- Email: [email protected]
Eigenstructure Assignment for Linear Quadratic Regulator
2a-042
Copyright © 1996 IFAC 13th Triennial World Congres."", San Francisco, USA
EIGENSTRUCTURE ASSIGNMENT FOR LINEAR QUADRATIC REGULATOR Y. Ochi and K. Kanai National De/ense Academy Yokosuka, Kanagawa, 239 Japan email:[email protected]
Abstract: This paper presents a design method for a linear quadratic regulator (LQR) with a prescribed eigenstrnctore. Specifically, a non-negative definite weighting matrix that provides the LQR with the eigenstructure is found by continuonsly shiftiug closed-loop eigenvalues, eigenvalues of the weighting matrix, and some selected elements of the closedloop eigenvectors. A set of differential equations for the weighting matrix are derived from characteristic equations of the Hamilton matrix and the weighting matrix and eigenvaluel eigenvector equations. The differential equations are integrated to obtain the desired eigenstructure. To illustrate the effectiveness of the method, a numerical example is shown. Keywords: Eigenstructure assignment, Linear quadratic regnlator, State feedback, Control system design
1. INTRODUCTION The optimal regulator or linear quadratic regulator (LQR) is a state-feedback control system with good properties such as robustness. A problem with the LQR is how to choose weighting matrices of the cost function. An approach to this problem is pole assignment for the LQR; namely assigning closed-loop eigenvalues(CLEs) determines a weighting matrix. Many papers on this problem have been published(Amin, 1985; Anderson and Moore, 1971; Fujii, 1987; Graupe, 1972; H.rvey and Stein, 1978; K.wasaki .nd Shimemur., 1983; Medanic, et aI., 1988; Shieh, et al., 1986; Solheim, 1972; Saif, 1989; Sugimotoand Yamamoto, 1989).
The authors also proposed a method of pole placement for the LQR(Ochi and Kanai, 1995a; Ochi, et al., 1995b). The basic approach of the method is to obtain a weighting matrix by continuously shifting the CLEs from CLEs for a given weighting matrix to desired CLEs. For this purpose, a set of differential equations of the weighting matrix are derived from the characteristic equation of the Hamilton matrix, and integrated from the given CLEs to the desired ones. Originally, the weighting matrix was limited to a diagonal one, and the resulting matrix was not necessarily nODncgativc dcfinitc(Ochi and Kanai, 1995.). In order to obtain a non-negative definite weighting matrix, the method was improved by combining differential equations for eigenvalues of the weighting matrix with those for CLEs(Ochi, et al., 1995b). In the improved method, all the elements of the weighting matrix are adjusted. Although
1098
much computation is required for the numerical integration, this method is so flexible that eigenvalues are shifted aJbiuarily and exactly except for repeated eigenvalues, from reaUcomplex numbers to real/complex numbers. In fact, in a case where a non-negative definite weighting matrix was not obtained by Saifs metllod(1989), it was obtained by this metltod(Ochi, et al., 1995b). Generally, however, tile resulting weighting matrix is not unique, since tile derivative matrix of tile differential equations is not square; tltat is, tile number of rows, which corresponds to the number of tile characteristic equations is smaller than the number of columns, which corresponds 10 tile number of adjustable elements of the weighting matrix. This means that the differential values of tIlose elements cannot be determined uniquely. To determine tile values, a pseudo-inverse of tile derivative matrix was used(Ochi, et aI. , 1995b). However, the non-uniqueness also implies tltat more equations tltat reflect other specifications desired 10 be satisfied can be taken into account. In tIlis paper. tile authors take advantage of this remaining freedom to assign part of tile closed-loop eigenvectors. This is what is called an eigenstructure assignment problem(Andry. et. al., 1983). Specifically, a set of differential equations are derived from eigenvalueleigenvector equations(EEE) for the Hantilton matrix; the EEE is a definition equation of eigenvalues and eigenvectors: (U-A)v=O. where A.nd v is an eigenvalue and an eigenvector of tile matrix A, respectively. The derived differential equations are combined with tile differential equations for the CLEs and those for the eigenv.lues of the weighting matrix.
The layout of this paper is as follows. In Section 2, following tile derivation of the differential equations, it is shown tltat tile metllod can be used to modily an indefinite weighting matrix into a non-negative definite one without changing the CLEs. In Section 3, eigenvector assignment is described. To illustrate the effectiveness of tile proposed metllod, in Section 4, results for a simple example are shown and discussed. Finally, conclusions are given in Section 5.
J = J.~ (x T Qx + UT Ru)dt
can be minimized. where xERJiI is a stale vector and "ER" a control vector. Here, AER~ and BER'~ are constanl matrices. QER'~ is a non-negative definite matrix, and RER'~ is a positive definite matrix. (,4, B) is assumed to be a controllable pair. For the closed-loop system to be stable. (A . CQ) is assumed to be a detectable f.air, where CQ is an appropriate matrix tltat satisfies Q- CQ CQ and has the same rank as Q.
2.2. Characteristic equation of the Hamilton matrix The following matrix, HER" " ', is called the Hamilton matrix:
H-
A -BR-IB'] , [-Q -A
(3)
The characteristic equation for H is (4)
I(A, Q) • det(1J" - H) - 0
where det(·) stands for a determinant and 1" a 2n x2n identity matrix.
2.3. Differential equations Let desired CLEs be Ai' (i-I, .. ., n) and initial CLEs for an initial weighting matrix Qo, which is given appropriately. be "'" (i-I, .... n) . Consider a small deviation of the eigenvalue. tlA" and define it as fjJ",:(N' -"",)IN. where N is a large integer. Further. define fjJ" _ m,in lA), and kF
,I
fjJ",IAA.
