Pole placement in a specified region based on a linear quadratic regulator (kawasaki 1988)

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Pole placement in a specified region based on a linear
quadratic regulator
a

NAOYA KAWASAKI & ETSUJIRO SHIMEMURA
a

b

Department of Education, Kochi University, 2-5-1 Akebono-cho, Kochi 780, Japan

b

Department of Electrical Engineering, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo
160, Japan
Version of record first published: 29 Oct 2007.

To cite this article: NAOYA KAWASAKI & ETSUJIRO SHIMEMURA (1988): Pole placement in a specified region based on a linear
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Pole placement in a specified region based on a linear quadratic
regulator

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NAOYA KAWASAKlt and ETSUJIRO SHIMEMURAS
A linear optimal quadratic regulator problem (an LQ-problem) is applied to assign
all poles of the multivariable continuous-time system in a suitable region of the lefthalf complex plane. In particular, two design methods based on an LQ-problem for
pole assignments in a truncated sector region of the left-half complex plane, which is
given as a common area of a half plane Re ,Is -1<0 and an open sector
tan-'llm )./Re 115 $0, are proposed. Each design method is given for the cases
where 0 2 i n and 0 5 t n respectively.As these two design methods are derived from
two basically different ideas, they will prove more useful if each method can be
applied according to the demands of the system's dynamical characteristics.

Introduction
Ever since it was proved that the closed-loop system constructed by utilizing a n
LQ-problem has considerable advantages in dynamical characteristics, LQ-problems
have been studied from various aspects (Johnson 1987). In particular, the desirable
properties of the LQ-problem, for example superiority in sensitivity problem, gain and
phase margins, transient responses etc., have interested us in applying an LQ-problem

as a practical design method for a feedback control law (Safonov and Athans 1977,
Lehtomaki et al. 1981, Kobayashi and Shimemura 1981). But when we construct a
closed-loop system by utilizing an LQ-problem, the weighting matrices of the
quadratic cost function must often be decided by trial and error to obtain the best
responses, because very little is known about the relation between the quadratic
weights and the dynamical characteristics or the closed-loop system (Harvey and
Stein 1978, Kouvaritakis 1978, Champetier 1983). As the dynamical characteristics of
a linear system are influenced by the pole locations of the system, the aim is to locate
all poles to specified positions in order to get good responses. But we know that for
many design purposes, it is sufficient to assign all poles in a suitable region of the lefthalf complex plane instead of assigning them exactly to their desired respective
positions. Then, if the closed-loop system constructed by utilizing a suitable poleassignment technique simultaneously has the properties of the system constructed by
utilizing an LQ-problem, the responses of the system will be expected to improve and
should be more than just asymptotically stable.
It is generally quite difficult to obtain a feedback control law which is not only an
optimal solution of an LQ-problem but also a law assigning poles to the prescribed
positions. But the above problem, an optimal pole-assignment problem, has been
studied by some technical procedures. In the case of a single-input linear system, the
problem has been generally solved as a kind of inverse optimal problem (Widodo
1972, Buelens end Hellinckx 1974). In the case of a multi-input linear system, two
1.

Received 6 July 1987.

t Department of Education, Kochi University, 2-5-1

Akebono-cho, Kochi 780, Japan.
Okubo Shinjuku-ku,

$ Department of Electrical Engineering, Waseda University, 3-4-1

Tokyo 160, Japan.


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226

N . Kawasaki and E. Shimemura

types of procedure have been mainly discussed. One type deals with the poles of the
system as a whole and assigns them to the exact locations (Solheim 1972, Eastman
and Bossi 1984, Kawasaki and Shimemura 1983 a) or assigns them in a specified
region which is given as, for example, an area on the left-hand side of a parallel line
distant from the imaginary axis (Anderson and Moore 1969, Kawasaki and Shimemura 1979) or a truncated open sector area symmetric with the negative real axis in
the left-half complex plane (Kawasaki and Shimemura 1983 b). The other type is
mainly given by iterative procedures for determining the weighing matrices of the
quadratic performance index where the procedures utilize a kind of sensitivity
technique to locate some dominant poles to the desired positions (Graupe 1972, BarNess 1978, Broussard 1982). We have previously proposed a design method belonging
to the former type to locate all poles in a specified region edged by a hyperbola in the
left-half complex plane ( ~ a w a s a kand
i Shimemura 1983 b).
In this paper, we further discuss our design method and increase the flexibility
concerning the shape of the desired specified region. Specifically, we give two design
methods for optimal pole assignments in the specified region given by the common
area of a halfplane Re 1. 5 - I < 0 and an open sector tan-' 1 Im >,/Re 1.1 5 QB (Fig. I).
One method is presented for the case where 0, the angle of the desired sector region, is
greater than Qn,and the other concerns the case where 0 is less than Qn. As the angle of
the sector region is generally considered to be deeply related to the transient
responses, especially overshooting or undershooting phenomena, it is important for
practical control design to develop various procedures which can be properly applied

depending upon the demands of the system's dynamical characteristics (Babary and
Hiriri 1986). It is then expected that optimal pole-assignment methods will be more
powerful in designing the practical control laws, if the methods proposed here are
taken into consideration.

