© 1983 International Federation of Automatic Control Brief Paper Determining Quadratic Weighting Matrices to Locate Poles in a Specified Region* NAOYA KAWASAKD" and ETSUJIRO SHIMEMUR
Trang 1Printed in Great Britain Pergamon Press Ltd
© 1983 International Federation of Automatic Control
Brief Paper
Determining Quadratic Weighting Matrices to Locate Poles in
a Specified Region*
NAOYA KAWASAKD" and ETSUJIRO SHIMEMURA:~
Key Words Linear optimal regulator; pole placement; state feedback; feedback control; multivariable control
systems; control system design; computer-aided design
Abstract A new procedure of selecting weighting matrices in
linear quadratic optimal control problems (LQ-problems) is
proposed In LQ-problems, the quadratic weights are usually
decided on trial and error to get good responses But using the
proposed method, the quadratic weights are decided in such a
way that all poles of the closed loop system are located in the
desired region for good response as well as for stability As the
system constructed by this method has merits of an LQ-problem
as well as a pole-assignment problem, this procedure will be
useful for designing a linear feedback system
1 Introduction
THE closed-loop system constructed by utilizing an LQ-problem
has some merits (Safonov and Athans, 1977; Kobayashi and
Shimemura, 1981) But when we construct a closed-loop system
by utilizing the LQ-problem, the weighting matrices of the
quadratic cost function must be decided on by trial and error to
get the good responses, because only very little is known about
the relation between the quadratic weights and the dynamical
characteristics of the closed-loop system (Harvey and Stein,
1978; Stein 1979; Francis, 1979) The dynamical characteristics
of a linear system are influenced by the location of poles of the
system Therefore to get good responses, it is necessary to locate
all poles in the desired positions But we know that it is sufficient
to place all poles in a suitable region instead of placing them in
their desired respective positions
In this paper, we give a new method of selecting the quadratic
weights in LQ-problems by which all poles of the closed-loop
system are located in the specified region for good response as
well as for stability As the system constructed by this method has
the merits of an LQ-problem as well as a pole-assignment
problem, this method will be useful for constructing linear
feedback systems Conceptually this decision method may be
considered to derive from the so-called inverse optimal control
problems (Thau, 1967; Yokoyama and Kinnen, 1972; Moylan
and Anderson, 1973) But it will be dit~cult to derive the concrete
result such as obtained in this paper from the arguments about
the inverse optimal control problems
2 Problem formulation
Now we consider a linear multivariable system (1) and a
quadratic cost function (2)
*Received 28 December 1981; revised 19 August 1982; revised
26 January 1983 The original version of this paper was presented
at the 8th IFAC Congress on Control Science and Technology
for the Progress of Society which was held in Kyoto, Japan
during August 1981 The published proceedings of this IFAC
meeting may be ordered from Pergamon Press Ltd, Headington
Hill Hall, Oxford OX3 0BW, U.K This paper was recommended
for publication in revised form by associate editor D H Jacobson
under the direction of editor H Kwakernaak
~'Department of Education, Kochi University, 2-5-1 Akebono-
cho, Kochi 780, Japan
~:Department of Electrical Engineering, Waseda University, 3-4-
1 Okubo Shinjuku-ku, Tokyo 160, Japan
557
where A, B are n x n, n x r constant matrices, Q and R are n x n, , x r positive definite symmetric matrices respectively, x is an n- dimensional state vector, u is an r-dimensional input vector, and (A, B) is a controllable pair Now we consider the method of deciding quadratic weights by which all poles of the closed-loop system are located in the hatched region of Fig 1 We know by experience, if all poles are located in the region of Fig 1, the responses converge to the steady state at appropriate speed and
