systems March 3, 1999 Abstract polynomial equations. The algorithms which fall in this category make up the engines in symbolic algebra

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1 CDC99 - REG0098 Computer algebra in the control of singularly perturbed dynamical systems J.W. Helton æ F. Dell Kronewitter y March 3, 1999 Abstract Commutative Groebner basis methods have proven to be a useful tool in the manipulation of sets of polynomial equations. The algorithms which fall in this category make up the engines in symbolic algebra packages' Solveë ë commands. Most algebraic calculations which one sees in linear systems theory, for example IEEE TAC, involve block matrices so are highly noncommutative. Thus the more conventional commutative computer algebra does not address them. The noncommutative Groebner theory is more recent, ëmor86ë, but many of the symbolic computational algorithms have been implemented, ëhws98ë, ëkfg98ë. The noncommutative versions, though not nearly as well understood as their commutative counterparts, have also been shown to be useful in the manipulation of some noncommutative sets of polynomial equations arising in control theory, ëhs97aë and ëhws98ë. Here we investigate the usefulness of noncommutative computer algebra in a particular area of control theory, singularly perturbed dynamic systems, where the noncommutative polynomials involved are especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. For example, the methods introduced here take the most standard textbook singular perturbation calculation out one step further than has been done previously. This represents the ærst step in the production of noncommutative computer algebra tools to assist with singular perturbation calculations. Contents 1 Introduction The idea behind Groebner computer algebra The Standard State Feedback Singular Perturbation Problem The System The Near-Optimal LQR Problem Decomposing the Problem The Slow System The Fast System Decomposing the Output Computer algebra vs. the standard singular perturbation problem The zero-th order term of the Riccati equation ècoeæcients of æ 0 è Computer algebra jargon Computer algebra ænds the basic equation The output and it's analysis Other algebraic identities which are known to hold The output The zero-th order term of the controller The order æ term of the Riccati equation Extracting the coeæcients of æ Solving for the unknowns The order æ 2 term of the Riccati equation æ University of California, San Diego, Math Dept., 9500 Gilman Dr., San Diego, CA , helton@math.ucsd.edu, Partially supported by the AFOSR and the NSF y University of California, San Diego, Math Dept., 9500 Gilman Dr., San Diego, CA , dell@ieee.org, Partially supported by the AFOSR and the NSF 1

2 4 Perturbing singular solutions of the Information State Equation The General Problem The linear case Computer algebra works Introduction Singular perturbation is a commonly used technique in the analysis of systems whose dynamics consist of two pieces. One piece might be slow, the other fast, or one might be known where the other is somewhat uncertain. Extensive analysis has been done of this type of plant for the LQR and H 1 control problems, for example ëkko86ë, ëpb93ë, ëpb94ë. Typically one has an equation with some coeæcients depending on a parameter 1. One postulates an expansion " for the solutions x " to the equation, èaè substitutes x " into the equation, èbè sorts by powers of " and ècè solves for successive terms of the expansion x ". The sorting in èbè can be tedious and the business of solving ècè can be very involved. Indeed our method carries out the expansion one step further than has previously been done, see Section 3.3. This article concerns methods we are developing for doing both of these steps automatically. The software runs under NCAlgebra, ëhms96ë, the most widely distributed Mathematica package for general noncommuting computations. As we shall illustrate NCAlgebra constructions and commands easily handle steps èaè and èbè, thereby