Figure I.

Desired region where poles of the closed-loop system are to be located.

2. Problem formulation
We now consider a linear multivariable system (1) and a quadratic cost function
(2):
i=Ax+Bu
(1)


227

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Pole placement in a specijed region

where A , B are n x n, n x r constant matrices, Q and R are n x n positive semi-definite
and r x r positive definite symmetric matrices respectively, x is an n-dimensional state
vector, u is a n r-dimensional input vector, and ( A , B ) is a controllable pair. Here we
discuss the design method for optimal pole assignment, namely the method for
designing a state feedback control law which satisfies the following two conditions.
Then the control law assigns all poles of the closed-loop system in the hatched region
of Fig. I and corresponds to an optimal control law obtained from the LQ-problem
( I ) and (2'). In Fig. 1, I( >.O)represents the prescribed degree of relative stability and 0

gives the angle of the open sector region which is concerned with the damping ratios.
We consider specifically the case where 0 lies between $n and n in 9 3, and discuss
the case where 0 is less than f n in 54. We therefore discuss two distinct design
methods, one corresponding to each case. But strictly speaking, in 5 3 we cannot take
the whole region of Fig. I as the specified region where all poles can be practically
assigned by this proposed method. Instead of the region of Fig. 1, the hatched region
of Fig. 2 or Fig. 3 is regarded as approximately the desired region for this design

Figure 2. Region where poles of the closed-loop system are able
Method I.

located

Decision

Figure 3. Region where the poles of the closed-loop system can be located by Decision
Method I.


228

N. Krrwusaki and E. Shimemura

method because of its attainability without technical complications. Furthermore, in

5 4, note that the angle of sector region must be discretely given as 0 = nlk (k = 2, 3, ...),

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which cannot be continuously varied. But the hatched region of Fig. I is exactly equal

to the pole-assignable region by this proposed method except that the angle 0 must be
given as above.

3. Optimal pole assignments in the region whose sector angle lies between tr and n
3.1. Some preliminary lemmas
In this section, we give a design method for optimally assigning poles in the
specified region of Fig. 1 whose sector angle 0 is greater than t n . But, as mentioned
above, we regard the hatched region of Fig. 2 or Fig. 3 as approximately the desired
region for this design method, because of its attainability without technical complications. Before giving the result, some preliminary lemmas are presented.
Lemma 1
Among the eigenvalues of matrix A, we represent eigenvalues of A in the hatched
region of Fig. 4, edged by a hyperbola -(Re 1. - 11)~
+ (In1 1.)' = m2, by i.,, and
eigenvalues outside this region by 2,. Then eigenvalues -(I.,- h)'- m2 of -(A
- hl )2 - m21, which correspond to I., of matrix A, exist in the left half-plane, and -,(;
- h)' - mZ,which correspond to ljof matrix A, exist in the right half-plane, where m
is an arbitrary non-negative real number.
Now we consider a solution of a n algebraic Riccati equation:

We obtain the following lemmas relating to this equation (Kawasaki and Shimemura
1983 b).
Lemma 2
Let 1;. I ; , ..., I; be the left half-plane eigenvaluest of A and 5;, 5;, ..., 5; be the
corresponding eigenvectors. If a positive semidefinite symmetric matrix Q of (3)
satisfies the following equation:
the closed-loop system matrix A - BR-'BTK+ formed by the maximum solution:
K + has the eigenvalue I.; and the corresponding eigenvector 5;.
Lemma 3
Let 1; and (; (i = 1,2, ..., p ) be the same as above. The maximum solution K + of
the equation

KBR-IBTK - K A - A ~ =KO
(5)

t We call the eigenvalues in the left half-plane containing the imaginary-axis the le/r haljplone eigenvalrres. Similarly, the eigenvalues in the right half-plane not containing the imagmary
axis are called the rigbr hulj-plane eigenualues. Furthermore we call the eigenvalues which lie in
the leR half-plane not containing the imaginary axis the pure left halfplane eigenvulues.
f The relation K, - K , 2 0 (positive semidefinite matrix) is written as K , 2 K,. Equation
(3) has many real symmetric solutions. If a solution K + satisfies K + 2 K, where K is an
arbitrary solution, K + is called the nrnrrmunt soltrrio~~.


