MINIMUM-ENERGY RIPPLE-FREE DEAD-BEAT CONTROL
OF TYPE-I SECOND-ORDER PLANTS
C. A. Barbargires and C. A. Karybakas
Aristotle University of Thessaloniki
Department of Physics, Section of Electronics & Computers
Thessaloniki, 540 06, Greece (e-mail: barbargires@physics.auth.gr)
ABSTRACT
The design of digital controllers for minimum-energy ripple-free dead-beat control of type-I
second-order plants is considered. Two recently introduced necessary and su¢cient conditions for ripple-free dead-beat response are utilized for the solution of the optimization problem posed, thus leading to analytical expressions for the digital controller
pulse transfer function. It is readily seen that a linear increment in the overall system order results to a
proportional increase in the settling-time and an exponential decrease not only in the energy of the control signal needed to attain the desirable response,
but the maximum control signal, as well.
1.
INTRODUCTION
Dead-beat (DB) control was introduced over four
decades ago by Bergen and Ragazzini [1] and has
been intensively studied since then within the framework of the discrete-time control theory. But for
years past, designers have avoided the DB controller
due to problems with physical realizability and incomplete pole-zero cancellation. However, with the
advent of DSP systems, many of these problems can
now be avoided.
By de…nition, the DB response implies zero error response at the sampling instants after a speci…ed …nite settling-time, irrespective of the intersample response of the system [2]. Nevertheless,
the DB control of certain classes of continuous-time
plants may present problems, because of the remaining inter-sample activity, which is highly undesirable. The term ripple-free (RF) DB control refers
to the elimination of this inter-sample activity, that
is, the accomplishment of zero error response for
all the continuous time after the …nite settling-time
[3]. During the last years, there has been noticed
an extensive research on RF-DB control systems [35], and various schemes have been proposed, aiming
at the application of such modern techniques to the
control of widely used plants [5-8].
In this work, the minimum energy (ME) RF-DB
control of type-I second-order plants is addressed,
using a generalized approach for the speci…c plant
and the selection of the sampling period. The design
of the digital controller through this optimization
procedure is greatly facilitated by taking advantage
of two recently introduced necessary and su¢cient
conditions for RF-DB response [9], stemming from
the study of the control sequence, which is restricted
to be constant, and especially zero for the plant considered here, after the …nite settling-time. Systems
with greater settling-time than the minimum one
required for RF-DB response are optimized with respect to the control sequence, the energy of which is
minimized. For these cases, analytical expressions
are extracted for the controller pulse transfer functions and the control sequence.
2.
DISCRETE-TIME MODEL
The continuous-time model of a type-I secondorder plant (that may represent, for example, a
servo motor) can be mathematically expressed by
the transfer function
G(s) =
K
s(s+a)
(1)
with a > 0 . The discrete-time model of this plant
in cascade with a zero-order hold is
Gh (z) =
A(z¡B)
(z¡1)(z¡C)
with the parameters A, B and C de…ned as
¡
¢
:
A = aK2 aT ¡ 1 + e¡aT
:
:
)e¡aT
C = e¡aT
B = 1¡(1+aT
1¡aT ¡e¡aT ;
(2)
(3)
where T is the sampling period. Obviously, the
discrete-time plant is critically stable, because its
two poles reside at z1 = 1 and z2 = C 2 (0; 1),
and also inversely stable, because its zero resides at
z3 = B 2 (¡1; 0). Since this zero takes negative real
values inside the unit circle of the z-plane, its cancellation by a controller pole will lead to ripple [4].
Thus, for RF response this zero must be present in
the closed-loop pulse transfer function of the system.
3.
RF-DB CONTROL
For DB response, the pulse transfer function of
the closed-loop system must be in the form of a …nite
polynomial in terms of powers in z ¡1
P
¡i
(4)
F (z) = N
i=1 fi z
where the coe¢cients fi , i = 1; 2; : : : ; N , constitute
the impulse response sequence of the system [9]. For
DB response to step inputs these coe¢cients must
additionally satisfy the linear condition
PN
(5)
i=1 fi = 1
which means that the DC gain of the closed-loop system has to be equal to unity, in order that the error
sequence be zero after the …nite settling-time [9]. If
the pulse transfer function of the closed-loop system with the desirable response characteristics has
been computed by some method, the pulse transfer
function of the corresponding digital controller in a
unity feedback scheme is given by [2]
Since the control sequence coe¢cients uk are already
expressed through Eq.(9) as a function of the impulse response coe¢cients fi of the closed-loop system, the optimization of the system’s performance
according to the speci…ed objective function will be
achieved through the minimization of J with respect
to the coe¢cients fi , with the subsidiary constraints
expressed by Eqs.(5) and (10) for these coe¢cients.