producing the èlongè list of equations which must be solved. This is straightforward and can save one engaged in singular perturbation calculations considerable time. Solving complicated systems of equations is always tricky business; thus there is no way of knowing in advance if noncommutative Groebner basis methods will be eæective for èreducing to simple formè equations found in large classes of singular perturbation problems. This is the focus of experiments we have been conducting and on which we report in this paper. Most of the paper shows how one can treat the most classic of all singular perturbation problems using computer algebra. Ultimately we see that Mora's Groebner basis algorithms are very eæective on the equations which result. Then we sketch another newer H 1 estimation problem called the ëcheap sensor" problem èsee ëhjm98ëè. On this our computer techniques proved eæective and a longer will report these results. 1.1 The idea behind Groebner computer algebra The Groebner basis algorithm ègbaè, due to F. Mora ëmor86ë, can be used to systematically eliminate variables from a collection èe.g., fp j èx1;::: ;x n è=0:1çjçk1gè of polynomial equations so as to put it in triangular form. One speciæes an order on the variables èx1 éx2éx3é:::éx n è 1 which corresponds to ones priorities in eliminating them. Here a GBA will try hardest to eliminate x n and try the least to eliminate x1. The output from it is a list of equations in a ëcanonical form" which is triangular: 2 q1èx1è = 0 è1è q2èx1;x2è = 0 è2è q3èx1;x2è = 0 è3è q4èx1;x2;x3è = 0 è4è. q k2 èx1;::: ;x n è = 0: è5è Here, the set of solutions to the collection of polynomial equations fq j =0:1çjçk2gequals the set of solutions to the collection of polynomial equations fp j =0:1ç jç k1g. This canonical form greatly 1 From this ordering on variables one induces an order on monomials in those variables which is referred to as lexicographic order. 2 There need not be ç n equations in this list and there need not be any equation in just one variable. 2

3 simpliæes the task of solving the collection of polynomial equations by facilitating back-solving for x j in terms of x1;::: ;x j,1. The eæect of the ordering is to specify that variables high in the order will be eliminated while variables low in the order will not be eliminated. If the variables commute, the Groebner basis is always ænite and can be generated by Buchberger's algorithm if one waits long enough. In the noncommutative case, which is the subject of this paper, the Groebner basis is usually inænite and the GBA could fail to halt given inænite computational resources. Nevertheless, the solution set of the output of a terminated èsay k iterationè GBA, fq j g, is always equivalent to the solution set of the input, fp i g, and this partial GBA often proves to be useful in computations as will be shown below. Groebner basis computer runs can be notoriously memory and time consuming. Thus their eæectiveness on any class of problems can only be determined by experiment. Implementations of the GBA include the NCGB component of NCAlgebra, ëhs97bë, and OPAL, ëkfg98ë. Our computer computations were performed with NCGB on a Sun Ultra I with one 166 Mhz processor and 192MB of RAM. 2 The Standard State Feedback Singular Perturbation Problem The standard singularly perturbed linear time-invariant model consists of a diæerential state equation; which depends on some perturbation parameter, æ; and an output equation. The general control problem is to design some feedback law which speciæes the input as a function of the output so that the controlled system will satisfy some performance objective. 2.1 The System Here we study the two time scale dynamic system previously analyzed in ëkko86ë: ç dx ç ç çç ç ç dt A11 æ dz = A12 x +ç B1 u dt A21 A22 z B2 y = æ M1 M2 æç x z ç è6è è7è where x 2 R n, z 2 R m, u 2 R p, and y 2 R q. 2.2 The Near-Optimal LQR Problem The inænite-time optimal linear regulator problem is to ænd a control, uètè; t 2 ë0; 1ë which minimizes the quadratic cost Z 1 J = èy T y + u T Ruèdt è8è 0 where R is a positive deænite weighting matrix. It is well known that the solution to this problem is of the form ç ç ç ç u æ =,R,1 B T x x Kèæè = Gèæè è9è z z where Kèæè is a solution to the Algebraic Riccati Equation èareè with ç A11 A = A 21 æ KA + A T K, KBR,1 B T K + M T M =0 A12 A 22 æ ç ;B = ç B1 B 2 æ ç è10è ; and M = æ M1 M2 æ è11è Kèæè =K0+æK1 + æ 2 K2 + ::: è12è 3

