Chapter 11

Dynamic Behavior and Stability of
Closed-Loop Control Systems
• In this chapter we consider the dynamic behavior of processes that are operated using feedback control.
• This combination of the process, the feedback controller, and the instrumentation is referred to as a feedback control loop or a closed-loop system.

Block Diagram Representation
To illustrate the development of a block diagram, we return to a previous example, the stirred-tank blending process considered in earlier chapters.
1

Chapter 11
Figure 11.1 Composition control system for a stirred-tank blending process.
2

Chapter 11

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.

Process
In section 4.3 the approximate dynamic model of a stirred-tank blending system was developed:

K
K
′ ( s ) = ⎛ 1 ⎞ X1 ( s ) + ⎛ 2 ⎞W2 ( s )
′
X
⎜
⎟
⎜
⎟ ′
⎝ τs + 1 ⎠
⎝ τs + 1 ⎠

(11-1)

1− x
K2 = w (11-2)

where

Vρ τ= , w w1
K1 = , and w 3

Chapter 11

Figure 11.2 Block diagram of the process.

4

Chapter 11

Composition Sensor-Transmitter (Analyzer)
We assume that the dynamic behavior of the composition sensortransmitter can be approximated by a first-order transfer function:
′
Xm (s)
Km
=
(11-3)
X ′ ( s ) τm s + 1

Controller
Suppose that an electronic proportional plus integral controller is used. From Chapter 8, the controller transfer function is
P′ ( s )

⎛
1 ⎞
= Kc ⎜1 +
⎟
E (s) τI s ⎠
⎝

(11-4)

where P′ ( s ) and E(s) are the Laplace transforms of the controller output p′ ( t ) and the error signal e(t). Note that p′ and e are electrical signals that have units of mA, while Kc is dimensionless.
The error signal is expressed as
5

′
′
e ( t ) = xsp ( t ) − xm ( t )

(11-5)

or after taking Laplace transforms,

Chapter 11

′
′
E ( s ) = X sp ( s ) − X m ( s )

(11-6)

The symbol xsp ( t ) denotes the internal set-point composition
′
expressed as an equivalent electrical current signal. This signal
′
is used internally by the controller. xsp ( t ) is related to the actual
′
composition set point xsp ( t ) by the composition sensortransmitter gain Km:
′
′ xsp ( t ) = K m xsp ( t )
Thus

′
X sp ( s )
′
X sp ( s )

= Km

(11-7)

(11-8)
6

Chapter 11
Figure 11.3 Block diagram for the composition sensortransmitter (analyzer).

7

Chapter 11

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:
Pt′( s )
= K IP
(11-9)
P′ ( s )

Control Valve
As 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 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 modeled as
W2′ ( s )
Kv
=
Pt′( s ) τv s + 1

(11-10)
8

Chapter 11

Figure 11.5 Block diagram for the I/P transducer.

Figure 11.6 Block diagram for the control valve.

9

Chapter 11
Figure 11.7 Block diagram for the entire blending process composition control system.
10

Closed-Loop Transfer Functions

Chapter 11

The block diagrams considered so far have been specifically developed for the stirred-tank blending system. The more general block diagram in Fig. 11.8 contains the standard notation:
Y = controlled variable
U = manipulated variable
D = disturbance variable (also referred to as load variable) P = controller output
E = error signal
Ym = measured value of Y
Ysp = set point
Ysp = internal set point (used by the controller)
11

Chapter 11

Figure 11.8 Standard block diagram of a feedback control system.

12

Yu = change in Y due to U
Yd = change in Y due to D

Chapter 11

Gc = controller transfer function
Gv = 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

13

Block Diagram Reduction

Chapter 11

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:
X1 = G1U
X 2 = G2 X1

(11-11)

X 3 = G3 X 2

By successive substitution,
X 3 = G3G2G1U

(11-12)

X 3 = GU

(11-13)

or

where G

G3G2G1.
14

Chapter 11

Figure 11.10 Three blocks in series.

Figure 11.11 Equivalent block diagram.

