- Email: [email protected]
Pulse Controller Design for Linear Time-Delay Systems
IFAC
[:0[>
Copyright ID IFAC System Structure and Control , Prague , Czech Republic , 200 I
Publications www.elsevier.comllocateli fac
PULSE CONTROLLER DESIGN FOR LINEAR TIME-DELAY SYSTEMS A. Fattouh' O. Sename* J.M. Dion *
* Laboratoire d'Automatique de Grenoble
INPG-CNRS (UMR5528) ENSIEG-BP 46 38402 Saint Marlin d 'Heres Cedex, FRANCE Fax: (33) 4.76.82.63.88, email : [email protected]
Abstract: The problem of digital redesign of control laws containing distributed time delays for linear systems with delayed state and control is addressed in this paper. The continuous controller with distributed time delays is approximated by a discrete one with only point time delays using a set of block-pulse functions . Copyright © 2001 [FAG. Keywords: Linear systems, Time-delay, Digital control
time-delay in order to control the Mach number in a wind tunnel. In order to simulate those control laws, the integral term has been approximated by a finite sum and the resulting control laws gave satisfying results. However, it has been mentioned that this approximation may not be hold for systems with large time delay (the time delay in this example is 0.33 sec.) .
1. INTRODUCTION The control of time-delay systems has been thoroughly studied in the last ten years (see for example (Dugard and , Eds) and the references therein) . Generally, a time-delay system has an infinite number of eigenvalues. Under the finite spectrum assignability assumption, control laws have been developed containing distributed timedelay such that a finite number of eigenvalues of the corresponding closed-loop system can be located at an arbitrarily preassigned set of points in the complex plane while the others are automatically eliminated (Manitius and Olbrot, 1979) and (Watanabe, 1986). However, the "finite" integral in those control laws has to be approximated in order to implement those control laws.
This problem has also been discussed by VanAssche et al. (1999) through an example. It has been shown that if the distributed control law is replaced by a weighted sum of point delays only using the composite trapezoidal rule, then the closed-loop system may be unstable. Then Mondie and Santos (2000) have analyzed this example and they have shown that the closed-loop system with the ideal control law is a time-delay system which becomes of neutral type using the approximated control law and so it needs more complicated conditions to ensure his stability.
A simple method for realizing this finite integral on a form of differential-difference equation is to differentiate and then to integrate by part those control laws. However Manitius and Olbrot (1979) have shown that this realization may be unstable and that the stability of the closed-loop system is the result of unstable poles-zeros cancellation.
In this paper, a method for solving this problem is proposed which is based on the following observation: if the kernel of a distributed time delay is constant, then this distributed time delay is implementable numerically. So if the integral in a distributed time delay is divided into several
As an illustration, Manitius (1984) has proposed four feedback control laws containing distributed
437
The next theorem provides a necessary and sufficient condition for finite spectrum assignability of system (1) as well as an explicit form of a control law which realizes a finite closed-loop spectrum. Note that this necessary and sufficient condition is, in fact , the spectral controllability assumption.
intervals and the kernel is approximated in those intervals by some constants using any adequate approximation method, then the distributed time delay can be approximated by a finite sum of distributed time delays with constant kernels. In other words, a control law with distributed time delays can be approximated by a finite sum of control laws containing distributed time delays with constant kernels. These distributed time delays can be then easily implemented numerically. The advantage of this method is that the closed-loop system with the approximated control law is not a neutral one.
Theorem 1. (Watanabe, 1986; Nobuyama and Kitamori, 1990) Consider system (1). This system is finite spectrum assignable if and only if:
rank
In this paper the kernel of a distributed time delay
[SIn - ~ Aje- sih
~ Bie- sih1= n
I
for all complex number s.
is approximated by a finite sum of functions using a set of block-pulse functions. So the resulting control law is said to be ';a pulse control law". Note that a set of block-pulse functions has been used by Kraus and Schaufelberger (1990) in order to develop an identification method for linear delayfree systems. Ahn et al. (1999) have also used this set of functions in order to implement the differential operation which has been used later to design unknown input observers.
Moreover, the control law which assign the spectrum of the closed-loop system is given by:
J
m
u(t) ==
0
L KiX(t -
ih)
;=0
+
F( -O)x(t
+ O)dO
-Tt
o
+
The paper is organized as follows. The problem statement is presented in Section 2. Section 3 is devoted to summaries the use of a set of block-pulse functions in order to approximate an integrable function. A pulse control law for those systems is designed in Section 4. An illustrative example is given in Section 5. Finally, the paper concludes with Section 6.
