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Implementation of a distributed control law for a class of systems with delay
IFAC
Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001
~
Publications www.elsevier.com/locate/ifac
IMPLEMENTATION OF A DISTRIBUTED CONTROL LAW FOR A CLASS OF SYSTEMS WITH DELAY V. Van Assche* M. Dambrine** J.-F. Lafay* J .-P. Richard **
* IRCCyN, UMR CNRS 6597, E. C. Nantes, 1, rue de la Noe,
B.? 92101, 44321 Nantes Cedex 03, France ** LAIL, U?RESA CNRS 8021, E. C. Lille, B.? 48, 59651 Villeneuve d'Ascq Cedex. France
Abstract: The use of distributed delays in the control law of time-delay systems has been proposed by several authors. The implementation of such controllers is not trivial, and recent publication shown that replacing the distributed delay operator by an approximation computed through pointwise delay operators was unsafe with respect to the stability of the system. In this paper the use of digital controller is addressed, for the control of systems with an input delay. Practically, this means replacing an integral by a recurrent system. The first approach was to use a method of numerical approximation of an integral to build the recurrence law. In this paper it is shown that a such controller, built with the Simpson method, leads to an unstable closed-loop system. A second approach leads to the construction of a control law which realizes a sampled pole assignment, in the same way as the distributed control law realizes a pole assignment in continuous time. Copyright © 20011FAC
1. INTRODUCTION
poles in the complex plane. The control laws will have the form:
This paper deals with some pole assignment methods for systems with delays. Throughout the paper, we will consider systems with commensurate delays, that is all delays are integer multiple of a real noted h. We define \7, the delay operator, such that, for any integer k, \7 k x(t) = x (t - kh). A linear system with commensurate delays will be written the following way: X (t) = A (\7) x(t)
+ B (\7) u(t)
(2)
where Ut (.) and Xt (.) are functions defined on the interval [-7,0] by ut(B) = u(t+B) and xt(B) = x(t+ B), for e E [-h,O]. Very often, the study and the control design of timedelay systems deal only with punctual delays, i.e. of the form x( t - 7), excluding integrals over the past of the command and state, this allows to work in a nice algebraic framework. In (Morse, 1976), it is shown that such a law allows a finite spectrum assignment only if the system fulfills the rather restrictive condition of strong controllability. Manitius and Olbrot (1979) show that spectral controllability is a sufficient condition for finite spectrum assignment with a control law of the form
(1)
A (\7) and B (\7) being two matrices of polynomials in \7. The Laplace transform of \7 k x(t) is e- khs X(s), where X (s) is the Laplace transform of x( t). Precisely, we are interested in method of finite spectrum assignment, Le. methods allowing to reduce the number of poles of a system with delays to a finite number and to assign arbitrarily the location of these
195
a right half-plane pole-zero cancellation. In order to avoid such a phenomenon, Manitius and Olbrot recommend to compute directly z(t) using a numerical quadrature method. Van Assche et al. (1999) for the example above, and Mondie et al. (August 2001) for the generalization to the case of systems with input delay, show that an approximation of the distributed delay with a finite sum of pointwise delays may give an unstable closed-loop system. Moreover, several examples of numerical regulators computed to approximate the continuous theoretical control law are given in (Van Assche et al., 1999), with either the rectangular, trapezoidal and Simpson quadrature methods. The first two methods give a stable closedloop system, although the last one, which is supposed to be the most precise, gives an unstable system, even when changing the sampling period.
J +J t
+
u(t) = F(V)x(t)
j(8)x(8)d8
t-r
t
+H(V)u(t)
h(8)u(8)s8.
t-r
The system (1) is said to be spectrally controllable if, and only if, for any SEC, the matrix (sI - A(e- S ) B(e- S )) is of rank n. In (Watanabe et al., 1983), the necessity of this condition is proven for SISO systems, and in (Watanabe, 1986), for MIMO systems. Recently, an algebraic background has been formalized in (BretM and Loiseau, 1996) and (Brethe, 1997) for systems with distributed delays. The authors define the ring of the Laplace transform of both distributed delays and punctual delays. This formalism leads to a n-assignation algorithm which is much simpler than those of (Watanabe et al., 1983) and (Watanabe, 1986).