no objectionable vibrating modes appear on the responses In the following, we regard the hatched region of Fig 2 edged by a hyperbola as the desired region in which all poles are to be located instead of the region of Fig 1 because the region of Fig 2 could become a good approximation of the region of Fig 1 by choosing m of a hyperbola (Re 2) 2 - (Ira 2) 2 = m 2 appropriately
In the next section we consider the method of deciding quadratic weights by which all poles of the closed-loop system are located in the hatched region of Fig 2
3 A method of deciding quadratic weights 3.1 Some preliminary lemmas In this section a new method of
deciding quadratic weights of LQ-problem is given Before showing the result, some preliminary lemmas are prepared for obtaining the method
Lerama 1 Among eigenvalues of matrix A, we represent
eigenvalues of A in the hatched region of Fig 3, edged by a hyperbola (Re ~.)2 _ (Im 2) 2 = m 2, by 2~, and eigenvaiues outside
this region by 2j Then eigenvalues - 2 2 + m 2 of - A 2 + m2I,
" , ( 4 5 "~
N
N
N
~ - h / / / / / /
Im
0
FIG 1 A desired region where the poles of the closed-loop system
are to be located for good responses
Trang 2558 Brief Paper
Im
I
FIG 2 A region where the poles of the closed-loop system are able
to be located by this method
/ / -ITI /
/ _-fq, /
/
J
J
Im
:m 302= m 2
FIG 3 A region where the eigenvalues of A satisfying
R e 2 ( - A 2 + m21) < 0 exist
which correspond to 2~ of matrix A, exist in the left
half plane, and - ~ + m 2, which correspond to '~i of matrix A,
exist in the right half plane, where m is an arbitrary nonnegative
real number
a controllable pair if and only if (A, B) is a controllable pair
Lemma 3 Let matrix A have no eigenvalues on the imaginary
axis Then (A 2, B) is a controllable pair if and only if (A, B) is a
controllable pair
Now we give some lemmas about the solution of Riccati
equation
of A and ~ i , ~ ~- be the corresponding eigenvectors If a
positive semidefinite symmetric matrix Q of the (3) satisfies the
following equation
Q~T = 0, iE{1,2 /z} (4)
the closed-loop system matrix A - B R - ~ B T K + formed by the
maximum solutioni" K + has the eigenvalue 2~-
and the corresponding eigenvector (~-
expressed as Q = CTC, where C ~ R q x , a n d q is the rank of Q The
null space of K÷, denoted by null (K+), satisfies the following
relation (Molinari, 1977)
null (K+) = 7oC~{.La,~Aa,} (5)
where 70, Aa and Lgi denote the unobservable subspace of(C, A),
stable and imaginary modes of A, respectively This establishes
*We call the eigenvalues in the left half plane containing the
imaginary axis the left half plane eigenvalues Similarly the
eigenvaiues in the right half plane not containing the imaginary
axis are called the right half plane eigenvalues Furthermore we
call the eigenvalues which lie in the left half plane not containing
the imaginary axis the pure left plane eigenvalues
~'The relation Ka - K2 > 0 (positive semidefinite matrix) is
written as K t > K z Equation (3) has many real symmetric
solutions If a solution K+ satisfies K+ > K, where K is an
arbitrary solution, K+ is called the maximum solution
In the above lemma, the special case of Q = 0 is stated as follows
Lemma 5 Let 27 and ~7 (i = 1, ,/z) be the same as before The maximum solution K+ of the equation
K B R - 1 B T K - K A - A T K = 0
satisfies
null (K +) = span (~i-, ~- ~- )
(6)
(7) where span (~ i , ~ ~ ~ ~ ) denotes the linear subspace spaned
by vectors ~ , ~ i ~ Furthermore the eigenvalues of
are 2(A - r B R - I B T K + ) =
{2i-, 2f 2- and n - / z pure left half plane eigenvalues} ,u (8) where r is an arbitrary real number satisfying r > ½
proof of Lemma 4, only the latter part is proved As was proved