J
G( -O)u(t + O)dO
(2)
where Ki E IRT xn, 71, 72 are multiple numbers of h, F( -8) and G( -0) are continuous and integrable matrix-valued functions with appropriate dimension satisfying:
with: 2. STATE OF THE ART
m
All (s)
Consider the following linear time-delay system: x(t) == {
~ AiX(t -
= sIn -
i=0
m
ih)
+ BiU(t -
A12(S)
ih) (1)
L Aie- sih
=- L
Bie- sih
i=O
m
x(t) == 'Ij;(t);
A21(S)
t E [-rnh,O]
=-
J 0
LKie- Sih i=O
F(-O)e- s8 dO
-Tt
o
where x(t) E IRn is the state vector, u(t) E IRr is the input control vector, 'Ij;(t) E C[-h,O] is the initial-value function vector, h E IR+ is the delay duration and Ai and B i , i == 0, I, ... ,rn, are real matrices with appropriate dimensions with some positive integer rn.
A22(S) :;;: Ir -
J
G(-O)e- s8 d8
-T2
and o(s) is a stable polynomial.
0
The control law (2) cannot be implemented directly since the distributed time-delay term has to be approximated. However, it has been shown in (Manitius, 1984; Van-Assche et al., 1999; Mondie and Santos, 2(00) that some of approximation methods involve problems and the stability of the closed-loop system (1) with an approximated control law of (2) may not be kept.
In general system (1) has infinite spectrum. Controlling the location of an infinite number of eigenvalues is not practically feasible. Finite spectrum assignment has been considered by Olbrot (1978); Manitius and Olbrot (1979); Watanabe (1986) and others where feedback control laws containing integrals over the past values of state or control trajectory have been proposed such that finite spectral points are assigned while the others are automatically eliminated.
The objective of this paper is to digitally redesign the control law (2) by using a set of block-pulse
438
functions. The following section presents a summary of the use of a set of block-pulse functions in order to approximate an integrable function . This set of functions will be used in the next section in order to approximate the functions F ( -9) and G (-9) in the control law (2).
kT
F~ = ~
J(t - ih)dt
/ kT-1
= ~2 (f(kT where k
ih)
In this section, a set of block-pulse functions is firstly presented. Then, an integrable function is approximated using this set.
= !,n.
ih
= aiT + f3i
where ai is an integer and 0 0) , then (8) becomes:
A set of block-pulse functions
. F;
1
= 2 (f (kT + J(kT -
? :s kT
= {I,
kT - T t 0, otherW2se
k
k
(3)
(9)
:s f3i < T (ao =
f30 =
ai T - f3d
(10)
T - aiT - f3i»
where k = !,n. Because only the sampled values of the function J(t) are available, but not the values J(kT - aiTf3i) and J(kT-T-aiT-(3i) in (10), these ones need to be approximately evaluated using the values of J(t) at (kT - aiT - T, kT - aiT) and (kT - aiT 2T, kT - aiT - T) respectively as shown in Fig. 1.
",r+ tl n E 1"1' , T =-
= -1, n,
(8)
Notice that any time delay i h can be written as follows :
3. A SET OF BLOCK-PULSE FUNCTIONS
+ J(kT - T - ih»
n
where T is the sampling period and n is the number of samples. Using (3), an integrable function in the interval [0 , t I) can be expanded in block-pulse function series such that the mean-square error between the function and its approximation is minimized as follows: n
J(t)
=L
(4)
Fk
k=l
where Fk is the coefficient of
Fk =
~
/
J(t)dt,
k = !,n
(5)
k1-1
Fig. 1. The a.pproxima.tion of f(t) a.t
It should be noted that the equality (4) is not mathematically correct as the mean-square approximation error should be taken into considemtion. However, in order to simplify the presentation, this approximation error will be omitted throughout this paper.
Using the linear interpolation (Ralston, 1965), one can get: J(kT - ih)
Using the trapezoidal rule (Virk, 1991), Fk in (5) can be calculated as follows :
Fk =
J(kT - T - ih)
1
2 (f(kT) + J(kT -
T»,
k =!,n
(6)
Now, using (3), the integrable function J(t- ih) in the interval [0, t I) can be expanded in block-pulse function series as follows:
=L
F~
= f3i J(kT -
aiT - T)
t - (3 + T J(kT -
aiT)
=
f3i J(kT - aiT - 2T) T
+
TJ(kT-aiT-T)
(11)
T-f3
Replacing (11) in (10) gives:
. 21(T -
n
J(t - ih)
tl = kT - T - fJi and
tz = kT - fJi
~=
(7)
f3i
-T-J(kT - aiT)
+ J(kT -
aiT - T)
k=l
+~ J(kT -
where F; is the coefficient of
439
aiT - 2T») ,
k =!,n
(12)
t
t
In the next section, the formulae (4)-(6) and (7)(12) will be used in order to digitally redesign the control law (2).
vet) =
J
wet)
x(r)dr,
=
J
u(r)dr
(17)
o
o
Using (17) , the equality (16) becomes:
4. PULSE CONTROLLER DESIGN
m
In this section, the set of block-pulse functions presented in the previous section is used in order to get a discrete control law from (2).
u(t)
=L
KiX(t - ih)
i=O nl
+
L
+
L
Al [vet + (kl
-
ndTd
4.1 Continuous implementation Jorm n2
Following the previous section, the integrable functions F( -8) and G( -8) in the control law (2) can be expanded in block-pulse function series in the intervals [-r1,0) and [-r2,0) respectively as follows:
gk2 [wet + (k2 - n2)Tz)
-wet + (k2 - n2 - I)T2)]
(18)
The equations (18) with (17) represent a linear dynamical systems with point delays only. This form is said to be "a continuous implementation" of the control law (2) as this continuous form is digitally implementable.
