Chapter 9 - Stability Analysis of Closed-loop System (1)

ECB4034 - Chemical Process Instrumentation and Control*Chapter 9STABILITY OF CLOSED-LOOP CONTROL SYSTEMS

ECB4034 - Chemical Process Instrumentation and Control

ECB4034 - Chemical Process Instrumentation and Control*Chapter ObjectivesEnd of this chapter, you should be able to:Explain the concepts of feedback control systemsDetermine the stability of control systems using Routh Hurwitz Criteria


ECB4034 - Chemical Process Instrumentation and Control*IntroductionAn important consequence of feedback control is that it can cause oscillatory responses. Under certain circumstances, the oscillations may be undamped or even have amplitude that increases with time until a physical limit is reached. In these situations, the closed-loop system is said to be unstable.


ECB4034 - Chemical Process Instrumentation and Control*Control system:Consider the feedback control system with the following transfer functions:


(9.1)

_1134563459.unknown

_1134563493.unknown

_1134563517.unknown

_1134563361.unknown

Consider a step change in set-point

.

The transfer function for set-point changes is:

(9.2)

_1133677621.unknown

_1134563584.unknown

Substituting (9.1) in (9.2), and rearranging

gives us

(9.3)

_1134563886.unknown

ECB4034 - Chemical Process Instrumentation and Control* After Kc is specified, y(t) can be obtained.Fig. below demonstrates that as Kc increases, the response become more oscillatory and is unstable for Kc = 15.


ECB4034 - Chemical Process Instrumentation and Control*General Stability Criterion:Most industrial processes are stable without feedback controllers. They are said to be open-loop stable or self-regulating.Definition of stability:An unconstrained linear system is said to be stable if the output response is bounded for all bounded inputs. Otherwise, it is said to be unstable.By a bounded input, we mean an input variable that stays within upper and lower limits for all values of time. The term unconstrained refer to the ideal situation where there is no physical limits on the input and output variables.


ECB4034 - Chemical Process Instrumentation and Control*Characteristic equation


For the block diagram shown above, we have developed that

(9.4)

where GOL is the open-loop transfer function, GOL = GcGpGvGm.

_1133677641.unknown

Consider set-point changes only. Eq. (9.4) reduces to

(9.5)

_1134565665.unknown

If GOL is a ratio of polynomials in s, then the closed-loop transfer function in (9.5) is also a rational function. Then, it can be factored into poles (pi) and zeroes (zi) as

(9.6)

_1134566089.unknown

ECB4034 - Chemical Process Instrumentation and Control*


where K is a multiplicative constant selected to give the correct steady-state gain. To have a physically realizable system, the number of poles must be greater than or equal to the number of zeroes.

Comparing (9.5) and (9.6) indicates that the poles are also the roots of the following equation, referred to as the characteristic equation of the closed-loop system:

(9.7)

The characteristic equation plays a decisive role in determining system stability.

_1134566487.unknown



Taking the inverse Laplace transform of (9.9) gives

(9.10)

_1134629018.unknown

For a unit step change in set-point, (9.6) becomes

(9.8)

_1206852666.unknown

If there are no repeat roots (all distinct poles), then the partial fraction expansion of (9.8) has the form

(9.9)

_1206852681.unknown



Suppose that one of the poles is a positive real number; i.e., pk > 0.

Then it is clear from (9.10) that y(t) is unbounded and thus the closed-loop system is unstable.

If pk is a complex number,

, with a positive real part, then the system is also unstable.

If all the poles are negative (or have negative real parts) then the system is stable.

_1134629301.unknown

ECB4034 - Chemical Process Instrumentation and Control*General stability criterion:A feedback control system is stable if and only if all roots of the characteristic equation are negative or have negative real parts. Otherwise, the system is unstable.


ECB4034 - Chemical Process Instrumentation and Control*Graphical interpretation of stability criterion: Roots of1 + GcGvGpGm(Each test is for different value of Kc)


ECB4034 - Chemical Process Instrumentation and Control*If the characteristic equation is either first-order or second-order, we can find the roots analytically. For higher-order polynomials, this is not possible.


ECB4034 - Chemical Process Instrumentation and Control*Routh Stability Criterion Uses an analytical technique for determining whether any roots of a polynomial have positive real parts.Characteristic equation


(9.11)

where an >0. According to the Routh criterion, if any of the coefficients a0, a1, aK, an-1 are negative or zero, then at least one root of the characteristic equation lies in the RHP, and thus, the system is unstable.

_1134633733.unknown

ECB4034 - Chemical Process Instrumentation and Control*Routh Array On the other hand, if all of the coefficients are positive, then one must construct the Routh Array shown below: For stability, all elements in the first column must be positive.


ECB4034 - Chemical Process Instrumentation and Control* The first two rows of the Routh Array are comprised of the coefficients in the characteristics equation. The elements in the remaining rows are calculated from coefficients from the using the formulas:(n+1 rows must be constructed n = order of the characteristic eqn.)


_1134634191.unknown

_1134634221.unknown

_1134634247.unknown

_1134634164.unknown

ECB4034 - Chemical Process Instrumentation and Control*Routh Stability Criterion: A necessary and sufficient condition for all roots of the characteristic equation to have negative real parts is that all of the elements in the left column of the Routh array are positive. Solution: Because the s term is missing, its coefficient is zero. Thus the system is unstable.


Example 1: Determine the stability of a system that has the characteristic equation

_1134634879.unknown

ECB4034 - Chemical Process Instrumentation and Control*Example 2


Find the values of the controller gain Kc that make the feedback control system with the following characteristic equation unstable.

_1134635116.unknown

All coefficients are positive provided that 1+Kc > 0 or Kc > -1.

ECB4034 - Chemical Process Instrumentation and Control* To have a stable system, each element in the left column must be positive, Kc < 12.6 and Kc > -1. Thus we conclude -1 < Kc < 12.6.The Routh array is:


ECB4034 - Chemical Process Instrumentation and Control*Conclusion!Concept of stabilityCriteria for stabilityRouth Hurwitz Stability Analysis


Documents

Chapter 9 - Stability Analysis of Closed-loop System (1)