Chapter 11: Dynamic Behavior & Stability of Closed-loop Control System.

Block Diagram

Next, we develop a transfer function for each of the five elements in the feedback control loop. For the sake of simplicity, flow rate w1 is assumed to be constant, and the system is initially operating at the nominal steady rate.

ProcessIn section 4.1 the approximate dynamic model of a stirred-tank blending system was developed:

Where

The symbol denotes the internal set-point composition expressed as an equivalent electrical current signal. is related to the actual composition set point by the composition sensor-transmitter gain Km:

Current-to-Pressure (I/P) TransducerThe transducer transfer function merely consists of a steady-state gain KIP:

Control ValveAs discussed in Section 9.2, control valves are usually designed so that the flow rate through the valve is a nearly linear function of the signal to the valve actuator. Therefore, a first-order transfer function is an adequate model

Composition Sensor-Transmitter (Analyzer)We assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function, but τm is small so it can be neglected.

ControllerSuppose that an electronic proportional plus integral controller is used.

where and E(s) are the Laplace transforms of the controller output and the error signal e(t). Kc is dimensionless.

Summer

Comparator

Block

• Blocks in Series

“Closed-Loop” Transfer Functions• Indicate dynamic behavior of the controlled process

(i.e., process plus controller, transmitter, valve etc.)• Set-point Changes (“Servo Problem”)

Assume Ysp ¹ 0 and D = 0 (set-point change while disturbance change is zero)

• Disturbance Changes (“Regulator Problem”)Assume D ¹ 0 and Ysp = 0 (constant set-point)

*Note same denominator for Y/D, Y/Ysp.

EXAMPLE 1: P.I. control of liquid level

Assumptions

1. q1, varies with time; q2 is constant.

2. Constant density and x-sectional area of tank, A.

3. q3 ≠f(h) (for uncontrolled process)

4. The transmitter and control valve have negligible dynamics (compared with dynamics of tank).

5. Ideal PI controller is used (direct-acting).

For these assumptions, the transfer functions are:

The closed-loop transfer function is:

Substitute,

Simplify,

Characteristic Equation:

Recall the standard 2nd Order Transfer Function:

To place Eqn. (4) in the same form as the denominator of the T.F. in Eqn. (5), divide by Kc, KV, KM :

Comparing coefficients (5) and (6) gives:

Substitute,

For 0 < z < 1 , closed-loop response is oscillatory. Thus decreased degree of oscillation by increasing Kc or tI (for constantKv, KM, and A).

• unusual property of PI control of integrating system• better to use P only

Stability of Closed-Loop Control Systems

Proportional Control of First-Order ProcessSet-point change:

Set-point change = M

Offset =

Closed-Loop Transfer function approach:

First-order behavior closed-loop time constant

(faster, depends on Kc)

General Stability Criterion

Most industrial processes are stable without feedback control. Thus, they are said to be open-loop stable or self-regulating. An open-loop stable process will return to the original steady state after a transient disturbance (one that is not sustained) occurs. By contrast there are a few processes, such as exothermic chemical reactors, that can be open-loop unstable. 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.

Effect of PID Control on a Disturbance Change

For a regulator (disturbance change), we want the disturbance effects to attenuate when control is applied. Consider the closed-loop transfer function for proportional control of a third-order system (disturbance change).

Kc is the controller function, i.e.,

Let

If Kc = 1,

Since all of the factors are positive, , the step response will be the sum of negative exponentials, but will exhibit oscillation.

If Kc = 8,

Corresponds to sine wave (undamped), so this case is marginally stable.

If Kc = 27

is unspecified)(sD

Since the sign of the real part of the root is negative, we obtain a positive exponential for the response. Inverse transformation shows how the controller gain affects the roots of the system. Offset with proportional control (disturbance step-response; D(s) =1/s )

Therefore, if Kc is made very large, y(t) approaches 0, but does not equal zero. There is some offset with proportional control, and it can be rather large when large values of Kc create instability.

Integral Control:

For a unit step load-change and Kc=1,

no offset

(note 4th order polynomial)

PI Control:

no offset

adjust Kc and tI to obtain satisfactory response (roots of equation which is 4th order).

PID Control: (pure PID)

No offset, adjust Kc, tI , tD to obtain satisfactory result (requires solving for roots of 4th order characteristic equation). Analysis of roots of characteristic equation is one way toanalyze controller

Rule of Thumb:Closed-loop response becomes less oscillatory and more stable bydecreasing Kc or increasing tI .

General Stability CriterionConsider the “characteristic equation,”

Note that the left-hand side is merely the denominator of the closed-loop transfer

function.

The roots (poles) of the characteristic equation (s - pi) determinethe type of response that occurs:

Complex roots Þ oscillatory responseAll real roots Þ no oscillations***All roots in left half of complex plane = stable system

Stability Considerations

• Feedback control can result in oscillatory or even unstable closed-loop responses. • Typical behavior (for different values of controller gain, Kc).•

Routh Stability Criterion

Characteristic equation

Where an >0 . According to the Routh criterion, if any of the coefficients a0, a1, …, 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. 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.

The first two rows of the Routh Array are comprised of thecoefficients in the characteristic equation. The elements in theremaining rows are calculated from coefficients by using theformulas:

(n+1 rows must be constructed; n = order of the characteristic eqn.)

Application of the Routh Array:

Characteristic Eqn is

We want to know what value of Kc causes instability, I.e., at leastone root of the above equation is positive. Using the Routh array,

Conditions for Stability

The important constraint is Kc<8. Any Kc ³8 will cause instability.

Flowchart for performing a stability analysis.