Anis Atikah Ahmad anisatikah@unimap.edu.my CHAPTER 11: Dynamic Behaviour & Stability of Closed-Loop Control Systems Anis Atikah Ahmad anisatikah@unimap.edu.my

Outline Block Diagram Representation Closed-Loop Transfer Function Closed-Loop Response of Simple Control Systems Stability of Closed-Loop Control Systems

Block Diagram Representation Figure 11.8: Standard block diagram of a feedback control system

Block Diagram Representation disturbance variable controlled variable controller output error signal manipulated variable measured value of Y Figure 11.8: Standard block diagram of a feedback control system

Block Diagram Representation Ysp = Ysp  set point internal set point (used by the controller) Yu = Yd = Gc = Gv = change in Y due to U change in Y due to D controller transfer function transfer function for final control element (including KIP, if required) Gp = process transfer function Gd = disturbance transfer function Gm = transfer function for measuring element and transmitter Km = steady-state gain for Gm

Block Diagram Representation Figure 11.9 : Alternative form of the standard block diagram of a feedback control system.

Block Diagram Representation Example: Control Objective : to regulate the tank composition, x, by adjusting the mass flow rate w2. Disturbance MV Flow rate w1 is assumed to be constant & the system is initially operating at the nominal steady rate CV Figure 11.1 Composition control system for a stirred-tank blending process.

 X1 s   W2 s  τs 1   τs 1   1 x Block Diagram Representation 1. Process The approximate dynamic model of a stirred- tank blending system was developed (Example 4.4):  K   K  X s    X1 s   W2 s 1 2 (11-1)  τs 1   τs 1  where,  1 x   Vρ , K  w1 , and K (11-2) 1 2 w w w

Block Diagram of the process Figure 11.2: Block diagram of the process

2. Sensor-Transmitter (Analyzer) We assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function: X m s Km  X s τ s 1 (11-3) m [mass fraction] [mA] Figure 11.2: Block diagram of the process

3. Controller Suppose that an electronic proportional plus integral controller is used. The controller transfer function is (See Chapter 8): where Ps and E(s) are the Laplace transforms of the controller output p(t) and error signal e(t). Note that pand e are electrical signals that have units of mA, while Kc is dimensionless.

(11-6) (11-5) xsp t   Km xsp t  (11-7) 3. Controller The error signal is expressed as: et   xsp t   xm t  or after taking Laplace transforms, E s  X sp s  X m s The symbol xsp denotes the internal set-point composition expressed as an equivalent electrical current signal. This signal is used internally by the controller xsp t  is related to the actual composition set point by the composition sensor-transmitter gain Km: (11-5) (11-6) xsp t   Km xsp t  (11-7)

xsp t   Km xsp t  (11-7) X sp s  K (11-8) X  s 3. Controller xsp t   Km xsp t  (11-7) X sp s  K Thus, (11-8) X  s m sp Fig 11.4: Block diagram for the controller

4. Current to Pressure (I/P) Transducer Because transducers are usually designed to have linear characteristics and negligible (fast) dynamics, we assume that the transducer transfer function merely consists of a steady-state gain, KIP [mA] [psi] Figure 11.5 Block diagram for the I/P transducer.

W2 s Kv  (11-10) Ps τ s 1 5. Control Valve 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 usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady state. Thus, we assume that the control valve can be modelled as W2 s Kv  (11-10) Ps τ s 1 t v

5. Control Valve The block diagram of a control valve is shown in Fig. 11.16 [psi] [kg/min]

Block diagram for the entire blending process Combining the individual block diagram: Figure 11.7 : Block diagram for the entire blending process (composition control system)

Closed Loop Transfer Function 1. Block diagram reduction In deriving closed-loop transfer functions, it is often convenient to combine several blocks into a single block. For example, consider the three blocks in series in Fig. 11.10. The block diagram indicates the following relations: Figure 11.10 Three blocks in series.

