Proportional control involves the selection of a DC gain value that corresponds to a time response satisfying design specifications. As in s-domain design, a satisfactory time response is obtained by tuning the gain to select a dominant closed-loop pair in the appropriate region of the complex plane. Analytical design is possible for low-order systems but is more difficult than its analog counterpart. The following example illustrates the design of proportional digital controllers.
example 6.3
Design a proportional controller for the digital system described in Example 6.2 with a sampling period T = 0.1 s to obtain
1. A damped natural frequency of 5 rad/s 2. A time constant of 0.5 s
3. A damping ratio of 0.7
solution
After some preliminary calculations, the design results can be easily obtained using the rlocus command of MATLAB. The following calculations, together with the information pro- vided by a cursor command, allow us to determine the desired closed-loop pole locations:
1. The angle of the pole is wd T = 5 × 0.1 = 0.5 rad or 28.65º.
2. The reciprocal of the time constant is zwn = 1/0.5 = 2 rad/s. This yields a pole mag- nitude of e−zωnT=0 82. .
3. The damping ratio given can be used directly to locate the desired pole.
Using MATLAB, we obtain the results shown in Table 6.4. The corresponding sampled step response plots obtained using the command step (MATLAB) are shown in Figure 6.8.
As expected, the higher gain designs are associated with a low damping ratio and a more oscillatory response.
Table 6.4 results can also be obtained analytically using the characteristic equation for the complex conjugate poles of (6.4). The system’s closed-loop characteristic equation is
z2−1 5. z+ +K 0 5. =z2−2cos(ωdT e) −zωnT z e+ −2zωnT
table 6.4 Proportional Control Design Results
Design gain z wn rad/s
(a) 0.23 0.3 5.24
(b) 0.17 0.4 4.60
(c) 0.10 0.7 3.63
FiguRe 6.8
Time response for the designs of Table 6.4: (a) , (b) +, (c) *.
15 20 25 30 35
Time s
Output
0 5 10
0 0.2 0.4 0.6 1.4 1.2 1 0.8
Equating coefficients gives the two equations
z T e
z K e
d T
T n
n 1
0 2
1 5 2 0 5
: . cos
: .
= ( )
+ =
−
−
ω zω
zω
1. From the z1 equation,
zωn=T ωdT
( )
= ( ) =
1 1 5
2 10 1 5
2 0 5 1 571
ln .
cos ln .
cos . .
In addition,
ωd2=ωn2(1−z2)=25 Hence, we obtain the ratio
ω zω
z z
d n 2
2
2
2 2
1 25
1 571
( ) =
− =
( . )
This gives a damping ratio z = 0.3 and an undamped natural frequency wn = 5.24 rad/s. Finally, the z0 equation gives a gain
K =e−2zωnT−0 5. =e− ×2 1 571 0 1. × . −0 5 0 23. = . 2. From (6.11) and the z1 equation, we obtain
zω τ
ω zω
n
d
T
T
e n e
= = =
=
− = −
1 1
0 5 2
1 1 5
2 10 0 75
1 1 0 2
.
cos .
cos . .
rad s
(( )=4 127. rad s
Solving for z gives
ω zω
z z
d n 2
2
2 2
2 2
1 4 127
( ) = 2
− =( . )
which gives a damping ratio z = 0.436 and an undamped natural frequency wn = 4.586 rad/s. The gain for this design is
K =e−2zωnT−0 5. =e− × ×2 2 0 1. −0 5 0 17. = .
3. For a damping ratio of 0.7, the z1 equation obtained by equating coefficients remains nonlinear and is difficult to solve analytically. The equation now becomes
1 5 2. = cos .(0 0714ωn)e−0 07. ωn
The equation can be solved numerically by trial and error with a calculator to obtain the undamped natural frequency wn = 3.63 rad/s. The gain for this design is
K=e−0 14 3 63. × . −0 5 0 10. = .
This controller can also be designed graphically by drawing the root locus and a segment of the constant z spiral and finding their intersection. But the results obtained graphically are often very approximate, and the solution is difficult for all but a few simple root loci.
example 6.4
Consider the vehicle position control system of Example 3.3 with the transfer function G s( )= s s
+
( )
1 5
Design a proportional controller for the unity feedback digital control system with analog process and a sampling period T = 0.04 s to obtain
1. A steady-state error of 10% due to a ramp input 2. A damping ratio of 0.7
solution
The analog transfer function together with a DAC and ADC has the z-transfer function
G z z
z z
ZAS( )= × ( + )
( − )( − )
7 4923 10− 0 9355
1 0 8187
. 4 .
. and the closed-loop characteristic equation is
1 2 1 8187 7 4923 10 4 0 8187 7 009 104
2
+ ( )= −( − × ) + − ×
=
− −
KG z z K z K
z
ZAS . . . .
−−2cos(ωdT e) −zωnTz e+ −2zωnT
The equation involves three parameters z, wn, and K. As in Example 6.3, equating coeffi- cients yields two equations that we can use to evaluate two unknowns. The third parameter must be obtained from a design specification.
1. The system is type 1, and the velocity error constant is
K T
z
z KG z K
K
v
z
= − ( )
= × ( + )
( )( − )
=
=
−
1 1
7 4923 10 1 0 9355 0 04 1 0 8187
1
. 4 .
. .
55
This is identical to the velocity error constant for the analog proportional control system. In both cases, a steady-state error due to ramp of 10% is achieved with
K K
v e 5
100 100
10 10
= =
∞
= =
( )%
Hence, the required gain is K = 50.
2. As in Example 6.3, the analytical solution for constant z is difficult. But the design results are easily obtained using the root locus cursor command of any CAD program.
As shown in Figure 6.9, moving the cursor to the z = 0.7 contour yields a gain value of approximately 11.7
FiguRe 6.9
Root locus for the constant z design.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Real Axis
Imaginary Axis
0 0.5 0.6
0.1 0.2 0.3 0.4 0.7 0.8 0.9
.3 31.4
23.6
15.7
7.85 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System: gd Gain: 11.7 Pole: 0.904 + 0.0889i Damping: 0.699 Overshoot (%): 4.63 Frequency (rad/sec): 3.42
FiguRe 6.10
Root locus for K = 50.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Real Axis
Imaginary Axis
0 0.5 0.6
0.1 0.2 0.3 0.4 0.7 0.8 0.9
.3 31.4
23.6
15.7
7.85 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System: gd Gain: 50 Pole: 0.89 + 0.246i Damping: 0.284 Overshoot (%): 39.4 Frequency (rad/sec): 7.09
System: gd Gain: 268 Pole: 0.807 + 0.592i Damping: –0.000758 Overshoot (%): 100 Frequency (rad/sec): 15.8
The root locus of Figure 6.10 shows that the critical gain for the system Kcr is approxi- mately 268, and the system is therefore stable at the desired gain and can meet the design specifications for both (1) and (2). However, other design criteria, such as the damping ratio and the undamped natural frequency, should be checked. Their values can be obtained using a CAD cursor command or by equating characteristic equation coefficients as in Example 6.3. For the gain of 50 selected in (1), the root locus of Figure 6.10 and the cursor give a damping ratio of 0.28. This corresponds to the highly oscillatory response of Figure 6.11, which is likely to be unacceptable in practice. For the gain of 11.7 selected in (2), the steady-state error is 42.7% due to a unit ramp. It is therefore clear that to obtain the steady-state error specified together with an acceptable transient response, proportional control is inadequate.