0005-1098183 $3.00 + 0.00 Pergamon Press Ltd © 1983 International Federation of Automatic Control Automattca, Vol 19, No 5, 557-560, 1983 Printed in Great Britain Brief Paper Determining Quadratic Weighting Matrices to Locate Poles in a Specified Region* NAOYA KAWASAKD"and ETSUJIRO SHIMEMURA:~ KeyWords Linear optimal regulator; pole placement; state feedback; feedback control; multivariable control systems; control system design; computer-aided design Abstract A new procedure of selecting weighting matrices in linear quadratic optimal control problems (LQ-problems) is proposed In LQ-problems, the quadratic weights are usually decided on trial and error to get good responses But using the proposed method, the quadratic weights are decided in such a way that all poles of the closed loop system are located in the desired region for good response as well as for stability As the system constructed by this method has merits of an LQ-problem as well as a pole-assignment problem, this procedure will be useful for designing a linear feedback system J = (xTQx + uTRu) dt (2) where A, B are n x n, n x r constant matrices, Q and R are n x n, , x r positive definite symmetric matrices respectively, x is an ndimensional state vector, u is an r-dimensional input vector, and (A, B) is a controllable pair Now we consider the method of deciding quadratic weights by which all poles of the closed-loop system are located in the hatched region of Fig We know by experience, if all poles are located in the region of Fig 1, the responses converge to the steady state at appropriate speed and no objectionable vibrating modes appear on the responses In the following, we regard the hatched region of Fig edged by a hyperbola as the desired region in which all poles are to be located instead of the region of Fig because the region of Fig could become a good approximation of the region of Fig by choosing m of a hyperbola (Re 2) - (Ira 2) = m2 appropriately In the next section we consider the method of deciding quadratic weights by which all poles of the closed-loop system are located in the hatched region of Fig Introduction THE closed-loop system constructed by utilizing an LQ-problem has some merits (Safonov and Athans, 1977; Kobayashi and Shimemura, 1981) But when we construct a closed-loop system by utilizing the LQ-problem, the weighting matrices of the quadratic cost function must be decided on by trial and error to get the good responses, because only very little is known about the relation between the quadratic weights and the dynamical characteristics of the closed-loop system (Harvey and Stein, 1978; Stein 1979; Francis, 1979) The dynamical characteristics of a linear system are influenced by the location of poles of the system Therefore to get good responses, it is necessary to locate all poles in the desired positions But we know that it is sufficient to place all poles in a suitable region instead of placing them in their desired respective positions In this paper, we give a new method of selecting the quadratic weights in LQ-problems by which all poles of the closed-loop system are located in the specified region for good response as well as for stability As the system constructed by this method has the merits of an LQ-problem as well as a pole-assignment problem, this method will be useful for constructing linear feedback systems Conceptually this decision method may be considered to derive from the so-called inverse optimal control problems (Thau, 1967; Yokoyama and Kinnen, 1972; Moylan and Anderson, 1973) But it will be dit~cult to derive the concrete result such as obtained in this paper from the arguments about the inverse optimal control problems A method of deciding quadratic weights 3.1 Some preliminary lemmas In this section a new method of deciding quadratic weights of LQ-problem is given Before showing the result, some preliminary lemmas are prepared for obtaining the method Lerama Among eigenvalues of matrix A, we represent eigenvalues of A in the hatched region of Fig 3, edged by a hyperbola (Re ~.)2 _ (Im 2) = m2, by 2~,and eigenvaiues outside this region by 2j Then eigenvalues - 2 + m2 of - A + m2I, Im Problem formulation Now we consider a linear multivariable system (1) and a quadratic cost function (2) ".,(45 "~ N Y¢ = A x + Bu N (1) N N ~ *Received 28 December 1981; revised 19 August 1982; revised 26 January 1983 The original version of this paper was presented at the 8th IFAC Congress on Control Science and Technology for the Progress of Society which was held in Kyoto, Japan during August 1981 The published