A method for computing quadratic Brunovsky forms

Transcription

1 Electronic Journal of Linear Algebra Volume 13 Volume 13 (25) Article 3 25 A method for computing quadratic Brunovsky forms Wen-Long Jin wjin@uciedu Follow this and additional works at: Recommended Citation Jin, Wen-Long (25), "A method for computing quadratic Brunovsky forms", Electronic Journal of Linear Algebra, Volume 13 DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information, please contact scholcom@uwyoedu

2 A METHOD FOR COMPUTING QUADRATIC BRUNOVSKY FORMS WEN-LONG JIN Abstract In this paper, for continuous, linearly-controllable quadratic control systems with a single input, an explicit, constructive method is proposed for studying their Brunovsky forms, initially studied in [W Kang and A J Krener, Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, SIAM Journal on Control and Optimization, 3: , 1992] In this approach, the computation of Brunovsky forms and transformation matrices and the proof of their existence and uniqueness are carried out simultaneously In addition, it is shown that quadratic transformations in the aforementioned paper can be simplified to prevent multiplicity in Brunovsky forms This method is extended for studying discrete quadratic systems Finally, computation algorithms for both continuous and discrete systems are summarized, and examples demonstrated Key words Continuous quadratic systems, Discrete quadratic systems, Linearly-controllable control systems, Quadratic transformations, Quadratically state feedback equivalence, Quadratic Brunovsky forms AMS subject classifications 93B1, 15A4, 93B4 1 Introduction Linear control systems can be continuous in time t, (11) ξ = Aξ + bµ, or discrete (12) ξ(t +1)=Aξ(t)+bµ(t), where coefficients A R n n and b R n 1 are generally constant, and state variable, ξ( R n ), and control variable, µ( R), are continuous or discrete respectively When controllable, both (11) and (12) admit the Brunovsky form [1], which is derived from controller form, under the following linear change of coordinates and state feedback (13) ξ = Tx, µ = u + x T v, where x and u are new state and control variables respectively, and T R n n and v R n 1 are the transformation matrices (vectors) (refer to Chapter 3 of [2]) In the linear Brunovsky form of (11) and (12), A and b have the following forms: (14) A = 1 1 1, b = Received by the editors 4 April 23 Accepted for publication 26 January 25 Handling Editor: Michael Neumann Institute of Transportation Studies, University of California, 522 Social Science Tower, Irvine, CA , USA (wjin@uciedu) 4 1

3 Quadratic Brunovsky Forms 41 In an attempt to extending the Brunovsky forms for non-linear systems, which generally do not even have controller forms, Kang and Krener [3] studied continuous linearly controllable quadratic control systems with a single input, which can be written as (15) ξ = Aξ + bµ + F [2] (ξ)+gξµ + O(ξ,µ) 3, where F [2] (ξ) =(ξ T F 1 ξ,,ξ T F n ξ) T is a vector of n quadratic terms with symmetric n n matrices F i s (i =1,,n), Gξµ =(G 1 ξµ,,g n ξµ) T is a vector of n bilinear terms with G R n n,ando(ξ,µ) 3 includes all terms ξ a µ b with a + b 3 (We use different notations from [3] for the purpose of clearly presenting our method of computation) Moreover, the following additional assumptions are made for this system First, the coefficients in (15) and (19) A, b, F i (i =1,,n), G and h are assumed to be time-invariant Second, the two systems are assumed to be linearly controllable; by linearly controllable we mean that the linear parts of the two quadratic systems are controllable and, as a result, the linear parts can be transformed into the Brunovsky form with (13) Kang and Krener [3] first defined quadratically state feedback equivalence up to second-order, or quadratic equivalence for short, under the following quadratic change of coordinates and state feedback, (16) ξ = x + P [2] (x)+o(x) 3, ν = µ + x T Qx + rxµ + O(x, µ) 3, in which P [2] (x) = (x T P 1 x,,x T P n x) T is a vector of n quadratic terms, and transformation matrices include symmetric P i R n n (i = 1,,n), symmetric Q R n n,andr R 1 n We can see that (16) are equivalent to (17) ξ = x + P [2] (x)+o(x) 3, µ = ν x T Qx rxν + O(x, ν) 3, and hereafter refer to transformations of [3] as (17) Then, from all quadratically equivalent systems of a general system (15), two types of Brunovsky forms were defined in [3] In type I forms, the nonlinear terms are reduced into a number of quadratic terms x 2 i ; ie, there are no the cross terms in x i and x j (i j) or bilinear terms, xν In type II forms, only bilinear terms are kept In both types of Brunovsky forms, the number of non-zero nonlinear terms is n()/2, compared to n 2 (n+3)/2 for a general quadratic system In this paper, we propose a new method for studying the Brunovsky forms, first for (15) under transformations (17) This method can be carried out in three steps: first, we find the relationships between the coefficients of (15), the coefficients of its quadratically equivalent systems, and the corresponding transformation matrices; second, from these relationships, we derive a mapping from the coefficients of (15) and its equivalent systems to the transformation matrix P 1, which can be considered as the necessary condition that all equivalent systems of (15) should satisfy; third, we show how to compute, from the necessary condition, Brunovsky forms as well as the corresponding transformations With this method, we find that (15) admits the same two types of Brunovsky forms defined in [3] In contrast, our approach, constructive in