Equivalent block diagram The block diagram indicates the following relations: X1  G1U X 2  G2 X1 X3 G3 X 2 By successive substitution, X3 G3G2G1U or X3  GU where G = G3G2G1 Equivalent block diagram

Closed Loop Transfer Function 2. Set-Point Changes The closed-loop system behavior for set-point changes is also referred to as the servomechanism (servo) problem in the control literature. We assume for this case that no disturbance change occurs and thus D = 0. From Fig. 11.8, it follows that: Combining gives Y  GpU Y  Yd  Yu Yd  Gd D  0 (because D  0) Yu  GpU (11-14) (11-15) (11-16)

Block Diagram Figure 11.8: Standard block diagram of a feedback control system

Closed Loop Transfer Function 2. Set-Point Changes Figure 11.8 also indicates the following input/output relations for the individual blocks: U  Gv P P  Gc E E  Ysp  Ym Ysp  KmῩsp Ym  GmY (11-18) (11-19) (11-20) (11-21) (11-22)

Closed Loop Transfer Function 2. Set-Point Changes Combining the above equations gives Y  GpGv P  GpGvGc E  GpGvGc Ysp Ym   GpGvGc KmYsp GmY  Rearranging gives the desired closed-loop transfer function, (11-23) (11-24) (11-25) KmGcGvGp Y  (11-26) 1 GcGvGpGm Ysp

Closed Loop Transfer Function 3. Disturbance Changes Now consider the case of disturbance changes, which is also referred to as the regulator problem since the process is to be regulated at a constant set point From Fig. 11.8, Substituting (11-18) through (11-22) gives, Y  Gd D  GpU Because Ysp = 0 we can arrange (11-28) to give the closed-loop transfer function for disturbance changes: Y  Yd Yu  Gd D  GpU (11-27)  GdD  GpGvGc(KmYsp GmY  (11.28) Y G  d (11-29) D 1 GcGvGpGm

Closed Loop Transfer Function 3. Disturbance Changes A comparison of Eqs. 11-26 and 11-29 indicates that both closed-loop transfer functions have the same denominator, 1 + GcGvGpGm. The denominator is often written as 1 + GOL where GOL is the open-loop transfer function; GOL ΔGcGvGpGm KmGcGvGp Y Servo Problem (11-26)  Ysp 1 GcGvGpGm Y G Regulator Problem  d (11-29) D 1 GcGvGpGm

Closed Loop Transfer Function Suppose that D ≠ 0 and Ysp ≠ 0, as would be the case if a disturbance occurs during a set-point change. To analyze this situation, we rearrange Eq. 11-28 and substitute the definition of GOL to obtain: Thus, the response to simultaneous disturbance variable and set-point changes is merely the sum of the individual responses, as can be seen by comparing Eqs. 11-26, 11-29, and 11-30. Gd KmGcGvGp D  Y Y  (11-30) sp 1 G 1 G OL OL

Closed Loop Transfer Function 4. General Expression for Feedback Control System Closed-loop transfer functions for more complicated block diagrams can be written in the general form: Z  f  (11-31) Zi 1 e where Z is the output variable or any internal variable within the control loop Zi is an input variable (e.g., Ysp or D)  f = product of the transfer functions in the forward path from Zi to Z = product of every transfer function in the feedback loop e

Example 11.1 Find the closed-loop transfer function Y/Ysp for the complex control system in Figure 11.12. Notice that this block diagram has two feedback loops and two disturbance variables feedback loops and two disturbance variables. Figure 11.12 Complex control system.

Solution Using the general rule in (11-31), we first reduce the inner loop to a single block as shown in Fig. 11.13 Figure 11.13 Block diagram for reduced system.

Solution Figure 11.13 Block diagram for reduced system. To solve the servo problem, set D1 = D2 = 0 Because Fig. 11.13 contains a single feedback loop, use (11-31) to obtain Fig. 11.14a. Fig. 11.13 Figure 11.13 Block diagram for reduced system. Fig. 11.14

Km1Gc1Gc2G1G2G3 Ysp 1 Gc2G1Gm2  Gc1G2G3Gm1Gc2G1 Y  Solution The final block diagram is shown in Fig. 11.14b with Y/Ysp = Km1G5. Substitution for G4 and G5 gives the desired closed-loop transfer function: Figure 11.14b: Final block diagrams for Example 11.1. Y Km1Gc1Gc2G1G2G3  Ysp 1 Gc2G1Gm2  Gc1G2G3Gm1Gc2G1

Closed-Loop Response of Simple Control Systems Consider the liquid-level control system shown in Fig. 11.15. Figure 11.15 Liquid-level control system. Figure 11.15 Liquid-level control system.

