Closed-loop stability

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

The first objective can be reached by using a general property of discretized system namely, if for the system (23) there exists a matrix Kd such that Ad + BdKd has all its eigenvalues inside the unit circle, then the controller u k) = Kdx(k) will also stabilize the continuous system. Thus, the closed-loop stability can be then assured by properly assigning the discrete poles inside the unit circle. 

The controller has tuning parameters related to proportional, integral, derivative, lag, dead time, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This relationship is accomplished, first,... 

The challenge posed by establishing and maintaining communication between distributed controllers has also stimulated research in the area of networked process control (El-Farra et al. 2005, Mhaskar et al. 2007, Sun and El-Farra 2008, 2010). The central issue of maintaining closed-loop stability in the presence of bandwidth constraints and limitations in transmitter battery longevity is typically addressed by a judicious distribution of computation and communication burdens between local/distributed control systems and a centralized supervisory controller. 

We call this high sensitivity of the recycle flowrates to small disturbances the snowball effect. We illustrate its occurrence in the simple example below, It is important to note that this is not a dynamic effect it is a steady-state phenomenon. But it does have dynamic implications for disturbance propagation and for inventory control. It has nothing to do with closed-loop stability. However, this does not imply that it is independent of the plant s control structure. On the contrary, the extent of the snowball effect is very strongly dependent upon the control structure used. 

Start with one isolated unit operation. Get its control system installed, tuned, and tested for closed-loop stability and robustness. Then move on to the next unit and repeat. Build up the entire plant... 

Example 6—Closed-loop stability for a nonminimum-phase process. 

Example 11—Sensitivity of closed-loop stability to small variations in controller parameters. For the stable transfer function... 

As in the linear case, a proof for closed-loop stability can be constructed, if it can be guaranteed that... 

Equation (110) can be satisfied if fhe moving horizon length, p, is chosen to be large enough, or if the constraint in Eq. (110) is directly incorporated in the on-line optimization problem. In either case, a closed-loop stability proof can be constructed as follows. 

On the other hand, feedback control is rather insensitive to all three of these drawbacks but has poor performance for a number of systems (multicapacity, dead time, etc.) and raises questions of closed-loop stability. Table 21.1 summarizes the relative advantages and disadvantages of the two control systems. 

The closed-loop stability of the two noninteracting loops depends on the roots of their characteristic equations. Thus if the roots of the two equations... 

Safety, chemical process, 2 Sampled data, 559, 571, 572-76 Sampled representation of continuous signals, 571-76 Sampler, 558, 559, 571 ideal impulse, 575, 576 Sampling, 539, 571, 372-73 of closed-loop responses, 639-40 effect on closed-loop stability, 638-39 of oscillatory signals, 573-74... 

There are several approaches that can be used to tune PID controllers, including model-based correlations, response specifications, and frequency response (Smith and Corripio 1985 Stephanopoulos 1984). An approach that has received much attention recently is model-based controller design. Model-based control requires a dynamic model of the process the dynamic model can be empirical, such as the popular first-order plus time delay model, or it can be a physical model. The selection of the controller parameters Kc, ti, to) is based on optimizing the dynamic performance of the system while maintaining closed-loop stability. 

Two methods are commonly used to determine closed-loop stability ... 

Under the assessment of closed-loop stability analysis results the following typical review... 

Chapter 7 discusses robust control. This allows for the inclusion of uncertainty of process parameters in the control design. The concept of robustness refers to the preservation of closed-loop stability under allowable variations in system parameters. General stability results and integrity results are given for the LQR problem. 

Known approaches to this problem either use indieators of I/O-controllability (e.g. [22, 25]) or include the controller in the overall optimization. This has been done either by parameterizing fixed controller structures (e.g.[2]) or by optimizing over the inputs [32], The assumption of a specific control structure has two disadvantages firstly it restricts the control performance and secondly, it renders the optimization problem to be non-convex and thus a lot more difficult. The optimization of the inputs in an open-loop fashion on the other hand does not reflect the issues of closed-loop stability and of robustness correctly. 

In this subsection a geometric feedforward plus SF feedback scheme is developed, according to a separation principle-like reasoning the combination of the SF controller (Eq. 17) and the compatible open-loop estimator (28a,b) are designed separately, their combination yields the measurement-driven controller, and the closed-loop stability is assessed a... 

The closed-loop stability of the batch motion can be established with the application of the standard singular perturbation [25] or small gain theorems [8, 10] available in the nonlinear dynamical systems literature, in eonjunction with the definition (7) of finite-time motion stability. In a chemical process context this closed-loop stability assessments can be seen in the cascade control of a continuous reactor [22], the cascade control of a continuous distillation [21, 24], and in the calorimetric estimation [15] of a batch polymer reactor. The closed-loop motion stability is ensured if the observer gain ( o) is tuned slower than the characteristic frequency ( j) of the fastest unmodeled dynamics, and the observer ( ), secondary ( ,.), and primary ( p) gains are sufficiently separated. This is. 

The formal analysis of the closed-loop dynamics can be done with an extension of the nonlocal closed-loop stability analysis of a continuous reactor with temperature cascade controller presented before [22] in conjunction with the stability definitions (Eq. 7) given in section 2. Here it suffices to mention that, the closed-loop motion is stable if the filter and estimator gains are chosen not faster than the characteristic frequency Oj of the jacket hydraulic dynamics [26], and the secondary and primary control gains are chosen so that there is adequate dynamic separation between coj, coj and 0). This is,(Eq. 30),... 

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