4 42 W-L Jin nature, is capable of computing the Brunovsky forms and the transformation matrices simultaneously We further show that quadratic transformations, (17), can be simplified by setting r =to (18) ξ = x + P [2] (x)+o(x) 3, µ = ν x T Qx + O(x, ν) 3 Still defining the quadratic equivalence in the sense of [3], the new transformations prevent multiple solutions of type I or type II Brunovsky forms Ie, the Brunovsky form of each type and the corresponding transformation matrices P i (i =1,,n) and Q are uniquely determined by the original system Moreover, we apply the same method, but with the simplified transformations defined in (18), and study the following discrete system (19) ξ(t +1)=Aξ(t)+bµ(t)+F [2] (ξ(t)) + Gξ(t)µ(t)+hµ 2 (t)+o(ξ,µ) 3, where, similarly, F [2] (ξ(t)) and Gξ(t)µ(t) are a vector of quadratic terms and a vector of bilinear terms respectively 1 We find that for (19) there exists only one type of Brunovsky form consisting of n(n + 1)/2 bilinear terms, which corresponds to type II Brunovsky forms of continuous systems The rest of this paper is organized as follows After reviewing the Brunovsky form of linear systems (Section 2), we study the Brunovsky forms of continuous quadratic systems in Section 3 and of discrete quadratic systems in Section 4 In Section 5, we summarize our method into two computation algorithms: one for continuous systems, and the other for discrete systems We conclude our paper with Section 6 2 Review: computation of the Brunovsky form of a controllable linear control system We first review the computation of the Brunovsky form of a continuous controllable linear control system (11) 2 The procedure and the results also apply to the discrete system (12) Computation of the Brunovsky form for (11) comprises of two steps: first, the linear system is transformed into the controller form with a linear change of coordinates given by the first equation in (13); second, the controller form is further reduced into the Brunovsky form with a state feedback given by the second equation in (13) The control matrix for (11) is defined as (21) C =[A b Ab b] Since (11) is controllable, rank(c) isn and, therefore, C is invertible Let d denote the first row of C 1, then we can compute the transformation matrix T as (22) T = d da da n 2 da 1 Note that the discrete system (19) contains a term quadratic in the control variable, hµ 2 (t), where h R n 1,and(19)hasn 2 (n +3)/2 +n non-zero nonlinear terms 2 For further explanation, refer to [1] and Chapter 3 of [2],