(13)
By defining the following delay operators: with:
~1 (F(klTl) + F(klTl -
AI =
T1 » (14)
{
2 (G(k2T2) + G(k2T2 -
!Jk2 =
T2»
u(t) = K(V')x(t)
Replacing (13) in (2) gives: m
=L
= J(t = J(t -
h)
+ Tl) r2 + T2)
rl
the control law (18)-(17) can be rewritten as follows:
whereki=l,ni, niEN+, Ti=~' i=I,2.
u(t)
= J(t -
V' J(t) V'II(t) { V'z/(t)
KiX(t - ih)
+ F(V'dv(t) + G(V'2)W(t) (19)
where:
i=O
o
Jt A, ~kl + Jt gk2~k2(8)u(t+8)d8
+
(8)x(t
-Tl
o
+
-T2
m
n
K(V')
8)d8
=L
Ki V'i
i=0
kl=1
nl
F(V'I)
n
=
L
Al (V'~1
- V'~1 -1)
(20)
kl=1
(15)
n2
G(V'Z) =
k2=1
L
gI<2(V'~2 - V'~2-1)
k2=1 As
The closed-loop system of system (1) with the control law (19)-(20) is given by:
Xe(t)
m
u(t) =
L KiX(t -
ih)
where xe(t):= [x'(t) v'(t) w'(t)]' and
i=0
+
f:
T
-TI"I
l
AIX(t + 8)d8
kl=I_ T1 +kl TI-Tl 2T + -T/k 2 !Jk 2U (t kz=I_Tz+kzTz-T2
f
Let
US
= Aexe(t)
A('J)
A." =
+ B('J)K('J)
B('J)F('Jt} B('J)G('J2)
In
0
0
K('J)
F('Jt}
G('J2)
1
[
+ 8)d8 (16)
It should be noted that the above system is a linear system with point time delays only and it is not a neutral one.
define the following functions :
440
4.2 Discrete implementation form The control law (18)-(17) at t
= kT + T equals to:
T2 must be chosen as multiples of the sampling period T. Otherwise, one has to use an interpolation method in order to get a discrete control law with only one sampling period.
m
u (kT + T) =
L KiX(kT + T - i h) i=O
nl
+
L A, [v( kT + T -
(nl - kdTd
- v(kT+T+ (k l -nl -l)Td] n2
+
L
9"2 [w(kT + T - (n2 - k2)T 2)
- w(kT + T + (k2 - n 2 - l)Tz)]
(21)
In order to approximate the inter-sample values of x (t) the time delay decomposition (9) and the linear interpolation (see Fig. 1) can be used which lead to: x(kT + T - ih) = x(kT + T - aiT - f3i) f3i T - f3i = T"X(kT - aiT) + -T-X(kT + T - air) (22)
In addition, using the Euler's integration method, the integral in (17) earl by evaluated as follows: v(kT + T) = _T_ x(kT + T) q-l { w(kT + T)
(23)
= q ~ 1 u(kT + T)
Remark 2. Distributed time-delay can also be used in order to construct deadbeat controllers and observers or to improve the stability of the closed-loop system for delay-free systems. The proposed method can be used in order to digitally redesign this type of controllers or observers, to get a discrete observer or a discrete observer-based controller containing distributed time delays, or to get a discrete-time representation of a time-delay system including distributed time-delays.
5. ILLUSTRATIVE EXAMPLE
In order to illustrate the proposed method, let us consider the following unstable time-delay system: x(t)
°
For :$ h < 1 a memoryless controller u(t) = kx(t) can stabilize the system while for h 2: 1 this type of controller does not work (Olbrot, 1978). For h 2: 1 an optimal controller which minimize OO the functional J = JO [3x2(t) + u 2(t)]dt is given by Olbrot (1978) : u(t)
u(kT + T) = K(q)x(kT + T) T
+ 2(q _ 1) {F(ql)X(kT + T) +G(q2)u(kT + T)}
(24)
where
=
f
(26)
h)
o
where q is the shift operator defined for any discrete sequence y by qy(k) = y(k + 1). Replacing (14) , (22) and (23) in (21) gives the following discrete control law:
K(q)
= x(t) + u(t -
Ki(~ qO"i+l + T ~f3i qO"i)
= -3e
h
x(t) - 3 / e-eu(t + B)d8 (27) -h
and the closed-loop system eigenvalue is s
= -2.