Closed-Loop Response of Simple Control Systems A second inlet flow rate q1 is the disturbance variable. Assume: The liquid density  and the cross-sectional area of the tank A are constant. The flow-head relation is linear, q3 = h/R. The level transmitter, I/P transducer, and control valve have negligible dynamics. An electronic controller with input and output in% is used (full scale = 100%).

Closed-Loop Response of Simple Control Systems Derivation of the process and disturbance transfer functions directly follows Example 4.4. Consider the unsteady-state mass balance for the tank contents: Substituting the flow-head relation, q3 = h/R, and introducing deviation variables gives dh ρA  ρq1  ρq2  ρq3 dt (11-32) dh h A  q1  q2  dt R (11-33)

Closed-Loop Response of Simple Control Systems Thus, we obtain the transfer functions where Kp = R and τ = RA. Note that Gp(s) and Gd(s) are identical because q1 and q2 are both inlet flow rates and thus have the same effect on h. H s  G s  K p (11-34) Q2 s p τs 1 K p H s  G s  (11-35) Q1 s d τs 1

Closed-Loop Response of Simple Control Systems Figure 11.16: Block diagram for level control system.

Kc Kv K p Km / τs 1 H s  H  s 1 K K K K / τs 1 Closed-Loop Response of Simple Control Systems 1. Proportional Control and Set-Point Changes If a proportional controller is used, then Gc(s) =Kc. From Fig. 11.16, it follows that the closed-loop transfer function for set-point changes is given by H s Kc Kv K p Km / τs 1  H  s 1 K K K K (11-36) / τs 1 sp c v p m Recall: For a set point change;

KOL K1  1 K H s K1  Hsp s τ1s 1 where τ τ  1 K Closed-Loop Response of Simple Control Systems 1. Proportional Control and Set-Point Changes This relation can be rearranged in the standard form for a first-order transfer function, H s  K1 (11-37) Hsp s τ1s 1 KOL (11-38) where K1  1 K OL τ τ  (11-39) 1 1 K OL

Unit step change of magnitude M in set point Closed-Loop Response of Simple Control Systems 1. Proportional Control and Set-Point Changes From Eq. 11-37 it follows that the closed- loop response to a unit step change of magnitude M in set point is given by, t / τ1     h t  K1M 1 e (11-41) Change to time domain form K1 M H s = X τ1s 1 s Unit step change of magnitude M in set point

Closed-Loop Response of Simple Control Systems 1 Closed-Loop Response of Simple Control Systems 1. Proportional Control and Set-Point Changes Note that a steady-state error or offset exists because the new steady-state value is K1M rather than the desired value of M. Figure 11.17: Step response for proportional control (set-point change).

  h'  offset Δ M offset  M  K1M  1 K (11-43) Closed-Loop Response of Simple Control Systems 1. Proportional Control and Set-Point Changes The offset is defined as For a step change of magnitude M in set point, h’sp(∞). From (11-41), it is clear that h’(∞)=K1M. Substituting these values and (11-38) into (11-42) gives (11-38)   h'  offset Δ (11-42) h' sp M offset  M  K1M  1 K (11-43) OL

Closed-Loop Response of Simple Control Systems 2 Closed-Loop Response of Simple Control Systems 2. Proportional Control and Disturbance Changes The closed-loop transfer function for disturbance changes with proportional control (Gc=Kc) is: Rearranging gives, where Comparing with setpoint changes (11.37)-(11-39), both are first order and have same time constant, but have different steady state gains. Recall: For a disturbance change;