Then, by applying the Taylor series expansion to
f{Q, Ai) around the nontinal Q and N. and taking Eq. (4) into account, the following equations are obtained using firstorder approximation:
f{ aqgaj, Ilqg f( aqJ'aj,
f.j
2. EIGENVALUE ASSIGNMENT BY CONTlNUOUS EIGENVALUE-SHIFTlNG :1.1. The LQR problem
(2)
+
f.j
+
at, )AqJ'}-- aI, k,IlA aA,
aq~
i=l . 2... ., n
(5)
The characteristic equation for an eigenvalue of the weighting matrix Ae is
For a linear system described by (6)
x _ Ax+ Bu
(I)
a control law is determined so tltat the performance index
The differenlial equation for Ae is derived in the same way as Eq. (5):
1099
where m is the number of negative eigenvalues of Q, hi indicates h('AQi, Q) in Eq. (6), and kei is defined as kQi=('AQ:-~i)/(NI!.A), where 'AQi' and )'QOi are desired and initial values of'AQi, respectively. Define ll=[f/, h,Yand G=1g", geY, wherej;=(fi, .... ,/.1', h,=[h" .... , hmI', q=[qll, .. ,q", .. ,q=f (j=I, ... ,n; j<>k9l),
g=[of. k" ..... , of. k.t, oA,
oA.
determine I!.q/I!.'J.. that exactly satisfies Eqs. (5) and (7). In this section, differential equations are derived from the EEE for the Hamilton matrix and combined with Eq. (8). Thereby part of the closed-loop eigenvectors as well as CLEs and 'AQ can be assigned to desired values. 3.1. EEE equation for the Hamilton matrix
Let an eigenvector for the eigenvalue A of the Hamilton where vERn and wERn • Then, the EEE matrix be [VT, for the Hamilton matrix can be written as
wY,
(U - Hl[v T w T )'
ge =[ oh, kep ..... ' Oh. ke,,X, OA e,
_
0
(9)
OA Q•
Then, from Eqs. (5) and (7) the differential equation for q is derived as
The optimal feedback gain matrix is given by (10)
(8) where the superscript + indicates a pseudo·inverse matrix. If Q is non-negative definite, Eq. (7) is not included in Eq. (8), that is, m=(J. The weighting matrix R is assumed to be kept a nominal matrix.
where W=[w" .... , wnl and V=[v" .... , vnl. Note that [wt,
vt)" (i=I, ... , n) is an eigenvector for a stable eigenvalue Ai of H. Suppose that [w', vY in Eq. (9) is one of the stable eigenvectors. Meanwhile, generally the feedback gain matrix that provides the closed-loop system with the CLEs {A" .... , hn ) is given by the modal matrix V, as I K--WV, ,
1.4. Modification of the weighting matrix
There are two ways of using the differential equations for eigenvalues of the weighting matrix. The ftrsl way is to combine the equations with those for the CLEs, when 'AQ is about to become negative at a point of the integral contour, and thereafter the eigenvalue is kept zero or a small positive constant value by setting gQ a null vector(Ochi, et al., 1995b). The second is to integrate the differential equations for negative eigenvalues of the weighting matrix from the negative values to zero or small positive values. In this paper, taken is the second way, which can be used to modify an indefinite weighting matrix obtained by the authors' previous method(Ochi and Kanai, 1995a) or other methods such as Saifs(1989), without changing the CLEs. The assigned CLEs do not change by setting g a null vector. In this case, I!.'J.. is chosen as ,<, h _ min 1,<, A ",I· I'
,
where W, is a matrix that satisfies [Al- A: BllJ':T w.Tf = 0 (i=I, ... , n)(Andry, et al., 1983). Comparing Eq. (10) with Eq. (11), it is observed that v is an eigenvector of the closed-loop system. The vectors v and w or the matrices V and W bave the
relation: (12) where P is a solution of the algebraic Riccati equation: ATp+PA_PBR"BTp+Q_O
(13)
Substituting w in Eq. (12) into Eq. (9) and eliminating P yields
{iJ -A-BR-'BT(lI +ATt'Q}v =0
3. EIGENSTRUCTURE ASSIGNMENT In Eq. (8), the matrix illj/aqT has n+m rows and n(n+l)!2 columns. Since 0"""91, n(n+I)I2-n-m",n(n-3)/2. Hence, if n",3, then the number of the rows of illj/aqT is equal to or smaller than that of the columns. This means that if the matrix is offull rank, the number of equations in Eq. (8) can be increased until all/aqT becomes a square matrix to
(11)
(14)
In the following, differential equations are derived for the case of real eigenvalues/eigenvectors and the case of complex ones, separately.