4 If K is the solution to this optimal state feedback control problem it also may be used to express the optimal cost as a function of the initial state of the system as æ æ ç ç J æ xè0è = x T è0è z T è0è K : è13è zè0è Notice that the solution as presented involves solving equation è10è which is a n+mdimensional Riccati. Since we have noticed the partitioned structures present in the system è6è it seems likely that we might decompose the problem and signiæcantly reduce the complexity of the linear algebra while deriving an only slightly sub-optimal controller. Indeed it is standard to divide the problem of solving the n + m dimensional Riccati, è10è, into solving two smaller Riccatis, one n dimensional, the other m dimensional, and then using these solutions to obtain a control law which gives a performance J nearly equal to the optimal performance J æ. This regulator is known as the near-optimal regulator. The noncommutative Groebner computer algebra which is the subject of our investigations is well suited for manipulating polynomials into a desired form. In the next two subsections we review this decomposition. This is mostly a matter of setting our notation, which in fact is the same notation as ëkko86ë. 2.3 Decomposing the Problem Here, we decompose our two time scale system into it's fast parts and slow parts by ærst setting the fast subsystem to it's steady state. In doing so, we will introduce standard notation which will help make the formulas we derive a little more readable The Slow System We will begin our analysis of this perturbed system by considering this system in it's slow state. The dynamics of the slow system can be found by setting æ to zero, often referred to as the quasi-steady state of the system. This transforms the equation involving æ in system è6è into an algebraic equation rather than a diæerential equation where dx s = A0x s ètè+b0u s ètè; x s èt0è=x 0 è14è dt z s ètè =,A,1 22 èa 21x s ètè+b2u s ètèè è15è A0, A11, A12A,1 22 A 21; B0, B1, A12A,1 22 B 2: Here the subscript s indicates that the vectors in equations è14è-è15è are the slow parts of the vectors in è6è The Fast System The fast system has dynamics dz f dt = A 22z f ètè+b2u f ètè; z f èt0è=z 0,z s èt0è è16è where z f = z, z s and u f = u, u s : 2.4 Decomposing the Output We may then also decompose è7è into it's slow and fast parts y = M1x + M2z = M1ëx s + Oèæèë + M2ë,A,1 22 èa 21x s + B2u s è+z f +Oèæèë = y s ètè+y f ètè+oèæè è17è 4

5 deæning y s = M0x s + N0u s and y f = M2z f è18è where M0, M1, M2A,1 22 A 21 and N0,,M2A,1 22 B 2: è19è 3 Computer algebra vs. the standard singular perturbation problem The matrix Kèæè described above in è12è must be partitioned compatibly with the states, x and z, and is the limit of the power series which is conventionally written in the form, NX ç kè1;iè K N èæè = æ i i=0 æk T è2;iè ækè2;iè ækè3;iè ç : è20è The remainder of this section will be devoted to ænding formulas for the kèj;iè for j 2f1;2;3gand i ç The zero-th order term of the Riccati equation ècoeæcients of æ 0 è We begin our investigations of the perturbed control problem by searching for the ærst term of the series è20è consisting of matrices, kè1;0è, kè2;0è, and kè3;0è. We can ænd an order of æ approximation to the optimal K, è20è, by taking only the zero-th order terms in æ of the equations given in è10è. This is problem èbè mentioned in the introduction. In the next section we will show how this may be done with the assistance of a computer. This ænally brings us to the subject of our investigations, the manipulation of matrix polynomials with computer algebra methods Computer algebra