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REGULATOR PROBLEM FOR LINEAR SYSTEMS WITH CONSTRAINTS ON CONTROL AND ITS INCREMENT
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
www.elsevier.com/locate/ifac
REGULATOR PROBLEM FOR LINEAR SYSTEMS WITH CONSTRAINTS ON CONTROL AND ITS INCREMENT Fouad MESQUINE , Fernando TADEO , and Abdellah BENZAOUIA
University Cadi Ayyad, Faculty of Sciences Department of Physics, Research unit Constrained Robust Regulation B.P. 2390, Marrakesh Morocco E-mail: [email protected], [email protected] Departamento de Ingenieria de Sistemas y Automatica Univesidad de Valladolid, 47005 Valladolid, Spain. E-mail: [email protected]
Abstract: This paper discusses the problem of constraints on both control and its increment for linear systems in state space form, in both the continuous and discrete time domains. For autonomous linear systems with constrained increment, necessary and sufficient conditions are derived, such that the evolution of the system respects the incremental constraints. It is also derived a pole placement technique to solve inverse problem, deriving stabilizing state feedback controllers which respect constraints on both control and its increment. An illustrative example shows the application of the method. Copyright © 2002 IFAC
Keywords: Keywords: Linear Systems, Constraints, Control, Increment, Positive Invariance, Pole assignment
work has been published on incremental constraints using state space representations. Henceforth, this paper investigates the problem of stabilizing linear continuous and discrete time systems with constraints on both control and control increment. Necessary and sufficient conditions of positive invariance for incremental domains with respect to autonomous systems are given. Furthermore, a link is done between pole assignment procedure and these conditions to find stabilizing controllers by state feedback.
1. INTRODUCTION Usually, real plants or physical plants are subject to constrained variables. The most frequent constraints are of saturation type, that is, limitations on the magnitude of certain variables. Hence, this topic is of continuing interest and one could cite, not exhaustively, (Benzaouia, and Burgat, 1988; Benzaouia, and Hmamed, 1993; Benzaouia, and Mesquine, 1994; Blanchini 1990;1999 and the references Therein). Other type of constraints were introduced while considering predictive control (GPC) and practical applications that is incremental constraints (Dion, et al.,1987; Clarke et al. 1987; Warwick, and Clarke, 1988). In fact, for some processes, the rate of variables change is limited within certain bounds. These limits can arise from physical constraints that, if exceeded, could damage the process. From our knowledge, no
1.1 Notations:
If is a vector, denotes its derivative with respect to time in the continuous time case or in the discrete-time case. Further, for a scalar we define and , and then we note that
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Consider also that the state increment is constrained as follows:
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2. PROBLEM STATEMENT
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Consider a linear time invariant system represented in the state space by:
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where is the state of the system, is the input constrained to evolve in the following domain "
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We give also a technical lemma that will be related to a pole placement procedure to find stabilizing controllers for systems with constrained control and increment.
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Lemma 2. The evolution of the autonomous system (6) respects incremental constraints if and only if matrix satisfies: q
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for the discrete-time case
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for the continuous-time case
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The control increment is constrained as follows: where
i) For discrete-time systems:
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ii) For continuous-time systems: Z
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Proof. If part Consider the discrete time case and
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(4)
assume that condition (10) is satisfied. Hence, it is possible to write:
We denote
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the problem studied in this paper is the following: find a stabilizing state feedback as
but it is known that p
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(5) Next, decompose matrix multiplying (12) by and
ensuring closed-loop asymptotic stability of the system with non saturating controls that also respects incremental constraints.
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pre, respectively, gives
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Positive invariance conditions have already been proposed in (Benzaouia, and Burgat, 1988; Benzaouia and Hmamed, 1993). Using these conditions it is possible to derive the following result:
(14)
addition of inequalities (13) and (14) enables us to write: |
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Lemma 3. Domain (7) is positively invariant with respect to motion of system (6) and increment constraints (3) are respected if and only if
according to condition (10)
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for the discrete-time case
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}
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°
for the continuous-time case
which is equivalent to
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¢
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Proof. For the increment constraints, conditions can be derived from the previous lemma, and the positive invariance conditions are given in (Benzaouia, and Burgat, 1988; Benzaouia, and Hmamed, 1993).