15

Set-Point Changes

Chapter 11

Next we derive the closed-loop transfer function for 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.
Y = Yd + Yu

(11-14)

Yd = Gd D = 0 (because D = 0)

(11-15)

Yu = G pU

(11-16)

Y = G pU

(11-17)

Combining gives

16

Figure 11.8 also indicates the following input/output relations for the individual blocks:
(11-18)

P = Gc E

Chapter 11

U = Gv P

(11-19)

E = Ysp − Ym

(11-20)

Ysp = K mYsp

(11-21)

Ym = GmY

(11-22)

Combining the above equations gives
Y = G p Gv P = G p Gv Gc E

(11-23)

= G p GvGc Ysp − Ym

(11-24)

(
)
= G p GvGc ( K mYsp − GmY )

(11-25)
17

Rearranging gives the desired closed-loop transfer function,

Chapter 11

K mGc GvG p
Y
=
Ysp 1 + GcGvG p Gm

(11-26)

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,
Y = Yd + Yu = Gd D + G pU

(11-27)

Substituting (11-18) through (11-22) gives

(

Y = Gd D + G pU = Gd D + G p Gv Gc K mYsp − GmY

)

(11-28)
18

Because Ysp = 0 we can arrange (11-28) to give the closed-loop transfer function for disturbance changes:

Chapter 11

Gd
Y
=
D 1 + GcGvG p Gm

(11-29)

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 GcGv G p Gm .
At different points in the above derivations, we assumed that
D = 0 or Ysp = 0, that is, that one of the two inputs was constant.
But 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
19

Chapter 11

K mGc GvG p
Gd
Y=
D+
Ysp
1 + GOL
1 + GOL

(11-30)

Thus, the response to simultaneous disturbance variable and setpoint changes is merely the sum of the individual responses, as can be seen by comparing Eqs. 11-26, 11-29, and 11-30.
This result is a consequence of the Superposition Principle for linear systems.

20

General Expression for Feedback Control Systems

Chapter 11

Closed-loop transfer functions for more complicated block diagrams can be written in the general form:
Πf
Z
=
Zi 1 + Π e

(11-31)

where:
Z is the output variable or any internal variable within the control loop
Zi is an input variable (e.g., Ysp or D)
Π f = product of the transfer functions in the forward path from
Zi to Z
Π e = product of every transfer function in the feedback loop
21

Chapter 11

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. This configuration arises when the cascade control scheme of Chapter
16 is employed.

Figure 11.12 Complex control system.

22

Chapter 11

Figure 11.13 Block diagram for reduced system.

23

Chapter 11

Figure 11.14 Final block diagrams for Example 11.1.
24

Chapter 11

Solution

Using the general rule in (11-31), we first reduce the inner loop to a single block as shown in Fig. 11.13. 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. 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:
K m1Gc1Gc 2G1G2G3
Y
=
Ysp 1 + Gc 2G1Gm 2 + Gc1G2G3Gm1Gc 2G1

Closed-Loop Responses of Simple
Control Systems
In this section we consider the dynamic behavior of several elementary control problems for disturbance variable and setpoint changes.

25

Chapter 11

The transient responses can be determined in a straightforward manner if the closed-loop transfer functions are available.
Consider the liquid-level control system shown in Fig. 11.15. The liquid level is measured and the level transmitter (LT) output is sent to a feedback controller (LC) that controls liquid level by adjusting volumetric flow rate q2. A second inlet flow rate q1 is the disturbance variable. Assume:
1. The liquid density ρ and the cross-sectional area of the tank A are constant.
2. The flow-head relation is linear, q3 = h/R.
3. The level transmitter, I/P transducer, and control valve have negligible dynamics.
4. An electronic controller with input and output in % is used (full scale = 100%).
26

Chapter 11

Figure 11.15 Liquid-level control system.

27

Chapter 11

Derivation of the process and disturbance transfer functions directly follows Example 4.4. Consider the unsteady-state mass balance for the tank contents: dh ρA = ρq1 + ρq2 − ρq3 dt (11-32)

Substituting the flow-head relation, q3 = h/R, and introducing deviation variables gives dh′ h′
′ ′
A
= q1 + q2 − dt R

(11-33)

Thus, we obtain the transfer functions
Kp
H ′( s )
= Gp ( s) =
′
τs + 1
Q2 ( s )

(11-34)

28

Chapter 11

Kp
H ′( s)
= Gd ( s ) =
′
τs + 1
Q1 ( s )

(11-35)

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.

