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

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

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

4. 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

5. 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

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

7. 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.

8.  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

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

10. 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

11. 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.

12. (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)

13. 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

14. 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.

15. 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

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

17. 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)

18. 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
The block diagram indicates the following relations:
Figure Three blocks in series.

19. 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

20. 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)

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

22. 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)

23. 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

24. 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

25. Closed Loop Transfer Function
3. Disturbance Changes
A comparison of Eqs and 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

26. 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 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 , 11-29, and Gd
KmGcGvGp
D  Y
Y 
(11-30) sp
1 G
1 G OL OL

27. 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

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

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

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

31. Km1Gc1Gc2G1G2G3 Ysp 1 Gc2G1Gm2  Gc1G2G3Gm1Gc2G1 Y  Solution
The final block diagram is shown in Fig b 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

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

33. 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%).

34. 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)

35. 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

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

37. 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 , 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;

38. 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

39. 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 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

40. 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).

41.   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

42. 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;

43. 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:

44. 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)

45. 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;

46. 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;

47. 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

48. 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

49. 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

50. 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 = Kc

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

52. 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.

53. 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

54. 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.

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

56. 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.

57. 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

58. 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;

59. 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.

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