5 Quadratic Brunovsky Forms 43 and with the linear change of coordinates ξ = Tx, (11) can be transformed into the following controller form, (23) in which (24) Ā = T 1 AT = 1 1 ẋ = Āx + bµ, 1 v 1 v 2 v n, b = T 1 b = 1 Then, using a linear state feedback µ = u + x T v,inwhich, v 1 v 2 (25) v =, v v n we can transform the controller form (23) into the following Brunovsky form, 1 1 ẋ = x + (26) u 1 1 From the derivation above, we can see that the Brunovsky form exists and is unique, and the transformations can be explicitly computed In addition, this procedure can be easily integrated in applications related to the Brunovsky form In the same spirit, we propose a method for directly computing quadratic Brunovsky forms and the corresponding transformations in the following sections 3 Computation of continuous quadratic Brunovsky forms Note that, with linear transformations in (13), quadratic systems (15) and (19) will not change their forms except their coefficients Without loss of generality, therefore, we hereafter assume that A and b in (15) (and also in (19)) have already been transformed into the forms defined by (14) In this section, we study Brunovsky forms of the continuous system (15) under transformations (17), in a constructive manner First, we find relationships between coefficients of (15), coefficients of its quadratically equivalent systems, and the corresponding transformation matrices Second, from these relationships, we derive a mapping from the coefficients of (15) and those of the equivalent systems to transformation matrix P 1 ; this mapping is a necessary condition that all the equivalent systems should satisfy Third, we show how to obtain two types of Brunovsky forms as well as the corresponding transformations

6 44 W-L Jin 31 Quadratically equivalent system of (15) Definition 31 Assume A R n n is in the linear Brunovsky form given by (14), we define a linear operator L : R n n R n n as (31) L P = P, LP = A T P + PA, L i+1 P = L L i P, i =, 1, Properties of L The linear operator L has the following properties: 1 L i P =wheni 2n 1 2 The nullity of L is n; ifp ker(l); ie, LP =,thenp can be written as {, i+ j n, P ij = ( 1) i p i+j n, otherwise, in which p 1,,p n are independent 3 L is not invertible 4 If P is symmetric, LP is symmetric Theorem 32 The continuous quadratic system (15) is equivalent, in the sense of [3], under the quadratic transformations given by (17), to a quadratic system, whose ith (i =1,,n)equationis (32) ẋ i = x i+1 + b i ν + x T Fi x + Ḡixν + O(x, ν) 3, where x n+1 (t) is a dummy state variable, F 1,, F n are symmetric, Ḡi is the ith rowof the matrix Ḡ, and the coefficients F i (i =1,,n)andḠ are defined by (33) (34) F i = F i + P i+1 LP i b i Q, Ḡ i = G i 2b T P i b i r, where P n+1 R n n is a zero dummy transformation matrix, and b i is the ith element of b defined in (14) Proof Plugging the transformations defined in (17) into (15), we obtain ẋ + d dt P[2] (x) =Ax + bν + F [2] (x)+gxν +AP [2] (x) bx T Qx brxν + O(x, ν) 3, of which the ith (i =1,,n) differential equation can be written as ẋ i + d dt (xt P i x)=x i+1 + b i ν + x T F i x + G i xν +x T P i+1 x b i x T Qx b i rxν + O(x, ν) 3 After plugging ẋ = Ax + bν + O(x, ν) 2 into the term d dt (xt P i x) and collecting all terms whose orders are higher than two, we then obtain the equivalent system (32) with the coefficients given in (33) and (34)

7 Quadratic Brunovsky Forms 45 Remarks Equations 33 and 34 present the relationships between the coefficients of (15), those of its quadratically equivalent system (32), and the corresponding transformation matrices We further simplify these relationships in the following subsections 32 A necessary condition for quadratically equivalent systems Lemma 33 The mapping from the coefficients, G and Ḡ, andr to the transformation matrices P i (i =1,,n) is given by [P 1 ] (n) [P 2 ] (n) (35) = 1 2 G 1 2Ḡ 1 2 br, [P n ] (n) where the operator [ ] (n) takes the nth rowfrom its object Proof Sinceb i =(i =1,,n 1) and b n = 1, we obtain, from (34), [P i ] (n) = b T P i = 1 2 G i 1 2Ḡi 1 2 b ir Thus we have (35) Lemma 34 The mapping from the coefficients F i and F i (i =1,,n)tothe transformation matrices P i (i =1,,n)andQ is given by i 1 i 1 (36) P i+1 = L i P 1 L j F i j + L j Fi j, i =1,,n 1, (37) j= j= Q = L j (F n j F n j ) L n P 1 j= Proof Wheni =1,,n 1, from (33), we have P i+1 = LP i F i + F i After iterating this equation with respect to i, we obtain (36) When i = n, (33) can be written as Q = F n F n LP n Since P n is given in (36), we then have (37) Definition 35 With A R n n and L defined in (14) and Definition 31 respectively, we define a series of linear operators X i : R n n R n n (i =, 1, ) by (38) X P = [L P ] (n) [L 1 P ] (n) [L P ] (n), X ip =(A T ) i X P

