Control Theory and Applications 10.3 SISO System Controls 247
Part A | 10.3
Step response (Fig. 10.17)
y step .t/ D 1 e
t cosh .at/
a
e
t sinh .at/ ;
(10.99)
where D Re fp 1;2 g D D! 0 and a D ! 0
p
2 1.
The Bode plots for various positive values of > 0
(therefore asymptotically stable cases) are given in
Fig. 10.11.
Example 10.3 Linearized AUV Forward Speed Model
In this example, we will analyze the linearized AUV
forward speed model introduced in Example 10.2.
Given d D 20, M D 300, c T D 0:2, ˇ D 50, ˛ D
0:04, u 0 D 1, ˝ D 10, the linearized model is
300P u C 40u D 4˝ ;
P
˝ C 0:8˝ D 50 C 4
and the transfer functions are expressed as follows using (10.67)
H U .s/ D
U.s/
.s/
D
200
.300s C 40/ .s C 0:8/
;
H ˝ .s/ D
˝.s/
.s/
D
50
.s C 0:8/
;
where u.t/ $ U.s/, ˝.t/ $ ˝.s/, .t/ $ .s/. These
models are simulated in MATLAB/Simulink with a unit
step of 0:08 Nm as the input torque. Figure 10.18 shows
the simulated responses based on the nonlinear and linearized models at four different initial conditions.
One can clearly see that the linearized model becomes more accurate as the initial conditions move
toward the point of linearization (u 0 D 1, ˝ 0 D 10).
Also, the propeller response is visually more accurate
with a shorter time constant as compared to the speed
response.
10.3 SISO System Controls
This chapter presents some fundamental techniques
for the control of LTI–SISO systems and basic performance criteria typically used in controller design.
The techniques consist of (a) ON/OFF control; (b) PID
(proportional-integral-derivative) control, and (c) controller design by use of the root locus.
The closed-loop control system given above is encountered in many practical, industrial plants and processes. The objective in introducing feedback control
is to maintain the process response signal y as close
as possible to the reference signal (setpoint) r. Deflection of the process output signal from the reference one
is generated because either there is significant disturbance present, manifested by signal d, or the setpoint
is modified or both. The first problem is referred to as
the regulation problem while the second as the tracking
problem.
In any of these cases, however, the controller (control system) is inserted in order to reduce the error
signal, e D r y, that is, the deviation of the output from
the reference signal, to the lowest possible value; this
can be achieved by appropriately processing the error
signal in order to generate the driving control signal,
that will compensate the effect of the disturbance. The
block diagram in Fig. 10.19 is similar to the one shown
in Fig. 10.2; for the sake of simplicity, however, the
sensor and actuator blocks have been omitted, or equivalently included in the controller and process block,
respectively, and the plant is assumed to be SISO.
In the complex frequency domain, the output signal
is given by the relationship
Y.s/ D G.s/ ŒU.s/ C D.s/
U.s/ D K.s/E.s/ D K.s/ ŒR.s/ Y.s/
)
) Y.s/ D
G.s/K.s/
1 C G.s/K.s/
R.s/
C
G.s/
1 C G.s/K.s/
D.s/ :
(10.100)
The above can be alternatively demonstrated by separately applying the general equation (10.61) for each
one of the signals r and d and setting the other one identically equal to zero.
10.3.1 Performance Criteria
There are three primary criteria for which the performance of a system can be assessed. They are: transient
response, steady-state response, and disturbance rejection response. These criteria are generally used for the
setpoint control objective.
A transient response corresponds to how a system
reacts to a new desired output. The associated criteria
are usually defined in terms of:
Delay time (t d ): the time it takes to reach 30% of the
setpoint level
Part A | 10.3
Step response (Fig. 10.17)
y step .t/ D 1 e
t cosh .at/
a
e
t sinh .at/ ;
(10.99)
where D Re fp 1;2 g D D! 0 and a D ! 0
p
2 1.
The Bode plots for various positive values of > 0
(therefore asymptotically stable cases) are given in
Fig. 10.11.
Example 10.3 Linearized AUV Forward Speed Model
In this example, we will analyze the linearized AUV
forward speed model introduced in Example 10.2.
Given d D 20, M D 300, c T D 0:2, ˇ D 50, ˛ D
0:04, u 0 D 1, ˝ D 10, the linearized model is
300P u C 40u D 4˝ ;
P
˝ C 0:8˝ D 50 C 4
and the transfer functions are expressed as follows using (10.67)
H U .s/ D
U.s/
.s/
D
200
.300s C 40/ .s C 0:8/
;
H ˝ .s/ D
˝.s/
.s/
D
50
.s C 0:8/
;
where u.t/ $ U.s/, ˝.t/ $ ˝.s/, .t/ $ .s/. These
models are simulated in MATLAB/Simulink with a unit
step of 0:08 Nm as the input torque. Figure 10.18 shows
the simulated responses based on the nonlinear and linearized models at four different initial conditions.
One can clearly see that the linearized model becomes more accurate as the initial conditions move
toward the point of linearization (u 0 D 1, ˝ 0 D 10).
Also, the propeller response is visually more accurate
with a shorter time constant as compared to the speed
response.
10.3 SISO System Controls
This chapter presents some fundamental techniques
for the control of LTI–SISO systems and basic performance criteria typically used in controller design.
The techniques consist of (a) ON/OFF control; (b) PID
(proportional-integral-derivative) control, and (c) controller design by use of the root locus.
The closed-loop control system given above is encountered in many practical, industrial plants and processes. The objective in introducing feedback control
is to maintain the process response signal y as close
as possible to the reference signal (setpoint) r. Deflection of the process output signal from the reference one
is generated because either there is significant disturbance present, manifested by signal d, or the setpoint
is modified or both. The first problem is referred to as
the regulation problem while the second as the tracking
problem.
In any of these cases, however, the controller (control system) is inserted in order to reduce the error
signal, e D r y, that is, the deviation of the output from
the reference signal, to the lowest possible value; this
can be achieved by appropriately processing the error
signal in order to generate the driving control signal,
that will compensate the effect of the disturbance. The
block diagram in Fig. 10.19 is similar to the one shown
in Fig. 10.2; for the sake of simplicity, however, the
sensor and actuator blocks have been omitted, or equivalently included in the controller and process block,
respectively, and the plant is assumed to be SISO.
In the complex frequency domain, the output signal
is given by the relationship
Y.s/ D G.s/ ŒU.s/ C D.s/
U.s/ D K.s/E.s/ D K.s/ ŒR.s/ Y.s/
)
) Y.s/ D
G.s/K.s/
1 C G.s/K.s/
R.s/
C
G.s/
1 C G.s/K.s/
D.s/ :
(10.100)
The above can be alternatively demonstrated by separately applying the general equation (10.61) for each
one of the signals r and d and setting the other one identically equal to zero.
10.3.1 Performance Criteria
There are three primary criteria for which the performance of a system can be assessed. They are: transient
response, steady-state response, and disturbance rejection response. These criteria are generally used for the
setpoint control objective.
A transient response corresponds to how a system
reacts to a new desired output. The associated criteria
are usually defined in terms of:
Delay time (t d ): the time it takes to reach 30% of the
setpoint level