Pole placement in a specijed region

229

satisfies

where null ( K , ) and span (t;, 5 ; , . . . , t idenote
)
the null space of K + and the linear
subspace spanned by vectors 5 ; , 5 ; , 5; respectively. Furthermore, the eigenvalues
of A - r B R - ' B T K +, which are denoted by ).(A - r B R - ' B T K + ) , are given by

I ( A - r B R - 'B'K +) = { I ; , I ; , ..., I.; and n - p pure left half-plane eigenvalues} (7)

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where r is an arbitrary real number satisfying r >%.
The relation (7) states that all the other eigenvalues of the matrix A r B R - lBTK + except I ; , I.;, ..., I.; also exist in the left half-plane not containing the
imaginary axis for an arbitrary real number r > $ .

3.2. A design method of optimal pole assignments
In this section we discuss a design method of optimal pole assignments in the
region of Fig. 2 or Fig. 3. First we discuss the pole assignments in the region of Fig. 2;
then we give a fundamental theorem which is important in deriving the decision
method. Let I . , , I.,, ..., I., and l , ,&, ..., in-,be eigenvalues of A in the hatched
region and outside the hatched region of Fig. 4 respectively. Furthermore, among the
eigenvalues A,, I.,, ..., I.,, let I.,, I.,, ..., I., ( q 5 p) lie in the hatched region of Fig. 2.
Consider the following matrix equation:

Here m and h are arbitrary non-negative real numbers where h satisfies h $. Re % ( A ) .
Subsequently consider the following matrix equation with the maximum solution K +
of (8)t:

Figure 4. Region where eigenvalues of A satisfying Re i.( - ( A - h1)2 - m 2 1 ) $ 0 exist.

t The assumptions that ( A , B ) is controllable and h
maximum solution of (8).

+ Re E.(A) guarantee the existence of the


230

N. Kawusaki and E. Shimemura

where I and r are arbitrary real number satisfying I 2 0 and r > respectively. Then
from the previous lemmas, we can obtain the following theorem.
Theorem 1
The following relation holds with respect to eigenvalues of the closed-loop system
matrix I(A - BR -'BTP+), which is formed by the maximum solution P + of (9):

H[I.(A - BR -lBTP+)] = { I , , I.,, ..., I., and, at least, one more
(or a complex conjugate pair of) eigenvalue(s)} (10)

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Here H[I.(A)] denotes the set of eigenvalues of matrix A in the hatched region of
Fig. 2.
Proof
Let [ , , 5,, ..., 5, and f , , f 2 , ...,
be eigenvectors of A corresponding to the
eigenvalues I , , I.,, ..., I., (in the region of Fig. 4) and I , , I , , ..., In-,(outside the
region of Fig. 4). From Lemma 1 and Lemma 3, the eigenvalues of -(A - hi), - m21
- rBR - ' BTK+, where K + is the maximum solution of (8), are given as follows:

[.-,

..., -(I.,-

h),

- m2

and n - p pure

left half-plane eigenvalues}

(11)

Furthermore, from Lemma 3 and the equivalence of eigenvectors of A and -(A
- 111)~

- tn21, the maximum solution K + satisfies

Consider the matrix A - BR-'BTP+ where P + is the maximum solution of (9). From
Lemma 2, we see that I.,, I.,, ..., R, and t , , 5,, ..., 5, are included in the set of
eigenvalues and eigenvectors of A -BR-'BTP+ respectively. Next, consider the
remaining eigenvalues of A - BR-'BTP+ except I,, I.,, ..., 2,. We write those
eigenvalues as a , , a,, ..., a,-,. After a simple calculation, also using (9), we have the
following relation:

As (ABR-'BT-BR-'ETAT)
is a skew symmetric matrix, tr(ABR-'BT
- BR-'BTAT)P+ = 0. Hence from (13), the following relation holds:

Since a , , a,, ..., a,-, are the remaining n - q eigenvalues of A - BR -'BTP+ except
I , , A,, ..., I,, the eigenvalues of -(A - BR -'BTP+ - hi), - m21 are -(I., - 11)~
- m2, -(I, - h)' - m2, ..., -(I., - h)' - m2 and -(al - h)' - m2, -(a2 - h)' - m2,
..., -(a,-,- h)' -m2. The relation tr (BR-'BTP+) > 0 and comparing the above
fact with the relation (I I) and (14) give:


Pole placement in a specijed region

23 1

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The relation (15) shows that at least one (or one complex conjugate pair) of {-(u,
- h)2 - m2} (i = 1,2,. ..,n - q) exist in the left half-plane. Namely, at least, one (or one
complex conjugate pair) of a, (i = 1,2, . .. , n - q) exists in the hatched region of Fig. 4,
and simultaneously, all u,, n,, ..., a,-, exist on the left-hand side of a line Re I = - 1

because A + 11 - BR-'BTP+ becomes a stable matrix from (9). As a result, it is found
that a t least one (or one complex conjugate pair) of a, (i = 1, 2, ..., n - q) exists in the
0
hatched region of Fig. 2.
From the above theorem, we can see that the eigenvalues of A - BR-'BTP+ are
located in the following way: (i) the eigenvalues of A in the hatched region of Fig. 2 are
the eigenvalues of A - BR-'BTP+; (ii) a t least one (or one complex 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 number of iterated applications of the theorem,
all eigenvalues of the closed-loop system matrix can be located in the hatched region
of Fig. 2. Besides, from the proof of the theorem, h is allowed to be negative if h satisfies
only the condition h + I ? 0 without satisfying h 2 0.
Remark
Strictly speaking, in the case oTq < p, p - q eigenvalues of u, (i = 1.2, ...,n - q) are
given by -%,+, - 21, -i.,+, -I, ..., -i., - 21 which undoubtedly exist in the region
of Fig. 2. This fact is immediately obtained from considering a version of (9) such that
the right half-plane eigenvalues of A + I1 whose corresponding eigenvectors belong to
the subspace null ( K + ) are shifted to their corresponding symmetric positions with
respect to the imaginary axis as the eigenvalues of A + 11 - BR-lBTP (the so-called
mirror-image shift, Molinari 19771.
Here we consider the design method of optimal pole assignments in the region of
Fig. 2. First it should be pointed out that we can omit Step 0 in the following design
method. But if Step 0 is carried out once, it is guaranteed that all eigenvalues of the
system matrix will always exist in a half-plane Re i. < - 1throughout the iterations. In
other words, it corresponds to the case q = p, contrary to the above remark. Step 0 is
also recommended for the numerical stability of the computation of the algebraic
Riccati equations. We therefore consider that Step0 is obligatory in the present design
method. After deciding the appropriate non-negative numbers 1, / I , m which characterize the desired region, we can obtain the control law as follows.
Decision method for optimal pole assignments I
Step 0 (may be skipped if desired)

Calculate the maximum solution P: of the following Riccati equation for
arbitrary Q, > 0 and R > 0):

and obtain a closed-loop system matrix A

- BR-'BTP:

Step I
Let A, = A , - , - BR-'BTP' ( i = 1, 2, ..., where A, = A), and calculate the
maximum solution K; of the equation
K,BR-'BTK,

+ Ki{(Ai- h1)2 + m21} + {(A, - h1)2 + nz21}TKi= O

(17)


N . Kawasaki and E. Shimemura

232

Step 2
If K' is equal to zero, then go to Step 3. Otherwise choose an arbitrary real
number ri satisfying ri > i ,and calculate the maximum solution P A , of the equation

Subsequently update i = i

+ l and go back to Step 1.

Step 3

If the maximum solution K f satisfies K f = 0 for some integer j, this algorithm is
completed. Then all eigenvalues of A - B R - ' B T ( P : + P i + ... + P f ) exist in the
hatched region of Fig. 2. This system matrix A - B R - ' B T ( P : PT + ... P f ) is
equal to the system matrix A - B R - ' B T P + which is formed after solving an LQproblem once for the system (A, B ) with the quadratic weights (Q, + r , K
r, K i
+ ... + r , - , K j + _ , + 2 1 ( P : + P : + ... + P f ) , R).

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+

+
:+

Remark
I f Step 0 is not carried out, the first sentence of Step 2 will not be necessarily
correct only when i = 1. Namely, if all eigenvalues of A exist in the region of Fig. 4 and
even if some of them exist on the right-hand side of a line Re E. = - I, K becomes
equal to zero. But we can go on with the iterations without paying special attention to
that case, because Step 2 has the same effect as Step 0. Once Step 2 is carried out, all
eigenvalues of the closed-loop system matrix will always exist on the left-hand side of
= - 1 all through the iterations. Then for i 2 2, the first sentence of Step 2
a line Re i.
will be always correct even if Step 0 is skipped.
Since K , K T ,... , K jf_, and P : , P: , ..., P f are all maximum solutions, they are
all positive semidefinite matrices and the sum (Q, + r , K T + r , K : + . .. r j - I K f - ,
+ 2 1 ( P : + P i ... P f ) ) is a positive definite matrix. Furthermore it should be
noted that K f satisfies K f = 0 for some positive integer j (where j 2 2 if Step 0 is
skipped) ifand only if all eigenvalues of A, = A - B R - ' B T ( P : + P i + ... + P f ) exist
in the region of Fig. 2. Note, by the way, that we can make the hatched region of Fig. 2

exactly the truncated open sector region of Fig. 1 whose sector angle 0 is f n by
choosing m = 0 and h = 0.
In this section, we have discussed the design method of optimal pole assignment in
the hatched region of Fig. 2. The design method for the region of Fig. 3 can be
similarly obtained by replacing (17) of Step I with the following Riccati equation:

:

:

+ +

+

Details of this design method are omitted for lack of space.
4. Optimal pole assignments in the region whose sector angle is less than f x
4.1. Some preliminary lemmas
In this section, we discuss the design method for optimal pole assignments in the
region of Fig. I whose sector angle is less that i n . As mentioned before, the sector
angle 0 is discretely given as n/k ( k = 2,3, . ..) here. Before showing the results, we give
some preliminary lemmas which are necessary to obtain the design method.