jargon There are several terms we will use which, though simple conceptually,maynot be familiar to the control engineer. A product of variables, x æ1 1 æ xæ2 2 æææxæn n where æ k 2 N, is called a monomial. Apolynomial, f, is a ænite C -linear combination of monomials, mx j=1 a j x æj 1 1 æ 2 xæj 2 æææxæj n n where a j 2 C. We call a k the coeæcient of the term a k x æk 1 1 æ x æk 2 2 æææxæk n n. A relation is a polynomial which is assumed to be 0. That is, we may write the equation, 3x 2 yz = yz +4z 2, as a relation, 3x 2 yz, yz, 4z 2. Our computer calls will take either relations or equations. We will slip back and forth between the two notations and our meaning should be clear from the context Computer algebra ænds the basic equation Groebner basis theory suggests that to ænd a polynomial in k10, A0, B0, M0, N0, R0, and other variables with a minimal number of occurances of k10 we should create a Groebner basis for all polynomial relations known to hold under the following order. N0 ém0 ér0 éa0 éb0 çk10 ç other variables Now we will list the input to our computer algebra program which will generate all polynomial relations known to hold. First, we deæne the block matrices ç inç è11è. èwhat follows is the NCAlgebra notation for the matrices described a b in è11è and è20è. The matrix,, is represented as ffa,bg,fc,dgg. The suæx, ëëi; jëë, extracts the i; j- c d th element from a matrix. MatMultë ë performs the matrix multiplication operation, tpmatë ë performs the symbolic transpose operation, and ** indicates noncommutative multiplication.è è21è 5

6 A =ffa11,a12g,f1èep**a21,1èep**a22gg; B = ffb1g,f1èep**b2gg; M =ffm1,m2gg; We also deæne a K0. K0 = ffk10,ep**k20g,fep**tpëk20ë,ep**k30gg; è22è è23è The following Mathematica function takes as an argument a matrix K and generates the Riccati è10è. RiccatiëKë := MatMultëK,Aë + MatMultëtpMatëAë,Kë - MatMultëK,B,InvëRë,tpMatëBë,Kë + MatMultëtpMatëMë,Më The NCAlgebra command NCTermsOfDegreeë ëtakes 3 arguments: a ènoncommutativeè polynomial, a list of variables, and a set of indices. The command returns an expression such that each term is homogeneous of degree given by the indices. For example, the call NCTermsOfDegreeë A**B + B**C**C + B**A**C + C**D, fcg, f1g ë returns B**A**C + C**D The following Mathematica commands will extract the 0-th order terms in æ, creating the polynomials in è29è, è30è, and è31è. è24è è25è è26è Ep10 = NCTermsOfDegreeë RiccatiëK0ëëë1,1ëë, fepg,f0gë Ep20 = NCTermsOfDegreeë RiccatiëK0ëëë1,2ëë,fepg,f0gë Ep30 = NCTermsOfDegreeë RiccatiëK0ëëë2,2ëë,fepg,f0gë è27è The output and it's analysis Input è27è creates three polynomials, a typical one of which is k30**a22+tpëa22ë**k30+tpëm2ë**m2-k30**b2**invërë**tpëb2ë**k30. When all three are output in TeX, which is done easily by NCAlgebra, we get that RiccatiëK0ë = 0 to the equations è28è corresponds 0 = k10 A11 + A T 11 k 10 + k20 A21 + A T 21 kt 20 + M 1 T M 1, k10 B1R,1 B1 T k 10, k20 B2 R,1 B1 T k 10, k10 B1R,1 B2 T kt 20 + k20 B2 R,1 B2 T kt 20 è29è 0 = k20 A22, k20 B2 R,1 B2 T k 30 + k10 A12 + A T 21 k 30 + M1 T M 2, k10 B1R,1 B2 T k 30 è30è 0 = k30 A22 + A T 22 k 30 + M2 T M 2, k30 B2 R,1 B2 T k 30 è31è Notice that è31è, the beautiæed form of è28è, contains only one unknown, k30, and has the Riccati form. Thus one k30 is determined by this equation. Equation è29è contains two unknowns, k10 and k20, and equation è30è contains all three unknowns, k10, k20, and k30. To solve for k20 we use equation è30è and call NCCollectë Ep20, k20 ë to get the following relation k20**èa22-b2**invërë**tpëb2ë**k30è+tpëa21ë**k30+tpëm1ë**m2- k10**b1**invërë**tpëb2ë**k30+k10**a12 Upon examination of è32è it is immediate that we may give k20 explicitly in terms of k10 and k30 by assuming the invertibility of the parenthetic expression in the above relation, A22, B2R,1 B T 2 k 30. Wehave è32è k20 = æ,k10a12 + k10b1r,1 B T 2 k 30, A T 21 k 30, M T 1 M 2 æ èa22, B2R,1 B T 2 k 30è,1 : è33è Use this deænition of k20 to change è29è into an equation involving k10 and k30. The unknown matrices, k i0, could then be found by ærst using è31è to solve for k30, then using our transformed è29è to solve for k10, and ænally using è33è to obtain k20. We will show in the next section how è29è maybechanged into an equation involving only one unknown, k10, so that k30 and k10 may be computed concurrently using two independent Riccati equations. 6

