Unstable process

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. 

Example 7.1 Consider the unstable process function Gp =, which may arise from a... 

In the last chapter we used Laplace-domain techniques to study the dynamics and stability of simple closedloop control systems. In this chapter we want to apply these same methods to more complex systems cascade control, feedforward control, openloop unstable processes, and processes with inverse response. Finally we will discuss an alternative way to look at controller design that is called model-based control. 

A. FIRST-ORDER OPENLOUP UNSTABLE PROCESS. Suppose we have a first-order process with the openloop transfer function... 

Can we make the system stable by using feedback control That is, can an openloop unstable process be made closedloop stable by appropriate design of the feedback controller Let us try a proportional controller = The... 

B. SECOND-ORDER OPENLOOP UNSTABLE PROCESS. Consider the process given in Eq. (11.44) with a first-order lag added. 

Root locus curves for openloop unstable processes (positive poles). 

C. THIRD-ORDER OPENLOOP UNSTABLE PROCESS. If an additional lag is added to the system and a proportional controller is used, the closedloop characteristic equation becomes... 

Make root locus plots of first- and second-order openloop unstable processes with PI feedback controllers. 

Find the closedloop stability requirements for a third-order openloop unstable process with a proportional controller ... 

Find the value of feedback controller gain K, that gives a closedloop system with a damping coeflicient of 0.707 for a second-order openloop unstable process with... 

An openloop unstable process has a transTer function containing a positive pole at... 

Keep in mind that we are talking about closedloop stability and that we are studying it by making frequency-response plots of the total openlaop system transfer function. We are also considering openlocp stable systems most of the time. We will show how to deal with openloop unstable processes in Sec. 13.4. 

The Nyquist stabihty criterion can be used for openloop unstable processes, but we have to use the complete, rigorous version with P (the number of poles of the closedloop characteristic equation in the RHP) no longer equal to zero. 

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

SAMPLED-DATA CONTROL OF OPENLOOP UNSTABLE PROCESSES... 

Example 20.11. A second-order opienloop unstable process has the pulse transfer function given below ... 

Characterization of an unstable process impurity in the protease inhibitor Tipranavir... 

We can similarly stabilise an unstable process using P or PD action—but not PI action. This situation is unlikely, however, as most process systems met by the chemical engineer are operated under conditions where the process is inherently stable. 

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. 

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. 

Unstable process model. In that case, an FIR process model cannot be used. A state-space model such as... 

Rawlings and Muske (1993) have shown that this idea can be extended to unstable processes. In addition to guaranteeing stability, their approach provides a computationally efficient method of on-line implementation. Their idea is to start with a finite control (decision) horizon but an infinite prediction (objective function) horizon, i.e., m < < and p = , and then use the principle of optimality and results from optimal control theory to substitute the infinite prediction horizon objective by a finite prediction horizon objective plus a terminal penalty term of the form... 

Unstable processes pose the additional challenge that stabilization is possible only if the state x k) lies in a certain domain, so that, even though the input may be constrained [Eq. (5)] enough control action can be available. If the state is not in the stabilizability domain, then nothing can be done to steer the state to the setpoint. 

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