1100
3.2. Case 1: real eigenvaluesleigerwecfors
v.;(i=l, ... , $). The derivative matrix of the left-hand side of (20) has n+m+n$ rows. However, even if n+m+ns
Let IdER"" be a vector that consists of desired values to be assigned to the eigenvector for a selected eigenvaiue. For exact assignment, md must be equal to or smaller than r. The assignable eigenvector is given by(Andry, et al. 1983) (15) where L=4L/(LdL/t' and 4 =(IJ -At'B. Here, Ld is an mdxr matrix that consists of the rows of L,. The rows of Ld correspond to the elements of the eigenvector to which desired values are assigned. Ld is assumed to be of full rank. Substituting Eq. (15) into Eq. (14) yields
(C + DQ)Lld
=
0
that Eq. (4) is equivalent to Eq. (14). This means that the derivative matrix, [&f,Jaq D,t)', has the rank ofB. Hence, in order to make a full-rank derivative matrix, one has only to choose rank(B)-1 independent rows of D" and combine them with &T)/&qT. This fact implies that the number of assignable eigenvectors $ does not exceed {n(n-I)I2m }/(rank(B)-I). Let the row vectors chosen from D" be d" and the corresponding element of GLI be gLl, and replace D a and GLa in Eq. (20) with Db=[d,/, .... , d,.Y and GLb=[gLl, .... , &..1', respectively. Then, the differential equation for q is obtained as
(16)
where C=1J-A and D=-BKIBT(1J+ATrl. Since the CLEs are not shifted, Eq. (16) is a function of Q and Id. By applying the Taylor series expansion to the left-hand side of Eq. (16) and using first-order approximation, the following equation is obtained: (17)
3.3. Case 2: complex eigenvaluesieigenvec/ors Let a pair of complex conjugate eigenvalues and eigenvectors of H be ruifll and yrj&, respectively. Defining i\=diag{ a+}II, a-}II} and V=[y+}&, y-j&], the EEE for H can then be written as
Let the ith column vector of D be d,ERn (i=I, .... , n) and definep=[PI, .... ,p.f=ud . Thenusingq~=q;.,DQLldcanbe written as
HV=Vi\
(22)
Bya linear transformation Eq. (22) can be rewritten as (18)
Derme a small deviation of Id as Ma=kLI!J.., where kL ER"" is determined from initial and fiual vaiues of Id, as k, or ~, is done. Then, from Eqs. (17) and (18), the differential equation for the eigenvectors is obtained as Aq D ---GL p
A)"
][Y]
al,. - H -fil,. =0 [ fil,. al,. - H b
(23)
As in Case I, the eigenvectors have the following structure: y-[v', (Pv)")' and &_[wT, (Pw)y, where vERn and wERn. By substituting these y and b into Eq. (23) and elintinating P, the follOwing equation is obtained:
(19)
where D,=(Pldl , Pld2 +p,dlo ... , p~+p,d" ... , Pndn] E R·""n+m and GL=(C+DQ)LkL . Thus, the equations can be made for selected eigenvectors v" (i=l, ... , $). Putting those equations together and combining them with Eq. (8) yields
(20)
(24)
where C _ [al. - A
,
and D , -
Ill.
[BWIBT 0
-Ill. aln-A
0 BW'BT
I,
fL _ [Q 0
][-{ala +AT) -Ill.
Ill.
-{al. +AT)
]-1
The assignable eigenvector corresponding to the one in Eq. (15) is given by (25)
whereDa=[D,/ Dp/ .... DpsT)' and GLa=[GLl Tar./ .... GuT)'; D" and GLI are Dp and GL in Eq. (19) for the eigenvector
1101
where Icr[/,,,,.", 1",,,/1' ER 2.... is a vector that consists of desired values to be assigned to the eigenvector; l.-d~ ERm, and I,dim ER .... are the real part and the imagiruuy part of the desired vector, respectively; L, is defined by L,= L"Lj(L,;LJr l ; L~ is defined by
L _
[alIll. . - al.Ill.
]-1
D",
1
(30)
Gd),
T T and GcU--fgeLl, T .... , geLsTJT. where Dcb=[depI T, ••.•• dcps] Here, the number of the selected eigenvectors 2s must satisfY 2ss{ n(n-I )/2-m }/(rank(B)-I).
[B 0]; and Lod is a 2mdx2r -A 0 B full-rank matrix that consists of the rows of L~, which correspond to the elements of the eigenvector to which desired values are assigned. Regarding [v,,T, v..,T), as v" Eq. (24) becomes
If selected eigenvectors include both real and complex numbers, then from Eqs. (21) and (30) the differential equation becomes
(26)
(31)
"
A
I1q __ [d'lldqTj'[ G
!lA
In the same way as Eq. (17), the following equation is derived from Eq. (26): Thus, a non-negative definite weighting matrix can be obtained by integrating the differential equations, Eq. (21), Eq. (30) or Eq. (31), from given initial CLEs, eigenvalues of the weighting matrix, if negative, and values of specified elements of the selected eigenvectors to their desired values.
D/2,LJoo can be written as
D,Q,LJ", -
~ {(Pn,d'j + p"",d,.,j)qjj 4. EXAMPLE
~(P",d" + p"",d~" + p~Aj + P"",dm,)qj,}
+
(28)
The proposed method has been applied to the system(Saif, 1989), where the matrices in Eq. (I) are
2n
where [Prd, .. " Prrn. Piml, ...• Pimnf=LJcaER. and Dc=[dch •.. _, den, dcn+l, ... " dc2n]ER2n><2n, Define a small deviation of I,d as Mcrkd.dA, where k,
D I1q q. I1A
= -(0_. ~
(29)
where D "P=[Pr,] del +Piml dd, P,.,Zdd +P im2dCfl+1 +Prd de2 .I JER2n .n('" lY' • and Gcl.+Pimldcn+2 • .....• p,.,nt/cn+Pimn"c21l (C,+D,Q,)L,kd.' As in Case 1, when D" is combined with aTJ/aqT, the rank of the derivative matrix is not 3n+m but n+m+2(rank(B)-I). Defining [D".", D"l),=D,p, where D cpIlERnxn(n-I-l)12 and D cp ;ER"xn(n-I-l)l2, one has only to choose rank(B)-1 independent rows from D 'P" and as many independent rows fromD"I, and combine them with ~/dqT to make a derivative matrix with full rank. Let the matrix composed of the chosen rows of DCPi , which is D" for a selected eigenvector Vsi and its conjugate one v.rI O-;;:1, .. " s), be d,pi, and let the vector composed of the corresponding elements of Ga.i be ga.i. Then in the same way as Eq. (21), the differential equation for q is obtained as
-I -I2 02 1and B - [ll -I' I1 [-2 -2 -3 l I
A_ 0
The eigenvalues of A are -3 and -I±2j. The desired weighting matrix that places CLEs at {-4, -5, -7} is given by Saif(I989). The eigenvalues of the weighting matrix are -9.5667, 9.4457, and 90.9670, which indicates that the weighting matrix is indefinite. Zero is chosen as a desired value for the negative eigenvalue . Then, since the number of the CLEs is 3 and that of the negative eigenvalue of the weighting matrix is I, the number of assignable eigenvectors is at most 2. In this example, let us select the eigenvectors for the eigenvalues -4 and -5 and consider assigning one and zero to the first and the second elements of the eigenvectors, respectively. This assignment means that the modes of the eigenvalues -4 and -5 do not appear in the second element of the state vector, x. The weighting matrix R is an identity matrix. Table I summarizes the initial and desired terminal conditions for integration. The results are summarized in Table 2.