In the continuous time case,
following the same reasoning, replacing matrix by matrix , and condition (10) by condition (11), it is easy to obtain }
Relating the previous lemma to a pole placement procedure makes possible to solve the problem stated above. Recall the pole assignment procedure used in the so-called Inverse Procedure for constrained control (Benzaouia, 1994). Consider the time invariant system given by (1). Without loss of generality (see Remark below), assume that matrix possesses stable eigenvalues. Resolution of equation
|
Only if part, Consider the continuous time case. Assume that the derivative of respects the constraints and that condition (11) is not satisfied for an index such that
|
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(15)
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The following state vector for the system can be selected
if
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µ
(16)
gives us a state feedback assigning spectrum of matrix ( together with the stable part of spectrum of matrix in closed loop. For this equation to have a valid solution, matrix must satisfy the following conditions:
µ
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if ¢
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(17)
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It is easy to check that is an admissible state for component of the the system. Calculation of the derivative of this state gives
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are linearly independent ¶
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¾ ¾
such that , that is eigenvectors for of matrix There exits a unique solution to equation (16) if and only if
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are linearly independent, where are eigenvectors associated to stable eigenvalues of matrix and are computed by
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taking into account inequality (15), it is possible to write
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Hence, the solution is given by:
which contradicts the assumption. The discrete time part could be easily deduced replacing matrix by matrix in the necessary part.
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Evolution of the autonomous system (6) will respect both constraints on the state and constraints on its increment if domain given by (7) is positively invariant and conditions given in the previous lemma are satisfied. ®
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Remark 4. Without loss of generality, it was assumed that the system presents stable eigenvalues. If the system matrix does not satisfy such requirement,
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it is always possible to augment the representation as follows: let be a vector of fictitious inputs such that Ø
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If the state does not leave the domain (23), the control signal does not violate the constraints. That is, the set is positively invariant with respect to motion of system (20). This gives the following result: ô
Ý
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Ñ
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where and are freely chosen constraints on the fictitious inputs. In this case, vectors and become: É
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×
â
ã â
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Proposition 5. System (20) with state feedback (21)(22) is asymptotically stable at the origin with constraints on both the control and its increment if there exists a matrix satisfying conditions (17) such that:
ã
ã ã
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É Ô
Õ
Ê
×
Ø
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ä
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The augmented system is then given by Ü
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(19) è
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It is true that this augmentation limits the domain of linear behavior of the closed-loop system, but it is always possible to soften the fictitious limitations to enlarge the domain. It is easy to see that for the stable square system obtained the problem of eigenvalues is eliminated and controllability is not changed.
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for the discrete-time case
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for the continuous-time case
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4. MAIN RESULTS
í
(25)
where
With this background, we are now able to solve the problem stated in section 2. Consider a stabilizable linear time invariant system with constraints on both control and increments of the control, that is
Ñ
×
Ø
for all initial state
Î
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Ò
Ó Î
Ü
Ï
Ð
å
æ
ç
è
Ø
é
å
æ
ç
è
ê
ì
æ
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Proof. Introduce the following change of coordinates:
(20) è
æ
ç
è
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Ë
æ
ç
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ý
å
it is possible to write
is the state of the system, where the input constrained to evolve in the following domain å
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å
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å
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(26) è
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ö
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æ
ç
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æ
ç
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With this transformation, domain is transformed to domain given by (7). Further, with conditions (24) and (25), it is easy to note that domain is positively invariant with respect to the system (26) while the constraints on the increment of the control are respected. Bearing in mind that and that the linear behaviour is guaranteed, one can conclude to the asymptotic stability of the closed-loop system. ô
ø
Î
Ï
Ð
Ì
Î
Ò
Ó
Ê
ù
Ë
Ý
Ü
Ü
Ü
Ü
and the increment of the control is constrained as follows: Ý
Ü
÷
ô
Ü
Ô
Î
Ï
Ð
Ñ
æ
ú
ê
û
æ
è
Í
æ
ú
è
Ñ
Ô
Î
Ò
Ó
For continuous-time systems Ü
Í
Ô
è
Î
Ø
Ï
Ð
Ñ
ü
æ
ú
è
Ñ
Ô
ý
å
æ
ç
è
Ì
ý
Ê
Î
Ò
è
Ê
Algorithm:
þ Ý
é
ý
ÿ
ß
á
Ü
æ
ì
Remark 6. It is worth noting that conditions (24) and (25) do not affect the set of positive invariance . However, they present an additional constraint on the pole assignment problem.