In order to implement the control law (27), VanAssche et al. (1999) and Mondie and Santos (2000) have considered some conventional numerical approximation of the integral term. They have shown that if the distributed time delay in the control law (27) is replaced by a weighted sum of point delays only then the closed-loop system is unstable. We will show in the following that this is can be done using the proposed method.
For h = 1, according to (2), we have:
i=0
nl
F(qd
=
L [F(klTd + F(klTl - Td] (25)
=
L [G(k2T2) + G(k2T2 - T2)]
with qi f(kT) = f(kT - iT), qU(kT) = f(kT - iTd, qU(kT)
= 1,
7"2
= 1 and G( -B) = -3e e
G( -B)
= f(kT -
Ko
= -3e l ,
K1
= 0,
F( -B)
== 0,
Choosing n2 = 1, we have T2 = 1 sec. and the function G(-B) in the interval [-1 ,0] can be approximated by (13)-(14) as follows :
n2
G(q2)
m
iT2) '
It should be noted that the sampling periods Tland
441
= 91 ~l(B) = 0.5(G(I) + G(O» = -5.5774
Replacing those values in (18) gives the following continuous implementable control law which corresponds to the form (16) :
o
u(t)
= -8.1548x(t) -
5.5773
not change the nature of the closed-loop system.
J
It will be interesting to study the robustness of
u(t + 8)dB (28)
the proposed method.
-1
or in the form (18): w(t) = -5.5773w(t) { u(t)
References
+ 5.5773w(t -
1) - 8.1548x(t)
P. Ahn, M.-H. Kim, and D.-S. Ahn. A novel approach to unknown input observer design via block pulse function 's differential operation. In Proc. 14th Triennial World Congress IFAC, Beijing, China, 1999. 1. Dugard and E. 1. Verriest (Eds) . Stability and control of time-delay systems, volume 228 of Lecture Notes in Control and Information Sciences. Springer Verlag, 1998. Z. H. Jiang and W. Schaufelberger. Recursive formula for the multiple integral using blockpulse functions. Int. Journal of Control, 41(1) : 271-279, 1985. F. Kraus and W . Schaufelberger. Identification with block pulse functions, modulating functions and differential operators. Int. Journal of Control, 51(4):931-942, 1990. A. Z. Manitius. Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation. IEEE Trans. on Automatic Control, 29(12):1058-1068,1984. A. Z. Manitius and A. W. Olbrot. Finite spectrum assignment problem for systems with delays. IEEE Trans. on Automatic Control, 24(4):541553,1979. S. Mondie and O. Santos. Une condition necessaire pour l'implantation de lois de commandes a retards distribues. In Premiere Conference Internationale Francophone d'Automatique, pages 201-206, Lille - France, 5-8 Juillet, 2000. E. Nobuyama and T. Kitamori. Spectrum assignment and parameterization of all stabilizing compensators for time-delay systems. In Proc. 29th Confer. on Decision and Control, pages 3629-3634, Honolulu, Hawaii, 1990. A. W. Olbrot. Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans. on Automatic Control, 23(5):887-890, 1978. A. Ralston. A first course in numerical analysis. New York: McGraw-Hill, 1965. V. Van-Assche, M. Dambrine, and J.-F. Lafay. Some problems arising in the implementation of distributed-delay control laws. In Proc. 38th IEEE Confer. on Decision f3 Control, pages 4668-4672, Phoenix, Arizona, USA, 1999. G. S. Virk. Digital computer control systems. MacmWan New Electronics, Introductions to Advanced Topics, 1991. K. Watanabe. Finite spectrum assignment and observer for multi variable systems with commensurate delays. IEEE Trans. on Automatic Control, 31(6):543-550, 1986.
(29)
= w(t )
The control law (29) at t = kT + T equals to: u(kT + T) = -3e 1 x(kT + T) _
3(~ + e1~T {u(kT + T) _ 2q-1
u(kT+ T -1)}(30)
Fig. 2 illustrates the convergence of the closedloop system with the continuous implementation (29) and the discrete implementation (30) respectively for T = O.lsec . Cklsed-loop system (x(I)) using contnuous and disaete controllers O.05~-----;r------'---.-----'---'-----'
-{)25 -{)3'OL_ _- L_ _---'-_ _- ' -_ _-'----_ _:---_~
CoIiinuous and disaele cot
1.2,------;,.----,----,----,---,----
-{)2L-_~
__
~
__
~
_ _- L_ _
-L_~
o
Fig. 2. Closed loop system and control laws
6. CONCLUSION
In this paper we have developed a method for digital redesign of control laws containing distributed time-delay for linear systems with delayed state and control. Indeed, distributed time delays are not implementable directly and they need to be approximated. However, some approximation methods lead to instability problems. The idea of this paper is based on the use of a set of block-pulse functions in order to approximate the kernel of a distributed time delay which leads to some implementable distributed time delays. The advantage of the proposed method lies in the fact that the digital redesign of the control law dose
442
[:0[>
Copyright ID IFAC System Structure and Control , Prague , Czech Republic , 200 I
Publications www.elsevier.comllocateli fac
PULSE CONTROLLER DESIGN FOR LINEAR TIME-DELAY SYSTEMS A. Fattouh' O. Sename* J.M. Dion *
* Laboratoire d'Automatique de Grenoble
INPG-CNRS (UMR5528) ENSIEG-BP 46 38402 Saint Marlin d 'Heres Cedex, FRANCE Fax: (33) 4.76.82.63.88, email : [email protected]
Abstract: The problem of digital redesign of control laws containing distributed time delays for linear systems with delayed state and control is addressed in this paper. The continuous controller with distributed time delays is approximated by a discrete one with only point time delays using a set of block-pulse functions . Copyright © 2001 [FAG. Keywords: Linear systems, Time-delay, Digital control
time-delay in order to control the Mach number in a wind tunnel. In order to simulate those control laws, the integral term has been approximated by a finite sum and the resulting control laws gave satisfying results. However, it has been mentioned that this approximation may not be hold for systems with large time delay (the time delay in this example is 0.33 sec.) .