1102
References
Table 1 Initial and desjred values
CLEs Eigenvalue of Q Assigned elements of selected eigenvectors
Initial -4,-5,-7 -9.5667 I, -O.4709(),,--4) I, -O.274Q(A.--5)
Terminal -4,-5, -7
o 1, 0 1,0
Table 2 Design results WeildltinJ( matrix Q -17.7117 10.7695
-17.7117 -11.7310
30.0002 22.7116
-11.7310 22.7116 26J925
Elgenvalues of Q 0,7.0770, 59.8852
Modal matrix I I -I
0 0 3 5
3 I
CLEs -4, -5, -7
The results show that all the specifications are satisfied. Results in the case of complex eigenvaluesieigenvectors, which are not shown in this paper, have also successfully been obtained.
5. CONCLUSIONS The proposed method makes it possible to design a linear quadratic regulator with a prescribed eigenstructure. However, as"gmng eigenstructure while retaining optimality of the LQR is so demanding that the desired weighting matrix cannot always be obtained, depending on specified desired eigenvaluesieigenvectors. Specifically, as integration proceeds, the derivatives of the differential equations become large, so that it becomes difficult to retain the precision of numerical integration; as a result, at a point the integration does not proceed any more. This implies that in practical systems th"re may not be so many cases where physically meaningful eigenstructure assignment such as decoupling is possible. Meanwhile, one can give another specification to the optimal regulator in the same way as eigenstructure, if the specification is given by equations described by differentiable functions of the weighting matrix and some parameters that reflect the specification.
AmiD, M H. (1985). Optimal Pole Shifting for Continuous Multivariable Systems. International Journal 0/ Control, Vol. 41, No. 3, pp. 701·707. Anderson, B. D. O. andJ. B. Moore(1971). Linear Optimal Control, Chapters 4 and 7, Prenctice·Hall, Englewood Cliffs, NJ. Andry, Jr. A. N., E. Y. Shapiro, and J. C. Chung (1983). Eigenstructure Assigmoent for Linear Systems. IEEE Trans., VoL AES·19, No. 5, pp. 711·728. Fujii, T. (1987). A New Approach to the LQ Design from the Viewpoint of the Inverse Regulator Problem. IEEE Trans., VoL AC·32, No. 11, pp. 995·1004. Graupe, D. (1972). Derivation of Weighting Matrices towards Satisfying Eigenvalue Requirements. International Journal a/Control, Vol. 16, No. 5, pp. 881·888. Harvey, C A. and G. Stein (1978)., Quadratic Weights for Asymptotic Regulator Properties. IEEE Trans., Vol. AC·23, No. 3, pp. 378·387. Kawasaki, N. and E. Shimemura (1983). Determining Quadratic Weighting Matrices to Locate Poles in a Specific Region. Automatica, Vol. 19, No. 5, pp.
557·560. Medanic, J., H., S. Tharp, and W. R. Perkins (1988). Pole Placement by Performance Criterion Modification. IEEE Trans., Vol. AC·33, No. 5, pp. 469-472. Oehi, Y. and K. Kanai (1995a). Pole Placement in Optimal Regulator by Continuous Pole·Shifting. AL4A Journal a/Guidance, Control, and Dynamics , Vol. 18, No. 6, pp. 1254·1258. Oehi, Y., K. Kanai, and B. Wie (1995b). Pole Assigmoent for Optimal Regulator with Non·Negative Definite Weighting Matrix. Proceedings 0/ AL4A Guidance, Navigation, and Control Conference, Baltimore, MD, August 7.9, pp. 1148·1156. (Also see AIAA Paper 95.3300.) Saif, M. (1989). Optimal Linear Regulator Pole·Placement by Weight Selection. International Journal a/Control, Vol. 50, No. 1, pp. 399·414. Shieh, L. S., H. M. Dib, and B. C. Mcinnis (1986). Linear Quadratic Regulators with Eigenvaluc Placement in a Vertical Strip. IEEE Trans., Vol. AC·31. No. 3, pp. 241·243. Solheim, O. A. (1972). Desjgn of Optimal Control Systems with Prescribed Eigenvalues. International Journal of Control, VoL 15, No. 1, pp. 143·160. Sugimoto, K. and Y. Yamamoto (1989). On Successive Pole Assignment by Linear-Quadratic Optimal Feedbacks. Linear Algebra and its Application, VoL 1221123/124, pp. 697·723.
