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

This article was downloaded by: [University of Tokyo] On: 04 April 2013, At: 01:00 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcon20 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 quadratic regulator, International Journal of Control, 48:1, 225-240 To link to this article: http://dx.doi.org/10.1080/00207178808906171 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material Pole placement in a specified region based on a linear quadratic regulator Downloaded by [University of Tokyo] at 01:00 04 April 2013 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 and comparing the above fact with the relation (I I) and (14) give: Pole placement in a specijed region 23 Downloaded by [University of Tokyo] at 01:00 04 April 2013 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 = - 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 hatched region of Fig 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 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 moves into the hatched region of Fig 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 Besides, from the proof of the theorem, h is allowed to be negative if h satisfies only the condition h + I ? without satisfying h 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 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 First it should be pointed out that we can omit Step in the following design method But if Step 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 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 (may be skipped if desired) Calculate the maximum solution P: of the following Riccati equation for arbitrary Q, > 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 If K' is equal to zero, then go to Step 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 Step If the maximum solution K f satisfies K f = 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 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 + _ , + ( P : + P : + + P f ) , R) Downloaded by [University of Tokyo] at 01:00 04 April 2013 + + :+ Remark I f Step is not carried out, the first sentence of Step will not be necessarily correct only when i = Namely, if all eigenvalues of A exist in the region of Fig 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 has the same effect as Step Once Step is carried out, all eigenvalues of the closed-loop system matrix will always exist on the left-hand side of = - all through the iterations Then for i 2, the first sentence of Step a line Re i will be always correct even if Step 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 - , + ( P : + P i P f ) ) is a positive definite matrix Furthermore it should be noted that K f satisfies K f = for some positive integer j (where j 2 if Step 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 Note, by the way, that we can make the hatched region of Fig exactly the truncated open sector region of Fig whose sector angle is f n by choosing m = and h = In this section, we have discussed the design method of optimal pole assignment in the hatched region of Fig The design method for the region of Fig 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 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 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 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 + + Downloaded by [University of Tokyo] at 01:00 04 April 2013 for some non-negative integers g ( 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 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 is satisfied with the eigenvalues to be shifted Lemma Let k be an arbitrary positive integer Consider the open sector regions of Fig 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 , 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 Consider an algebraic Lyapunov equation PA ATP = - Q If Q > and Re R ( A ) < 0, there exists a symmetric positive definite solution P Conversely, if Q > and the solution P > 0, A must be a stable matrix, namely Re A ) < + N Kawasaki nnd E Shimemura 234 Lemma Let P , be a solution of the algebraic Riccati equation: where R > and Q > When m is an arbitrary number satisfying m > 1, then a matrix PI = mPl also becomes a solution of the Riccati equation: + Downloaded by [University of Tokyo] at 01:00 04 April 2013 where {mQ m(m - l ) P l BR-'BTP1} > 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 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) < 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) < If the above equation has no real root, an arbitrary real number m satisfies r,(m) < From now on, we call the m, in Lemma 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 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 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 ( 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: Downloaded by [University of Tokyo] at 01:00 04 April 2013 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 > That is, then the matrix (A,+,,+ I I ) - ~ B R - ' B ~becomes Re i.(A,+,, - ~ B R - ' B ~ P +=) Re %(A,+,, - m P ) < - is satisfied for an arbitrary m > 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 = in L e m y , 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) +( ~ : + h ) ~ ' + (33) The negative definiteness of matrix H,(m) shows that the matrix (A,+,, - m B t ) , is stable from Lemma Then if m satisfies m > max (1, m,), all the eigenvalues of matrix A,+,, - mBF exist in the hatched region of Fig whose sector angle is f n Furthermore, the following lemma is satisfied for an arbitrary integer k 2 236 N Kawasaki and E Shimemura Lemma 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) Downloaded by [University of Tokyo] at 01:00 04 April 2013 If m satisfies m > max (1, m,), all the eigenvalues of matrix A,+,, hatched region of Fig 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), > 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 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 - 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 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 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 Region where eigenvalues of A satisfying Re A{(- l ) ' + ' A k } $ 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 ) ' - ~ } ~ mk+, Here let A =A,+,, - mBF, then the relation Downloaded by [University of Tokyo] at 01:00 04 April 2013 < 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 Corollarv For k = or k = 3, all eigenvalues of A,+,, - mBF would exist in the desired region of Fig for an arbitrary m only satisfying both tn > and Hk(nl)< For k 6, the same holds by replacing the above condition m > I with m > max (I, rn,(minj), and for g k 9, the same holds by replacing it with rn > max ( I , m,(min)) Note that for k 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 Consider the Riccati equation where Ql = L ~ & L= L ~ { + ~ m(m Q - l j P + BR -'BTP+ + m l P + } ~ 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 stay at the same places respectively; (ii) among the eigenvalues outside the region of Fig I, ( g + 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 as well as the asymptotic stability After deciding the appropriate non-negative numbers and k, we can obtain the control law as follows Downloaded by [University of Tokyo] at 01:00 04 April 2013 Decision nterhod of opritnnl pole assignments I1 Step 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 ( ) by utilizing some left eigenvectors of Ai which correspond to some specified eigenvalues outside the desired region Scep Calculate A,,+,,i and B, = L i B like ( ) ,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 By utilizing Lemma 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+ + m l P + } ~ , + Step 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 ([$T qTITB)= I holds with respect to the eigenvectors corresponding to the specified eigenvalues, we can continue the same iterations as above with a slight modification in Steps and 3: define L, = [$T vT]T in Step I , and define m, as a positive number satisfying tr { ( - I)*+ ' ( A , + , , - r n ~ E ) m, in Step 3, instead of H,(m) < of Lemma 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 ( ) 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 ( ) 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 can become the maximum solution of the Riccati equation ( ) where Q is given by Q = P E R -'BTP - P(A,+ 11) - ( A g + , *+ 1 ) ~>p Then the following lemma is obtained with respect to the maximum solution of (28) ,,+ Downloaded by [University of Tokyo] at 01:00 04 April 2013 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,+,, > + 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 ( ) , Q of ( ) becomes positive definite (42) Lemma 10 shows that any positive definite symmetric matrix can be the maximum solution of ( ) 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 ( ) one by one in Step 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 Let A,+,, + 11 be an asymptotically stable matrix An arbitrary positive definite symmetric matrix P satisfying P(A,+,, + 11) + ( A , + , , + I I ) ~