Pole placement in a specijed region

233

Lemma 4
Let ( A , B ) be a controllable pair. Among eigenvalues of A, let y , , ..., y,, and
cc, +jb, , ..., a, jb, be p real and q complex conjugate pairs of eigenvalues which are

arbitrarily selected, and let c p , , ..., cp, and $, k j q l . ..., $, JV, be left-eigenvectors of
A corresponding respectively to the above eigenvalues. Then, In most cases, there exists
satisfying:
the matrix L E R(R+Zh)xn

+

+

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for some non-negative integers g ( 5 p) and h ( < q ) , where L is given as:

q:ITB) = I holds regarding the
In particular cases, the relation rank ([I):
eigenvectors corresponding to a, kjSi. The main arguments to follow are not
appropriate for such eigenvalues, because Lemma 4 cannot be satisfied by any means.
However, our design method will subsequently be shown to be applicable to even such
eigenvalues; for the present, we discuss only the case where Lemma 4 is satisfied with
the eigenvalues to be shifted.

Lemma 5
Let k be an arbitrary positive integer. Consider the open sector regions of Fig. 5
where each sector angle is n/k. The regions are symmetric with the real axis, and
necessarily contain the negative real axis in the complex plane. Among eigenvalues of
matrix A, we represent eigenvalues in these hatched regions by E.;, and eigenvalues
outside these regions by , f j . Then eigenvalues ( 1': of matrix ( - I)"+'Ak, which
correspond to i.; of matrix A, exist in the left half-plane, and ( - l ) ' + ' , f : , which
correspond to 1
, of matrix A, exist in the right half-plane.

Furthermore, the following holds with respect to the algebraic matrix Lyapunov
equation and the Riccati equation.
Lemma 6
Consider an algebraic Lyapunov equation PA ATP = - Q. If Q > 0 and Re R ( A )
< 0, there exists a symmetric positive definite solution P. Conversely, if Q > 0 and the
solution P > 0, A must be a stable matrix, namely Re 4 A ) < 0.

+


N. Kawasaki nnd E. Shimemura

234

Lemma 7
Let P , be a solution of the algebraic Riccati equation:

where R > 0 and Q > 0. When m is an arbitrary number satisfying m > 1, then a matrix
PI = mPl also becomes a solution of the Riccati equation:

+

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where {mQ m(m - l ) P l BR-'BTP1} > 0. If P I is the maximum solution (positive
definite solution) of (22), then Bl is also the maximum solution of (23).
With respect to the negative definiteness of the matrix polynomial, we can obtain
the following lemma.
Lemma 8
Consider the symmetric matrix

where S,, S k - , , ..., So are symmetric matrices and S, is positive definite. Then there
exists such a positive number m, that the relation T,(m) < 0 holds for an arbitrary
number m satisfying m > m,.
Proof
Let p, be the minimum eigenvalue of S, and let p,-,, ..., po be the maximum
eigenvalues of S,-,, ..., So respectively. When we write the maximum root of the
following algebraic equation:

as m,, an arbitrary number m satisfying m > m, satisfies the relation T,(m) < 0. If the
above equation has no real root, an arbitrary real number m satisfies r,(m) < 0.
From now on, we call the m, in Lemma 8 as the kth-degree pole assignment
number (k - p a n . ) of r,(m). Note that the maximum root of (25) is merely one of the
k-p.a.n.'s of T,(m). Generally the minimum k - p a n . of T,(m) is considered to be rather
less than the maximum root of (25) for almost all cases.
4.2. A design methodJor optimal pole assignment
Now we discuss the design method for optimal pole assignment in the region of
Fig. 1. First let y , , ..., y, and a , + j f l , , ..., a, +jflq be the real and the complex
conjugate pairs of eigenvalues of A outside the reglon of Fig. 1 respectively, and let cp,,
,.,,cp,, and 4, j q , , ..., fjq, be the corresponding left eigenvectors of A
respectively. From Lemma 1, we can compose thesamematrix L E R(=+Z h ) x n(21)
a ~ by
utilizing some of the above left eigenvectors where the matrix L satisfies rank (LB) = g
+ 211 for some non-negative integers g ( 5 p ) and h ( S q). After this, for-simplicity of
notation, we regard the suffixes p , , ..., p, and v,, ..., v, in L as 1, ..., g and 1, ..., h