1. INTRODUCTION The control of time-delay systems has been thoroughly studied in the last ten years (see for example (Dugard and , Eds) and the references therein) . Generally, a time-delay system has an infinite number of eigenvalues. Under the finite spectrum assignability assumption, control laws have been developed containing distributed timedelay such that a finite number of eigenvalues of the corresponding closed-loop system can be located at an arbitrarily preassigned set of points in the complex plane while the others are automatically eliminated (Manitius and Olbrot, 1979) and (Watanabe, 1986). However, the "finite" integral in those control laws has to be approximated in order to implement those control laws.
This problem has also been discussed by VanAssche et al. (1999) through an example. It has been shown that if the distributed control law is replaced by a weighted sum of point delays only using the composite trapezoidal rule, then the closed-loop system may be unstable. Then Mondie and Santos (2000) have analyzed this example and they have shown that the closed-loop system with the ideal control law is a time-delay system which becomes of neutral type using the approximated control law and so it needs more complicated conditions to ensure his stability.
A simple method for realizing this finite integral on a form of differential-difference equation is to differentiate and then to integrate by part those control laws. However Manitius and Olbrot (1979) have shown that this realization may be unstable and that the stability of the closed-loop system is the result of unstable poles-zeros cancellation.
In this paper, a method for solving this problem is proposed which is based on the following observation: if the kernel of a distributed time delay is constant, then this distributed time delay is implementable numerically. So if the integral in a distributed time delay is divided into several
As an illustration, Manitius (1984) has proposed four feedback control laws containing distributed
437
The next theorem provides a necessary and sufficient condition for finite spectrum assignability of system (1) as well as an explicit form of a control law which realizes a finite closed-loop spectrum. Note that this necessary and sufficient condition is, in fact , the spectral controllability assumption.
intervals and the kernel is approximated in those intervals by some constants using any adequate approximation method, then the distributed time delay can be approximated by a finite sum of distributed time delays with constant kernels. In other words, a control law with distributed time delays can be approximated by a finite sum of control laws containing distributed time delays with constant kernels. These distributed time delays can be then easily implemented numerically. The advantage of this method is that the closed-loop system with the approximated control law is not a neutral one.
Theorem 1. (Watanabe, 1986; Nobuyama and Kitamori, 1990) Consider system (1). This system is finite spectrum assignable if and only if:
rank
In this paper the kernel of a distributed time delay
[SIn - ~ Aje- sih
~ Bie- sih1= n
I
for all complex number s.
is approximated by a finite sum of functions using a set of block-pulse functions. So the resulting control law is said to be ';a pulse control law". Note that a set of block-pulse functions has been used by Kraus and Schaufelberger (1990) in order to develop an identification method for linear delayfree systems. Ahn et al. (1999) have also used this set of functions in order to implement the differential operation which has been used later to design unknown input observers.
Moreover, the control law which assign the spectrum of the closed-loop system is given by:
J
m
u(t) ==
0
L KiX(t -
ih)
;=0
+
F( -O)x(t
+ O)dO
-Tt
o
+
The paper is organized as follows. The problem statement is presented in Section 2. Section 3 is devoted to summaries the use of a set of block-pulse functions in order to approximate an integrable function. A pulse control law for those systems is designed in Section 4. An illustrative example is given in Section 5. Finally, the paper concludes with Section 6.
