Closed-loop unstable system

These simulation results indicate that the reflux-ratio structure in this column produces a closed-loop unstable system. The reason for this unexpected behavior appears to be the competing effects of changes in reflux and changes in distillate. Increasing reflux decreases temperatures. Increasing distillate increases temperatures. 

The effect of adding a lag or a pole is to pull the root locus plot toward the unstable region. The two curves that s ait ats=— Jand.s=—1 become complex conjugates and curve off into the RHP. Therefore this third-order system is closed-loop unstable if is greater than = 20. This was the same result that we obtained in Example 10.5,... 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

Since FCC units are usually operated at their middle unstable steady state, extensive efforts are needed to analyze the design and dynamic behavior of open loop and closed loop control systems to stabilize the desirable middle steady state. 

We have chosen the steady state with Yfa = 0.872 and FCD = 1.0 giving a dense phase reactor temperature of Yrd = 1.5627 (Figure 7.14(b) and (c)) and a dense-phase gasoline yield of x-id = 0.387 (Figure 7.14(a)). This is the steady state around which we will concentrate most of our dynamic analysis for both the open-loop and closed-loop control system. We first discuss the effect of numerical sensitivity on the results. Then we address the problem of stabilizing the middle (desirable, but unstable) steady state using a switching policy, as well as a simple proportional feedback control. 

What makes controller design challenging is that all real processes can be made closed loop unstable when a controller is implemented to steer the process to specified operating conditions. In other words, a process which is open loop stable and therefore will come to a new, although not the desired, steady state after a disturbance may become unstable when a controller is implemented to steer the process towards the desired steady state. Stability is therefore of vital concern in all control systems. 

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... 

Thus the linear analysis predicts that the system will be closed-loop unstable when... 

If any of the roots of the above equation are in the RHP the system is closed-loop unstable. The treatment of this problem by Nyquist or root locus plot is can be found in Luyben (1990). 

This example demonstrates several extremely important faets about sampled-data control. This simple first-order system, which could never be made closed-loop unstable in a continuous control system, can become closedloop unstable in a sampled-data system. This is an extremely important difference between continuous control and sampled-data control. It points out that continuous control is almost always better than sampled-data control ... 

The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for GqiXs), a polar plot of its frequency response characteristics (see Chapter 14). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of co or cOg. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. 

Several parameters come into the relation between density and equivalence ratio. Generally, the variations act in the following sense a too-dense motor fuel results in too lean a mixture causing a potential unstable operation a motor fuel that is too light causes a rich mixture that generates greater pollution from unburned material. These problems are usually minimized by the widespread use of closed loop fuel-air ratio control systems installed on new vehicles with catalytic converters. 

When the loop is closed, will the system in (iv) be stable or unstable Solution... 

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

Our objective is simple. We want to make sure that the controller settings will not lead to an unstable system. Consider the closed-loop system response that we derived in Section 5.2 (p. 5-7) ... 

The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jco in the closed-loop characteristic equation to find the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this specific proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. 

The system in (e) can become unstable, while a proper addition of an open-loop zero, as in (f), can help stabilize the system (Fig. E7.6c). In (e), the two loci from -1 and -2 approach each other (arrows not shown). They then break away and the closed-loop poles become unstable. The... 

From the root locus plots, it is clear that the system may become unstable when x = 0.05 s. The system is always stable when = 5 s, but the speed of the system response is limited by the dominant pole between the origin and -0.2. The proper choice is xt = 0.5 s in which case the system is always stable but the closed-loop poles can move farther, loosely speaking, away from the origin. 

In both cases the open-loop system is unstable but the location of the poles makes the second one more difficult to control. Sketching the root locus for the transfer function in Eq.(33), it is easy to verify that the system is conditionally stable. There is only a range of controller gain, K G K nn, K ,ax), leading to a closed-loop stable reactor. 

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

This indicates that the oscillation, once set in motion, will be maintained with constant amplitude around the closed-loop for =. % = 0. If, however, the open-loop gain or AR of the system is greater than unity, the amplitude of the sinusoidal signal will increase around the control loop, whilst the phase shift will remain unaffected. Thus the amplitude of the signal will grow indefinitely, i.e. the system will be unstable. 

Although by definition the system is stable, the phase margin is so close to zero and the gain margin so close to unity that any slight variation in any of the control system parameters or, indeed, in the process conditions, could make the system unstable, i.e. could cause a pole or poles of the system closed-loop transfer function to move into the right half of the complex plane (Section 7.10.1). 

The control loop affects both the static behavior and the dynamic behavior of the system. Our main objective is to stabilize the unstable saddle-type steady state of the system. In the SISO control law (7.72) we use the steady-state values Yfass = 0.872 and Yrdss = 1.5627 as was done in Figures 7.14(a) to (c). A new bifurcation diagram corresponding to this closed-loop case is constructed in Figure 7.20. 

The static bifurcation characteristics of the resulting closed loop system have also been discussed in the previous section and we have seen that the bifurcation diagram of the reactor dense-phase dimensionless temperature, namely a plot of Yrd versus the controller gain Kc is a pitchfork. Such bifurcations are generally structurally unstable when any of the system parameters are altered, even very slightly. 

The transition mode is an unstable process regime for conventional deposition systems. Closed-loop control concepts or modified chamber designs outlined in Sect. 5.3.4.2 are necessary for transition mode process control. 

In contrast to constrained MPC of stable plants, constrained MPC of unstable plants has the complication that the tightness of constraints, the magnitude and pattern of external signals, and the initial conditions all affect the stability of the closed loop. The following simple example illustrates what may happen with a simple unstable system. 

Then one can state the following criterion for the stability of a closed-loop system A feedback control system is stable if all the roots of its characteristic equation have negative real parts (i.e., are to the left of the imaginary axis). If any root of the characteristic equation has a real positive part (i.e., is on or to the right of the imaginary axis), the feedback system is unstable. 

The second important bifurcation that is connected with a stability change in a stationary state is the /fop/bifurcation. At a Hopf bifurcation, the real parts of two conjugate complex eigenvalues of J vanish, and as Hopf s theorem ensures, a periodic orbit or limit cycle is bom. A limit cycle is a closed loop in phase space toward which neighboring points (of the kinetic representation) are attracted or from which they are repelled. If all neighboring points are attracted to the limit cycle, it is stable otherwise it is unstable (see Ref. 57). The periodic orbit emerging from a Hopf bifurcation can be stable or unstable and the existence of a Hopf bifurcation cannot be deduced from the mere fact that a system exhibits oscillatory behavior. Still, in a system with a sufficient number of parameters, the presence or absence of a Hopf bifurcation is indicative of the presence or absence of stable oscillations. 

Simultaneously satisfying each of these objectives is never possible therefore, tuning is a compromise. For example, tuning for minimum deviation from setpoint for normal disturbances is contrary to tuning the controller to remain stable for major disturbances. That is, if the controller is tuned for normal disturbances, the closed-loop system may go unstable when a major disturbance enters the process. On the other hand, if the controller is tuned for the largest possible disturbance, control performance is likely to be excessively sluggish for normal disturbance levels. 

In Section 9.4 we concluded that if G(s) has a pole with positive real part, it gives rise to a term C t ept which grows continuously with time, thus producing an unstable system. The transfer function G(s) can correspond to an uncontrolled process or it can be the closed-loop transfer function of a controlled system (e.g., GSp or Gioad). Therefore,... 

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