by Safonov and Athans (1977), the insertion of linear constant gain r > ½ into the feedback loops of the respective controls leaves the dosed loop system constructed by using an LQ-problem asymptotically stable in the large Namely the matrix
number r>½, where K+ is the maximum solution of (3) By applying their result to the case Q = 0 and considering the relation (7), we can obtain the relation (8) [] The relation (8) states that all the other eigenvalues of the matrix A - r B R - ~ B r K + except 2i-,2~, , 2~ also exist in the left half plane not containing the imaginary axis for an arbitrary real number r>½
3.2 A f u n d a m e n t a l theorem In this section, we shall give a fundamental theorem which is important to derive the decision method Let A satisfy Re 2(A)<0 F u r t h e r m o r e let 2~, 22 2p and ~1, ~2 ~,_p be eigenvalues of A in the hatched region and outside the hatched region of Fig 2, respectively Consider the following matrix equation
where m is an arbitrary nonnegative real number Subsequently consider the following matrix equation with the maximum solution K+ of (9)
(10)
Trang 3Brief Paper 559
where r is an arbitrary real n u m b e r satisfying r > ½ Then from the
previous lemmas, we can obtain the following theorem
eigenvalues of the dosed-loop system matrix A - B R - 1 B r P + ,
which is formed by the maximum solution P+ of (10)
H [)`CA - B R - t B r P + )] = {)`1, )`2 )`p and, at least, one more
(or a complex conjugate pair) eigenvalue} (11)
Here H [)`(A)] denotes the set of eigenvalues of matrix A in the
hatched region of Fig 2
corresponding to the eigenvalues )`1, )`2 )`p (in the region) and
~-1, ~2 ~,_p (outside the region) Now consider the maximum
solution K+ of (9) Because Lemmas 2 and 3 show ( - A 2 + m21,
B) is controllable, there exist K + Hence from Lemmas 1 and 5,
the eigenvalues of - A 2 + m2I - r B R - 1 B r K + are given by
) ` ( - A 2 d- m 2 I r B R - I B T K + ) = { _ ) ` 2 d- m 2, _)`2 + m 2
_ )`2 + m 2 and n - p pure left half plane eigenvalues} (12)
Furthermore from Lemma 5 and equivalence of the eigenvectors
of A and - A 2 + m21, the maximum solution K+ satisfies
null (K+) = span (~1, ~2 ~p)' (13)
Consider the matrix A - B R - 1BTP+ formed by P+, which is the
maximum solution of (10) Then from Lemma 4 we see that )`1,
)`2, :, )`p and ~1, ~2 ~p are included in the set of eigenvalues
and eigenvectors of A - B R - 1 B r p + respectively Next, consider
the remaining eigenvalues of A - B R - I B T P + except 21,
)`2, -,)`p We write those eigenvalues as al, a2, ,a,_~ After a
simple calculation, also using (10), we have the following relation
trace { - (A - B R - 1 B T p + ) 2 + mZl} =
trace ( - - A 2 + mZl r B R - 1 B r K + ) (15) holds Since trl, a2 ~r._p are the remaining n - p eigen-
values of A - B R - I B r P + except )`1, )`2 2p, the eigen-
values of - ( A - B R - 1 B r P + ) 2 + m2I are - 2 2 + m 2,
_)`22 + m 2 _)`2 + m 2 and - a ~ + m 2, -tr22 + m z,
- a 2 _ p + m 2 Comparing this fact with the relation (12) and
(15) gives
The relation (16) shows that at least one (or one complex
conjugate pair) of ( - tr 2 + m 2) (i - 1,2 n - p) exists in the left
half plane Namely, at least, one (or one complex conjugate pair)
of at (i = 1,2 n - p) exists in the hatched region of Fig 2,
because A - B R - 1 B r P + which is obtained by using an LQ-
problem with quadratic weights (rK+, R ) is an asymptotically
F r o m the above theorem, we can see if A satisfies Re )` (A) < 0
the eigenvalues of A - B R - 1 B r p + are located in the following
way: (a) the eigenvalues of 4 in the hatched region of Fig 2 are
the eigenvalues of A - B R - I B r P + , and (b) at least, one (or one
comi~lex conjugate pair) of eigenvalues of A outside the hatched
region of Fig 2 moves into the hatched region of Fig 2 Therefore
after a finite n u m b e r of iterated application of the theorem, all
eigenvalues of the closed loop system matrix can be located in the
hatched region of Fig 2
In Theorem 1 it is assumed the system matrix A satisfies