1103
Copyright © 1996 IFAC 13th Triennial World Congres."", San Francisco, USA
EIGENSTRUCTURE ASSIGNMENT FOR LINEAR QUADRATIC REGULATOR Y. Ochi and K. Kanai National De/ense Academy Yokosuka, Kanagawa, 239 Japan email:[email protected]
Abstract: This paper presents a design method for a linear quadratic regulator (LQR) with a prescribed eigenstrnctore. Specifically, a non-negative definite weighting matrix that provides the LQR with the eigenstructure is found by continuonsly shiftiug closed-loop eigenvalues, eigenvalues of the weighting matrix, and some selected elements of the closedloop eigenvectors. A set of differential equations for the weighting matrix are derived from characteristic equations of the Hamilton matrix and the weighting matrix and eigenvaluel eigenvector equations. The differential equations are integrated to obtain the desired eigenstructure. To illustrate the effectiveness of the method, a numerical example is shown. Keywords: Eigenstructure assignment, Linear quadratic regnlator, State feedback, Control system design
1. INTRODUCTION The optimal regulator or linear quadratic regulator (LQR) is a state-feedback control system with good properties such as robustness. A problem with the LQR is how to choose weighting matrices of the cost function. An approach to this problem is pole assignment for the LQR; namely assigning closed-loop eigenvalues(CLEs) determines a weighting matrix. Many papers on this problem have been published(Amin, 1985; Anderson and Moore, 1971; Fujii, 1987; Graupe, 1972; H.rvey and Stein, 1978; K.wasaki .nd Shimemur., 1983; Medanic, et aI., 1988; Shieh, et al., 1986; Solheim, 1972; Saif, 1989; Sugimotoand Yamamoto, 1989).
The authors also proposed a method of pole placement for the LQR(Ochi and Kanai, 1995a; Ochi, et al., 1995b). The basic approach of the method is to obtain a weighting matrix by continuously shifting the CLEs from CLEs for a given weighting matrix to desired CLEs. For this purpose, a set of differential equations of the weighting matrix are derived from the characteristic equation of the Hamilton matrix, and integrated from the given CLEs to the desired ones. Originally, the weighting matrix was limited to a diagonal one, and the resulting matrix was not necessarily nODncgativc dcfinitc(Ochi and Kanai, 1995.). In order to obtain a non-negative definite weighting matrix, the method was improved by combining differential equations for eigenvalues of the weighting matrix with those for CLEs(Ochi, et al., 1995b). In the improved method, all the elements of the weighting matrix are adjusted. Although
1098
much computation is required for the numerical integration, this method is so flexible that eigenvalues are shifted aJbiuarily and exactly except for repeated eigenvalues, from reaUcomplex numbers to real/complex numbers. In fact, in a case where a non-negative definite weighting matrix was not obtained by Saifs metllod(1989), it was obtained by this metltod(Ochi, et al., 1995b). Generally, however, tile resulting weighting matrix is not unique, since tile derivative matrix of tile differential equations is not square; tltat is, tile number of rows, which corresponds to the number of tile characteristic equations is smaller than the number of columns, which corresponds 10 tile number of adjustable elements of the weighting matrix. This means that the differential values of tIlose elements cannot be determined uniquely. To determine tile values, a pseudo-inverse of tile derivative matrix was used(Ochi, et aI. , 1995b). However, the non-uniqueness also implies tltat more equations tltat reflect other specifications desired 10 be satisfied can be taken into account. In tIlis paper. tile authors take advantage of this remaining freedom to assign part of tile closed-loop eigenvectors. This is what is called an eigenstructure assignment problem(Andry. et. al., 1983). Specifically, a set of differential equations are derived from eigenvalueleigenvector equations(EEE) for the Hantilton matrix; the EEE is a definition equation of eigenvalues and eigenvectors: (U-A)v=O. where A.nd v is an eigenvalue and an eigenvector of tile matrix A, respectively. The derived differential equations are combined with tile differential equations for the CLEs and those for the eigenv.lues of the weighting matrix.
The layout of this paper is as follows. In Section 2, following tile derivation of the differential equations, it is shown tltat tile metllod can be used to modily an indefinite weighting matrix into a non-negative definite one without changing the CLEs. In Section 3, eigenvector assignment is described. To illustrate the effectiveness of tile proposed metllod, in Section 4, results for a simple example are shown and discussed. Finally, conclusions are given in Section 5.
J = J.~ (x T Qx + UT Ru)dt
can be minimized. where xERJiI is a stale vector and "ER" a control vector. Here, AER~ and BER'~ are constanl matrices. QER'~ is a non-negative definite matrix, and RER'~ is a positive definite matrix. (,4, B) is assumed to be a controllable pair. For the closed-loop system to be stable. (A . CQ) is assumed to be a detectable f.air, where CQ is an appropriate matrix tltat satisfies Q- CQ CQ and has the same rank as Q.
2.2. Characteristic equation of the Hamilton matrix The following matrix, HER" " ', is called the Hamilton matrix:
H-
A -BR-IB'] , [-Q -A
(3)
The characteristic equation for H is (4)
I(A, Q) • det(1J" - H) - 0
where det(·) stands for a determinant and 1" a 2n x2n identity matrix.
2.3. Differential equations Let desired CLEs be Ai' (i-I, .. ., n) and initial CLEs for an initial weighting matrix Qo, which is given appropriately. be "'" (i-I, .... n) . Consider a small deviation of the eigenvalue. tlA" and define it as fjJ",:(N' -"",)IN. where N is a large integer. Further. define fjJ" _ m,in lA), and kF
,I
fjJ",IAA.