Pole placement in a specified region

respectively. L is thus composed as follows:

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L=

Then L satisfies:

When we write B = LB, we consider the following Riccati equation for an arbitrarily
positive definite symmetric matrix Q E R ' ~ + ~ ~ ) ~ ( ~ + ~ ~ ' :
Let P+ be the maximum solution of the above equation (namely, P+ > 0). From
Lemma 4, mP+ is also the maximum solution of some Riccati equation for m > 1, and
~ + a stable one for m > 1. That is,
then the matrix (A,+,,+ I I ) - ~ B R - ' B ~becomes
Re i.(A,+,, - ~ B R - ' B ~ P +=) Re %(A,+,, - m 6 P ) < - 1 is satisfied for an arbitrary
m > 1 where = R - l B T P + . O n the other hand, we consider the following symmetric
matrix H,(m):

Corresponding to the case of k = 2 in L e m y 8 , the 2-p.a.n. of matrix H,(m) exists
P 0.+ Then the matrix H,(m)
because of the relation S , = P + ( B ~ ) {~( B F ) ~ } ~ >
becomes negative definite for an arbitrary m satisfying m > m,. Here m , can be
obtained by an appropriate way; for example, obtained as the maximum root of the
following equation:

+

Here

p2=Lmin{P+(BE~P)+(BEBE)TP+}=2i.mi,(B+B~-1BTP+~~-1BTP+)>O
(31)

p , = L,,,{P+(BEA,+~~
PO = i.max{-

(P+

+ A , + ~ ~ B F ' )+ ( B E A , + ~+~ A
f ( ~ i + Z h ) ~ ~ +='&tin{
; ;

P+

, + ~ ~ B ~ ) ~ ~ + }(32)

+( ~ : + 2 h ) ~ ' + 1

(33)

The negative definiteness of matrix H,(m) shows that the matrix (A,+,, - m B t ) , is
stable from Lemma 3. Then if m satisfies m > max (1, m,), all the eigenvalues of matrix
A,+,, - mBF exist in the hatched region of Fig. 1 whose sector angle 0 is f n.
Furthermore, the following lemma is satisfied for an arbitrary integer k 2 2.


236

N. Kawasaki and E. Shimemura

Lemma 9
Let m, be the k-p.m. of the following matrix H,(m):
H,(tn) = ( - I ) ~ + ' ~ + ( A , +-, n, l ~ F+) (~- I ) ~ + ~ { ( A ,-+M~B~F ) ~ } ~ B + (34)

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If m satisfies m > max (1, m,), all the eigenvalues of matrix A,+,,
hatched region of Fig. 1 whose sector angle O is nlk.

-

~ B exist
F in

the

Pr ooJ'
There exists an rn,, namely a k - p a n . of matrix H,(m), because B+(BF), > 0.
Therefore if m satisfies m > m,, the matrix ( '(A,.,, - m ~ p ) becomes
,
stable, i.e.
all the eigenvalues ofmatrix A,,,, - mBEexist at least in the hatched regions of Fig. 5
where each sector angle is nlk. Here we pay attention to only one of the hatched
regions of Fig. 5, namely the most interesting hatched region containing the negative
real axis. Suppose that all the eigenvalues of A,+,,-m6F
exist in the above
interesting hatched region whose sector angle is n/k - 1 for an arbitrary number m
satisrying m > m,- , where m,-, is a k - I-p.a.n. If m also satisfies m > m,, namely
m > max (m,-,, tn,), all the eigenvalues of A,+,,-~BF' must exist in the above
interesting region whose sector angle is nlk. As the matrix H,(m) always maintains
negative definiteness for a n arbitrary m satisfying m > m, because of the definition
of m,, then for such an m no eigenvalues of A , + , , - ~ B F ' shift to the outside of
hatched regions of Fig. 5 where each sector angle is nlk a t all. As a result, this

means that for an arbitrary m satisfying m > m, all eigenvalues of A,+,, - m 6 F exist in
only the above region, namely the region containing the negative real axis and
whose sector angle is nlk. As it would be easily proved that the above assumption
is satisfied in case of k = 1, and Re I(A,+,, - nzBF) ')< -1 is satisfied for an arbitrary
m > 1, then the proof is completed.
We can immediately obtain the following corollaries from the above lemma

Figure 5. Region where eigenvalues of A satisfying Re A{(-

l ) ' + ' A k } $ 0 exist

(for k = 5).