J
G( -O)u(t + O)dO
(2)
where Ki E IRT xn, 71, 72 are multiple numbers of h, F( -8) and G( -0) are continuous and integrable matrix-valued functions with appropriate dimension satisfying:
with: 2. STATE OF THE ART
m
All (s)
Consider the following linear time-delay system: x(t) == {
~ AiX(t -
= sIn -
i=0
m
ih)
+ BiU(t -
A12(S)
ih) (1)
L Aie- sih
=- L
Bie- sih
i=O
m
x(t) == 'Ij;(t);
A21(S)
t E [-rnh,O]
=-
J 0
LKie- Sih i=O
F(-O)e- s8 dO
-Tt
o
where x(t) E IRn is the state vector, u(t) E IRr is the input control vector, 'Ij;(t) E C[-h,O] is the initial-value function vector, h E IR+ is the delay duration and Ai and B i , i == 0, I, ... ,rn, are real matrices with appropriate dimensions with some positive integer rn.
A22(S) :;;: Ir -
J
G(-O)e- s8 d8
-T2
and o(s) is a stable polynomial.
0
The control law (2) cannot be implemented directly since the distributed time-delay term has to be approximated. However, it has been shown in (Manitius, 1984; Van-Assche et al., 1999; Mondie and Santos, 2(00) that some of approximation methods involve problems and the stability of the closed-loop system (1) with an approximated control law of (2) may not be kept.
In general system (1) has infinite spectrum. Controlling the location of an infinite number of eigenvalues is not practically feasible. Finite spectrum assignment has been considered by Olbrot (1978); Manitius and Olbrot (1979); Watanabe (1986) and others where feedback control laws containing integrals over the past values of state or control trajectory have been proposed such that finite spectral points are assigned while the others are automatically eliminated.
The objective of this paper is to digitally redesign the control law (2) by using a set of block-pulse
438
functions. The following section presents a summary of the use of a set of block-pulse functions in order to approximate an integrable function . This set of functions will be used in the next section in order to approximate the functions F ( -9) and G (-9) in the control law (2).
kT
F~ = ~
J(t - ih)dt
/ kT-1
= ~2 (f(kT where k
ih)
In this section, a set of block-pulse functions is firstly presented. Then, an integrable function is approximated using this set.
= !,n.
ih
= aiT + f3i
where ai is an integer and 0 0) , then (8) becomes:
A set of block-pulse functions
. F;
1
= 2 (f (kT + J(kT -
? :s kT
= {I,
kT - T t 0, otherW2se
k
k
(3)
(9)
:s f3i < T (ao =
f30 =
ai T - f3d
(10)
T - aiT - f3i»
where k = !,n. Because only the sampled values of the function J(t) are available, but not the values J(kT - aiTf3i) and J(kT-T-aiT-(3i) in (10), these ones need to be approximately evaluated using the values of J(t) at (kT - aiT - T, kT - aiT) and (kT - aiT 2T, kT - aiT - T) respectively as shown in Fig. 1.
",r+ tl n E 1"1' , T =-
= -1, n,
(8)
Notice that any time delay i h can be written as follows :
3. A SET OF BLOCK-PULSE FUNCTIONS
+ J(kT - T - ih»
n
where T is the sampling period and n is the number of samples. Using (3), an integrable function in the interval [0 , t I) can be expanded in block-pulse function series such that the mean-square error between the function and its approximation is minimized as follows: n
J(t)
=L
(4)
Fk
k=l
where Fk is the coefficient of
Fk =
~
/
J(t)dt,
k = !,n
(5)
k1-1
Fig. 1. The a.pproxima.tion of f(t) a.t
It should be noted that the equality (4) is not mathematically correct as the mean-square approximation error should be taken into considemtion. However, in order to simplify the presentation, this approximation error will be omitted throughout this paper.
Using the linear interpolation (Ralston, 1965), one can get: J(kT - ih)
Using the trapezoidal rule (Virk, 1991), Fk in (5) can be calculated as follows :
Fk =
J(kT - T - ih)
1
2 (f(kT) + J(kT -
T»,
k =!,n
(6)
Now, using (3), the integrable function J(t- ih) in the interval [0, t I) can be expanded in block-pulse function series as follows:
=L
F~
= f3i J(kT -
aiT - T)
t - (3 + T J(kT -
aiT)
=
f3i J(kT - aiT - 2T) T
+
TJ(kT-aiT-T)
(11)
T-f3
Replacing (11) in (10) gives:
. 21(T -
n
J(t - ih)
tl = kT - T - fJi and
tz = kT - fJi
~=
(7)
f3i
-T-J(kT - aiT)
+ J(kT -
aiT - T)
k=l
+~ J(kT -
where F; is the coefficient of
439
aiT - 2T») ,
k =!,n
(12)
t
t
In the next section, the formulae (4)-(6) and (7)(12) will be used in order to digitally redesign the control law (2).
vet) =
J
wet)
x(r)dr,
=
J
u(r)dr
(17)
o
o
Using (17) , the equality (16) becomes:
4. PULSE CONTROLLER DESIGN
m
In this section, the set of block-pulse functions presented in the previous section is used in order to get a discrete control law from (2).
u(t)
=L
KiX(t - ih)
i=O nl
+
L
+
L
Al [vet + (kl
-
ndTd
4.1 Continuous implementation Jorm n2
Following the previous section, the integrable functions F( -8) and G( -8) in the control law (2) can be expanded in block-pulse function series in the intervals [-r1,0) and [-r2,0) respectively as follows:
gk2 [wet + (k2 - n2)Tz)
-wet + (k2 - n2 - I)T2)]
(18)
The equations (18) with (17) represent a linear dynamical systems with point delays only. This form is said to be "a continuous implementation" of the control law (2) as this continuous form is digitally implementable.