8 46 W-L Jin Properties of X i The linear operators X i (i =, 1, ) have the following properties: 1 X transforms a diagonal matrix into a skew-triangular matrix of the following structure For a kth (k = n +1,,n 1) diagonal matrix P,wedenotethekth diagonal elements by p l = P ( k k)/2+l,( k +k)/2+l (l =1,,n k ) Then X P becomes a diagonal matrix with the following properties: (X P ) n ( k k)/2 l+1,( k +k)/2+l = p l,all(x P ) i,j for i + j = n + k +1+2m, wherem 1andn + k +1+2m 2n, are determined by p l, and the other entries are zeros 3 2 From the preceding property, X transforms a lower-triangular matrix into a full matrix and transforms an upper-triangular matrix into a lower skew-triangular matrix, defined by (39) ij =, when i + j n +1 3 From Properties 1 and 2, we have that the nullity of X is Therefore, X is invertible 4 From the definition of L, X i =wheni n 5 From the definition of X i (i =1,,n 1), X i P can be obtained by shifting X P down by i rows and replacing the first i rows by zeros From Property 1, therefore, X i transforms a diagonal matrix P,whosemain diagonal elements are p 1,,p n into a lower skew-triangular matrix of the following structure: (X i P ) n l+i+1,l = p l for l = i +1,,n,and all other elements except (X i P ) i,j for i + j = n + i +2m, whereinteger m 1andn + k +1+2m 2n, are zeros Theorem 36 The mapping from F i, G, F i, Ḡ (i =1,,n 1), and r to P 1 is given by (31) P 1 = X 1 ( ) X i F i G X i Fi 1 2 Ḡ 1 2 br Proof From (36) we can find the nth row of P i+1 (i =1,,n 1) as i 1 i 1 [P i+1 ] (n) =[L i P 1 ] (n) [L j F i j ] (n) + [L j Fi j ] (n) j= j= Hence, we have (311) [P 1 ] (n) [P 2 ] (n) [P n ] (n) = X P 1 X i F i + X i Fi 3 Note that, for n = 2, there is no solution to m In this case, all entries are zeros except (X P ) ij for i + j = n + k +1

9 Quadratic Brunovsky Forms 47 From (35) and (311), we obtain X P 1 = X i F i G X i Fi 1 2 Ḡ 1 2 br Multiplying both sides by the inverse of X, we then have (31) Remarks Note that P 1 is assumed to be symmetric Therefore quadratically equivalent coefficients F i (i =1,,) and Ḡ have to ensure the symmetry of the right hand side of (31) Thus (31) constitutes a necessary condition for all equivalent systems of (15), including the Brunovsky forms 33 Computation of the Brunovsky forms and the transformation matrices In this section, given r as well as the coefficients of (15), F i (i =1,,n) and G, weshowhowtochoose F i (i =1,,n)andḠ in Brunovsky forms, which, first, satisfy the necessary condition (31) and, second, have the smallest number of non-zero terms Since F n does not appear in the necessary condition, (31), we simply let F n = in Brunovsky forms To determine other coefficients, we decompose the terms related to original system (15) and r on the right hand side of (31) as (312) X 1 ( ) X i F i G 1 2 br = L + D + U, where L, D, U are strictly lower triangular, diagonal, and strictly upper triangular matrices respectively In order to add X 1 ( X i F 1 i + 2Ḡ) to the right hand side of (31) to obtain a symmetric matrix, we have, according to the property of the decomposition, the following three cases 1 If L U T =, L + D + U is already symmetric, and we simply set F i (i =1,,) and Ḡ to be In this case, the Brunovsky form of (15) is a linear system; ie, (15) can be linearized 2 When L U T,wecansetX 1 ( X i F 1 i + 2Ḡ) to a lower-triangular matrix, L U T Inthiscase,P 1 = U T + D + U According to the properties of X, X i F i + 1 2Ḡ is a full matrix 3 When L U T, we can also set X 1 ( X i F 1 i + 2Ḡ) to be an uppertriangular matrix, U L T, X i F i Ḡ is a triangular matrix defined by (39), which has n(n 1)/2 non-zero terms as U L T In this case, we have (313) X i Fi + 1 2Ḡ = X (U L T ) 1, and the first transformation matrix is given by (314) P 1 = L + D + L T 4 In addition to the aforementioned cases, X 1 ( X i F 1 i + 2Ḡ) can be also either L U T or U L T plus an arbitrary symmetric matrix