This is a problem of static optimization under linear
constraints that can be solved using the Lagrange
method of undetermined multipliers. For this, the
extended function is de…ned as
¶
µN
N
P
P
:
(12)
fi ¡ 1 + ¹
B N¡i fi
I =J +¸
i=1
D(z) =
1
Gh (z)
¢
F (z)
1¡F (z)
(6)
The z-transform of the control signal is easily
shown to be [2]
U (z) = F (z)R(z)=Gh (z)
(7)
and assuming a step input with R(z) = z=(z¡1), the
inversion integral method gives the control sequence
¶
µN
I
1
1 z¡C P
N¡i
fi z
z k¡N dz (8)
uk =
2¼j ¡ A z ¡ B i=1
where ¡ is any closed curve that encloses the isolated
poles of the integrand at z1 = B and z2 = 0 for
k < N . The computation of the above integral leads
to the closed-form relation
( P
k
1
k¡i
B¡C
fi + B¡C
fk+1 ; k < N
i=1 B
uk =
¢
P
N
A
k¸N
B k¡N i=1 B N¡i fi ;
(9)
That is, the condition for RF response is extracted
as a simple linear equation
PN
N¡i
(10)
fi = 0
i=1 B
that makes zero the control sequence for k ¸ N .
First- and second-order overall systems have
been thoroughly studied in the past [7]. The …rstorder overall system constitutes a minimum prototype system for step inputs, but it presents ripple at
the system output. The second-order overall system
is a non-minimum prototype one for step inputs, but
it presents the minimum settling-time for RF-DB response to step inputs.
4.
OPTIMIZATION
The response of the system can be improved by
optimizing the control sequence. This can be accomplished by minimizing the energy of this sequence,
that is, the sum of the squares of the control sequence coe¢cients, by choosing the following cost
function
PN¡1
(11)
J = k=0 u2k
i=1
where ¸ and ¹ are the Lagrange multipliers. The
solution of the minimization problem is achieved by
solving the following set of equations
(
@I
j = 1; 2; : : : ; N
@fj = 0;
(13)
@I
@I
=
=
0
@¸
@¹
The …rst equation of this set can be written [6]
(B ¡ C)
+
j¡1
P
i=1
k
N¡1
P P
B 2k¡j¡i fi +
k=j i=1
B j¡1¡i fi + fj = ¡
N¡1
P
B k¡j fk+1
k=j
¢
¡
A2
¸ + ¹B N¡j
2(B ¡ C)
(14)
and the solution of this linear system gives the impulse response coe¢cients of the closed-loop system
as a function of, among other variables, the Lagrange multipliers ¸ and ¹. Elimination of these
parameters is achieved using Eqs.(5) and (10). As
a result, the coe¢cients fi are …nally expressed as a
function of the variables A, B, C and N, only, and
the pulse transfer function of the digital controller
can be derived using Eq.(6).
5.
DESIGN EXAMPLE
As a speci…c plant, the DC servo motor appearing in [5] is considered with K = ¡59:6 and a = 6:0.
The discrete-time transfer function of the plant in
cascade with a zero-order hold is found using Eq.(2)
Gh (z) = ¡0:06758 ¢
z+0:90490
(z¡1)(z¡0:74082)
(15)
where the sampling period has been taken T =
0:05 sec.
The pulse transfer function of the digital controller for the second-order overall system with the
RF-DB requirement is given from
D2 (z) = ¡7:76838 ¢
z¡0:74082
z+0:47504
(16)
This …rst-order controller cancels only the stable
pole of the plant at z = 0:74082 and presents a pole
at z = ¡0:47504. The system response is RF-DB,
and this is achieved at the expense of an increased
settling-time ts2 = 0:10 sec. The maximum control
signal appearing at the plant is umax 2 = 7:76838,
while the energy of the control signal takes the value
of J2 = 93:46729.