In this paper, it is shown that a numerical regulator constructed through an approximation of the control law (4) will give an unstable closed-loop system regardless of the sampling period. Then, our contribution is to provide a safe way to implement a stabilizing control law of the type (2) using a numerical regulator without loss of the stability for the closed-loop system.
The problem we want to deal with in this article is the implementation of the distributed delays. As the integral term in (2) cannot be exactly computed with a numerical computer, to use these command laws implies to approximate it. We will here show that the use of quadrature methods involves stability problems especially when delays are on the input variables.
2. SYSTEM WITH INPUT DELAY 2.1 Continuous system with an input delay
As an illustration of this problem, let us consider the very simple unstable system:
x(t) = x(t)
+ u(t -
1).
In this paper, we consider systems with a delay on the input. The state space description of such systems is:
(3)
x(t) = Ax(t)
With the control law
+ Ad) (e1x(t)
u(t) = - (1
J
e IJ u (t - 8) d8) ,
(5)
The dimension of the vectors x(t) and u(t) are noted respectively nand m.
1
+
+ V Bu(t).
(4)
o
Remark 1. A larger class of systems can be turned into a system with delay on the input. Indeed, if a system correspond to the scheme 1, it can be written in state space form:
which is a Volterra equation of second kind, the closed-loop system has one pole in -Ad. With Ad > 0 , the system is stable, but, as we stated above, the problem we are faced with is the way of realizing the integral term
Xl(t) = Al1Xl(t) + Bu(t), X2(t) = VA21Xl(t) + A22X2(t),
J
(6)
1
z(t) =
where the vectors Xl(t) and X2(t) are of dimensions nl and n2, and u(t) is a real valued function, and where All, A 21 and A 22 , and B are matrices of convenient dimensions over the field lR of real numbers.
eIJu(t - 8)d8.
o A simple way of proceeding is to differentiate the previous expression. We obtain the differential equation
i(t) = z(t)
+ u(t) -
eu(t - 1).
Fig. 1. System (6)
~
But, as explained in (Manitius and Olbrot, 1979), this realization is unsafe since unstable, and in this case the stabilization of the process is the result of
196
SI
I
xl
r::::l~2
.,~
S2
~2
_ _.....J
Such a system can be turned into a system with a delay only on the input with x~ (t) = \7Xl(t) :
(9) can be rewritten
x((k +
1)~) = q>(~)x(k~)
+ +
(7)
x(t)=(x~(t)) X2(t)
If the system (5) is spectrally controllable, one can apply the finite spectrum assignment methods. This gives the following control law:
and
(All A A
0 )
21
22
.
=
2.2 Sampling of the system (5)
+ v(t),
see (Manitius and Olbrot, 1979).
J eA(k~+~-T) BdTU(k~).
We propose here a numerical computation of the integral, resulting in a discrete controller, which is clearly implementable. Moreover, the discrete system obtained with this method is similar to the system one gets when discretizing the theoretical continuous system obtained with the control law (13).
k~
For a system with a delay of value h on the input, the expression above is directly adapted if the delay is a multiple of the sampling period, i.e. if there exists an integer p with h = p~ :
Before stating this original result, we first prove that the use of a quadrature method is not a good approach to solve this problem.
x(k~ +~) = eA~x(k~)+
JeA(~-T) BdTU(k~ ~
_ p~). (8)
o
3.1 Instability due to a Simpson's quadrature
If the delay is not a multiple of the sampling period, we note h = (p -1)~ + h' and, following Wittenmark (1985), one has
In a first approach, we compute the integral term through quadrature methods such as the rectangular method, the trapezoidal method and the Simpson's rule, see (Van Assche et al., 1999). This last method is the one of higher order. Meanwhile, it gives the worst result in our simulations.
= eA~x(k~)
JeA(~-h'+T) ~ p)~) J eATdTu((k-p+1)~). h'
BdTU((k
o
(13)
(14)
k~+~
+
eA(-h-T) Bu(t - T)dT
where H is a real matrix of dimension 1 x n. The spectrum of the closed-loop system realized by (5) controlled by (13) is the spectrum of the matrix
x(k~ +~) = eM'x(k~)
+ 1)~)
-F
o -Fx(t)
In order to implement a control law with a numerical device, the system has to be sampled. The sampling period is noted ~, and the input u(t) is constant between the sampling times. For a system without delay, it is well known that the equation of the sampled system is given by:
x((k
J h
u(t)
+
- h')r(h')u((k - p)~) (12) - h')u((k - p + 1)~).