proceedings of this IFAC meeting may be ordered from Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW, U.K This paper was recommended for publication in revised form by associate editor D H Jacobson under the direction of editor H Kwakernaak ~'Department of Education, Kochi University, 2-5-1 Akebonocho, Kochi 780, Japan ~:Department of Electrical Engineering, Waseda University, 3-41 Okubo Shinjuku-ku, Tokyo 160, Japan N -h////// FIG A desired region where the poles of the closed-loop system are to be located for good responses 557 558 Brief Paper Im Im / o / / / / / J J , -ITI _-fq, :m 302=m2 I FIG A region where the eigenvalues of A satisfying R e ( - A + m21) < exist FIG A region where the poles of the closed-loop system are able to be located by this method In the above lemma, the special case of Q = is stated as follows which correspond to 2~ of matrix A, exist in the left half plane, and - ~ + m 2, which correspond to '~i of matrix A, exist in the right half plane, where m is an arbitrary nonnegative real number L e m m a Let k be an arbitrary real number, then (A + kI, B) is a controllable pair if and only if (A, B) is a controllable pair Lemma Let matrix A have no eigenvalues on the imaginary axis Then (A 2, B) is a controllable pair if and only if (A, B) is a controllable pair Now we give some lemmas about the solution of Riccati equation KBR-IBTK - K A - A T K - Q = (3) L e m m a Let i, ~ ~ be the left half plane eigenvaiues* of A and ~i, ~ ~- be the corresponding eigenvectors If a positive semidefinite symmetric matrix Q of the (3) satisfies the following equation Q~T = 0, iE{1,2 /z} (4) the closed-loop system matrix A - B R - ~ B T K + formed by the maximum solutioni" K+ has the eigenvalue 2~and the corresponding eigenvector (~- Proof: Any positive semidefinite symmetric matrix Q can be expressed as Q = CTC, where C ~ R q x , a n d q is the rank of Q The null space of K÷, denoted by null (K+), satisfies the following relation (Molinari, 1977) null (K+) = 7oC~{.La,~Aa,} (5) where 70, Aa and Lgi denote the unobservable subspace of(C, A), stable and imaginary modes of A, respectively This establishes the lemma [] *We call the eigenvalues in the left half plane containing the imaginary axis the left half plane eigenvalues Similarly the eigenvaiues in the right half plane not containing the imaginary axis are called the right half plane eigenvalues Furthermore we call the eigenvalues which lie in the left half plane not containing the imaginary axis the pure left plane eigenvalues ~'The relation Ka - K2 > (positive semidefinite matrix) is written as K t > K z Equation (3) has many real symmetric solutions If a solution K+ satisfies K+ > K, where K is an arbitrary solution, K+ is called the maximum solution Lemma Let 27 and ~7 (i = 1, ,/z) be the same as before The maximum solution K+ of the equation KBR- 1BTK - KA - ATK = (6) null (K +) = span (~i-, ~- ~- ) (7) satisfies where span (~ i, ~ ~ ~ ~ ) denotes the linear subspace spaned by vectors ~ , ~ i ~ Furthermore the eigenvalues of A - r B R - B r K + , which are denoted by (A - r B R - B r K + ), are 2(A - r B R - I B T K + ) = {2i-, 2f 2- and n - / z pure left half plane eigenvalues} ,u (8) where r is an arbitrary real number satisfying r > ½ Proof: As the former part of the lemma is obvious from the proof of Lemma 4, only the latter part is proved As was proved by Safonov and Athans (1977), the insertion of linear constant gain r > ½into the feedback loops of the respective controls leaves the dosed loop system constructed by using an LQ-problem asymptotically stable in the large Namely the matrix A - r B R - ~ B r K + is asymptotically stable for an arbitrary real number r>½, where K+ is the maximum solution of (3) By applying their result to the case Q = and considering the relation (7), we can obtain the relation (8) [] The relation (8) states that all the other eigenvalues of the matrix A - r B R - ~ B r K + except 2i-,2~, , 2~ also exist in the left half plane not containing the imaginary axis for an arbitrary real number r>½ 3.2 A fundamental theorem In this section, we shall give a fundamental theorem which is important to derive the decision method Let A satisfy Re 2(A) ½ Then from the previous lemmas, we can obtain the following theorem Theorem The following relation holds with respect to eigenvalues of the dosed-loop system matrix A - B R - B r P + , which is formed by the maximum solution P+ of (10) H [)`CA - B R - t B r P + )] = {)`1, )`2 )`p and, at least, one more (or a complex conjugate pair) eigenvalue} (11) Here H [)`(A)] denotes