Proportional Control and Set-Point Changes
If a proportional controller is used, then Gc(s) = Kc. From Fig.
11.6 and the material in the previous section, it follows that the closed-loop transfer function for set-point changes is given by
K c K v K p K m / ( τs + 1)
H ′( s )
=
′
H sp ( s ) 1 + K c K v K p K m / ( τs + 1)

(11-36)

29

Chapter 11

Figure 11.16 Block diagram for level control system.

30

This relation can be rearranged in the standard form for a firstorder transfer function,

Chapter 11

H ′( s )
K1
=
′
H sp ( s ) τ1s + 1

(11-37)

KOL
K1 =
1 + KOL

(11-38)

τ τ1 =
1 + KOL

(11-39)

where:

and the open-loop gain KOL is given by
KOL = K c K v K p K m

(11-40)

31

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

(

Chapter 11

h′ ( t ) = K1M 1 − e−t / τ1

)

(11-41)

This response is shown in Fig. 11.17. Note that a steady-state error or offset exists because the new steady-state value is K1M rather than the desired value of M. The offset is defined as offset ′ hsp ( ∞ ) − h′ ( ∞ )

(11-42)

′
For a step change of magnitude M in set point, hsp ( ∞ ) = M .
From (11-41), it is clear that h′ ( ∞ ) = K1M . Substituting these values and (11-38) into (11-42) gives offset = M − K1M =

M
1 + KOL

(11-43)
32

Chapter 11

Figure 11.17 Step response for proportional control (setpoint change).

33

Proportional Control and Disturbance Changes

Chapter 11

From Fig. 11.16 and Eq. 11-29 the closed-loop transfer function for disturbance changes with proportional control is
K p / ( τs + 1)
H ′( s)
=
′
Q1 ( s ) 1 + KOL / ( τs + 1)

(11-53)

H ′( s)
K2
=
′
Q1 ( s ) τ1s + 1

(11-54)

Rearranging gives

where τ1 is defined in (11-39) and K2 is given by
K2 =

Kp
1 + KOL

(11-55)

34

• A comparison of (11-54) and (11-37) indicates that both closedloop transfer functions are first-order and have the same time constant. Chapter 11

• However, the steady-state gains, K1 and K2, are different.
• From Eq. 11-54 it follows that the closed-loop response to a step change in disturbance of magnitude M is given by

(

h′ ( t ) = K 2 M 1 − e−t / τ1

)

(11-56)

′
The offset can be determined from Eq. 11-56. Now hsp ( ∞ ) = 0 since we are considering disturbance changes and h′ ( ∞ ) = K 2 M for a step change of magnitude M.
Thus,
offset = 0 − h′ ( ∞ ) = − K 2 M = −

K pM
1 + KOL

(11-57)
35

Chapter 11

Figure 11.18 Set-point responses for Example 11.2.

36

Chapter 11

Figure 11.19 Load responses for Example 11.3.

37

Chapter 11

PI Control and Disturbance Changes
For PI control, Gc ( s ) = K c (1 + 1/ τ I s ) . The closed-loop transfer function for disturbance changes can then be derived from Fig.
11.16:
K p / ( τs + 1)
H ′( s)
=
(11-58)
′
Q1 ( s ) 1 + KOL (1 + 1/ τ I s ) / ( τs + 1)
Clearing terms in the denominator gives
K pτ I s
H ′( s)
=
′
Q1 ( s ) τ I s ( τs + 1) + KOL τ I s

(11-59)

Further rearrangement allows the denominator to be placed in the standard form for a second-order transfer function:
H ′( s)
K3 s
= 2 2
′
Q1 ( s ) τ3 s + 2ζ 3 τ3 s + 1

(11-60)
38

where
(11-61)

1 ⎛ 1 + KOL ⎞ τ I ζ3 = ⎜
⎟
⎜ K
⎟ τ
2⎝
OL ⎠

Chapter 11

K3 = τ I / K c K v K m

(11-62)

τ3 = ττ I / KOL

(11-63)