10 48 W-L Jin Comparing Cases 2, 3, and 4, we can see that the solutions in Cases 2 and 4 have more non-linear terms than in Case 3 Therefore, in the Brunovsky forms, we set X 1 ( X i F i + 1 2Ḡ) tobeu LT, and from (313), we can obtain two types of Brunovsky forms as follows First, by setting Ḡ =, we have from (313) that X i F i = 1,whichis a lower skew-triangular matrix To obtain the smallest number of, ie, n(n 1)/2, non-zero items in the Brunovsky forms, we can select F i to be a diagonal matrix with main diagonal elements as (,,, f i,i+1,, f i,n ), which can be uniquely computed as follows From properties of X i (i =1,,n 1), we first have (315) f 1,j =(X 1 F1 ) n j+2,j =( 1 ) n j+2,j,j =2,,n, and (316) 2 = 1 X 1 F1 Then, we can compute f 2,j for j =3,,n from 2 and 3 = 2 X 2 F2 in the same fashion By repeating this process, we can obtain all non-zero elements in F i for i =1,,n 1 Since all these matrices are uniquely determined by 1, the original system is uniquely equivalent to the following system, (317) n ẋ i = x i+1 + b i ν + f ij x 2 j + O(x, ν) 3, j=i+1 i =1,,n, which is type I, complete-quadratic Brunovsky form in [3] Second, by setting F i =(i =1,,n 1), we have Ḡ =2 1 and the corresponding equivalent system, (318) n ẋ i = x i+1 + b i ν + Ḡ ij x j ν + O(x, ν) 3, j=n (i+1) i =1,,n, which is type II, bilinear Brunovsky form in [3] We can see that Ḡ is also uniquely determined by the original system Once we have the Brunovsky forms of a continuous quadratic system, we can solve the corresponding transformation matrices P i (i =1,,n)andQfrom (314), (36), and (37) These transformation matrices are also unique corresponding to each Brunovsky form 34 Discussions In the preceding subsection, we finished computing the two types of Brunovsky forms and the corresponding transformation matrices of a linearly controllable quadratic system, (15) Besides, we showed that the Brunovsky form of each type and the corresponding transformation matrices are unique However, the uniqueness is dependent on r That is, for different values of r, a system can have multiple solutions of type I or type II Brunovsky forms A straightforward strategy to prevent the multiplicity is to set r =, and we have a simplified version of transformations, defined by (18) That is, with (18), a quadratic system has unique