7 3.1.4 Other algebraic identities which are known to hold In light of the slow system terminology introduced above in Sections and 2.4 we make the following abbreviations. Abbreviations = f N0 == - M2**InvëA22ë**B2, M0 == M1 - M2**InvëA22ë**A21, A0 == A11 - A12 ** InvëA22ë**A21, B0 == B1 - A12**InvëA22ë**B2, R0 == R + tpën0ë**n0 g Several of the matrices are known to be self adjoint, and therefore the following relations must hold: SelfAdjoints = f k10 == tpëk10ë, k30 == tpëk30ë, R == tpërë, R0 == tpër0ë, InvëRë == tpëinvërëë, InvëR0ë == tpëinvër0ëë g è35è Several of the matrices or matrix polynomials in our problem are assumed to be invertible. It is common to take the matrices, A ii, to be of full rank, since otherwise a transformation could be applied to the original system to reduce the size of the state. The matrix A22, B2R,1 B2 T k 30 has already been assumed to be invertible to facilitate the deænition of k20 in è33è. The matrices R and R0 are positive deænite and so must be invertible. We generate the relations which result from these observations with the following command, Inverses = NCMakeRelationsëfInv, R,R0,A0,A11, A22, èa22 - B2**InvëRë**tpëB2ë**k30è gë, and combine all of our relations with Relations = UnionëEp10,Ep20,Ep30,Abbreviations,SelfAdjoints,Inversesë. We must also include the transposes of each of these relations: AllRelations = NCAddTpëRelationsë The order mentioned in è21è is speciæed next. NCSmartOrderë ffn0,m0,r0,a0,b0 g,fk10g, fb1,b2,m1,m2,r, A11, A12, A21, A22, InvëA22- B2**InvëRë**tpëB2ë**k30ë, tpëinvëa22- B2**InvëRë**tpëB2ë**k30ëëg fk30g,fk20gg, AllRelations ë è39è Finally, the call to make the Groebner basis is made. This call will create a four iteration Groebner basis from the polynomials included in AllRelations and the output will be stored in the æle, ëfindk10". NCProcessëAllRelations,4,"FindK10" ë è34è è36è è37è è38è è40è The output The output of this command is a set of polynomials which make up the the Groebner basis created from the polynomials in AllRelations under the order speciæed in è39è. The software we use actually does more than just create a Groebner basis. NCProcessë ë categorizes the output depending on how many unknowns lie in each relation. Then it automatically sets them in TeX and opens a xdvi window displaying them. In this case a category was found for k10 which consisted of a single relation, k10rel. After calling NCCollectë k10rel, k10 ë this polynomial takes the form The expressions with unknown variables fk10g and, knowns fa0; B0;M0;N0;A T 0 ;BT 0 ;MT 0 ;NT 0 ;R,1 0 g k10 A0, B0 R,1 æ, 0 N0 T M 0 + A T 0, M0 T N 0 R,1 æ 0 B0 T k10 + M0 T M 0, k10 B0 R,1 0 B0 T k 10, M0 T N 0 R,1 0 N0 T M 0 è41è This calculation is no easy feat by hand, as the substitutions and non-standard notation on pages of ëkko86ë will attest. The answer we found with the Groebner computer algebra is the same as derived there by hand. This computation took less than 3 minutes. 7

8 3.1.6 The zero-th order term of the controller Closer analysis of the ærst term of the optimal controller discovered in the previous section, Section 3.1, G =,R,1 h B T 1 B T 2 iç kè1;0è ækè2;0è æ ækè2;0è T ækè3;0è ç ; è42è reveals that an æ independent controller can be obtained by setting the upper right entry of K to zero. This gives us ç ç æ æç ç x x G = z G10 G20 =,R,1 èb T z 1 k è1;0èx + B2 T kt è2;0èè x + BT 2 k è3;0èzè è43è where kèi;0è is deæned by equations è41è, è33è, and è31è for i equal to 1,2, and 3 respectively. 