(13)
By defining the following delay operators: with:
~1 (F(klTl) + F(klTl -
AI =
T1 » (14)
{
2 (G(k2T2) + G(k2T2 -
!Jk2 =
T2»
u(t) = K(V')x(t)
Replacing (13) in (2) gives: m
=L
= J(t = J(t -
h)
+ Tl) r2 + T2)
rl
the control law (18)-(17) can be rewritten as follows:
whereki=l,ni, niEN+, Ti=~' i=I,2.
u(t)
= J(t -
V' J(t) V'II(t) { V'z/(t)
KiX(t - ih)
+ F(V'dv(t) + G(V'2)W(t) (19)
where:
i=O
o
Jt A, ~kl + Jt gk2~k2(8)u(t+8)d8
+
(8)x(t
-Tl
o
+
-T2
m
n
K(V')
8)d8
=L
Ki V'i
i=0
kl=1
nl
F(V'I)
n
=
L
Al (V'~1
- V'~1 -1)
(20)
kl=1
(15)
n2
G(V'Z) =
k2=1
L
gI<2(V'~2 - V'~2-1)
k2=1 As
The closed-loop system of system (1) with the control law (19)-(20) is given by:
Xe(t)
m
u(t) =
L KiX(t -
ih)
where xe(t):= [x'(t) v'(t) w'(t)]' and
i=0
+
f:
T
-TI"I
l
AIX(t + 8)d8
kl=I_ T1 +kl TI-Tl 2T + -T/k 2 !Jk 2U (t kz=I_Tz+kzTz-T2
f
Let
US
= Aexe(t)
A('J)
A." =
+ B('J)K('J)
B('J)F('Jt} B('J)G('J2)
In
0
0
K('J)
F('Jt}
G('J2)
1
[
+ 8)d8 (16)
It should be noted that the above system is a linear system with point time delays only and it is not a neutral one.
define the following functions :
440
4.2 Discrete implementation form The control law (18)-(17) at t
= kT + T equals to:
T2 must be chosen as multiples of the sampling period T. Otherwise, one has to use an interpolation method in order to get a discrete control law with only one sampling period.
m
u (kT + T) =
L KiX(kT + T - i h) i=O
nl
+
L A, [v( kT + T -
(nl - kdTd
- v(kT+T+ (k l -nl -l)Td] n2
+
L
9"2 [w(kT + T - (n2 - k2)T 2)
- w(kT + T + (k2 - n 2 - l)Tz)]
(21)
In order to approximate the inter-sample values of x (t) the time delay decomposition (9) and the linear interpolation (see Fig. 1) can be used which lead to: x(kT + T - ih) = x(kT + T - aiT - f3i) f3i T - f3i = T"X(kT - aiT) + -T-X(kT + T - air) (22)
In addition, using the Euler's integration method, the integral in (17) earl by evaluated as follows: v(kT + T) = _T_ x(kT + T) q-l { w(kT + T)
(23)
= q ~ 1 u(kT + T)
Remark 2. Distributed time-delay can also be used in order to construct deadbeat controllers and observers or to improve the stability of the closed-loop system for delay-free systems. The proposed method can be used in order to digitally redesign this type of controllers or observers, to get a discrete observer or a discrete observer-based controller containing distributed time delays, or to get a discrete-time representation of a time-delay system including distributed time-delays.
5. ILLUSTRATIVE EXAMPLE
In order to illustrate the proposed method, let us consider the following unstable time-delay system: x(t)
°
For :$ h < 1 a memoryless controller u(t) = kx(t) can stabilize the system while for h 2: 1 this type of controller does not work (Olbrot, 1978). For h 2: 1 an optimal controller which minimize OO the functional J = JO [3x2(t) + u 2(t)]dt is given by Olbrot (1978) : u(t)
u(kT + T) = K(q)x(kT + T) T
+ 2(q _ 1) {F(ql)X(kT + T) +G(q2)u(kT + T)}
(24)
where
=
f
(26)
h)
o
where q is the shift operator defined for any discrete sequence y by qy(k) = y(k + 1). Replacing (14) , (22) and (23) in (21) gives the following discrete control law:
K(q)
= x(t) + u(t -
Ki(~ qO"i+l + T ~f3i qO"i)
= -3e
h
x(t) - 3 / e-eu(t + B)d8 (27) -h
and the closed-loop system eigenvalue is s
= -2.