11 Quadratic Brunovsky Forms 49 type I and type II Brunovsky forms Further, one can follow the arguments of Kang and Krener [3] but with r = and prove that (18) still defines quadratic equivalence in the same sense 4 4 Computation of discrete quadratic Brunovsky forms In this section, we apply the method proposed in the preceding section and solve Brunovsky forms for a discrete quadratic control system (19), but with simplified transformations defined in (18) We carry out our study in the same three-step, constructive procedure as for the continuous system Compared to continuous system, the discrete system has one more term, and the relationships between coefficients and transformations and the resulted Brunovsky forms are fundamentally different from those of continuous systems 41 Quadratically equivalent systems of (19) Definition 41 Assume A R n n is in the linear Brunovsky form defined in (14), we define a linear operator L : R n n R n n by L P = P, LP = A T PA, L i+1 P = L L i P, i =, 1, Properties of L The linear operator L has the following properties: 1 L i P =wheni n 2 The nullity of L is 2n 1; if P ker(l), then P ij =when (n i)(n j) >, and elements in nth row and nth column can be arbitrarily selected 3 L is not invertible 4 If P is symmetric, LP is symmetric; but the inverse proposition may not be true Remarks Note that the linear operator L for the discrete system is different from that for continuous systems This fundamental difference leads to the difference in the relationships between coefficients and transformations and the Brunovsky forms Theorem 42 The Discrete system (19) is quadratically equivalent, in the sense of [3], under the quadratic transformations in (18), to a system, whose ith (i =1,,n)equationis (41) x i (t +1)=x i+1 (t)+b i ν(t)+x T (t) F i x(t)+ḡix(t)ν(t)+o(x, ν) 3, where x n+1 (t) is a dummy state variable, F 1,, F n are symmetric, Ḡi is the ith rowof the matrix Ḡ, and the coefficients F i and h i (i =1,,n)andḠ are defined by (42) (43) (44) F i = F i + P i+1 LP i b i Q, Ḡ i = G i 2b T P i A, h i =(P i ) nn, 4 Due to the difference in notations, r = is equivalent to saying β [1] = in [3]

12 5 W-L Jin where P n+1 is a zero dummy matrix, b i is the ith element of b, and(p i ) nn is the (n, n) entry of the matrix P i The proof is similar to that of Theorem 32 and, therefore, omitted Remarks Note that (43) is different from (34), and we have an additional equation (44) 42 A necessary condition for the equivalent systems Lemma 43 The mapping from the coefficients G and Ḡ to the transformation matrices P i (i =1,,n) is given by [P 1 ] (n) [P 2 ] (n) (45) A = 1 2 G 1 2Ḡ [P n ] (n) The proof is similar to that of Lemma 33 and, therefore, omitted Lemma 44 The mapping from the coefficients F i, F i (i =1,,n)andh to the transformation matrices P i (i =1,,n)andQ is given by (46) (47) i 1 i 1 P i+1 = L i P 1 L j F i j + L j Fi j, i =1,,n 1, j= j= Q = L j (F n j F n j ), j= and (48) (P 1 ) n i n i = i 1 j= (Lj F i j ) nn + h i+1, i =1,,n 1, (P 1 ) nn = h 1 Proof The derivation of (46) and (47) from (42) is similar to that in the proof of Lemma 34 and, therefore, omitted Here we only show how (48) can be derived as follows From (46), we have (i =1,,n 1) i 1 i 1 (P i+1 ) nn =(L i P 1 ) nn (L j F i j ) nn =(P 1 ) n i n i (L j F i j ) nn j= Comparing this with (44), we then obtain (48), from which we can see that the diagonal entries of P 1 are uniquely determined by the coefficients F i (i =1,,n) and h Definition 45 With A R n n and L given by (14) and Definition 41 respectively, we define a series of linear operators X i : R n n R n n (i =, 1, ) j=

13 Quadratic Brunovsky Forms 51 by (49) X P = [L P ] (n) [L 1 P ] (n) [L P ] (n) X i P = (A T ) i X P, Properties of X i The linear operators X i (i =, 1, ) have the following properties: 1 X i =wheni n 2 The rank of X is n(n +1)/2; denote the (i, j)th entry of P by P ij (i, j =1,,n), then the (i, j)th entry of X P canbewrittenas (41) (X P ) ij = { Pn i+1,j i+1, j i;, j < i 3 From Property 2, X is not invertible Note that its continuous counterpart in Definition 35 is invertible Theorem 46 The mapping from the coefficients F i, G, F i,andḡ (i =1,,n) to P 1 is given by (411) X P 1 A = X i F i A G X i Fi A 1 2 Ḡ The proof is similar to that of Theorem 36 and omitted Remarks According to (41), we can find the (i, j)th entry of X PA, (412) (X P 1 A) ij = { (P1 ) n i+1,j i, j > i; j i That is, X PA, the left hand side of (411), is an upper-triangular matrix, which contains all the non-diagonal entries of P 1 Therefore, Fi (i =1,,n)andḠ of a quadratically equivalent system of (19) must satisfy that the right hand side of (411) be also upper triangular Thus, (411) is a necessary condition for all equivalent systems of (19), including the Brunovsky form 43 Computation of the Brunovsky form and the transformation matrices Theorem 47 In the Brunovsky form of (19), the coefficients are (413) F i =, i =1,,n, Ḡ = 2(L + D), where L and D are the strictly lower triangular part and the diagonal part of X if i A G