Ó
(21) Ë
ô
such that
ê
ô
Ü
ç
é
Ü
Using the state feedback æ
ÿ
þ
For discrete-time systems Í
ê
ì
ý
è
Ê
(22)
Step 1. Check if matrix has ues, if not augment the matrix column. é
the following domain of linear behaviour is induced in the state space
88
æ
ò
Í
ó
è
ì
stable eigenvalwith null ò
Í
ó
Step 2. Choose matrix or , if the system is augmented, according to (17)-(24) or (25). Step 3. Compute the gain matrix or by using Step 4. Use or extracted from for the control.
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−5
x2
Example 7. Consider the double integrator in the continuous-time state space representation by (Gutman, and Hagander, 1985):
−10 '
(
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(
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,
4
5
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,
,
.
0
2
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Control constraints are given by; (
−15
(
6
6
,
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7
9
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−20 −20
Assume that control increments are constrained as follows: :
−10
−15
−5
0
Fig. 1. Set of positive invariance emanating from
:
W
\
(
;
>
(
;
(
;
5
S
T
U
A
25
20
15
10
5x1
30
with a trajectory
A
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,
9
@
7
I
As discussed in the Remark above, the system can be augmented with fictitious constrained inputs in domain . Select the following matrix :
5
=
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=
4.5
=
7
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4
;
3.5 (
, *
4
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,
,
4
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.
3 increment
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which satisfies all the required conditions (17). Further, the fictitious constraints are selected such that conditions (25) are satisfied. :
(
A
(
:
C
;
2.5
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2
( (
1.5 5 5
=
9
7
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# 9
4
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1
that is, D
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5
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5
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A
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20
30 time
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10
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40
60
50
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4
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Fig. 2. Evolution of the control increment I
;
,
5
5
5
> #
(
J
@
where and
I
Solution of equation (16) leads to the following augmented gain matrix :
,
6
Y
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^ =
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7
I
7
6
4 ^
Y
>
a
b
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Figure 1 represents the set of positive invariance and a trajectory emanating from the initial state , while Figure 2 shows the evolution of the control increment, which, it is possible to see, respects the constraints and .
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Note that the effective gain matrix from the previous matrix.
can be extracted
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It is easy to note here that . The obtained set of positive invariance, as shown in Figure 1, with the augmented matrix is given by: R
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In this paper, the regulator problem for linear systems with constraints on both control and its increment in the state space representation has been studied. Application of necessary and sufficient conditions, established for linear autonomous systems such that their motion respects incremental constraints, is the key to solve this problem. The link of the so called inverse procedure, the pole assignment method for
,
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;
,
5. CONCLUSION
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The closed-loop dynamics are given by: '
;
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[
89
Warwick, K., and D.W. Clarke (1988), Weighted input predictive controllers. IEE Proceedings Part D, Vol., 135, No 1.
constrained control, to the previous conditions is the cornerstone of this work. In fact, this link enables to give a simple algorithm to compute a regulator respecting constraints on both control and its increment.
ACKNOWLEDGEMENTS This work is the product of collaboration between Unversity Cadi Ayyad Morocco and University of Valladoli, Spain under grant N 23P/2000
6. REFERENCES Benzaouia, A. and C. Burgat (1988), Regulator problem for linear discrete-time systems with non symmetrical constrained control, Int. J. Control,Vol. 48, No. 6, pp. 2441-2451. Benzaouia, A., and A. Hmamed (1993), Regulator problem of linear continuous-time systems with constrained control, IEEE Trans. Aut. Control, Vol.38, No.10, pp. 1556-1560. Benzaouia, A. (1994), Resolution of equation XA+XBX=HX and the pole assignment problem, IEEE Trans. Aut. Control, Vol. 40, No. 10, pp. 2091-2095. Benzaouia, A., F. Mesquine (1994), Regulator problem for uncertain linear discrete time systems with constrained control, International Journal of Nonlinear and Robust Control, Vol. 4 pp. 387-395. Blanchini, F. (1990), Feedback control for linear systems with state and control bounds in the presence of disturbance, IEEE Trans. Aut. Control, Vol. 35, pp. 1131-1135. Blanchini, F. (1999), Set Invariance in control, Automatica Vol. 35, pp. 1747-1767. Dion, J.M., L. Dugard, & M. Tri (1987), Multivariable adaptive control with input-output constraints, IEEE Conference on Decision & Control, Los Angeles CA, December 1987. Gutman, P.O. and P. Hagander (1985), A new design of constrained controllers for linear systems, IEEE Trans. Aut. Control, Vol. 30, pp. 22-33. Clarke, D.W., C. Mohtadi, and P.S. Tuffs (1987) Generalised predictive control, Part 1: the Basic algorithm; Part 2: Extension and Interpretations. Automatica, Vol., 23, No 2, pp. 137-160.