The pulse transfer function of the digital controller for the third-order overall system with the
RF-DB requirement and the minimum control energy is given from Eq.(A-1)
D3 (z) =
¡3:88419(z+1)(z¡0:74082)
z 2 +0:73752z+0:23752
(17)
This second-order controller cancels only the stable
pole of the plant at z = 0:74082. It presents another
zero at z = ¡1 and two complex conjugate poles at
z = ¡0: 36876 § 0: 31865i. The system response
remains RF-DB, but with lower control signal than
the previous case, and this is achieved at the expense
of a further increased settling-time ts3 = 0:15 sec.
The maximum control signal appearing at the plant
is umax 3 = 3:88419, which is reduced to the half
compared to the previous case, while the energy of
the control signal takes the value of J3 = 24:38029:
The pulse transfer function of the digital controller for the fourth-order overall system with the
RF-DB requirement and the minimum control energy is given from Eq.(A-1)
D4 (z) =
¡2:33737(z 2 +1:32355z+1)(z¡0:74082)
z 3 +0:84205z 2 +0:49006z+0:14293
(18)
This third-order controller cancels only the stable
pole of the plant at z = 0:74082. It presents another
two complex conjugate zeros at z = ¡0: 66178 § 0:
7497i, a negative real pole at z = ¡0: 45524 and
two complex conjugate poles at z = ¡0: 19341 § 0:
52589i. The system response remains RF-DB, but
with lower control signal than the previous case, and
this is achieved at the expense of a further increased
settling-time ts4 = 0:20 sec. The maximum control
signal appearing at the plant is umax 4 = 2:33737,
which is further reduced compared to the previous
case, while the energy of the control signal takes the
value of J4 = 10:31894.
The previous, and further, numerical results for
the various response characteristics of this design example are tabulated for comparison reasons in Table
1. The …rst column of the table presents the overall
system order N , the second column the maximum
control signal umax , the third column the energy of
the control signal J, the fourth the settling-time of
the system ts (which equals the overall system order
times the sampling period) and the …fth column the
product between the control energy and the settlingtime. The last quantity can be regarded as a measure of the overall performance of the design technique presented here, since it incorporates the e¤ect
of the gradual settling-time increase when the system order increases. It is readily seen that with a
linear increment in the overall system order, there is
a proportional increase in the settling-time and an
exponential decrease not only in the energy of the
control signal needed to attain the desirable RF-DB
response, but also in the maximum magnitude of the
control signal.
N
umax
J
ts
2
3
4
5
6
7
8
7.76838
3.88419
2.33737
1.56727
1.12900
0.85610
0.67477
93.46729
24.38029
10.31894
5.52326
3.40971
2.31792
1.68863
0.10
0.15
0.20
0.25
0.30
0.35
0.40
J ¢ ts
9.34673
3. 65704
2. 06379
1. 38082
1. 022 91
0. 811 27
0. 675 45
Table 1: Response characteristics for the speci…c
plant of the design example.
6.
CONCLUSIONS
The design of digital controllers for type-I
second-order plants with minimum-energy ripplefree dead-beat response to step inputs was addressed. For the unity-feedback scheme studied, analytical expressions for the digital controller pulse
transfer functions together with the control sequences were derived.
It was clearly shown that RF-DB response to
step inputs cannot be attained with a minimum prototype system (…rst-order overall system), due to the
oscillating control signal that comes from the cancellation of the stable negative real zero of the plant
with a pole of the digital controller. The secondorder overall system was proved to have the minimum system order for RF-DB response to step inputs and that was achieved with a …rst-order digital controller that avoided the aforementioned polezero cancellation. For greater overall system orders,
there was proposed the minimization of the energy
of the control signal that resulted to a further improvement in the response characteristics. It was
shown that with a third-order overall system, there
was the half maximum control signal of that observed for a second-order system, independently of
the selection of the sampling period. Furthermore,
there was observed that a linear increment in the
overall system order resulted to a proportional increase in the settling-time and an exponential decrease in both the energy of the control signal needed
to attain the desirable ME-RF-DB response and the
maximum control signal.