In order to implement the control law (27), VanAssche et al. (1999) and Mondie and Santos (2000) have considered some conventional numerical approximation of the integral term. They have shown that if the distributed time delay in the control law (27) is replaced by a weighted sum of point delays only then the closed-loop system is unstable. We will show in the following that this is can be done using the proposed method.
For h = 1, according to (2), we have:
i=0
nl
F(qd
=
L [F(klTd + F(klTl - Td] (25)
=
L [G(k2T2) + G(k2T2 - T2)]
with qi f(kT) = f(kT - iT), qU(kT) = f(kT - iTd, qU(kT)
= 1,
7"2
= 1 and G( -B) = -3e e
G( -B)
= f(kT -
Ko
= -3e l ,
K1
= 0,
F( -B)
== 0,
Choosing n2 = 1, we have T2 = 1 sec. and the function G(-B) in the interval [-1 ,0] can be approximated by (13)-(14) as follows :
n2
G(q2)
m
iT2) '
It should be noted that the sampling periods Tland
441
= 91 ~l(B) = 0.5(G(I) + G(O» = -5.5774
Replacing those values in (18) gives the following continuous implementable control law which corresponds to the form (16) :
o
u(t)
= -8.1548x(t) -
5.5773
not change the nature of the closed-loop system.
J
It will be interesting to study the robustness of
u(t + 8)dB (28)
the proposed method.
-1
or in the form (18): w(t) = -5.5773w(t) { u(t)
References
+ 5.5773w(t -
1) - 8.1548x(t)
P. Ahn, M.-H. Kim, and D.-S. Ahn. A novel approach to unknown input observer design via block pulse function 's differential operation. In Proc. 14th Triennial World Congress IFAC, Beijing, China, 1999. 1. Dugard and E. 1. Verriest (Eds) . Stability and control of time-delay systems, volume 228 of Lecture Notes in Control and Information Sciences. Springer Verlag, 1998. Z. H. Jiang and W. Schaufelberger. Recursive formula for the multiple integral using blockpulse functions. Int. Journal of Control, 41(1) : 271-279, 1985. F. Kraus and W . Schaufelberger. Identification with block pulse functions, modulating functions and differential operators. Int. Journal of Control, 51(4):931-942, 1990. A. Z. Manitius. Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation. IEEE Trans. on Automatic Control, 29(12):1058-1068,1984. A. Z. Manitius and A. W. Olbrot. Finite spectrum assignment problem for systems with delays. IEEE Trans. on Automatic Control, 24(4):541553,1979. S. Mondie and O. Santos. Une condition necessaire pour l'implantation de lois de commandes a retards distribues. In Premiere Conference Internationale Francophone d'Automatique, pages 201-206, Lille - France, 5-8 Juillet, 2000. E. Nobuyama and T. Kitamori. Spectrum assignment and parameterization of all stabilizing compensators for time-delay systems. In Proc. 29th Confer. on Decision and Control, pages 3629-3634, Honolulu, Hawaii, 1990. A. W. Olbrot. Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans. on Automatic Control, 23(5):887-890, 1978. A. Ralston. A first course in numerical analysis. New York: McGraw-Hill, 1965. V. Van-Assche, M. Dambrine, and J.-F. Lafay. Some problems arising in the implementation of distributed-delay control laws. In Proc. 38th IEEE Confer. on Decision f3 Control, pages 4668-4672, Phoenix, Arizona, USA, 1999. G. S. Virk. Digital computer control systems. MacmWan New Electronics, Introductions to Advanced Topics, 1991. K. Watanabe. Finite spectrum assignment and observer for multi variable systems with commensurate delays. IEEE Trans. on Automatic Control, 31(6):543-550, 1986.
(29)
= w(t )
The control law (29) at t = kT + T equals to: u(kT + T) = -3e 1 x(kT + T) _
3(~ + e1~T {u(kT + T) _ 2q-1
u(kT+ T -1)}(30)
Fig. 2 illustrates the convergence of the closedloop system with the continuous implementation (29) and the discrete implementation (30) respectively for T = O.lsec . Cklsed-loop system (x(I)) using contnuous and disaete controllers O.05~-----;r------'---.-----'---'-----'
-{)25 -{)3'OL_ _- L_ _---'-_ _- ' -_ _-'----_ _:---_~
CoIiinuous and disaele cot
1.2,------;,.----,----,----,---,----
-{)2L-_~
__
~
__
~
_ _- L_ _
-L_~
o
Fig. 2. Closed loop system and control laws
6. CONCLUSION
In this paper we have developed a method for digital redesign of control laws containing distributed time-delay for linear systems with delayed state and control. Indeed, distributed time delays are not implementable directly and they need to be approximated. However, some approximation methods lead to instability problems. The idea of this paper is based on the use of a set of block-pulse functions in order to approximate the kernel of a distributed time delay which leads to some implementable distributed time delays. The advantage of the proposed method lies in the fact that the digital redesign of the control law dose
442