Feedback Control System Characteristics

Figure I l.lh shows the conventional feedback control system where a single controller senses the controlled variable and changes the manipulated variable M. The dosedloop characteristic equation for this system was developed in Chap. 10.

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. 

Analyze the stability characteristics of a feedback control system and learn how to design the appropriate feedback system to control a given process. 

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). 

While designing a feedback control system (i.e., selecting its components and tuning its controller), we are seriously concerned about its stability characteristics. Therefore, before we proceed with the particular details of designing a feedback control loop, we will study the notion of stability and analyze the stability characteristics of closed-loop systems. 

Consider the feedback control system of Example 15.2. The characteristic equation is... 

In Section 21.4 it was claimed that the stability characteristics of a feedforward-feedback control system are affected only by the feedback loop. Explain why. 

Cholesterol biosynthesis is affected by dietary and hormonal factors as well as by various external influences. Cholesterogenesis is enhanced by radiation, thyroid hormones, hypophysectomy, various metal ions and surface active agents. Biosynthesis is inhibited by fasting, thyroidectomy, vanadium salts, and by feeding of cholesterol or some of its steroid precursors. These influences were reviewed by Kritchevsky et al. (1960). In most cases cholesterol synthesis from acetate is more severely inhibited than is synthesis from mevalonate, suggesting that the inhibition occurs at an early step in cholesterol biosynthesis. The inhibition of cholesterol biosynthesis by cholesterol feeding was shown to possess the characteristics of a negative feedback control system (Bucher et al., 1959). 

Several authors have postulated a feedback control system that is modeled after the process. This kind of control action is known as complementary feedback, because the characteristics of the controller complement the dynamics of the process. A block diagram showing both process and complementary controller appears in Fig. 4.10. 

General Stability Criterion. The feedback control system in Fig. 11.8 is stable if and only if all roots of the characteristic equation are negative or have negative real parts. Otherwise, the system is unstable. 

The function of a feedback control system is to ensure that the closed-loop system has desirable dynamic and steady-state response characteristics. Ideally, we would like the closed-loop system to satisfy the following performance criteria ... 

In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions. This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis. Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function of any linear process. Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These metrics indicate how close to instability a control system is. A related issue is robustness, which addresses the sensitivity of... 

Frequency response techniques are powerful tools for the design and analysis of feedback control systems. The frequency response characteristics of a process, its amplitude ratio AR and phase angle, characterize the dynamic behavior of the process and can be plotted as functions of frequency in Bode diagrams. The Bode stability criterion provides exact stability results for a... 

In Chapter 11 it was shown that the roots of the characteristic equation completely determine the stabihty of the closed-loop system. Because Gf does not appear in the characteristic equation, the feedforward controller has no effect on the stability of the feedback control system. This is a desirable situation that allows the feedback and feedforward controllers to be tuned individually. 

From Eq. 16-11 the closed-loop time constant for the inner loop is 0.2 min. In contrast, the conventional feedback control system has a time constant of 1 min because in this case, Y2 s)IYsp2 s) = Gy = 5/(5 + 1). Thus, cascade control significantly speeds up the response of Y2. Using a proportional controller in the primary loop (G -i = K ), the characteristic equation becomes... 

The block diagram of a general feedback control system is shown in Fig. J.l. It contains three external input signals set point Ygp, disturbance D, and additive measurement noise, N. The noisy output is the sum of the noise N and the noise-free output Y. The following analysis illustrates the fundamental limitations and engineering tradeoffs that are inherent in achieving these characteristics. 

Feedback control systems have the property that as long as the loop gain is sufficiently high (and the system is stable), the transfer function of the system is the inverse of the feedback transfer function (Phillips and Harbor 1991). In the case of an inertial sensor, this means that the transfer function is determined by electronic components in the feedback network not the physical characteristics of the pendulum. Furthermore this same feedback holds the proof mass substantially at rest with respect to the frame of the sensor, greatly reducing the impact of nonlinearities in the suspension and transducers. 

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