency to stabilize the reactor. 

The second issue for cooled tubular reactors is how to introduce the coolant. One option is to provide a large flowrate of nearly constant temperature, as in a recirculation loop for a jacketed CSTR. Another option is to use a moderate coolant flowrate in countercurrent operation as in a regular heat exchanger. A third choice is to introduce the coolant cocurrently with the reacting fluids (Borio et al., 1989). This option has some definite benefits for control as shown by Bucala et al. (1992). One of the reasons cocurrent flow is advantageous is that it does not introduce thermal feedback through the coolant. It is always good to avoid positive feedback since it creates nonmonotonic exit temperature responses and the possibility for open-loop unstable steady states. 

Equation (5.11) represents a straight line in the diagram of fractional temperature rise versus reactor feed temperature. We show three such lines in Fig. 5.21. All lines intersect the temperature rise curve at least once (at a low temperature not shown in Fig. 5.21). It therefore appears that the reactor FEHE can have one, two, or three steady-state solutions for this particular set of reaction kinetics. Furthermore, the intermediate steady state, in the case of three solutions, is open-loop unstable due to the slope condition discussed in Chap. 4. This was verified by Douglas et al. (1962) in a control study of a reactor heat exchange system. 

What are the control implications of this analysis The first conclusion is that autothermal systems (no furnace) have two or more steady states. There is also a good chance that the normal operating point corresponds to the intermediate steady state that is open-loop unstable. This is certainly the case when the reactor is operated at less than 100 percent conversion. 

Figure 15.2 Curve a, open-loop unstable response curve / , closed-loop stable response with P control.

Root locus curves for open loop unstable processes (positive poles). 

Control Structure Selection for open loop unstable processes is the main theme of this paper. Hankel singular value has been used as a controllability measure for input-output selection. This method ensures feedback stability of the process with minimal control effort as well as it provides a quantitative justification for the controllability. Simulation results with Tennesse-Eastman test-bed problem justify the proposed theory. 

Glover (1986) studied the robust stabilization of a linear multivariable open-loop unstable system modelled as (G+A) where G is a known rational transfer function and A is a perturbation (or plant uncertainty). G is decomposed as G/+G2, where G, is antistable and G2 is stable (Figure 1). The controller and the output of the feedback system are denoted by K and y respectively. Gi is strictly proper and K is proper. Glover (1986) argued that the stable projection G2 does not affect the stabilizability of the system, since it can be exactly cancelled by feedback. The necessary and sufficient condition for G to be robustly stabilized is to stabilize its antistable projection G/. 

For a given data set D the system (1) has a unique (possibly open-loop unstable in a sense to be defined) solution motion x(t), with unique output trajectories z(t) and y(t) ... 

The term open-loop unstable is also used to describe process behaviour. Some would apply it to any integrating process. But others would reserve it to describe inherently unstable processes such as exothermic reactors. Figure 2.21 shows the impact that increasing the reactor inlet temperature has on reactor outlet temperature. The additional conversion caused by the temperature increase generates additional heat which increases conversion further. It differs from most non-self-regulating processes in that the rate of change of PV increases over time. It often described as a runaway response. Of course, the outlet temperature will eventually reach a new steady state when aU the reactants are consumed however this may be well above the maximum permitted. 

The term open-loop unstable can also be applied to controllers that have saturated. This means that the controller output has reached either its minimum or maximum output but not eliminated the deviation between PV and SP. It can also be applied to a controller using a discontinuous on-stream analyser that fails. Such analysers continue to transmit the last measurement until a new one is obtained. If, as a result of analyser failure, no new measurement is transmitted then the controller no longer has feedback. 

Polymerization reactors are well known both theoretically and experimentally to exhibit multiple steady states [26, 27] and in some cases they may also exhibit oscillations in terms of monomer conversion and polymer particle diameter [28]. In other cases it may be necessary to choose an open-loop unstable state as the reactor operating point. Furthermore, polymerization reactors can be highly exothermic and result in reactor thermal runaway. 

Some processes are open-loop unstable, which means that controls must be added and must be kept in automatic for stable operation. 

Most industrial processes are stable without feedback control. Thus, they are said to be open-loop stable, or self-regulating. An open-loop stable process will return to the original steady state after a transient disturbance (one that is not sustained) occurs. By contrast, there are a few processes, such as exothermic chemical reactors, that can be open-loop unstable. These processes are extremely difficult to operate without feedback control. 

We conclude that the liquid storage system is open-loop unstable (or non-self-regulating) because a bounded input has produced an unbounded response. However, if the pump in Fig. 11.24 were replaced by a valve, then the storage system would be self-regulating (cf. Example 4.4). 

Consider a process, Gp = 0.2/(-5 + 1), that is open-loop unstable. If Gy = G = 1, determine whether a proportional controller can stabilize the closed-loop system. 

The question has been raised whether an open-loop unstable process can be stabilized with a proportional-only controller. 

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