Then, by applying the Taylor series expansion to
f{Q, Ai) around the nontinal Q and N. and taking Eq. (4) into account, the following equations are obtained using firstorder approximation:
f{ aqgaj, Ilqg f( aqJ'aj,
f.j
2. EIGENVALUE ASSIGNMENT BY CONTlNUOUS EIGENVALUE-SHIFTlNG :1.1. The LQR problem
(2)
+
f.j
+
at, )AqJ'}-- aI, k,IlA aA,
aq~
i=l . 2... ., n
(5)
The characteristic equation for an eigenvalue of the weighting matrix Ae is
For a linear system described by (6)
x _ Ax+ Bu
(I)
a control law is determined so tltat the performance index
The differenlial equation for Ae is derived in the same way as Eq. (5):
1099
where m is the number of negative eigenvalues of Q, hi indicates h('AQi, Q) in Eq. (6), and kei is defined as kQi=('AQ:-~i)/(NI!.A), where 'AQi' and )'QOi are desired and initial values of'AQi, respectively. Define ll=[f/, h,Yand G=1g", geY, wherej;=(fi, .... ,/.1', h,=[h" .... , hmI', q=[qll, .. ,q", .. ,q=f (j=I, ... ,n; j<>k9l),
g=[of. k" ..... , of. k.t, oA,
oA.
determine I!.q/I!.'J.. that exactly satisfies Eqs. (5) and (7). In this section, differential equations are derived from the EEE for the Hamilton matrix and combined with Eq. (8). Thereby part of the closed-loop eigenvectors as well as CLEs and 'AQ can be assigned to desired values. 3.1. EEE equation for the Hamilton matrix
Let an eigenvector for the eigenvalue A of the Hamilton where vERn and wERn • Then, the EEE matrix be [VT, for the Hamilton matrix can be written as
wY,
(U - Hl[v T w T )'
ge =[ oh, kep ..... ' Oh. ke,,X, OA e,
_
0
(9)
OA Q•
Then, from Eqs. (5) and (7) the differential equation for q is derived as
The optimal feedback gain matrix is given by (10)
(8) where the superscript + indicates a pseudo·inverse matrix. If Q is non-negative definite, Eq. (7) is not included in Eq. (8), that is, m=(J. The weighting matrix R is assumed to be kept a nominal matrix.
where W=[w" .... , wnl and V=[v" .... , vnl. Note that [wt,
vt)" (i=I, ... , n) is an eigenvector for a stable eigenvalue Ai of H. Suppose that [w', vY in Eq. (9) is one of the stable eigenvectors. Meanwhile, generally the feedback gain matrix that provides the closed-loop system with the CLEs {A" .... , hn ) is given by the modal matrix V, as I K--WV, ,
1.4. Modification of the weighting matrix
There are two ways of using the differential equations for eigenvalues of the weighting matrix. The ftrsl way is to combine the equations with those for the CLEs, when 'AQ is about to become negative at a point of the integral contour, and thereafter the eigenvalue is kept zero or a small positive constant value by setting gQ a null vector(Ochi, et al., 1995b). The second is to integrate the differential equations for negative eigenvalues of the weighting matrix from the negative values to zero or small positive values. In this paper, taken is the second way, which can be used to modify an indefinite weighting matrix obtained by the authors' previous method(Ochi and Kanai, 1995a) or other methods such as Saifs(1989), without changing the CLEs. The assigned CLEs do not change by setting g a null vector. In this case, I!.'J.. is chosen as ,<, h _ min 1,<, A ",I· I'
,
where W, is a matrix that satisfies [Al- A: BllJ':T w.Tf = 0 (i=I, ... , n)(Andry, et al., 1983). Comparing Eq. (10) with Eq. (11), it is observed that v is an eigenvector of the closed-loop system. The vectors v and w or the matrices V and W bave the
relation: (12) where P is a solution of the algebraic Riccati equation: ATp+PA_PBR"BTp+Q_O
(13)
Substituting w in Eq. (12) into Eq. (9) and eliminating P yields
{iJ -A-BR-'BT(lI +ATt'Q}v =0
3. EIGENSTRUCTURE ASSIGNMENT In Eq. (8), the matrix illj/aqT has n+m rows and n(n+l)!2 columns. Since 0"""91, n(n+I)I2-n-m",n(n-3)/2. Hence, if n",3, then the number of the rows of illj/aqT is equal to or smaller than that of the columns. This means that if the matrix is offull rank, the number of equations in Eq. (8) can be increased until all/aqT becomes a square matrix to
(11)
(14)
In the following, differential equations are derived for the case of real eigenvalues/eigenvectors and the case of complex ones, separately.