For a unit step change in disturbance, Q1 ( s ) = 1/ s , and (11-59)
′
becomes
K3
H ′( s ) = 2 2
(11-64)
τ3 s + 2ζ 3 τ3 s + 1
For 0 < ζ 3 < 1 , the response is a damped oscillation that can be described by h′ ( t ) =

2 e − ζ 3t / τ3 sin ⎡ 1 − ζ 3 t / τ3 ⎤
⎢
⎥
2
⎣
⎦
1 − ζ3

K3 τ3 (11-65)
39

PI Control of an Integrating Process
Consider the liquid-level control system shown in Fig. 11.22. This system differs from the previous example in two ways:

Chapter 11

1. the exit line contains a pump and
2. the manipulated variable is the exit flow rate rather than an inlet flow rate.
In Section 5.3 we saw that a tank with a pump in the exit stream can act as an integrator with respect to flow rate changes because
H ′( s)
1
= Gp ( s) = −
′
Q3 ( s )
As

(11-66)

H ′( s)
1
= Gd ( s ) =
′
Q1 ( s )
As

(11-67)
40

Chapter 11
Figure 11.22 Liquid-level control system with pump in exit line.
41

Chapter 11

If the level transmitter and control valve in Eq. 11.22 have negligible dynamics, the Gm(s) = Km and Gv(s) = Kv. For PI control, Gc ( s ) = K c (1 + 1/ τ I s ) . Substituting these expressions into the closed-loop transfer function for disturbance changes
H ′( s)
Gd
=
′
Q1 ( s ) 1 + GcGv G p Gm

(11-68)

and rearranging gives
H ′( s)
K4s
= 2 2
′
Q1 ( s ) τ 4 s + 2ζ 4 τ 4 s + 1

(11-69)

K 4 = − τ / Kc Kv K m

(11-70)

τ 4 = τ I / KOL

(11-71)

ζ 4 = 0.5 KOL τ I

(11-72)

where

And KOL = KcKvKpKm with Kp = - 1/A.

42

Stability of Closed-Loop Control Systems

Chapter 11

Example 11.4
Consider the feedback control system shown in Fig. 11.8 with the following transfer functions:
Gc = K c
1
G p = Gd =
5s + 1

1
Gv =
2s + 1
1
Gm = s +1

(11-73)
(11-74)

Show that the closed-loop system produces unstable responses if controller gain Kc is too large.
43

Chapter 11
Figure 11.23. Effect of controller gains on closed-loop response to a unit step change in set point (example 11.1).
44

Stability

Chapter 11

• 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.

45

Characteristic Equation

Chapter 11

As a starting point for the stability analysis, consider the block diagram in Fig. 11.8. Using block diagram algebra that was developed earlier in this chapter, we obtain
Y=

K mGcGv G p
1 + GOL

Gd
Ysp +
D
1 + GOL

(11-80)

where GOL is the open-loop transfer function,
GOL = GcGvGpGm. For the moment consider set-point changes only, in which case Eq. 11-80 reduces to the closed-loop transfer function,
K mGcGvG p
Y
=
(11-81)
Ysp
1 + GOL

46

Comparing Eqs. 11-81 and 11-82 indicates that the poles are also the roots of the following equation, which is referred to as the characteristic equation of the closed-loop system:

Chapter 11

1 + GOL = 0

(11-83)

General Stability Criterion. The feedback control system in Fig.
11.8 is stable if and only if all roots of the characteristic equation are negative or have negative real parts. Otherwise, the system is unstable.

Example 11.8
Consider a process, Gp = 0.2/-s + 1), and thus is open-loop unstable. If Gv = Gm = 1, determine whether a proportional controller can stabilize the closed-loop system.

47

Chapter 11

Figure 11.25
Stability regions in the complex plane for roots of the characteristic equation.

48

Chapter 11

Figure 11.26
Contributions of characteristic equation roots to closed-loop response.

49

Solution

The characteristic equation for this system is

Chapter 11

s + 0.2 K c − 1 = 0

(11-92)

Which has the single root, s = -1 + 0.2Kc. Thus, the stability requirement is that Kc < 5. This example illustrates the important fact that feedback control can be used to stabilize a process that is not stable without control.