3. CONTROL LAW FOR SPECTRUM ASSIGNMENT OF THE SYSTEM WITH AN INPUT DELAY
One notes then
A _ -
q>(~ r(~
(9)
We show that the closed-loop system is unstable for the closed-loop system (3)-(4), with h = 1 = p~, p being an even integer, and with Ad = 1.
~-h'
+
o
For this choice of h, becomes, with (8),
By writing
x((k + 1)~)
q>(t) =e At
(10)
J
(11)
t
r(t) =
eATdT,
~ p
and Ad, the system (3)
= e~x(k~) +
(e~
- 1) u(k~). (15)
Its z-transform is : zP
o
197
(z -
e~) x = (e~
- 1) u.
(16)
The discrete control law is :
Therefore,
~
u(k~) = -2"3w(k~) - 2elx(k~)
+ v(k~) (17)
with p/2-1
w(k~) = u(k~)
+
L
2e 21 ti. u ((k - 2l)~)
1=1 p/2
+
L 4e(21-1)ti. u ((k -
2l
+ 1)~)
and, for z
1=1 + elu((k - p)~).
=
-1,
The z-transform of (17) is:
de(z)u
= 2e l z px+ zPv
(18)
with
As p-l
~ "~" e(k+l)ti. = ti.~O Hm
k=O
zP (z - eti. ) de (z) - (eti. - 1) 2e 1 zP ,
lim d( -1) = -2
and the closed-loop system is stable if, and only if, the polynomial
As the degree of d(z) is p has for any value of ~,
(19)
L ekti.zp-k = eti.(zP -
ell,
(20)
k=l
and that
=
(d-l zP(l + 2"3) + 2"3 L 2e1ti. z(p-2l)
=
-00,
The computation above show that approximating the controller with a quadrature method is not always a safe approach to the problem. The next section of this work presents another approach, which leads to a safe implementation of the controller.
~
1=1 +
an odd number, one
hence, for ~ sufficiently small, d( z) has at least one root on the real axis, of magnitude strictly superior to 1. This show that the system with this controller is unstable, for any sufficiently small value of ~, i.e. for the most precise approximation of the integral term of (4).
p
de(z)
+ 1,
- 1) > O.
Z-------l>-OO
First, let us simplify the term dl(z) by remarking that
~
+ ~(el 3
lim d(z)
has all its root inside the unit disc.
(z - eti.)
,
0
ti.~O
d(z) =
1
One can check that
The denominator of the closed-loop system is then:
(z - eti.) de(z) - (eti. - 1) 2e l
1
JeT dT = e l -
Lp/2 2e(21-1)ti. z(p-21+1) + e ) l
.
1=1 With some computation, this leads to -)_ 3+2~ p+l Z d 1'" ( -
+ 4~
3 p-l
L
-
3-2~ ti. p ---e z 3
4. SAFE IMPLEMENTATION This parts deals with systems of the form (7), sampled into (12). The control law is also sampled with the hypothesis that u( t) is a bloc-pulse function, Le. u(t) = u(k~) for k~ ~ t < (k + 1)~. This gives
((_I)k e (k+l)ti. z n-k)
k=O 2~
1
2~ Hti.
- - e z+-e 3 3
.
198
u(k~) =
Right multiplying the matrix P(z) by (
J III
p
-FL
eA(-h+r)BdTU((k
1
+F
0) .
-l)~)
1=1 (1-1)Ll
p)~)
eA(-h+r) BdTU((k -
In ( -F I m
(21)
with
+ v(k~).
i(t) = Ax(t)
+ Bu(t)
Now, let us expand
D 1(z) = (Inz -
(22)
P
- (In z -
1=1
u(t) = Fx(t),
+
p
(
D 1(z) = Inzp+1 -
1=1
-
~)F.
With a remark similar to (20), and remarking that f( -~) - r(h' - ~) = -
Proof: With the definitions (10) and (11) the control law is rewritten:
~)u((k
~)Fz
-
up to a multiplication by a power of z.