90
www.elsevier.com/locate/ifac
REGULATOR PROBLEM FOR LINEAR SYSTEMS WITH CONSTRAINTS ON CONTROL AND ITS INCREMENT Fouad MESQUINE , Fernando TADEO , and Abdellah BENZAOUIA
University Cadi Ayyad, Faculty of Sciences Department of Physics, Research unit Constrained Robust Regulation B.P. 2390, Marrakesh Morocco E-mail: [email protected], [email protected] Departamento de Ingenieria de Sistemas y Automatica Univesidad de Valladolid, 47005 Valladolid, Spain. E-mail: [email protected]
Abstract: This paper discusses the problem of constraints on both control and its increment for linear systems in state space form, in both the continuous and discrete time domains. For autonomous linear systems with constrained increment, necessary and sufficient conditions are derived, such that the evolution of the system respects the incremental constraints. It is also derived a pole placement technique to solve inverse problem, deriving stabilizing state feedback controllers which respect constraints on both control and its increment. An illustrative example shows the application of the method. Copyright © 2002 IFAC
Keywords: Keywords: Linear Systems, Constraints, Control, Increment, Positive Invariance, Pole assignment
work has been published on incremental constraints using state space representations. Henceforth, this paper investigates the problem of stabilizing linear continuous and discrete time systems with constraints on both control and control increment. Necessary and sufficient conditions of positive invariance for incremental domains with respect to autonomous systems are given. Furthermore, a link is done between pole assignment procedure and these conditions to find stabilizing controllers by state feedback.
1. INTRODUCTION Usually, real plants or physical plants are subject to constrained variables. The most frequent constraints are of saturation type, that is, limitations on the magnitude of certain variables. Hence, this topic is of continuing interest and one could cite, not exhaustively, (Benzaouia, and Burgat, 1988; Benzaouia, and Hmamed, 1993; Benzaouia, and Mesquine, 1994; Blanchini 1990;1999 and the references Therein). Other type of constraints were introduced while considering predictive control (GPC) and practical applications that is incremental constraints (Dion, et al.,1987; Clarke et al. 1987; Warwick, and Clarke, 1988). In fact, for some processes, the rate of variables change is limited within certain bounds. These limits can arise from physical constraints that, if exceeded, could damage the process. From our knowledge, no
1.1 Notations:
If is a vector, denotes its derivative with respect to time in the continuous time case or in the discrete-time case. Further, for a scalar we define and , and then we note that
85
!
"
#
%
&
"
#
(
and )
"
*
%
&
"
*
(
for )
%
,
-
.
.
.
-
3. PRELIMINARY RESULTS /
+
Furthermore, for a matrix the Tilde Transforms are defined by %
&
2
)
3
(
2
/
4
5
-
%
0
,
-
.
.
.
-
/
Consider the linear time invariant autonomous system +
M
p
p
p
p
-
&
N
)
%
q
&
N
)
-
&
i
r
)
%
(6) r
p
#
where domain
*
7
0
%
0
8
0
*
#
0
:
T
is the state constrained to evolve in W
U
0
p
where and
%
&
2
)
#
3
and (
#
0
%
&
2
)
*
3
(
-
5
-
%
,
-
.
.
.
-
p
p
p
/
*
W %
Y
T
U
-
Z
[
\
]
^
&
p
0
N
)
^
[
_
`
p
+
J
s
W [
\
]
-
[
_
`
T
a
U
#
0
?
0
=
:
0
A
0
?