3.2 The order æ term of the Riccati equation In Section 3.1.6, a controller was presented which does not depend on the parameter æ. This is especially appropriate if æ represents some small unknown parameter. In fact, there are many circumstances when the parameter, æ, isknown. In such a case, even though the optimal controller is an inænite power series in æ one can make ann th order approximation to Gèæè in è9è and arrive at a controller with enhanced performance. A major obstruction to such an improved approach is the tedious computation required to generate formulas for the coeæcients of higher powers of æ. We did not ænd references where anyone generated formulas for coeæcients of æ higher than 1. The methods in this paper do, see Section 3.3. As done in ëkko86ë we will now obtain formulas for the matrices kè1;1è, kè2;1è, and kè3;1è described in è20è. Our approach will require considerably less work. Instead of truncating the series è20è to only one term as done in Section 3.1.6, input è22è, here we deæne symbolic entries for the second term of K as well. K1 = ff k10, ep**k20 g, fep**tpëk20ë, ep**k30 gg + ep** ff k11, ep**k21 g, fep**tpëk21ë, ep**k31 gg è44è We also append the following abbreviations for the controller discussed above in Section and deæned in equation è43è. These formulas are standard, ëkko86ë. Abbreviations = Unionë Abbreviations, f G10 == -InvëRë**ètpëB1ë**k10 + tpëb2ë**tpëk20ëè, G20 == -InvëRë**tpëB2ë**k30 g ë and, since A22 + B2G20 = A22, B2R,1 B T 2 k 30 which was previously assumed to be invertible, we also add the invertibility relation, è45è Inverses = Unionë Inverses, NCMakeRelationsëfInv, èa22 + B2**G20ègë ë. è46è Extracting the coeæcients of æ 1 The Riccati expression in K1, RiccatiëK1ë, generates equations quadratic in æ. As done in the last section the approach here is to equate coeæcients of æ. Of course, the coeæcients of æ 2 cannot be equated since the actual power series è20è would have kèi;2è which have not been introduced in computer input è44è. We can extract the coeæcients of æ in equation è10è with the following commands. Ep11 = NCTermsOfDegreeë RiccatiëK1ëëë1,1ëë, fepg,f1gë è47è creates the following polynomial ep k11**a11+ep k21**a21+ep tpëa11ë**k11+ep tpëa21ë**tpëk21ëep k10**b1**invërë**tpëb1ë**k11-ep k10**b1**invërë**tpëb2ë**tpëk21ëep k20**b2**invërë**tpëb1ë**k11-ep k20**b2**invërë**tpëb2ë**tpëk21ë- 8

9 ep k11**b1**invërë**tpëb1ë**k10-ep k11**b1**invërë**tpëb2ë**tpëk20ëep k21**b2**invërë**tpëb1ë**k10-ep k21**b2**invërë**tpëb2ë**tpëk20ë and Ep21 = NCTermsOfDegreeë RiccatiëK1ëëë1,2ëë, fepg,f1gë Ep31 = NCTermsOfDegreeë RiccatiëK1ëëë2,2ëë, fepg,f1gë give similar looking formulas. è48è è49è è50è Solving for the unknowns These valid relations can now be added to all relations known to hold, è22è, è24è, è34è, è35è, and è36è, with the following command. Since the output of the NCTermsOfDegreeë ë command includes the variable, ep, which we took the coeæcients of we must set the unwanted variable, ep, to1. 3 We do this by appending the Mathematica rule è.ep-é1 to expressions involving ep. AllRelations = UnionëEp11è.ep-é1,Ep21è.ep-é1,Ep31è.ep-é1, Ep10,Ep20,Ep30, Abbreviations,SelfAdjoints,Inversesë Considering the analysis done in Section 3.1.2, kè1;0è, kè2;0è, and kè3;0è èk10, k20, k30è can be regarded as known and we are now looking for formulas which describe the second term of the series è20è, made up of symbols k11, k21, and k31 introduced above, è44è. With this distinction between known variables and unknown variables the following order is appropriate è51è N0 ém0éréa0éb0éb1éb2ém1ém2ér0é A11 éa12 éa21 éa22 ég10 ég20 ék30 ék20 ék10 ç k11 ç other variables è52è This order is imposed with the command NCSmartOrderë ffn0,m0,r,a0,b0, A11,A12,A21,A22,B1,B2,M1,M2,R0,G10,G20,k30,k20,k10g, fk11g,fk31, k21g,f InvëA22- B2**InvëRë**tpëB2ë**k30ë,tpëInvëA22- B2**InvëRë**tpëB2ë**k30ëë gg, AllRelations ë; A three iteration Groebner basis is created with the NCProcessë ë command similar to è40è. On this input NCProcessë ë took less than 7 minutes. The output of this command contains a single relation with 24 terms involving the single unknown matrix kè1;1è, k11rel. Collecting around the k11 with the command è53è NCCollectë k11rel, k11 ë gives