the set of eigenvalues of matrix A in the hatched region of Fig Proof: Let ~1 ~2 ~ and ~1, ~2 ~,_p be eigenvectors of A corresponding to the eigenvalues )`1, )`2 )`p (in the region) and ~-1, ~2 ~,_p (outside the region) Now consider the maximum solution K+ of (9) Because Lemmas and show ( - A + m21, B) is controllable, there exist K + Hence from Lemmas and 5, the eigenvalues of - A + m2I - r B R - B r K + are given by ) ` ( - A d- m2I r B R - I B T K + ) = { _ ) ` d- m 2, _)`2 + m _ )`2 + m and n - p pure left half plane eigenvalues} (12) Furthermore from Lemma and equivalence of the eigenvectors of A and - A + m21, the maximum solution K+ satisfies null (K+) = span (~1, ~2 ~p)' (13) Consider the matrix A - B R - 1BTP+ formed by P+, which is the maximum solution of (10) Then from Lemma we see that )`1, )`2, :, )`p and ~1, ~2 ~p are included in the set of eigenvalues and eigenvectors of A - B R - B r p + respectively Next, consider the remaining eigenvalues of A - B R - I B T P + except 21, )`2, -,)`p We write those eigenvalues as al, a2, ,a,_~ After a simple calculation, also using (10), we have the following relation - (A - B R - B r P + ) = ( - A - r B R - B T K + ) + ( A B R - I B r - B R - B r A t ) P + (14) As ( A B R - 1Br - B R - IB'rAT) is a skew symmetric matrix, trace ( A B R " I B r - B R - t B T A r ) p + = Hence f r o m (14) trace { - (A - B R - B T p + ) + mZl} = trace ( A + mZl r B R - B r K + ) (15) holds Since trl, a2 ~r._p are the remaining n - p eigenvalues of A - B R - I B r P + except )`1, )`2 2p, the eigenvalues of - ( A - B R - B r P + ) + m2I are - 2 + m 2, _)`22 + m _)`2 + m and - a ~ + m 2, -tr22 + m z, - a _ p + m Comparing this fact with the relation (12) and (15) gives ( tr2 + m2) + ( tr2 -k m2) -k -k ( tr2_p + m2) < O (16) The relation (16) shows that at least one (or one complex conjugate pair) of ( - tr2 + m 2) (i - 1,2 n - p) exists in the left half plane Namely, at least, one (or one complex conjugate pair) of at (i = 1,2 n - p) exists in the hatched region of Fig 2, because A - B R - B r P + which is obtained by using an LQproblem with quadratic weights (rK+, R ) is an asymptotically stable matrix I'q From the above theorem, we can see if A satisfies Re )` (A) < the eigenvalues of A - B R - B r p + are located in the following way: (a) the eigenvalues of in the hatched region of Fig are the eigenvalues of A - B R - I B r P + , and (b) at least, one (or one comi~lex conjugate pair) of eigenvalues of A outside the hatched region of Fig moves into the hatched region of Fig Therefore after a finite number of iterated application of the theorem, all eigenvalues of the closed loop system matrix can be located in the hatched region of Fig In Theorem it is assumed the system matrix A satisfies Re )` (A) < for the simplicity of the proof and the sufficiency for the derivation of this decision method But even without this assumption, it is seen the same result holds with respect to the matrix A - B R - B r P + which is obtained after the same operations as Theorem This fact is derived from considering 559 the relation that right half plane eigenvalues of A in the hatched region of Fig are shifted to their corresponding symmetric positions with respect to the imaginary axis by the operation of Theorem [so-called mirror-image shift (Molinari, 1977)] Theorem 1' L e t A satisfy the condition of Lemma A relation similar to Theorem holds with respect to eigenvales of the matrix A - B R - I B T P + H [)`(A - B R - 1BTp+ )] = {)-1, ),2 )`p - ~1, - ~2 - ~q and at least, one more (or a complex conjugate pair) eigenvalue} (17) Here )`1, )`2 )`p and ~1, ~2 ,2q denote the left and fight half plane eigenvalues of A in the hatched regions of Fig 3, respectively 3.3 A method o f deciding quadratic weights Here we consider the decision method of the quadratic weights of an LQ-problem to locate the poles of the closed loop system in the hatched region of Fig In the previous section, we assumed Re )`(A) < for the reasons mentioned before But it should be pointed out that this proposed decision method can be utilized not only for A satisfying R e ) ` ( A ) < but also for any A That is, the only requirement on A is that (A, B) is controllable Namely in tlae following decision method, the closed-loop system matrix A1 = Ao - BR-1BTp~{ = A - B R - B T p ~ o b t a i n e d by step satisfies Re)`(AI) < if Qo is selected as Q0 > Therefore A1 satisfies the assumption of Lemma and Theorem 1, then we can supply Theorem to A instead of applying the theorem to A directly Step I is also recommended for the numerical