For discrete time systems:
where
p
2
3
%
C
and -
for
# 2
3
(
p
(8)
3
Z
?
b
Consider also that the state increment is constrained as follows:
A
8 %
0
0
(7)
<
5
%
E
%
0
C
d
[
\
]
^
&
e
P
,
)
Z
&
e
)
^
d
[
_
`
G
for
A
* 2
3
(
+
5
%
E
+
For continuous time systems: p
Z
Also, denotes the spectrum of matrix ; denotes the stability domain for eigenvalues (that is, the left half plane in the continuous-time case or the unit disk in the discrete-time case). &
d
[
\
]
^
h
&
i
)
^
d
[
_
(9) `
)
I
0
0
J
K
First, recall the definition of positive invariance of domain which is very useful for the sequel. J
s
given by (7) is positively Definition 1. Domain invariant with respect to motion of system (6) if for all initial condition the trajectory of the system for all J
s
p
p
p
r
2. PROBLEM STATEMENT
i
-
i
r
-
r
)
Consider a linear time invariant system represented in the state space by:
i
N
)
%
"
&
N
)
P
Q
R
&
N
(1) )
0
where is the state of the system, is the input constrained to evolve in the following domain "
&
N
)
T
U
V
R
&
N
)
T
U
t
i
r
s
We give also a technical lemma that will be related to a pole placement procedure to find stabilizing controllers for systems with constrained control and increment.
M
&
-
s
T
J
"
T
J
&
W
Lemma 2. The evolution of the autonomous system (6) respects incremental constraints if and only if matrix satisfies: q
W %
Y
R
&
N
)
T
U
-
Z
R
[
\
]
^
R
&
N
)
^
R
[
_
`
u
J
X
&
W R
[
\
]
-
R
[
_
`
T
.
#
q
Z
v
)
w
^
for the discrete-time case
(10)
for the continuous-time case
(11)
d
(2)
a
U
b
7
q
w
^
d
The control increment is constrained as follows: where
i) For discrete-time systems:
p
p
[
w
Z
d
[
\
]
^
R
&
e
P
,
)
Z
R
&
e
)
^
d
[
_
`
d
[
\
]
^
h
R
&
i
)
_
d `
8
-
(3) [
ii) For continuous-time systems: Z
%
\
]
d
%
[
_
`
8
d :
[
\
]
:
Proof. If part Consider the discrete time case and
^
d
[
_
`
(4)
assume that condition (10) is satisfied. Hence, it is possible to write:
We denote
p
p
p
p
j
p
&
R
%
[
_
d `
8
d
%
[
_
e
P
,
)
Z
&
e
)
%
q
[
\
]
d :
[
\
]
&
the problem studied in this paper is the following: find a stabilizing state feedback as
but it is known that p
Z
W &
N
)
%
m
"
&
N
)
-
m
T
U
o
V
q
)
Z
Z
v
&
)
&
e
e
)
)
:
%
R
e
p
%
R
&
`
8
z
&
p
[
\
]
e
)
-
p
^
&
e
)
^
[
_
(12) `
(5) Next, decompose matrix multiplying (12) by and
ensuring closed-loop asymptotic stability of the system with non saturating controls that also respects incremental constraints.
z
z
Z
86
z
#
z
Z
z *
p
[
\
]
^
z
#
Z
z
#
#
p
pre, respectively, gives
%
-
*
&
p
e
)
^
z
#
[
_
`
(13)
|
}
~
|
}
~
}
~
Positive invariance conditions have already been proposed in (Benzaouia, and Burgat, 1988; Benzaouia and Hmamed, 1993). Using these conditions it is possible to derive the following result:
(14)
addition of inequalities (13) and (14) enables us to write: |
}
|
}
~
}
}
}
~
Lemma 3. Domain (7) is positively invariant with respect to motion of system (6) and increment constraints (3) are respected if and only if
according to condition (10)
°
±
|
³
for the discrete-time case
|
|
}
|
}
~
}
¤
}
}
~
°
for the continuous-time case
which is equivalent to
´
¢
¤
|
|
Proof. For the increment constraints, conditions can be derived from the previous lemma, and the positive invariance conditions are given in (Benzaouia, and Burgat, 1988; Benzaouia, and Hmamed, 1993).