APPENDIX
The pulse transfer function of the digital
controller designed with the proposed energyminimization procedure for an N -th order closed-
loop system is given by [6]
DN (z) =
N¡2
X
SN;i (C)z i
z¡C
i=0
¢ N¡1
A
X
RN;i (B; C)z i
(A-1)
i=0
[8] C. A. Barbargires and C. A. Karybakas, “Optimal control of …rst-order plants by dead-beat
techniques”, Optimal Control Appl. Methods,
vol. 18, pp. 355–362, 1997.
[9] C. A. Karybakas and C. A. Barbargires, “Explicit conditions for ripple-free dead-beat control”, Kybernetika, vol. 32, pp. 601–614, 1996.
where the polynomial RN;i (B; C) takes the form
RN;i (B; C) = PN;i (C) ¡ B ¢ QN;i (C)
(A-2)
while for the polynomial SN;i (C) one symmetry relation holds, namely
SN;i (C) = SN;N¡2¡i (C)
7
and the polynomials SN;i (C) and QN;i (C) take the
form
P
j
(A-7)
SN;i (C) =
j xN;i;j C
P
j
(A-8)
QN;i (C) =
j yN;i;j C
8
j
0
1
2
3
4
5
6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
0
1
1
2
2
0
1
1
1
2
2
2
1
1
1
1
1
2
3
1
1
2
1
2
3
3
1
2
2
0
1
1
1
2
3
4
1
1
1
1
1
1
1
2
3
3
1
2
2
2
1
1
1
1
Table 2: The coe¢cients xN;i;j .
with the coe¢cients xN;i;j and yN;i;j taking the values given in Tables 2 and 3, respectively.
[1] A. R. Bergen and J. R. Ragazzini, “Sampleddata processing techniques for feedback control
systems”, Trans. AIEE (Industry and Applications), vol. 73, pp. 236–247, 1954.
[2] K. Ogata, Discrete-Time Control Systems.
Prentice-Hall, Englewood Cli¤s, NJ, 1987.
[3] S. Urikura and A. Nagata, “Ripple-free deadbeat control for sampled-data systems”, IEEE
Trans. Automat. Control, vol. AC-32, pp. 474–
482, 1987.
[4] E. Za…riou and M. Morari, “Digital controllers
for SISO systems: a review and a new algorithm”, Int. J. Control, vol. 42, pp. 855–876,
1985.
·
[5] S. H. Zak
and E. E. Blouin, “Ripple-free deadbeat control”, IEEE Control Systems Mag., vol.
13, pp. 51–56, 1993.
[6] C. A. Barbargires, “Study of Discrete-Time Control Systems with Dead-Beat Response to Polynomial Inputs”, Ph.D. Dissertation, Aristotle
University of Thessaloniki, 1994.
[7] C. A. Barbargires and C. A. Karybakas, “Ripplefree dead-beat control of DC servo motors”,
Proc. 2nd IEEE Mediterranean Symposium on
New Directions in Control and Automation,
Chania, Crete, 1994, pp. 469–476.
0
0
1
0
1
0
1
2
0
1
2
0
1
2
3
6
(A-4)
(A-5)
(A-6)
REFERENCES
i
3
4
5
(A-3)
Additionally, the following equations are valid
PN;0 (C) = 0
PN;i (C) = QN;i¡1 (C)
QN;N¡1 (C) = QN;N¡2 (C)
N
N
i
3
0
1
0
1
2
0
1
2
3
0
1
2
3
4
0
1
2
3
4
5
0
1
2
3
4
5
6
4
5
6
7
8
j
0
1
2
3
4
5
6
1
2
1
2
3
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1
2
3
4
5
6
7
1
3
4
0
1
2
2
1
3
5
7
8
0
1
2
3
4
4
1
3
5
7
9
11
12
1
2
3
1
2
3
4
1
3
6
8
9
1
2
4
5
7
8
1
3
6
9
12
14
15
1
3
5
7
8
0
1
2
3
4
4
1
3
6
10
13
15
16
1
2
3
4
5
1
2
3
4
5
6
1
3
6
9
12
14
15
1
3
5
7
9
11
12
1
2
3
4
5
6
7
Table 3: The coe¢cients yN;i;j .