+ p~)f(h' -Fx(k~) + v(k~).
-
+B(z)F.
sampled with the same sampling period and controlled by the feed back
-
D(z) = (Inz -
Proposition 1. The system with an input delay (5), sampled with a period ~, and controlled with the feedback (21) has the same characteristic polynomial as the linear discrete system without delay realized by the sampling of the system
u(k~) = F
gIves det P(z) = det(D(z))zm. p
(p-1)Ll+h' -Fx(k~)
la ~:-P)
p)~)
-
- h)r( -~)FzP
-f(~
- h')Fz -
Its z-transform is then: (23)
Hence, we have
with
D(z) = (Inz p
dc(z) = zP - F
-
- h)f( -~)F) zp.
The characteristic polynomial of the closed-loop system is then
(
1=1
+F
~).
zmp det(D(z)) = zmp det (zP (In z - e ALl
-J Ll
The z-transform of the system (12) is : (f(~
x =
- h')z +
eA(Ll-r)dTe- Ah BF )).
(24)
o As the sampling of the system (22) gives
The characteristic polynomial of the closed-loop system is the determinant of
x((k
P(_)=((Inz-
.<0
-B(z) ) (25) ImzP-HM'
+ 1)~) = e ALl x(k5)
JeA(Ll-r)dTe-AhBu(k~), Ll
+ with
o the proof is achieved. 0
P
M(z) = L
The continuous control law (13) was designed so that the spectrum of the system with delay is shifted to be the spectrum of the matrix A + e- Ah BF, hence, the spectrum of the system without delay (22) controlled
1=1
-
B(z) =
r(~ -
h')z +
199
by the feedback u(t) = Fx(t). The discrete control law we present here allow to retain this spectrum assignment property when sampling the system.
Van Assche, V., M. Dambrine, Lafay J.-F. and J.P. Richard (1999). Some problems arising in the implementation of distributed-delay control laws. In: Proc. 38th IEEE CDG. p. 4668. Watanabe, K. (1986). Finite spectrum assignment and observer for multivariable systems with commensurate delays. IEEE Trans. Automatic Control AC-31(6), 543-550. Watanabe, K., M. Ito, M. Kaneko and T. Ouchi (1983). Finite spectrum assignment problem of systems with delay in state variables. IEEE Trans. Automatic Control AC-28(4), 506-508. Wittenmark, B. (1985). Sampling of a system with a time delay. IEEE Transactions on Automatic Control 30(5), 507-510.
Remark 2. In order to control a system (6) when only Xl (t) is measured, and not x~ (t), one can remark that, as x~(t) = XI(t - h), one has for the sampled system X~ (k~) = eA(Ll-h')x«k - p)~) Ll-h'
+
J
eA(Ll-h'-T)drBu«k -
p)~).
o This equation allows to deduce x~ (k~) from the knowledge of x(t) and u(t) at the sampling times.
5. CONCLUSION We have shown above that our method for the numerical implementation of the control law with distributed delay (13) for a system with an input delay is equivalent to the numerical implementation of a stabilizing control law for a linear system without delay. One can remark that the numerical control law we obtain is similar to the control law one would get by using a spectrum assignment method for sampled system. Indeed, our approach allows to implement a spectrum assignment control law, but this can be done without computing the sampled system, and especially the exponential of the matrix A. This associated with the method of (Brethe and Loiseau, 1996) allow the design of a numerical regulator for system with input delay with a limited amount of computation.