Re )` (A) < 0 for the simplicity of the proof and the sufficiency for
the derivation of this decision method But even without this
assumption, it is seen the same result holds with respect to the
matrix A - B R - 1 B r P + which is obtained after the same
operations as Theorem 1 This fact is derived from considering
the relation that right half plane eigenvalues of A in the hatched region of Fig 3 are shifted to their corresponding symmetric positions with respect to the imaginary axis by the operation of Theorem 1 [so-called mirror-image shift (Molinari, 1977)]
similar to Theorem 1 holds with respect to eigenvales of the matrix A - B R - I B T P +
H [)`(A - B R - 1BTp+ )] = {)-1, ),2 )`p - ~1, - ~2 - ~q and
at least, one more (or a complex conjugate pair) eigenvalue}
(17) Here )`1, )`2 )`p and ~1, ~2 ,2q denote the left and fight half plane eigenvalues of A in the hatched regions of Fig 3, respectively
3.3 A method o f deciding quadratic weights Here we consider the decision method of the quadratic weights of an LQ-problem
to locate the poles of the closed loop system in the hatched region
of Fig 2 In the previous section, we assumed Re )`(A) < 0 for the reasons mentioned before But it should be pointed out that this proposed decision method can be utilized not only for A satisfying R e ) ` ( A ) < 0 but also for any A That is, the only requirement on A is that (A, B) is controllable Namely
in tlae following decision method, the closed-loop system matrix A1 = Ao - B R - 1 B T p ~ { = A - B R - 1 B T p ~ o b t a i n e d by step 1 satisfies Re)`(AI) < 0 if Qo is selected as Q0 > 0 Therefore A1 satisfies the assumption of Lemma 3 and Theorem 1, then we can supply Theorem 1 to A 1 instead of applying the theorem to A directly Step I is also recommended for the numerical stability of the computation even in case of Re2(A) < 0 In concluding the above discussions, we can summarize the decision method of the quadratic weights as follows
Decision method
Step 1 (May be skipped in case of Re)`(A) < 0.) Solve an LQ- problem for arbitrary quadratic weights (Qo, R) selected from the demand for the system's dynamical characteristics
and obtain a closed-loop system matrix A - B R - 1 B T P ~ i
Step2 Let At = A ~ - I - B R - 1 B r P - ~ (i = 1,2 ,where Ao = A), and calculate the maximum solution K , + of the equation
Step 3 If K ,+ is equal to zero, then go to step 4 Otherwise choose
an arbitrary real n u m b e r rt satisfying r~ > ½, and calculate the maximum solution P~++ 1 of the equation
and form a dosed-loop system matrix
Step 4 If the maximum solution K + satisfies K + = 0 for some integer j, this algorithm is completed Then all eigenvalues of
hatched region of Fig 2 This system matrix A - B R ~ t B T ( p ~
+ P ~ ' + + P+) is equal to the system matrix A
once for the system (A, B) with quadratic weights (Qo + rt K~-
+ r2K~ + + rj-IKj+.-1, R)
compute the solutions of the algebraic Riccati equations (ARE) (19) and (20) Computational algorithms of ARE are proposed
by many authors (Potter, 1966; Ku~era, 1972; Kwakernaak and Sivan, 1972; Callier and Willems, 1981)
Since K~, K~ Kj+_I are all the maximum solutions, they are all positive semidefinite matrices and the
definite matrix Furthermore it should be notlld that K ~ satisfies K~ = 0 if and only if all eigenvalues of A j = A - B R - 1 B T
(P~ + P~ + + P ~ ) exist in the region of Fig 2
Trang 4560 Brief Paper
4 Conclusions
In this paper, a decision method for quadratic weights of an
LQ-problem to locate all poles of the closed-loop system in the
specified region has been discussed This decision method has the
advantage that K ,+ satisfies K ~ = 0 after a finite number of, at
most n, iterations Furthermore it should be pointed out that
those poles which have been located once in the hatched region of
Fig 2 are not moved by the successive iterations of the algorithm
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