Closed-Loop Response of Simple Control Systems 2 Closed-Loop Response of Simple Control Systems 2. Proportional Control and Disturbance Changes The closed loop response to a step change in disturbance of magnitude M is given by: Change to time domain form:

Equal to 0 because this is a regulator problem (disturbance changes) Closed-Loop Response of Simple Control Systems 2. Proportional Control and Disturbance Changes The offset can be determined as follows: Offset = h'   h'  offset Δ (11-42) sp Equal to 0 because this is a regulator problem (disturbance changes)

Closed-Loop Response of Simple Control Systems 3 Closed-Loop Response of Simple Control Systems 3. PI Control and Disturbance Changes For PI control, Gc(s) = Kc (1+1/τIs) The closed-loop transfer function for disturbance changes can be derived as; Clearing the denominator terms gives;

Closed-Loop Response of Simple Control Systems 3 Closed-Loop Response of Simple Control Systems 3. PI Control and Disturbance Changes Further rearrangement in standard form of second-order transfer function; Where;

For a unit step change in disturbance, Closed-Loop Response of Simple Control Systems 3. PI Control and Disturbance Changes For a unit step change in disturbance, For 0<ζ<1, the response is a damped oscillation that can be described by; h’(∞)= 0 Thus, the addition of integral action eliminates offset for a step change in disturbance

Stability of Closed Loop Control Systems 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. Bounded input: input variable that stays within upper and lower limits for all values of time. unbounded & unstable

Stability of Closed Loop Control Systems General Stability Criterion: The feedback control system is stable if and only if all roots of characteristic equation are negative or have negative real parts. Otherwise, the system is unstable. Characteristic equation: 1+ GOL

Example 11.8 Consider a process, that is open-loop unstable. If Gv= Gm = 1, determine whether proportional controller can stabilize the closed loop system. The characteristic equation for this system is: The root, s = 1+ 0.2 Kc

Example 11.8 In order for this system to be stable; s must have a negative value (s<0), thus Kc < -5

Stability of Closed Loop Control Systems Routh Stability Criterion The Routh stability criterion is based on a characteristic equation that has the form of: All of the coefficients in the characteristic equation must be positive. If any coefficient is negative or zero, then at least one root of the characteristic equation lies on the right of/on the imaginary axis, and the system is unstable.

Stability of Closed Loop Control Systems If all the coefficients are positive, we next construct the following Routh array: Row 1 an an-2 an-4 … 2 an-1 an-3 an-5 3 b1 b2 b3 4 c1 c2 . n + 1 z1

Stability of Closed Loop Control Systems Routh Stability Criterion: A necessary and sufficient conditions 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.

Example 11.10 Find the values of controller gain, Kc that make the feedback control system with the following transfer function stable:

Example 11.10-Solution First, find the characteristic equation, 1+GOL Expanding and make it equal to zero, All of the coefficient is positive. Next, construct the Routh array.

Example 11.10-Solution The Routh array is: In order for this system to be stable, b1 and c1 must be positive, Thus, the system will be stable if 10 8 17 1+Kc b1 b2 c1

Example 11.11 Consider a feedback control system with Gc= Kc, Gv= 2, Gm =0.25, and Gp = 4e-s/(5s+1) The characteristic equation is 1+5s+2Kce-s=0 Because this characteristic equation is not a polynomial s,the Routh criterion is not directly applicable. However, if a polynomial approximation to e-s is introduced, such a Padé approximation, then the Routh criterion can be used to determine approximate stability limits. For simplicity, use the 1/1 Padé approximation;

Example 11.11 Consider a feedback control system with Gc= Kc, Gv= 2, Gm =0.25, and Gp = 4e-s/(5s+1) The characteristic equation is 1+5s+2Kce-s=0 Because this characteristic equation is not a polynomial s,the Routh criterion is not directly applicable. However, if a polynomial approximation to e-s is introduced, such a Padé approximation, then the Routh criterion can be used to determine approximate stability limits. For simplicity, use the 1/1 Padé approximation; and determine the stability limits for the controller gain.

Example 11.11-Solution The characteristic becomes Rearranging; The Routh array: 2.5 1+2Kc 5.5-Kc b1