Stability of a closed-loop system

Maximum value of the open-loop gain constant for the stability of a closed-loop 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. 

Example 15.6 demonstrated that the root locus of a system not only provides information about the stability of a closed-loop system but also informs us about its general dynamic response characteristics as Kc changes. Therefore, the root locus analysis can be the basis of a feedback control loop design methodology, whereby the movement of the closed-loop poles (i.e., the roots of the characteristic equation) due to the change of the proportional controller gain can be clearly displayed. 

Representing the roots of the characteristic equation in the complex domain offers a simple way to perform a stability analysis. The system is stable if and only if all the poles are located in the open left-half-plane (LHP). If there is at least one pole in the right-half-plane (RHP), the system is unstable. The representation is similar wiA the well-known root-locus plot used to evaluate the stability of a closed-loop system. 

The method is a necessary but not sufficient condition for stability of a closed-loop system with integral action. If the index is negative, the system will be unstable for any controller settings (this is called integral instability ). If the index is positive, the system may or may not be stable. Further analysis is necessary. 

Next, we state one of the most important results of frequency response analysis, the Bode stabihty criterion. It allows the stability of a closed-loop system to be determined from the open-loop transfer function. 

The preceding examples have demonstrated very vividly that the stability characteristics of a closed-loop system depend on the value of gain Kc. Thus in Example 15.1 we notice that the closed-loop system becomes stable when Kc > 1/10. Also, in Example 15.4, the system is stable when... 

The criterion of stability for closed-loop systems does not require calculation of the actual values of the roots of the characteristic polynomial. It only requires that we know if any root is to the right of the imaginary axis. The Routh-Hurwitz procedure allows us to test if any root is to the right of the imaginary axis and thus reach quickly a conclusion as to the stability of the closed-loop system without computing the actual values of the roots. 

Here we adopt the following definition valid for linear systems a system is stable if bounded input variations produce bounded output variations as t —>oo otherwise the system is unstable (Ogunnaike Ray, 1994). One of the main issues in designing feedback controllers is stability. Let consider the response of a closed-loop system under proportional control, as deviation in outputy vs. time (Fig. 12.5). If the controller gain is moderate then y goes to zero after some oscillations. By increasing gain... 

According to stability analysis of linear time invariant system, stability of the closed-loop system x= A- BK)x depends on the eigenvalue of eigenmatrix (A - BK). In other words, the condition that the stabifity is positive is all the eigenvalues of matrix (A - BK) are negative. The switching function of SMC is... 

Suppose that the temperature in an exothermic continuous stirred-tank reactor is controlled by manipulating the coolant flow rate using a control valve. A PID controller is used and is well-tuned. Which of these changes could adversely affect the stability of the closed-loop system Briefly justify your answers. 

In the remainder of this chapter, we analyze the stability characteristics of closed-loop systems and present several useful criteria for determining whether a system will be stable. Additional stability criteria based on frequency response analysis are discussed in Chapter 14. But first we consider an illustrative example of a closed-loop system that can become unstable. 

Two important conclusions can be drawn from these closed-loop relations. First, a set-point change in one loop causes both controlled variables to change because Ti2 and F2i are not zero, in general. The second conclusion concerns the stability of the closed-loop system. Because each of the four closed-loop transfer functions in Eqs. 18-15 to 18-18 has the same denominator, the characteristic equation is D s) = 0, or... 

Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and dead time in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely hmits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

In this statement, we have used "polar plot of G0l" to replace a mouthful of words. We have added G0L-plane in the wording to emphasize that we are using an analysis based on Eq. (7-2a). The real question lies in what safety margin we should impose on a given system. This question leads to the definitions of gain and phase margins, which constitute the basis of the general relative stability criteria for closed-loop systems. 

Once we understand the origin of Nyquist stability criterion, putting it to use is easy. Suppose we have a closed-loop system with characteristic equation 1 + GCGP = 0. With the point (-1,0) as a reference and the Gc(jco)Gp(jco) curve on a... 

Oxygen sensors, in low volume use as part of a closed loop emission control system for automotive applications since 1977, have seen wide-spread use starting with the 1981 model year. At the present time, a partially stabilized zirconia electrolyte using yttrium oxide as the stabilizer appears to be the most common choice for this application. 

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. 

The stability criterion stated above secures stable response of a feedback system independently if the input changes are in the set point or the load. The reason is that the roots of the characteristic equation are the common poles of the two transfer functions, GSp and Gioad, which determine the stability of the closed loop with respect to changes in the set point and the load, respectively. 

The notion of spillover is important with respect to neglected structural modes. Other modelling errors include parametric uncertainties, which are more difficult to model and may have a substantial impact on the stability and performance of the closed-loop system. 

The key idea is to use the Youla parameterization of all closed-loop systems which is affine in a free transfer matrix, and to optimize the free parameter for convex performance criteria. Each relationship between the external signals w(.s) and the external output z s) which is attainable by a stabilizing controller can be described as ... 

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