Routh Stability Criterion
The Routh stability criterion is based on a characteristic equation that has the form an s n + an−1s n−1 + … + a1s + a0 = 0

(11-93)
50

Chapter 11

Routh array:

Row
1
2
3
4

an an-1 b1 c1 n+1

z1

an-2 an-3 b2 c2 an-4 an-5 b3

where: an−1an−2 − an an−3 an−1 (11-94)

an−1an−4 − an an−5 b2 = an−1 (11-95)

b1 =

51

and:

Chapter 11

b1an−3 − an−1b2 c1 = b1 (11-96)

b1an−5 − an−1b3 c2 = b1 (11-97)

Routh Stability Criterion:

A necessary and sufficient condition for all roots of the characteristic equation in Eq. 11-93 to have negative real parts is that all of the elements in the left column of the Routh array are positive.

52

Example 11.9
Determine the stability of a system that has the characteristic equation Chapter 11

s 4 + 5 s 3 + 3s 2 + 1 = 0

(11-98)

Solution

Because the s term is missing, its coefficient is zero. Thus, the system is unstable. Recall that a necessary condition for stability is that all of the coefficients in the characteristic equation must be positive.

53

Example 11.10
Find the values of controller gain Kc that make the feedback control system of Eq. 11.4 stable.

Chapter 11

Solution

From Eq. 11-76, the characteristic equation is
10 s 3 + 17 s 2 + 8s + 1 + K c = 0

(11-99)

All coefficients are positive provided that 1 + Kc > 0 or Kc < -1.
The Routh array is
10
17 b1 c1

8
1 + Kc b2 54

To have a stable system, each element in the left column of the
Routh array must be positive. Element b1 will be positive if
Kc < 7.41/0.588 = 12.6. Similarly, c1 will be positive if Kc > -1.
Thus, we conclude that the system will be stable if

Chapter 11

−1 < K c < 12.6

(11-100)

Direct Substitution Method
• The imaginary axis divides the complex plane into stable and unstable regions for the roots of characteristic equation, as indicated in Fig. 11.26.
• On the imaginary axis, the real part of s is zero, and thus we can write s=jω. Substituting s=jω into the characteristic equation allows us to find a stability limit such as the maximum value of
Kc .
• As the gain Kc is increased, the roots of the characteristic equation cross the imaginary axis when Kc = Kcm.

55

Example 11.12
Use the direct substitution method to determine Kcm for the system with the characteristic equation given by Eq. 11-99.

Chapter 11

Solution

Substitute s = jω and Kc = Kcm into Eq. 11-99:

−10 jω3 − 17ω2 + 8 jω + 1 + K cm = 0 or (1 + K

) (

)

(11-105)

− 17ω2 + j 8ω − 10ω3 = 0 cm 56

Equation 11-105 is satisfied if both the real and imaginary parts are identically zero:

Chapter 11

1 + K cm − 17ω2 = 0

(

)

8ω − 10ω3 = ω 8 − 10ω2 = 0

(11-106a)
(11-106b)

Therefore, ω2 = 0.8 ⇒ ω = ±0.894

(11-107)

and from (11-106a),
K cm = 12.6

57

Root Locus Diagrams
Example 11.13

Chapter 11

Consider a feedback control system that has the open-loop transfer function,
4Kc
GOL ( s ) =
( s + 1)( s + 2 )( s + 3)

(11-108)

Plot the root locus diagram for 0 ≤ K c ≤ 20.
Solution

The characteristic equation is 1 + GOL = 0 or

( s + 1)( s + 2 )( s + 3) + 4 Kc = 0

(11-109)
58

Chapter 11

• The root locus diagram in Fig. 11.27 shows how the three roots of this characteristic equation vary with Kc.
• When Kc = 0, the roots are merely the poles of the open-loop transfer function, -1, -2, and -3.

59

Chapter 11
Figure 11.27 Root locus diagram for third-order system. X denotes an open-loop pole. Dots denote locations of the closedloop poles for different values of Kc. Arrows indicate change of pole locations as Kc increases.
60

Chapter 11

Figure 11.29. Flowchart for performing a stability analysis. 61