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

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International Journal of Control

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

NAOYA KAWASAKI a & ETSUJIRO SHIMEMURA b a

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

quadratic regulator, International Journal of Control, 48:1, 225-240

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regulator

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 left- half 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 1 5 $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

1 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

a s 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 left- half complex plane instead of assigning them exactly to their desired respective positions Then, if the closed-loop system constructed by utilizing a suitable pole- assignment technique simultaneously has the properties of the system constructed by utilizing an LQ-problem, the responses of the system will be expected t o improve and should be more than just asymptotically stable

It is generally quite difficult t o obtain a feedback control law which is not only an optimal solution of an LQ-problem but also a law assigning poles t o 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

Received 6 July 1987

t 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

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 Shime- mura 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, Bar- Ness 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 k i and 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 t a n - ' 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):

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, ), 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 compli- cations 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

- h l ) 2 - m21, which correspond to I., of matrix A, exist in the left half-plane, and -(;,

- h)' - mZ, which correspond to lj of 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

t We call the eigenvalues in the left half-plane containing the imaginary-axis the le/r halj- plone 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 I f a solution K + satisfies K + 2 K, where K is an arbitrary solution, K + is called the nrnrrmunt soltrrio~~

satisfies

where null ( K , ) and span (t;, 5 ; , , t i ) denote 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)

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 - h 1 ) 2 - m 2 1 ) $ 0 exist

t The assumptions that ( A , B ) is controllable and h + Re E.(A) guarantee the existence of the maximum solution of (8)

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 - ' B T P + ) , 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) 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

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 - B R - ' B T P + respectively Next, consider the remaining eigenvalues of A - B R - ' B T P + except I,, I.,, , 2, We write those eigenvalues as a , , a,, , a,-, After a simple calculation, also using (9), we have the following relation:

- 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 - ( a l - 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:

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

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 t o their corresponding symmetric positions with respect t o 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 < - 1 throughout the iterations In other words, it corresponds to the case q = p, contrary t o 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 character- ize 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)

232 N Kawasaki and E Shimemura 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 LQ-

problem 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 ) 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

a line Re i = - 1 all through the iterations Then for i 2 2, the first sentence of Step 2

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

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

the matrix L E R(R+Zh)xn satisfying:

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

In particular cases, the relation rank ([I): q : I T B ) = I holds regarding the 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