3Next, 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:Chapter 11where
5The 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:Chapter 11
6Chapter 11 Current-to-Pressure (I/P) Transducer Control Valve The transducer transfer function merely consists of a steady-state gain KIP:Chapter 11Control 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
7Chapter 11 Composition Sensor-Transmitter (Analyzer) Controller 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.Chapter 11where and E(s) are the Laplace transforms of the controller output and the error signal e(t). Kc is dimensionless.
11Chapter 11 “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 0 and D = 0 (set-point change while disturbancechange is zero)Chapter 11(11-26)Disturbance Changes (“Regulator Problem”)Assume D 0 and Ysp = 0 (constant set-point)(11-29)*Note same denominator for Y/D, Y/Ysp.
18EXAMPLE 1: P.I. control of liquid level Block Diagram:Chapter 11
19Chapter 11 Assumptions 1. q1, varies with time; q2 is constant. 2. Constant density and x-sectional area of tank, A.(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).Chapter 11For these assumptions, the transfer functions are:
20Chapter 11 The closed-loop transfer function is: (11-68) Substitute, (2)Chapter 11Simplify,(3)Characteristic Equation:(4)Recall the standard 2nd Order Transfer Function:(5)
21To 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:Chapter 11Substitute,For 0 < < 1 , closed-loop response is oscillatory. Thusdecreased degree of oscillation by increasing Kc or I (for constantKv, KM, and A).unusual property of PI control of integrating systembetter to use P only
22Stability of Closed-Loop Control Systems Chapter 11
23Chapter 11 Proportional Control of First-Order Process Set-point change:Chapter 11
24Chapter 11 Set-point change = M Offset = See Section 11.3 for tank example
25Chapter 11 Closed-Loop Transfer function approach: First-order behaviorclosed-loop time constant(faster, depends on Kc)
27Chapter 11 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.Chapter 11
28Chapter 11 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).Chapter 11is unspecifiedKc is the controller function, i.e.,
29LetIf Kc = 1,Chapter 11Since 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 ismarginally stable.
30If Kc = 27Since 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.Chapter 11Offset with proportional control (disturbance step-response; D(s) =1/s )
31Therefore, 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:Chapter 11For a unit step load-change and Kc=1,no offset(note 4th order polynomial)
32Chapter 11 PI Control: no offset adjust Kc and I to obtain satisfactory response (roots of equation which is 4th order).Chapter 11PID Control: (pure PID)No offset, adjust Kc, I , D to obtain satisfactory result (requires solving for roots of 4th order characteristic equation).Analysis of roots of characteristic equation is one way toanalyze controller behavior
33Rule 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 theclosed-loop transfer function.Chapter 11The roots (poles) of the characteristic equation (s - pi) determinethe type of response that occurs:Complex roots oscillatory responseAll real roots no oscillations***All roots in left half of complex plane = stable system
34Chapter 11Figure Stability regions in the complex plane for roots of the characteristic equation.
35Chapter 11 Stability Considerations Feedback control can result in oscillatory or even unstable closed-loop responses.Chapter 11Typical behavior (for different values of controller gain, Kc).
36Chapter 11 Roots of 1 + GcGvGpGm (Each test is for different value of Kc)(Note complex roots always occur in pairs)Figure Contributions of characteristic equation roots to closed-loop response.
38Chapter 11 Routh Stability Criterion Characteristic equation (11-93)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:
39Chapter 11For stability, all elements in the first column must be positive.
40Chapter 11 The first two rows of the Routh Array are comprised of the coefficients in the characteristic equation. The elements in theremaining rows are calculated from coefficients by using theformulas:(11-94)Chapter 11(11-95).(11-96)(11-97)(n+1 rows must be constructed; n = order of the characteristic eqn.)
41Chapter 11 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 StabilityThe important constraint is Kc<8. Any Kc 8 will cause instability.
42Figure 11.29 Flowchart for performing a stability analysis. Chapter 11
43Chapter 11 Additional Stability Criteria 1. Bode Stability Criterion Ch can handle time delays2. Nyquist Stability CriterionCh. 14Chapter 11
44Direct Substitution Method Imaginary axis is the dividing line between stable and unstable systems.Substitute s = jw into characteristic equationSolve for Kcm and wc(a) one equation for real part(b) one equation for imaginary partExample (cf. Example 11.11)characteristic equation: s + 2Kce-s = 0 (11-101)set s = jw jw + 2Kce-jw = 01 + 5jw + 2Kc (cos(w) – j sin(w)) = 0Chapter 11
45Chapter 11 Direct Substitution Method (continued) Re: 1 + 2Kc cos w = 0 (1)Im: 5w – 2Kc sin w = 0 (2)solve for Kc in (1) and substitute into (2):Chapter 11Solve for w: wc = rad/min (96.87°/min)from (1) Kcm = 4.25(vs. 5.5 using Pade approximation in Example 11.11)