In the continuous time case,
following the same reasoning, replacing matrix by matrix , and condition (10) by condition (11), it is easy to obtain }
Relating the previous lemma to a pole placement procedure makes possible to solve the problem stated above. Recall the pole assignment procedure used in the so-called Inverse Procedure for constrained control (Benzaouia, 1994). Consider the time invariant system given by (1). Without loss of generality (see Remark below), assume that matrix possesses stable eigenvalues. Resolution of equation
|
Only if part, Consider the continuous time case. Assume that the derivative of respects the constraints and that condition (11) is not satisfied for an index such that
|
¶
µ
·
·
¸
·
·
(15)
~
º
¹
The following state vector for the system can be selected
if
®
»
µ
(16)
gives us a state feedback assigning spectrum of matrix ( together with the stable part of spectrum of matrix in closed loop. For this equation to have a valid solution, matrix must satisfy the following conditions:
µ
¢
¡
if ¢
¢
¡
¼
¹
if
|
½
¹
¢
¤
¥
¤
¾
(17)
¤
¡
µ
¸
¿
¢
¶
£
¾
It is easy to check that is an admissible state for component of the the system. Calculation of the derivative of this state gives
¨
¦
=
are linearly independent ¶
¤
¤
¤
¤
¤
¾ ¾
¾ ¾
such that , that is eigenvectors for of matrix There exits a unique solution to equation (16) if and only if
§
¨
¤
À
Á
Â
Â
Â
Â
~
Ã
~
Ã
«
©
©
¬
©
© Â
Ä
are linearly independent, where are eigenvectors associated to stable eigenvalues of matrix and are computed by
¡
|
¶
¤
~
Â
¤
¶
¤
µ
taking into account inequality (15), it is possible to write
¤
¤
¨
¤
 ¸
À
³
|
~
¤
¾
¶
¨
µ
¤
¤
¤
©
Hence, the solution is given by:
which contradicts the assumption. The discrete time part could be easily deduced replacing matrix by matrix in the necessary part.
¾ Æ
¾
Ç
¢
¢
È
Ã
}
~ Â
Evolution of the autonomous system (6) will respect both constraints on the state and constraints on its increment if domain given by (7) is positively invariant and conditions given in the previous lemma are satisfied. ®
Â
Ç
Â
~
Â
Ã
~
©
Ã
È
©
(18)
©
Remark 4. Without loss of generality, it was assumed that the system presents stable eigenvalues. If the system matrix does not satisfy such requirement,
¯
87
|
¶
ô
it is always possible to augment the representation as follows: let be a vector of fictitious inputs such that Ø
ö
å
Ê
Ë
Ì
Í
Î
Ï
Ð
Ñ
ý
å
æ
ç
è
Ñ
Î
Ò
Ó
(23)
É
ù ø
Î
Ï
Ð
Ì
Î
Ò
Ó
Ê
Ë
á
Ü
Ü
If the state does not leave the domain (23), the control signal does not violate the constraints. That is, the set is positively invariant with respect to motion of system (20). This gives the following result: ô
Ý
É
Ê
Ë
Ì
Í
É
Î
Ï
Ð
Ñ
É
Ñ
É
Î
Ò
Ó
Ü
Í
Ô
Õ
Ñ
Î
Ï
Ö
É
Ñ
Ô
Õ
Ð
Î
Ò
Ó
where and are freely chosen constraints on the fictitious inputs. In this case, vectors and become: É
Î
Ï
Ð
É
Î
Ò
÷
Ü
Ó
×
â
ã â
Ô
Proposition 5. System (20) with state feedback (21)(22) is asymptotically stable at the origin with constraints on both the control and its increment if there exists a matrix satisfying conditions (17) such that:
ã
ã ã
Ô
É Ô
Õ
Ê
×
Ø
ÚÙ Ì
Ô
Ø
ÙÚ
ä
ÚÛ
Ü
Ý
É
Þ
Ý
ä
ß
Þ
ÚÛ
Ô
ß
Ý
Þ
ß
Õ Ý
Ô
Ý
Þ
Ý
à
á
ß
Ý
ß
i)
The augmented system is then given by Ü
Ë
Ý
à
á
Ý
à
á
Ý
à
á
ý
æ
Ö
å
æ
ç
è
Ø
é
å
æ
ç
è
ê
ë
ì
í
ï
ç
(19) è
ð
É
æ
ç
è
é
ê
ý
ì
ý
Ø
ý
iia)
ñ
It is true that this augmentation limits the domain of linear behavior of the closed-loop system, but it is always possible to soften the fictitious limitations to enlarge the domain. It is easy to see that for the stable square system obtained the problem of eigenvalues is eliminated and controllability is not changed.