stability of the computation even in case of Re2(A) < In concluding the above discussions, we can summarize the decision method of the quadratic weights as follows Decision method Step (May be skipped in case of Re)`(A) < 0.) Solve an LQproblem for arbitrary quadratic weights (Qo, R) selected from the demand for the system's dynamical characteristics P I B R - B T P - P t A - ATp1 Q0 = (18) and obtain a closed-loop system matrix A - B R - B T P ~ i Step2 Let At = A ~ - I - B R - B r P - ~ (i = 1,2 ,where Ao = A), and calculate the maximum solution K ,+ of the equation K t B R - B T K t + K t ( A - m21) + (A - m l ) r K t = O (19) Step If K ,+.is equal to zero, then go to step Otherwise choose an arbitrary real number rt satisfying r~ > ½, and calculate the maximum solution P~++1 of the equation P I + I B R - B T p t + I - P t + l A t - ATpt+I rtK + = and form a dosed-loop system (20) matrix At +1 = At - BR-1BTp++ Subsequently go back to step Step If the maximum solution K + satisfies K + = for some integer j, this algorithm is completed Then all eigenvalues of A - B R - B T ( p i " + P~- + + P+) exist in the hatched region of Fig This system matrix A - B R ~ t B T ( p ~ + P ~ ' + + P+) is equal to the system matrix A - B R - I B T p + which is formed after solving an LQ-problem once for the system (A, B) with quadratic weights (Qo + rt K~+ r2K~ + + rj-IKj+.-1, R) Remark For applying this decision method, it is necessary to compute the solutions of the algebraic Riccati equations (ARE) (19) and (20) Computational algorithms of ARE are proposed by many authors (Potter, 1966; Ku~era, 1972; Kwakernaak and Sivan, 1972; Callier and Willems, 1981) Since K~, K~ Kj+_I are all the maximum solutions, they are all positive semidefinite matrices and the sum Qo + rlK~i + r z K ~ + + r j - l K ~ - i is positive definite matrix Furthermore it should be notlld that K ~ satisfies K~ = if and only if all eigenvalues of A j = A - B R - B T (P~ + P~ + + P~) exist in the region of Fig 560 Brief Paper Conclusions In this paper, a decision method for quadratic weights of an LQ-problem to locate all poles of the closed-loop system in the specified region has been discussed This decision method has the advantage that K ,+ satisfies K ~ = after a finite number of, at most n, iterations Furthermore it should be pointed out that those poles which have been located once in the hatched region of Fig are not moved by the successive iterations of the algorithm References Call~er, F M and J L Willems (1981) Criterion for the convergence of the solution of the Riccati differential equation IEEE Trans Aut Control, AC-26, 1232 Francis, B A (1979) The optimal linear-quadratic timeinvariant regulator with cheap control IEEE Trans Aut Control, AC-24, 616 Harvey, C A and G Stein (1978) Quadratic weights for asymptotic regulator properties IEEE Trans Aut Control, AC-23, 378 Kobayashi, H and E Shimemura (1981) Some properties of optimal regulator and their applications Int J Control, 33, 587 Ku6era, V (1972) On nonnegative definite solutions to matrix quadratic equations Automatica, 8, 413 Kwakernaak, H and R Sivan (1972) Linear Optimal Control Systems John Wiley, New York Molinari, B P (1977) The time-invariant linear-quadratic optimal control problem Automatica, 13, 347 Moylan, P J and B D O Anderson (1973) Nonlinear regulator theory and an inverse optimal control problem IEEE Trans Aut Control, AC-18, 460 Potter, J E (1966) Matrix quadratic solutions SIAM J Appl Math., 14, 496 Safonov, M G and M Athans (1977) Gain and phase margin for multiloop LQG regulators IEEE Trans Aut Control, AC-22, 173 Stein, G (1979) Generalized quadratic weights for asymptotic regulator properties IEEE Trans Aut, Control, AC-24, 559 Thau, F E (1967) On the inverse optimum control problem for a class of nonlinear autonomous systems IEEE Trans Aut Control, AC-12, 674 Yokoyama, R and E Kinnen (1972) The inverse problem of the optimal regulator IEEE Trans Aut Control, AC-17, 497 ... solutions to matrix quadratic equations Automatica, 8, 413 Kwakernaak, H and R Sivan (1972) Linear Optimal Control Systems John Wiley, New York Molinari, B P (1977) The time-invariant linear -quadratic. .. optimal linear -quadratic timeinvariant regulator with cheap control IEEE Trans Aut Control, AC-24, 616 Harvey, C A and G Stein (1978) Quadratic weights for asymptotic regulator properties IEEE Trans... containing the imaginary axis are called the right half plane eigenvalues Furthermore we call the eigenvalues which lie in the left half plane not containing the imaginary axis the pure left plane