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Numerical Solution of Differential Algebraic Riccati Equations zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
P. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAKunkei* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Fakultiit fiir Mathematik
Universitiit Bielefeld
Postfach 8640
D-4800 Bielefeld 1, FRG
and
V. Mehrmann+
Fachbereich Mathematik
Universitiit Oldenburg
Postfach 2503
D-2900 Oldenburg, FRG
Submitted by Paul van Dooren
ABSTRACT
We consider Riccati matrix differential algebraic equations arising from singular
or descriptor control problems. We discuss the solvability of such equations under
different conditions. In order to apply numerical methods for differential algebraic
systems one has to transform the equation. Unfortunately, these equations then have a
linear part, which is described by a singular pencil, and thus the usual integration
methods do not apply. Under some conditions, which we discuss, these singularities
can be removed by a preprocessing algorithm, and the equation can then be solved by
well-known methods for differential algebraic systems like DASSL of L. Petzold or
LIMEX of Deuflhard, Hairer, and Zugck. We discuss the numerical procedures and
give some numerical examples.
1. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAINTRODUCTION
We study the differential algebraic Riccati equation (DARE)
(1.1) - Er(t)J$)E(t) = ET(t)X(t)A(t)+ AT(t)X(t)E(t)+ Q(t)
- ET(t>X(t>W(t>X(t)E(t)
* Email: UMATFIO8~BIUNI11.BITNET
tErnail: 015O40~OLUNI1.BITNET.
LINEAR ALGEBRA AND ITS APPLICATlONS 137/138:39-66 (1990) 39
0 Elsevier Science Publishing Co., Inc., 1990
655 Avenue of the Americas, New York, NY 10010 0024-3795/90/$3.50
40 P. KUNKEL AND V. MEHRMANN zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
with “terminal” condition
(1.2) ET(tr)X(tr)E(tr) = M.
Here X(t), E(t), A(t), Q(t), W(t), M E IX”,” are sufficiently smooth and
are symmetric in the interval [t,, tr] C R. The coefficient
X(t), Q(t), zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAW(t), M
matrices have to satisfy further conditions which will be discussed later on.
Such equations arise for example from optimal regulator problems with
differential algebraic equations (DAEs) or from optimal filters with differen-
tial algebraic equations. Consider for example the optimal control problem
subject to
(1.4) E(t)i(t)=A(t)r(t)+B(t)u(t), ix( to) = 2,
whereE(t),A(t),Q(t) E FP”, B(t) E Finam, R(t) E [w”,“‘, Q(t) = Q(tjT, R(t) =
R(tjT, R(t) is positive definite, and Q(t) is positive semidefinite. In the next
section we show how under some further assumptions the optimal solution of
(1.3), (1.4) can be obtained via the solutions of (l.l), (1.2).
For the case that the matrices E, A, Q, R, B are independent of t this is
shown in Mehrmann [15, 161. In the nonautonomous case where E(t) is
nonsingular, it follows from standard linear control theory, e.g. Athans and
Falb [l] or Knobloch and Kwakernaak [ 131. For the infinite horizon problem,
i.e. tf =m, previous work is by Arnold [2] and by Bender and Laub [3]; see
also Mehrmann [15] and the list of references therein.
In this paper we discuss the numerical solution of (l.l), (1.2) regardless
of whether they come from an optimal control problem or not. The obvious
approach is to vectorize this equation and, leaving out the extraneous
components of the symmetric matrix, to transform it into a standard differen-
tial algebraic equation in R n(n+ ‘r/‘. If E(t) is singular, then this approach
unfortunately produces an equation whose linear part is described by a
singular pencil, and thus the standard solvers for differential algebraic
equations like LIMEX [6] or DASSL [17] d o not apply. We will discuss under
which conditions these unfortunate singularities can be removed and how
DIFFERENTIAL ALGEBRAIC RICCATI EQUATIONS 41
this zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAcan be done numerically, so that the solution can then be obtained from
standard solvers.
We demonstrate the proposed method with some numerical examples.
For descriptor control problems, numerical methods for solving the
variational equations, which form a linear two point boundary value problem,
could also be used. It is possible to embed the Riccati differential equation in
such a boundary value problem even if it does not come from a control
application. For linear differential algebraic boundary value problems, nu-
merical methods have been examined recently by several authors. Reviews of
the known methods are given in [I91 and 121.
1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
2. PRELIMINARIES
In this section we introduce our notation, give some preliminary results,
and show how Equations (1.11, (1.2) can be obtained from the control
problem (1.31, (1.4). By [w”,” (fZ)n,n) we denote the real (complex) 72 x n
matrices. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
DEFINITION 2.1. Let A E [w”~“. Then we denote by
M(A) the nullspace of A;
H(A)l the orthogonal complement of Jy(A);
vet(A) the vector in [wnz obtained by concatenating the columns of A one
after the other into one big vector.
In the following we make frequent use of the Kronecker canonical form
of a matrix pencil aE - PA, which is as follows:
THEOREM 2.2. Let E, A E [w”~“. Then there exist P, Z E C”, * nonsingular
such that
(2.3) P((rE - PA)2
where
(a) L, is an ej X(ej + 1) bidiagonal pencil of the form
42 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
P. KUNKEL AND V. MEHRMANN
(b) ~~~ is an (qj + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA1) X qj zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAbidiagonal pencil of the fm zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
1 0
0 . . 1 ‘.
(2.5) o! : -P 0
. 1
0 1
: * zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA1 I
. . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(c) J~,(A j) in a pj x pj ./ordan block
hj 1
(2.6) (Y -P
1
1
(d) No, is a aj X aj nilpotent pencil of the form
Proof. See e.g. Gantmacher [8]. n
We then have the following definition:
DEFINITION 2.8. A pencil (YE - PA is called regular if no blocks of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
type (2.4) (2.5) occur in its Kronecker canonical form (2.3).
By ind_,( E, A) we denote the size of the largest block Ngj.
Two pencils crE - PA, aI? - /3A are called equivalent if there exist
P, z E Cn3”, nonsingular, such that
(2.9) a.@ - /3A = P(aE - PA)Z.
We now consider the control problem (1.3) (1.4). In the following, for
simplicity, we often leave out the dependence on t in the formulas. We give
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