Ü
æ
ó
ò
æ
Í
è
×
Ñ
for the discrete-time case
Ô
Ì
×
Ñ
×
(24) iib)
è
×
Ñ
for the continuous-time case
Ô
Ì
×
ô
4. MAIN RESULTS
í
(25)
where
With this background, we are now able to solve the problem stated in section 2. Consider a stabilizable linear time invariant system with constraints on both control and increments of the control, that is
Ñ
×
Ø
for all initial state
Î
Ü
Ò
Ó Î
Ü
Ï
Ð
å
æ
ç
è
Ø
é
å
æ
ç
è
ê
ì
æ
ç
Proof. Introduce the following change of coordinates:
(20) è
æ
ç
è
Ê
Ë
æ
ç
è
ý
å
it is possible to write
is the state of the system, where the input constrained to evolve in the following domain å
Ê
Ø
Ö
å
Ê
Ü
Ö
æ
ç
è
Ø
ý
Ö
å
æ
ç
è
Ë
Ø
ý
æ
é
ê
ì
ý
å
è
æ
ç
(26) è
á
ô
õ
Ü
Ø
ý
å
æ
ç
è
Ý ô
Ø
ô
Ø
ö
æ
ç
è
Ê
Ë
Ì
Í
Î
Ï
Ð
Ñ
æ
ç
è
Ñ
Î
Ò
æ
ç
è
Ó
With this transformation, domain is transformed to domain given by (7). Further, with conditions (24) and (25), it is easy to note that domain is positively invariant with respect to the system (26) while the constraints on the increment of the control are respected. Bearing in mind that and that the linear behaviour is guaranteed, one can conclude to the asymptotic stability of the closed-loop system. ô
ø
Î
Ï
Ð
Ì
Î
Ò
Ó
Ê
ù
Ë
Ý
Ü
Ü
Ü
Ü
and the increment of the control is constrained as follows: Ý
Ü
÷
ô
Ü
Ô
Î
Ï
Ð
Ñ
æ
ú
ê
û
æ
è
Í
æ
ú
è
Ñ
Ô
Î
Ò
Ó
For continuous-time systems Ü
Í
Ô
è
Î
Ø
Ï
Ð
Ñ
ü
æ
ú
è
Ñ
Ô
ý
å
æ
ç
è
Ì
ý
Ê
Î
Ò
è
Ê
Algorithm:
þ Ý
é
ý
ÿ
ß
á
Ü
æ
ì
Remark 6. It is worth noting that conditions (24) and (25) do not affect the set of positive invariance . However, they present an additional constraint on the pole assignment problem.
Ó
(21) Ë
ô
such that
ê
ô
Ü
ç
é
Ü
Using the state feedback æ
ÿ
þ
For discrete-time systems Í
ê
ì
ý
è
Ê
(22)
Step 1. Check if matrix has ues, if not augment the matrix column. é
the following domain of linear behaviour is induced in the state space
88
æ
ò
Í
ó
è
ì
stable eigenvalwith null ò
Í
ó
Step 2. Choose matrix or , if the system is augmented, according to (17)-(24) or (25). Step 3. Compute the gain matrix or by using Step 4. Use or extracted from for the control.
"
#
$
0
%
−5
x2
Example 7. Consider the double integrator in the continuous-time state space representation by (Gutman, and Hagander, 1985):
−10 '
(
*
,
(
*
,
4
5
#
,
,
.
0
2
.
#
Control constraints are given by; (
−15
(
6
6
,
#
7
9
#
−20 −20
Assume that control increments are constrained as follows: :
−10
−15
−5
0
Fig. 1. Set of positive invariance emanating from
:
W
\
(
;
>
(
;
(
;
5
S
T
U
A
25
20
15
10
5x1
30
with a trajectory
A
?