Corollary I
Let m,(min) be the minimum value of k-p.a.n. of the matrix H,(m). Here the
following relation is satisfied:


Pole placement in a specified region

237

Proof
Suppose that the relations ( - I ) ~ P + ( A- ~m+~ ~f ~) ~ - +l ( - l)k{(Ag+2h
-m ~ f ) ' - ~ } ~
and
( - I ) k + Z ~ + ( ~ g mBF)'+'
+ Z h - +(- 1)k~Z{(A,+2,
- m ~ f ) ' + ' } ~ P< +O are satisfied for an arbitrary m > mk+, . Here let A =A,+,,

- mBF, then the relation

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< 0 because A becomes a n asymptotically stable
yields ( - I)'+l(F+Ak +(Ak)T@+)
matrix ror an arbitrary n1 > m k + ,. Considering that the assumption is satisfied for
k = 2, then the proof is complete.
0
Corollarv 2
For k = 2 or k = 3, all eigenvalues of A,+,, - mBF would exist in the desired
region of Fig. 1 for an arbitrary m only satisfying both tn > 1 and Hk(nl)< 0. For
4 5 k 6 6, the same holds by replacing the above condition m > I with m > max (I,
rn,(minj), and for 7 g k 9, the same holds by replacing it with rn > max ( I , m,(min)).
Note that for k 2 10, we can obtain similar conditions form, but it is considered
unnecessary to discuss such a case so it is also omitted for lack of space.
From the above, the eigenvalues of matrix A,.,, - mBF = A,+,, - mnR -'BTP+
can be assigned in the desired region of Fig. I by choosing m as above. Note that
P+ = mP+ is given as the maximum solution of the following Riccati equation:
a

-

+

where & = { m ~m(m - I)P+BR-'BTP++ 2mlP+} > 0, Then the fact above yields
the rollowing theorem with respect to the original system (A, B).
Theorem 2
Consider the Riccati equation

where Ql = L ~ & L= L ~ { +
~ m(m
Q - l j P + BR -'BTP+ + 2 m l P + } 2
~ 0. Then one of
its solutions (not necessarily the maximum solution), represented by P I , is given as
P , = LTP+L and the eigenvalues of matrix A - BR-'BTPl are given as: (i) the
eigenvalues of A originally lying in the region of Fig. 1 stay at the same places
respectively; (ii) among the eigenvalues outside the region of Fig. I, ( g + 2 h )
eigenvalues specified by the above discussion move into the desired region and the
remaining ( p + 2q) - (g + 2h) eigenvalues d o not move from the original positions.
From the above, by regarding A - BR-'BTPl as the new value of A and iterating
the steps similarly, all the eigenvalues can be assigned in the desired region. Note that
the solution obtained at each iteration, Pi, is not necessarily the maximum solution of
each Riccati equation. But the final form P + = P , + P, + ... + P, corresponds to the
maximum solution of the Riccati equation:


238

N . Kawasaki and E. Shimemura

where Q = ZQ,, and the matrix A - BR- BTP+ achieves the desired eigenvalue
locations in the region of Fig. 1 as well as the asymptotic stability. After deciding the
appropriate non-negative numbers 1 and k, we can obtain the control law as follows.

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Decision nterhod of opritnnl pole assignments I1
Step 0
Solve an LQ-problem for arbitrary quadratic weights ( Q , , R) selected from the

demand for the system's dynamical characteristics and obtain a closed-loop system.
Regard the closed-loop system matrix A , as the new original system matrix and put
I =I .
Step I
Compose Li according to ( 2 6 ) by utilizing some left eigenvectors of Ai which
correspond to some specified eigenvalues outside the desired region.
Scep 2
Calculate A,,+,,i and B, = L i B like ( 2 7 ) ,and obtain the maximum solution P + of
the following Riccati equation for an arbitrarily positive definite symmetric matrix
R(R,+Z h d x ki+Zhil.
Q,

Step 3
By utilizing Lemma 9 or Corollary 2, decide an appropriate m according to the
desired sector angle nlk. Put Pi= mLTP+ L, and calculate Q , = L ? { ~ Q
m(nt - I ) P +BR -'BTB+ + 2 m l P + } ~ , .

+

Step 4
Put A , + , = Ai - BF, = A, - B R - ' B T P , .I f all the eigenvalues of matrix A , , , exist
in the desired region, the iterations are over. Otherwise update i = i I and go back to
Step I .
Here the final form of the system matrix, namely A - B(F, F, ...), is equal to
the system matrix A - B R - ' B T P + ,which is formed after solving an LQ-problem once
for the system ( A , B ) with quadratic weights ( Z Q , . R ) .

+
+ +

Remark
Even if rank ([$T qTITB)= I holds with respect to the eigenvectors corresponding to the specified eigenvalues, we can continue the same iterations as above with
a slight modification in Steps 1 and 3: define L, = [$T vT]T in Step I , and define m, as a
positive number satisfying tr { ( - I)*+ ' ( A , + , , - r n ~ E )<~ 0} for an arbitrary m > m, in
Step 3, instead of H,(m) < 0 of Lemma 9. In other words, even in the particular cases,
we can apply this design method by shifting one pair of such complex eigenvalues at
each iteration.
We discuss the interesting properties of the Riccati equation ( 2 8 ) in the next
section.