us the following relation è54è,1 k11 èa0, B0 R,1 0 BT 0 k 01, B0 R,1 0 N 0 T M 0è+èk01 B0 R,1 0 B0 T + M0 T N 0 R,1 0 B0 T, A T 0 è k11 + A T 0 k 02 A,1 22 A 21 + A T 21 AT,1 22 k02 T A 0, k01 B0 R,1 0 BT 0 k 02 A,1 22 A 21, k01 B0 R,1 0 B2 T AT,1 22 k02 T A 0, A T 0 k 02 A,1 22 B 2 R,1 0 B0 T k 01, A T 0 k 02 A,1 22 B 2 R,1 0 N0 T M 0, A T 21 AT,1 22 k02 T B 0 R,1 0 B0 T k 01, A T 21 AT,1 22 k02 T B 0 R,1 0 N0 T M 0, M0 T N 0 R,1 0 B0 T k 02 A,1 22 A 21, M0 T N 0 R,1 0 BT 2 AT,1 22 k02 T A 0 + k01 B0 R,1 0 B0 T k 02 A,1 22 B 2 R,1 0 B0 T k 01 + k01 B0 R,1 0 B0 T k 02 A,1 22 B 2 R,1 0 N 0 T M 0 + k01 B0 R,1 0 B2 T AT,1 22 k02 T B 0 R,1 0 B0 T k 01 + k01 B0 R,1 0 B2 T AT,1 22 k02 T B 0 R,1 0 N 0 T M 0 + M0 T N 0 R,1 0 B0 T k 02 A,1 22 B 2 R,1 0 B0 T k 01 + M0 T N 0 R,1 0 B0 T k 02 A,1 22 B 2 R,1 0 N 0 T M 0 + M0 T N 0 R,1 0 B2 T AT,1 22 k02 T B 0 R,1 0 B0 T k 01 + M0 T N 0 R,1 0 B2 T AT,1 22 k02 T B 0 R,1 0 N 0 T M 0 è55è The coeæcients of kè1;1è in equation è55è, the relations in parentheses, suggest that we make the following abbreviation 3 Notice that Mathematica output è26è contains the variable C. 9

10 F0 == A0-B0**InvëR0ë**ètpëN0ë**M0 + tpëb0ë**k01è 4 With computer input è56è appended to all relations known to hold, AllRelations, we can put F0 low in the order and run NCProcessë ë again, creating a new Groebner basis. The output of this command contains the aesthetically pleasing relation deæning kè1;1è,k11 F0, 1 F0 T k 11 + A T 21 èa22 + B2 G20è,T k20 T F 0 + F0 T k 20 èa22 + B2 G20è,1 A21 + F0 T k 20 èa22 + B2 G20è,1 B2 G10 + G T 10 BT 2 èa22 + B2 G20è,T k20 T F 0: è57è This is a simple Lyapunov equation whose solution, kè1;1è, is unique as long as F0, is Hurwitz. We will therefore regard kè1;1è as known from this point forward. Similar to the zero-th order case the equation deæning kè3;1è is an immediate consequence of è58è and takes the collected form è56è k31, A22, B2 R,1 B T 2 k 30 æ +, A T 22, k30 B2 R,1 B T 2 k T 20 B 1 R,1 B T 2 k 30 æ k31 + A T 12 k 20 + k02 T A 12, k30 B2 R,1 B1 T k 20, è58è Also similar to the zero-th order case we have an explicit formula for kè2;1è in terms of kè1;1è and kè3;1è. The following relatively simple formula was also in the output of the NCProcessë ë command which generated è57è. k21!,1k11 A12 A,1 22,AT 11 k 20 èa22 + B2 G20è,1,A T 21 k 31 èa22 + B2 G20è,1,G T 10 BT 1 k 20 èa22 + B2 G20è,1, G T 10 BT 2 k 31 èa22 + B2 G20è,1 + k11 B0 R,1, 0 N T 0 M2 A,1 æ 22, BT 2 AT,1 22 k30 è59è Expressions equivalent to è57è, è59è, and è58è can be found in ëkko86ë. Notice that a similar procedure could be done as that described in Section to derive an order æ controller. 3.3 The order æ 2 term of the Riccati equation At this point the tale is growing long and the weary reader can most likely guess what will be done in this section from the title. For the sake of presenting a formula which has not appeared before we create a three term approximation to Kèæè, K2 = ff k10, ep**k20 g, fep**tpëk20ë, ep**k30 gg + ep** ff k11, ep**k21 g, fep**tpëk21ë, ep**k31 gg + ep^2** ff k21, ep**k22 g, fep**tpëk22ë, ep**k32 gg, è60è For this problem a three iteration Groebner basis was created and we arrived at formulas deæning kè1;2è, kè2;2è, and kè3;2è which in total are 1,478 lines long in Mathematica notation èineæcientè. We used a version of NCProcessë ë which was specialized to display only equations involving the unknowns; kè1;2è and kè2;2è. èthe formula for kè3;2è was immediate from the order two Riccati as in the lower order cases.è This substantially speeds up run times. Still, our rather formidable conclusion took 21 and a half minutes. The formulas can be found at It is gratifying that our Groebner equation processing techniques prevailed on such a large problem. It leads us to think that many singular perturbation problems are well within the scope of our computer algebra techniques. 4 More analytic methods can be used, ëks72ë, to show that this expression is of the form A 0 + B 0 G 0 where G 0 is the optimal control for the slow part of the LQR optimal problem, è14è. The focus here is on the computer algebra and the expression, F 0,is discovered purely algebraically. 10