#
,
9
@
7
I
As discussed in the Remark above, the system can be augmented with fictitious constrained inputs in domain . Select the following matrix :
5
=
>
?
=
4.5
=
7
@
4
;
3.5 (
, *
4
5
?
,
,
4
5
.
3 increment
?
which satisfies all the required conditions (17). Further, the fictitious constraints are selected such that conditions (25) are satisfied. :
(
A
(
:
C
;
2.5
C
2
( (
1.5 5 5
=
9
7
=
5
# 9
4
9
7
1
that is, D
0.5 D
(
;
4
E
F
>
?
4
?
0 0 ;
5
G
5
,
A
;
4
F
>
20
30 time
?
@
(
10
,
#
;
;
I
40
60
50
J
;
5
4
#
5
#
@
;
Fig. 2. Evolution of the control increment I
;
,
5
5
5
> #
(
J
@
where and
I
Solution of equation (16) leads to the following augmented gain matrix :
,
6
Y
>
^ =
_
`
( @ I I
7
I
7
6
4 ^
Y
>
a
b
=
@ I
I
I
Figure 1 represents the set of positive invariance and a trajectory emanating from the initial state , while Figure 2 shows the evolution of the control increment, which, it is possible to see, respects the constraints and .
;
(
*
?
#
?
W
\
(
;
>
,
.
#
A
?
A
:
:
#
Note that the effective gain matrix from the previous matrix.
can be extracted
@
I
(
(
M
;
?
#
( P
, *
4
5
.
#
'
?
P
(
It is easy to note here that . The obtained set of positive invariance, as shown in Figure 1, with the augmented matrix is given by: R
"
%
R
"
%
2
V
W
W
X ?
Y
7
In this paper, the regulator problem for linear systems with constraints on both control and its increment in the state space representation has been studied. Application of necessary and sufficient conditions, established for linear autonomous systems such that their motion respects incremental constraints, is the key to solve this problem. The link of the so called inverse procedure, the pole assignment method for
,
;
?
(
;
,
5. CONCLUSION
U
O
2
T
7
?
?
S
(
5
The closed-loop dynamics are given by: '
;
?
J
Y
J
[
89
Warwick, K., and D.W. Clarke (1988), Weighted input predictive controllers. IEE Proceedings Part D, Vol., 135, No 1.
constrained control, to the previous conditions is the cornerstone of this work. In fact, this link enables to give a simple algorithm to compute a regulator respecting constraints on both control and its increment.
ACKNOWLEDGEMENTS This work is the product of collaboration between Unversity Cadi Ayyad Morocco and University of Valladoli, Spain under grant N 23P/2000
6. REFERENCES Benzaouia, A. and C. Burgat (1988), Regulator problem for linear discrete-time systems with non symmetrical constrained control, Int. J. Control,Vol. 48, No. 6, pp. 2441-2451. Benzaouia, A., and A. Hmamed (1993), Regulator problem of linear continuous-time systems with constrained control, IEEE Trans. Aut. Control, Vol.38, No.10, pp. 1556-1560. Benzaouia, A. (1994), Resolution of equation XA+XBX=HX and the pole assignment problem, IEEE Trans. Aut. Control, Vol. 40, No. 10, pp. 2091-2095. Benzaouia, A., F. Mesquine (1994), Regulator problem for uncertain linear discrete time systems with constrained control, International Journal of Nonlinear and Robust Control, Vol. 4 pp. 387-395. Blanchini, F. (1990), Feedback control for linear systems with state and control bounds in the presence of disturbance, IEEE Trans. Aut. Control, Vol. 35, pp. 1131-1135. Blanchini, F. (1999), Set Invariance in control, Automatica Vol. 35, pp. 1747-1767. Dion, J.M., L. Dugard, & M. Tri (1987), Multivariable adaptive control with input-output constraints, IEEE Conference on Decision & Control, Los Angeles CA, December 1987. Gutman, P.O. and P. Hagander (1985), A new design of constrained controllers for linear systems, IEEE Trans. Aut. Control, Vol. 30, pp. 22-33. Clarke, D.W., C. Mohtadi, and P.S. Tuffs (1987) Generalised predictive control, Part 1: the Basic algorithm; Part 2: Extension and Interpretations. Automatica, Vol., 23, No 2, pp. 137-160.
90