6. REFERENCES Brethe, D. (1997). Contribution a l'etude de la stabilisation des systemes lineaires a retards. These de doctorat. Ecole Centrale de antes. Brethe, D. and J. J. Loiseau (1996). A result that could bear fruit for the control of delaydifferential systems. Proc. IEEE MSCA pp. 168172. Manitius, A. Z. and W. Olbrot (1979). Finite spectrum assigment problem for systems with delays. IEEE Trans. Automatic Control AC24(4), 541-553. Mondie, S., M. Dambrine and O. Santos (August 2001). Approximations of control laws with distributed delays: a necessary condition for stability. To appear in IFAC SSSC 2001 (First Symposium on System Structure and Control, Prague). Morse, A. S. (1976). Ring models for delaydifferential systems. Automatica 12, 529-531. T
200
Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001
~
Publications www.elsevier.com/locate/ifac
IMPLEMENTATION OF A DISTRIBUTED CONTROL LAW FOR A CLASS OF SYSTEMS WITH DELAY V. Van Assche* M. Dambrine** J.-F. Lafay* J .-P. Richard **
* IRCCyN, UMR CNRS 6597, E. C. Nantes, 1, rue de la Noe,
B.? 92101, 44321 Nantes Cedex 03, France ** LAIL, U?RESA CNRS 8021, E. C. Lille, B.? 48, 59651 Villeneuve d'Ascq Cedex. France
Abstract: The use of distributed delays in the control law of time-delay systems has been proposed by several authors. The implementation of such controllers is not trivial, and recent publication shown that replacing the distributed delay operator by an approximation computed through pointwise delay operators was unsafe with respect to the stability of the system. In this paper the use of digital controller is addressed, for the control of systems with an input delay. Practically, this means replacing an integral by a recurrent system. The first approach was to use a method of numerical approximation of an integral to build the recurrence law. In this paper it is shown that a such controller, built with the Simpson method, leads to an unstable closed-loop system. A second approach leads to the construction of a control law which realizes a sampled pole assignment, in the same way as the distributed control law realizes a pole assignment in continuous time. Copyright © 20011FAC
1. INTRODUCTION
poles in the complex plane. The control laws will have the form:
This paper deals with some pole assignment methods for systems with delays. Throughout the paper, we will consider systems with commensurate delays, that is all delays are integer multiple of a real noted h. We define \7, the delay operator, such that, for any integer k, \7 k x(t) = x (t - kh). A linear system with commensurate delays will be written the following way: X (t) = A (\7) x(t)
+ B (\7) u(t)
(2)
where Ut (.) and Xt (.) are functions defined on the interval [-7,0] by ut(B) = u(t+B) and xt(B) = x(t+ B), for e E [-h,O]. Very often, the study and the control design of timedelay systems deal only with punctual delays, i.e. of the form x( t - 7), excluding integrals over the past of the command and state, this allows to work in a nice algebraic framework. In (Morse, 1976), it is shown that such a law allows a finite spectrum assignment only if the system fulfills the rather restrictive condition of strong controllability. Manitius and Olbrot (1979) show that spectral controllability is a sufficient condition for finite spectrum assignment with a control law of the form
(1)
A (\7) and B (\7) being two matrices of polynomials in \7. The Laplace transform of \7 k x(t) is e- khs X(s), where X (s) is the Laplace transform of x( t). Precisely, we are interested in method of finite spectrum assignment, Le. methods allowing to reduce the number of poles of a system with delays to a finite number and to assign arbitrarily the location of these
195
a right half-plane pole-zero cancellation. In order to avoid such a phenomenon, Manitius and Olbrot recommend to compute directly z(t) using a numerical quadrature method. Van Assche et al. (1999) for the example above, and Mondie et al. (August 2001) for the generalization to the case of systems with input delay, show that an approximation of the distributed delay with a finite sum of pointwise delays may give an unstable closed-loop system. Moreover, several examples of numerical regulators computed to approximate the continuous theoretical control law are given in (Van Assche et al., 1999), with either the rectangular, trapezoidal and Simpson quadrature methods. The first two methods give a stable closedloop system, although the last one, which is supposed to be the most precise, gives an unstable system, even when changing the sampling period.
J +J t
+
u(t) = F(V)x(t)
j(8)x(8)d8
t-r
t
+H(V)u(t)
h(8)u(8)s8.
t-r
The system (1) is said to be spectrally controllable if, and only if, for any SEC, the matrix (sI - A(e- S ) B(e- S )) is of rank n. In (Watanabe et al., 1983), the necessity of this condition is proven for SISO systems, and in (Watanabe, 1986), for MIMO systems. Recently, an algebraic background has been formalized in (BretM and Loiseau, 1996) and (Brethe, 1997) for systems with distributed delays. The authors define the ring of the Laplace transform of both distributed delays and punctual delays. This formalism leads to a n-assignation algorithm which is much simpler than those of (Watanabe et al., 1983) and (Watanabe, 1986).