239

Pole placement in a speciJied region

4.3. Some properties of the solution of the Riccati equation
Here we will discuss some useful properties of the solution of Riccati equation ( 2 8 )
for the design method proposed here. As the matrix Q of (28) can be arbitrarily
selected to be positive definite symmetric, an arbitrary positive definite symmetric
matrix p which satisfies PBR -'BTk - P(A,+,, + 11) - ( A , + , , + I I ) ~ >P 0 can
become the maximum solution of the Riccati equation ( 2 8 ) where Q is given by
Q = P E R -'BTP - P(A,+
11) - ( A g + , *+ 1 1 ) ~>p 0 . Then the following lemma is
obtained with respect to the maximum solution of (28).

,,+

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Lemma 10

Let K E R ( R + ~ ~ ~be~an( Parbitrary
+ ~ ~ ) positive definite symmetric matrix, and let y
be an arbitrary positive number satisfying
>yo=i.,,,

[ { K ( A , + , , + I I )+ ( A , + , , + I I ) ~ K } ( K B R - ~ B1-7
~K
I

)

~ I ) ~ } ( B R ~ B ~(41)
) ' ]

Then a positive definite symmetric matrix P = yk becomes the maximum solution of
the Ricatti equation (28)' where Q is given by Q = P E R -'BTP - P ( A , + , ~+ 11)
-(A,+,,
> 0.

+

Proof
By substituting the relation
following:

Q =y

{

r

~

~

=

~

y R into the above equation for Q, we obtain the

-- @A,+,,
l ~ T +~11) - ( A ~ + , , +
, II)~K)

If y satisfies the condition ( 4 1 ) , Q of ( 4 2 ) becomes positive definite.

(42)

0

Lemma 10 shows that any positive definite symmetric matrix can be the maximum
solution of ( 2 8 ) if it were only to be multiplied by an appropriate positive number.
Roughly speaking, every positive definite symmetric matrix multiplied by a 'certain
large' number becomes the maximum solution of (28) with a certain positive matrix Q.
This means that it is unnecessary to solve the Riccati equations ( 4 0 ) one by one in
Step 2 of Decision Method 11, if it is not considered desirable. The judgement whether
this idea may be useful or not should be left for the designer, as in many practical
cases.
The following corollary is immediately obtained rrom the above lemma.

Corollary 3
Let A,+,, + 11 be an asymptotically stable matrix. An arbitrary positive definite
symmetric matrix P satisfying P(A,+,, + 11) + ( A , + , , + I I ) ~

+

The assumption of the above corollary, namely Re ).(A,+,, 11) < 0 , presents no
difficulties when we apply the corollary to Decision Method 11. Namely, if we consider
an LQ-problem for the system ( A 11, B) instead of ( A , B) in Step 0 of Decision
Method 11, then the closed-loop system matrix A , satisfies Re % ( A , )< -1. Then the
assumption of the corollary will be satisfied all through the iterations, if such a
technique is executed once in Step 0 .

+


240

Pole plucement in u specified region

We can also obtain an interesting " asymptotic property with respect to the
eigenvalues of the matrix A,+ - mBF = A,,,, - ~ B -R' B T P + , which plays an
important role in Decision Method 11 proposed here.

,,

Lemtnu 1 1

Let a , , a,, ..., o,,,, be the eigenvalues of matrix BE = B R - ' B ~ P + .Here u t s are
all positive real numbers. As m tends to infinity, the eigenvalues of matrix A,+,,~ B F A,+,,=

~ B R - I B +~ Pasymptotically approach m a , , tna,, ..., m u p + 2 h
respectively.

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-

a

It is considered that Lemma 11 gives the basic rationale by which the design
method proposed here can be constructed. As is to be expected, Lemma I I is deeply
associated with a so-called high-gain feedback (Shaked and Kouvaritakis 1976, 1977);
the feedback gain constructed by utilizing this design method might become
somewhat large if the desired sector angle 0 is made small.

5. Conclusions
In this paper, two optimal pole-assignment methods have been discussed. The
former assigns all poles in the region whose sector angle is greater than fn, and the
latter assigns all poles in the region whose sector angle is less than )n. The former has
the advantage that the control law could be automatically obtained after a finite
number of a t most n iterations without calculating the eigenvalues and eigenvectors of
the system matrix at each iteration. The latter has the advantage that the control law
could be obtained after a finite number of, at most n, iterations without solving the
algebraic Riccati equations if desired.

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