1100
3.2. Case 1: real eigenvaluesleigerwecfors
v.;(i=l, ... , $). The derivative matrix of the left-hand side of (20) has n+m+n$ rows. However, even if n+m+ns
Let IdER"" be a vector that consists of desired values to be assigned to the eigenvector for a selected eigenvaiue. For exact assignment, md must be equal to or smaller than r. The assignable eigenvector is given by(Andry, et al. 1983) (15) where L=4L/(LdL/t' and 4 =(IJ -At'B. Here, Ld is an mdxr matrix that consists of the rows of L,. The rows of Ld correspond to the elements of the eigenvector to which desired values are assigned. Ld is assumed to be of full rank. Substituting Eq. (15) into Eq. (14) yields
(C + DQ)Lld
=
0
that Eq. (4) is equivalent to Eq. (14). This means that the derivative matrix, [&f,Jaq D,t)', has the rank ofB. Hence, in order to make a full-rank derivative matrix, one has only to choose rank(B)-1 independent rows of D" and combine them with &T)/&qT. This fact implies that the number of assignable eigenvectors $ does not exceed {n(n-I)I2m }/(rank(B)-I). Let the row vectors chosen from D" be d" and the corresponding element of GLI be gLl, and replace D a and GLa in Eq. (20) with Db=[d,/, .... , d,.Y and GLb=[gLl, .... , &..1', respectively. Then, the differential equation for q is obtained as
(16)
where C=1J-A and D=-BKIBT(1J+ATrl. Since the CLEs are not shifted, Eq. (16) is a function of Q and Id. By applying the Taylor series expansion to the left-hand side of Eq. (16) and using first-order approximation, the following equation is obtained: (17)
3.3. Case 2: complex eigenvaluesieigenvec/ors Let a pair of complex conjugate eigenvalues and eigenvectors of H be ruifll and yrj&, respectively. Defining i\=diag{ a+}II, a-}II} and V=[y+}&, y-j&], the EEE for H can then be written as
Let the ith column vector of D be d,ERn (i=I, .... , n) and definep=[PI, .... ,p.f=ud . Thenusingq~=q;.,DQLldcanbe written as
HV=Vi\
(22)
Bya linear transformation Eq. (22) can be rewritten as (18)
Derme a small deviation of Id as Ma=kLI!J.., where kL ER"" is determined from initial and fiual vaiues of Id, as k, or ~, is done. Then, from Eqs. (17) and (18), the differential equation for the eigenvectors is obtained as Aq D ---GL p
A)"
][Y]
al,. - H -fil,. =0 [ fil,. al,. - H b
(23)
As in Case I, the eigenvectors have the following structure: y-[v', (Pv)")' and &_[wT, (Pw)y, where vERn and wERn. By substituting these y and b into Eq. (23) and elintinating P, the follOwing equation is obtained:
(19)
where D,=(Pldl , Pld2 +p,dlo ... , p~+p,d" ... , Pndn] E R·""n+m and GL=(C+DQ)LkL . Thus, the equations can be made for selected eigenvectors v" (i=l, ... , $). Putting those equations together and combining them with Eq. (8) yields
(20)
(24)
where C _ [al. - A
,
and D , -
Ill.
[BWIBT 0
-Ill. aln-A
0 BW'BT
I,
fL _ [Q 0
][-{ala +AT) -Ill.
Ill.
-{al. +AT)
]-1
The assignable eigenvector corresponding to the one in Eq. (15) is given by (25)
whereDa=[D,/ Dp/ .... DpsT)' and GLa=[GLl Tar./ .... GuT)'; D" and GLI are Dp and GL in Eq. (19) for the eigenvector
1101
where Icr[/,,,,.", 1",,,/1' ER 2.... is a vector that consists of desired values to be assigned to the eigenvector; l.-d~ ERm, and I,dim ER .... are the real part and the imagiruuy part of the desired vector, respectively; L, is defined by L,= L"Lj(L,;LJr l ; L~ is defined by
L _
[alIll. . - al.Ill.
]-1
D",
1
(30)
Gd),
T T and GcU--fgeLl, T .... , geLsTJT. where Dcb=[depI T, ••.•• dcps] Here, the number of the selected eigenvectors 2s must satisfY 2ss{ n(n-I )/2-m }/(rank(B)-I).
[B 0]; and Lod is a 2mdx2r -A 0 B full-rank matrix that consists of the rows of L~, which correspond to the elements of the eigenvector to which desired values are assigned. Regarding [v,,T, v..,T), as v" Eq. (24) becomes
If selected eigenvectors include both real and complex numbers, then from Eqs. (21) and (30) the differential equation becomes
(26)
(31)
"
A
I1q __ [d'lldqTj'[ G
!lA
In the same way as Eq. (17), the following equation is derived from Eq. (26): Thus, a non-negative definite weighting matrix can be obtained by integrating the differential equations, Eq. (21), Eq. (30) or Eq. (31), from given initial CLEs, eigenvalues of the weighting matrix, if negative, and values of specified elements of the selected eigenvectors to their desired values.
D/2,LJoo can be written as
D,Q,LJ", -
~ {(Pn,d'j + p"",d,.,j)qjj 4. EXAMPLE
~(P",d" + p"",d~" + p~Aj + P"",dm,)qj,}
+
(28)
The proposed method has been applied to the system(Saif, 1989), where the matrices in Eq. (I) are
2n
where [Prd, .. " Prrn. Piml, ...• Pimnf=LJcaER. and Dc=[dch •.. _, den, dcn+l, ... " dc2n]ER2n><2n, Define a small deviation of I,d as Mcrkd.dA, where k,
D I1q q. I1A
= -(0_. ~
(29)
where D "P=[Pr,] del +Piml dd, P,.,Zdd +P im2dCfl+1 +Prd de2 .I JER2n .n('" lY' • and Gcl.+Pimldcn+2 • .....• p,.,nt/cn+Pimn"c21l (C,+D,Q,)L,kd.' As in Case 1, when D" is combined with aTJ/aqT, the rank of the derivative matrix is not 3n+m but n+m+2(rank(B)-I). Defining [D".", D"l),=D,p, where D cpIlERnxn(n-I-l)12 and D cp ;ER"xn(n-I-l)l2, one has only to choose rank(B)-1 independent rows from D 'P" and as many independent rows fromD"I, and combine them with ~/dqT to make a derivative matrix with full rank. Let the matrix composed of the chosen rows of DCPi , which is D" for a selected eigenvector Vsi and its conjugate one v.rI O-;;:1, .. " s), be d,pi, and let the vector composed of the corresponding elements of Ga.i be ga.i. Then in the same way as Eq. (21), the differential equation for q is obtained as
-I -I2 02 1and B - [ll -I' I1 [-2 -2 -3 l I
A_ 0
The eigenvalues of A are -3 and -I±2j. The desired weighting matrix that places CLEs at {-4, -5, -7} is given by Saif(I989). The eigenvalues of the weighting matrix are -9.5667, 9.4457, and 90.9670, which indicates that the weighting matrix is indefinite. Zero is chosen as a desired value for the negative eigenvalue . Then, since the number of the CLEs is 3 and that of the negative eigenvalue of the weighting matrix is I, the number of assignable eigenvectors is at most 2. In this example, let us select the eigenvectors for the eigenvalues -4 and -5 and consider assigning one and zero to the first and the second elements of the eigenvectors, respectively. This assignment means that the modes of the eigenvalues -4 and -5 do not appear in the second element of the state vector, x. The weighting matrix R is an identity matrix. Table I summarizes the initial and desired terminal conditions for integration. The results are summarized in Table 2.