In this paper, it is shown that a numerical regulator constructed through an approximation of the control law (4) will give an unstable closed-loop system regardless of the sampling period. Then, our contribution is to provide a safe way to implement a stabilizing control law of the type (2) using a numerical regulator without loss of the stability for the closed-loop system.
The problem we want to deal with in this article is the implementation of the distributed delays. As the integral term in (2) cannot be exactly computed with a numerical computer, to use these command laws implies to approximate it. We will here show that the use of quadrature methods involves stability problems especially when delays are on the input variables.
2. SYSTEM WITH INPUT DELAY 2.1 Continuous system with an input delay
As an illustration of this problem, let us consider the very simple unstable system:
x(t) = x(t)
+ u(t -
1).
In this paper, we consider systems with a delay on the input. The state space description of such systems is:
(3)
x(t) = Ax(t)
With the control law
+ Ad) (e1x(t)
u(t) = - (1
J
e IJ u (t - 8) d8) ,
(5)
The dimension of the vectors x(t) and u(t) are noted respectively nand m.
1
+
+ V Bu(t).
(4)
o
Remark 1. A larger class of systems can be turned into a system with delay on the input. Indeed, if a system correspond to the scheme 1, it can be written in state space form:
which is a Volterra equation of second kind, the closed-loop system has one pole in -Ad. With Ad > 0 , the system is stable, but, as we stated above, the problem we are faced with is the way of realizing the integral term
Xl(t) = Al1Xl(t) + Bu(t), X2(t) = VA21Xl(t) + A22X2(t),
J
(6)
1
z(t) =
where the vectors Xl(t) and X2(t) are of dimensions nl and n2, and u(t) is a real valued function, and where All, A 21 and A 22 , and B are matrices of convenient dimensions over the field lR of real numbers.
eIJu(t - 8)d8.
o A simple way of proceeding is to differentiate the previous expression. We obtain the differential equation
i(t) = z(t)
+ u(t) -
eu(t - 1).
Fig. 1. System (6)
~
But, as explained in (Manitius and Olbrot, 1979), this realization is unsafe since unstable, and in this case the stabilization of the process is the result of
196
SI
I
xl
r::::l~2
.,~
S2
~2
_ _.....J
Such a system can be turned into a system with a delay only on the input with x~ (t) = \7Xl(t) :
(9) can be rewritten
x((k +
1)~) = q>(~)x(k~)
+ +
(7)
x(t)=(x~(t)) X2(t)
If the system (5) is spectrally controllable, one can apply the finite spectrum assignment methods. This gives the following control law:
and
(All A A
0 )
21
22
.
=
2.2 Sampling of the system (5)
+ v(t),
see (Manitius and Olbrot, 1979).
J eA(k~+~-T) BdTU(k~).
We propose here a numerical computation of the integral, resulting in a discrete controller, which is clearly implementable. Moreover, the discrete system obtained with this method is similar to the system one gets when discretizing the theoretical continuous system obtained with the control law (13).
k~
For a system with a delay of value h on the input, the expression above is directly adapted if the delay is a multiple of the sampling period, i.e. if there exists an integer p with h = p~ :
Before stating this original result, we first prove that the use of a quadrature method is not a good approach to solve this problem.
x(k~ +~) = eA~x(k~)+
JeA(~-T) BdTU(k~ ~
_ p~). (8)
o
3.1 Instability due to a Simpson's quadrature
If the delay is not a multiple of the sampling period, we note h = (p -1)~ + h' and, following Wittenmark (1985), one has
In a first approach, we compute the integral term through quadrature methods such as the rectangular method, the trapezoidal method and the Simpson's rule, see (Van Assche et al., 1999). This last method is the one of higher order. Meanwhile, it gives the worst result in our simulations.
= eA~x(k~)
JeA(~-h'+T) ~ p)~) J eATdTu((k-p+1)~). h'
BdTU((k
o
(13)
(14)
k~+~
+
eA(-h-T) Bu(t - T)dT
where H is a real matrix of dimension 1 x n. The spectrum of the closed-loop system realized by (5) controlled by (13) is the spectrum of the matrix
x(k~ +~) = eM'x(k~)
+ 1)~)
-F
o -Fx(t)
In order to implement a control law with a numerical device, the system has to be sampled. The sampling period is noted ~, and the input u(t) is constant between the sampling times. For a system without delay, it is well known that the equation of the sampled system is given by:
x((k
J h
u(t)
+
- h')r(h')u((k - p)~) (12) - h')u((k - p + 1)~).