11 4 Perturbing singular solutions of the Information State Equation We would also like to mention that the techniques illustrated on the previous problem apply to other problems. In particular we mention a singular perturbation analysis of an important entity in the output feedback H 1 control problem, the information state. It corresponds not to fast and slow time scales but to sensors becoming increasingly accurate, for details see ëhjm98ë. 4.1 The General Problem Consider the system dx dt = A æ èxè+bèxèv è61è out = Cèxè+Dèxè: è62è An equation useful in estimation associated to this èdetails in ëhjm98ëè is the information state equation, ISE., dp dt = èa æ èxè+bèxèævètèè T r x p t èxè+èr x p t èxèè T Qèxèr x p t èxè è63è + ëcèxè, Dvètèë T JëCèxè, Dvètèë + 1 æ 2 kw ëvètè, xëk2 ; where J is self adjoint. It is written even more concisely as, dp dt = æèx; tèr x p t èxè+èr x p t èxèè T Qèxèr x p t èxè è64è + Jèx; tè+ 1 æ 2kWëvètè,xëk2 : 4.2 The linear case Assuming that the associated vector æeld is linear and æxing æ it is known that a solution exists of the form p t èxè = 1 2 èx,x æè T P æ èx,x æ è+ç e t ; è65è where P æ is a matrix which does not depend on t, but x and x æ do depend on t, or more simply as p t èxè = 1 2 ~xt P æ ~x+ç e t with ~x = x, x æ è66è We expand x æ to arrive at x æ =x e0+æx æ;1 + æ 2 x æ;2 + ::: è67è A few examples have led us to believe that P æ is an irrational function of æ which has a series form P æ = 1 æ P,1 + P0 + æp1 + æ 2 P2 + ::: è68è Then r x p = P æ èx, x æ è and noting that èa æ x + Bvètèè T rp is scalar, we can symmetrize the above information state equation, ISE, è63è, and arrive at, dp dt where R = W T W. = ëa æ x + B æ vètèë T P æ èx, x æ è+èx,x æ è T P æ ëa æ x+bævètèë è69è + èx, x æ è T P æ QP æ èx, x æ è+ëcx, Dvètèë T JëCx, Dvètèë + 1 æ 2 èvètè, xèt Rèvètè, xè: 11

12 4.3 Computer algebra works We applied NCAlgebra methods very similar to the ones demonstrated in Section 3 to the ISE singular perturbation problem just described and found them highly eæective. Results will be reported in the longer paper corresponding to this conference note. References ëcls92ë D. Cox, J. Little, and D. O' Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer-Verlag, Undergraduate Texts in Mathematics, ëhjm98ë J. W. Helton, M. R. James, and W. M. McEneaney. Nonlinear control: The joys of having an extra sensor. Proceedings of the 37th IEEE CDC,Tampa, Florida, USA, Dec 16-18, 4:3609í13, ëhms96ë J.W. Helton, R.L. Miller, and M. Stankus. NCAlgebra: A Mathematica Package for doing noncommuting Algebra. available from Beware that to perform the computations in this paper you need NCGB. ëhs97aë J.W. Helton and M. Stankus. Computer assistance in discovering formulas and theorems in system engineering. to appear in Journal of Functional Analysis, ëhs97bë J.W. Helton and M. Stankus. NCGB: Noncommutative Groebner bases. available from As of April'99 this requires Mathematica and the Solaris operating system. A Windows version is coming soon. ëhws98ë J.W. Helton, J.J. Wavrik, and M. Stankus. Computer simpliæcation of formulas in linear systems theory. IEEE Transactions on Automatic Control, 43:302í14, ëkfg98ë B. Keller, C.D. Feustel, and E. Green. The GRBèOPAL System. available from ëkko86ë Petar V. Kokotovic, Hassan K. Khalil, and John O'Reilly. Singular Perturbation Methods in Control: Analysis and Design. Academic Press, ëks72ë Huibert Kwakernaak and Raphael Sivan. Linear optimal control systems. Wiley Interscience., ëmor86ë ëpb93ë ëpb94ë F. Mora. Groebner bases for non-commutative polynomial rings. Lecture Notes in Computer Science, 229:353í362, Z.G. Pan and T. Basar. H 1 optimal control for singularly perturbed systems 1. perfect state measurements. Automatica, 29:401í423, Z.G. Pan and T. Basar. H 1 optimal control for singularly perturbed systems 2. imperfect state measurements. IEEE Transactions on Automatic Control, 39:280í299,