1102
References
Table 1 Initial and desjred values
CLEs Eigenvalue of Q Assigned elements of selected eigenvectors
Initial -4,-5,-7 -9.5667 I, -O.4709(),,--4) I, -O.274Q(A.--5)
Terminal -4,-5, -7
o 1, 0 1,0
Table 2 Design results WeildltinJ( matrix Q -17.7117 10.7695
-17.7117 -11.7310
30.0002 22.7116
-11.7310 22.7116 26J925
Elgenvalues of Q 0,7.0770, 59.8852
Modal matrix I I -I
0 0 3 5
3 I
CLEs -4, -5, -7
The results show that all the specifications are satisfied. Results in the case of complex eigenvaluesieigenvectors, which are not shown in this paper, have also successfully been obtained.
5. CONCLUSIONS The proposed method makes it possible to design a linear quadratic regulator with a prescribed eigenstructure. However, as"gmng eigenstructure while retaining optimality of the LQR is so demanding that the desired weighting matrix cannot always be obtained, depending on specified desired eigenvaluesieigenvectors. Specifically, as integration proceeds, the derivatives of the differential equations become large, so that it becomes difficult to retain the precision of numerical integration; as a result, at a point the integration does not proceed any more. This implies that in practical systems th"re may not be so many cases where physically meaningful eigenstructure assignment such as decoupling is possible. Meanwhile, one can give another specification to the optimal regulator in the same way as eigenstructure, if the specification is given by equations described by differentiable functions of the weighting matrix and some parameters that reflect the specification.
AmiD, M H. (1985). Optimal Pole Shifting for Continuous Multivariable Systems. International Journal 0/ Control, Vol. 41, No. 3, pp. 701·707. Anderson, B. D. O. andJ. B. Moore(1971). Linear Optimal Control, Chapters 4 and 7, Prenctice·Hall, Englewood Cliffs, NJ. Andry, Jr. A. N., E. Y. Shapiro, and J. C. Chung (1983). Eigenstructure Assigmoent for Linear Systems. IEEE Trans., VoL AES·19, No. 5, pp. 711·728. Fujii, T. (1987). A New Approach to the LQ Design from the Viewpoint of the Inverse Regulator Problem. IEEE Trans., VoL AC·32, No. 11, pp. 995·1004. Graupe, D. (1972). Derivation of Weighting Matrices towards Satisfying Eigenvalue Requirements. International Journal a/Control, Vol. 16, No. 5, pp. 881·888. Harvey, C A. and G. Stein (1978)., Quadratic Weights for Asymptotic Regulator Properties. IEEE Trans., Vol. AC·23, No. 3, pp. 378·387. Kawasaki, N. and E. Shimemura (1983). Determining Quadratic Weighting Matrices to Locate Poles in a Specific Region. Automatica, Vol. 19, No. 5, pp.
557·560. Medanic, J., H., S. Tharp, and W. R. Perkins (1988). Pole Placement by Performance Criterion Modification. IEEE Trans., Vol. AC·33, No. 5, pp. 469-472. Oehi, Y. and K. Kanai (1995a). Pole Placement in Optimal Regulator by Continuous Pole·Shifting. AL4A Journal a/Guidance, Control, and Dynamics , Vol. 18, No. 6, pp. 1254·1258. Oehi, Y., K. Kanai, and B. Wie (1995b). Pole Assigmoent for Optimal Regulator with Non·Negative Definite Weighting Matrix. Proceedings 0/ AL4A Guidance, Navigation, and Control Conference, Baltimore, MD, August 7.9, pp. 1148·1156. (Also see AIAA Paper 95.3300.) Saif, M. (1989). Optimal Linear Regulator Pole·Placement by Weight Selection. International Journal a/Control, Vol. 50, No. 1, pp. 399·414. Shieh, L. S., H. M. Dib, and B. C. Mcinnis (1986). Linear Quadratic Regulators with Eigenvaluc Placement in a Vertical Strip. IEEE Trans., Vol. AC·31. No. 3, pp. 241·243. Solheim, O. A. (1972). Desjgn of Optimal Control Systems with Prescribed Eigenvalues. International Journal of Control, VoL 15, No. 1, pp. 143·160. Sugimoto, K. and Y. Yamamoto (1989). On Successive Pole Assignment by Linear-Quadratic Optimal Feedbacks. Linear Algebra and its Application, VoL 1221123/124, pp. 697·723.
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