3. CONTROL LAW FOR SPECTRUM ASSIGNMENT OF THE SYSTEM WITH AN INPUT DELAY
One notes then
A _ -
q>(~ r(~
(9)
We show that the closed-loop system is unstable for the closed-loop system (3)-(4), with h = 1 = p~, p being an even integer, and with Ad = 1.
~-h'
+
o
For this choice of h, becomes, with (8),
By writing
x((k + 1)~)
q>(t) =e At
(10)
J
(11)
t
r(t) =
eATdT,
~ p
and Ad, the system (3)
= e~x(k~) +
(e~
- 1) u(k~). (15)
Its z-transform is : zP
o
197
(z -
e~) x = (e~
- 1) u.
(16)
The discrete control law is :
Therefore,
~
u(k~) = -2"3w(k~) - 2elx(k~)
+ v(k~) (17)
with p/2-1
w(k~) = u(k~)
+
L
2e 21 ti. u ((k - 2l)~)
1=1 p/2
+
L 4e(21-1)ti. u ((k -
2l
+ 1)~)
and, for z
1=1 + elu((k - p)~).
=
-1,
The z-transform of (17) is:
de(z)u
= 2e l z px+ zPv
(18)
with
As p-l
~ "~" e(k+l)ti. = ti.~O Hm
k=O
zP (z - eti. ) de (z) - (eti. - 1) 2e 1 zP ,
lim d( -1) = -2
and the closed-loop system is stable if, and only if, the polynomial
As the degree of d(z) is p has for any value of ~,
(19)
L ekti.zp-k = eti.(zP -
ell,
(20)
k=l
and that
=
(d-l zP(l + 2"3) + 2"3 L 2e1ti. z(p-2l)
=
-00,
The computation above show that approximating the controller with a quadrature method is not always a safe approach to the problem. The next section of this work presents another approach, which leads to a safe implementation of the controller.
~
1=1 +
an odd number, one
hence, for ~ sufficiently small, d( z) has at least one root on the real axis, of magnitude strictly superior to 1. This show that the system with this controller is unstable, for any sufficiently small value of ~, i.e. for the most precise approximation of the integral term of (4).
p
de(z)
+ 1,
- 1) > O.
Z-------l>-OO
First, let us simplify the term dl(z) by remarking that
~
+ ~(el 3
lim d(z)
has all its root inside the unit disc.
(z - eti.)
,
0
ti.~O
d(z) =
1
One can check that
The denominator of the closed-loop system is then:
(z - eti.) de(z) - (eti. - 1) 2e l
1
JeT dT = e l -
Lp/2 2e(21-1)ti. z(p-21+1) + e ) l
.
1=1 With some computation, this leads to -)_ 3+2~ p+l Z d 1'" ( -
+ 4~
3 p-l
L
-
3-2~ ti. p ---e z 3
4. SAFE IMPLEMENTATION This parts deals with systems of the form (7), sampled into (12). The control law is also sampled with the hypothesis that u( t) is a bloc-pulse function, Le. u(t) = u(k~) for k~ ~ t < (k + 1)~. This gives
((_I)k e (k+l)ti. z n-k)
k=O 2~
1
2~ Hti.
- - e z+-e 3 3
.
198
u(k~) =
Right multiplying the matrix P(z) by (
J III
p
-FL
eA(-h+r)BdTU((k
1
+F
0) .
-l)~)
1=1 (1-1)Ll
p)~)
eA(-h+r) BdTU((k -
In ( -F I m
(21)
with
+ v(k~).
i(t) = Ax(t)
+ Bu(t)
Now, let us expand
D 1(z) = (Inz -
(22)
P
- (In z -
1=1
u(t) = Fx(t),
+
p
(
D 1(z) = Inzp+1 -
1=1
-
~)F.
With a remark similar to (20), and remarking that f( -~) - r(h' - ~) = -
Proof: With the definitions (10) and (11) the control law is rewritten:
~)u((k
~)Fz
-
up to a multiplication by a power of z.
+ p~)f(h' -Fx(k~) + v(k~).
-
+B(z)F.
sampled with the same sampling period and controlled by the feed back
-
D(z) = (Inz -
Proposition 1. The system with an input delay (5), sampled with a period ~, and controlled with the feedback (21) has the same characteristic polynomial as the linear discrete system without delay realized by the sampling of the system
u(k~) = F
gIves det P(z) = det(D(z))zm. p
(p-1)Ll+h' -Fx(k~)
la ~:-P)
p)~)
-
- h)r( -~)FzP
-f(~
- h')Fz -
Its z-transform is then: (23)
Hence, we have
with
D(z) = (Inz p
dc(z) = zP - F
-
- h)f( -~)F) zp.
The characteristic polynomial of the closed-loop system is then
(
1=1
+F
~).
zmp det(D(z)) = zmp det (zP (In z - e ALl
-J Ll
The z-transform of the system (12) is : (f(~
x =
- h')z +
eA(Ll-r)dTe- Ah BF )).
(24)
o As the sampling of the system (22) gives
The characteristic polynomial of the closed-loop system is the determinant of
x((k
P(_)=((Inz-
.<0
-B(z) ) (25) ImzP-HM'
+ 1)~) = e ALl x(k5)
JeA(Ll-r)dTe-AhBu(k~), Ll
+ with
o the proof is achieved. 0
P
M(z) = L
The continuous control law (13) was designed so that the spectrum of the system with delay is shifted to be the spectrum of the matrix A + e- Ah BF, hence, the spectrum of the system without delay (22) controlled
1=1
-
B(z) =
r(~ -
h')z +
199
by the feedback u(t) = Fx(t). The discrete control law we present here allow to retain this spectrum assignment property when sampling the system.
Van Assche, V., M. Dambrine, Lafay J.-F. and J.P. Richard (1999). Some problems arising in the implementation of distributed-delay control laws. In: Proc. 38th IEEE CDG. p. 4668. Watanabe, K. (1986). Finite spectrum assignment and observer for multivariable systems with commensurate delays. IEEE Trans. Automatic Control AC-31(6), 543-550. Watanabe, K., M. Ito, M. Kaneko and T. Ouchi (1983). Finite spectrum assignment problem of systems with delay in state variables. IEEE Trans. Automatic Control AC-28(4), 506-508. Wittenmark, B. (1985). Sampling of a system with a time delay. IEEE Transactions on Automatic Control 30(5), 507-510.
Remark 2. In order to control a system (6) when only Xl (t) is measured, and not x~ (t), one can remark that, as x~(t) = XI(t - h), one has for the sampled system X~ (k~) = eA(Ll-h')x«k - p)~) Ll-h'
+
J
eA(Ll-h'-T)drBu«k -
p)~).
o This equation allows to deduce x~ (k~) from the knowledge of x(t) and u(t) at the sampling times.
5. CONCLUSION We have shown above that our method for the numerical implementation of the control law with distributed delay (13) for a system with an input delay is equivalent to the numerical implementation of a stabilizing control law for a linear system without delay. One can remark that the numerical control law we obtain is similar to the control law one would get by using a spectrum assignment method for sampled system. Indeed, our approach allows to implement a spectrum assignment control law, but this can be done without computing the sampled system, and especially the exponential of the matrix A. This associated with the method of (Brethe and Loiseau, 1996) allow the design of a numerical regulator for system with input delay with a limited amount of computation.
6. REFERENCES Brethe, D. (1997). Contribution a l'etude de la stabilisation des systemes lineaires a retards. These de doctorat. Ecole Centrale de antes. Brethe, D. and J. J. Loiseau (1996). A result that could bear fruit for the control of delaydifferential systems. Proc. IEEE MSCA pp. 168172. Manitius, A. Z. and W. Olbrot (1979). Finite spectrum assigment problem for systems with delays. IEEE Trans. Automatic Control AC24(4), 541-553. Mondie, S., M. Dambrine and O. Santos (August 2001). Approximations of control laws with distributed delays: a necessary condition for stability. To appear in IFAC SSSC 2001 (First Symposium on System Structure and Control, Prague). Morse, A. S. (1976). Ring models for delaydifferential systems. Automatica 12, 529-531. T
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