14 52 W-L Jin Proof The first two terms on the right hand side of (411) can be written as (414) X i F i A G = L + D + U, where L, D, U are strictly lower triangular, diagonal, and strictly upper triangular matrices respectively To ensure the right hand side of (411) to be an upper triangular matrix, we set X i F i A + 1 2Ḡ to be (415) X i Fi A + 1 2Ḡ = L + D, in which there are n(n +1)/2 non-zero elements at most, and we hence have (416) X P 1 A = U Due to the properties of X i,nosimple F i s with n(n+1)/2 non-zero elements can satisfy (415) We simply set the coefficients as in (413), and obtain the only type of Brunovsky form of (19): i (417) x i (t +1)=x i+1 (t)+b i ν(t)+ Ḡ ij x j (t)ν + O(x, ν) 3, j=1 i =1,,n Remarks The Brunovsky form of discrete systems corresponds to type II form of continuous systems and has n(n + 1)/2 bilinear terms After finding the Brunovsky form of a discrete quadratic system, we can solve the corresponding transformation matrices P i (i =1,,n)andQ as follows From (412) and (416), we can find the non-diagonal elements of P 1, and the diagonal elements from (48) Then P i (i =2,,n)andQ can be calculated from (46) and (47) 5 Algorithms and examples In this section, we summarize the algorithms for computing Brunovsky forms and the corresponding transformations of both continuous and discrete linearly controllable quadratic systems with a single input To prevent multiplicity in Brunovsky forms, we use the simplified version of transformations, (18), for both systems Thus, r for all formulas in Section 3 Following each algorithm, an example is given Algorithm 51 Computation ( of continuous quadratic Brunovsky forms Step 1 Compute X 1 ) X if i G and decompose it into L + D + U Step 2 Compute P 1 = L + D + L T and 1 = X (U L T ) Step 3 For type I Brunovsky forms, set Ḡ =and solve the equation X i Fi = 1 to compute F i (i =1,,n 1) as in (315) and (316) Go to Step 5

15 Quadratic Brunovsky Forms 53 Step 4 For type II Brunovsky forms, set F i =(i =1,,n)andḠ =2 1 Go to step 5 Step 5 Compute P i (i =2,,n)andQ from (36) and (37) Example 52 Find type I Brunovsky form of a continuous quadratic control system, which is already in type II Brunovsky form, (51) ξ 1 = ξ 2 + O(ξ,µ) 3, ξ 2 = µ + ξ 2 µ + O(ξ,µ) 3 Solutions In this system, n = 2, and coefficients are F 1 = F 2 =, [ G = 1 Following Algorithm 51, we find type I Brunovsky form, ] (52) ẋ 1 = x x2 2 + O(x, ν)3, ẋ 2 = ν + O(x, ν) 3, and the corresponding transformations, (53) ξ 1 = x 1, ξ 2 = x x2 2, µ = ν Reversely, if given (52), we find the transformations for obtaining (51), x 1 = ξ 1, x 2 = ξ ξ2 2, ν = µ, which are the inverses of (53) Algorithm 53 Computation of discrete quadratic Brunovsky form Step 1 Compute X if i A + 1 2G and decompose it into L + D + U Step 2 Compute Ḡ =2(L + D) Step 3 Compute the non-diagonal elements of P 1 from (412) and (416) Step 4 Compute the diagonal elements of P 1 from (48) Step 5 Compute P i (i =2,,n)andQ from from (46) and (47) Example 54 Find the Brunovsky form of the following discrete system: ξ 1 (t +1)=ξ 2 (t)+ξ1 2 (t)+ξ2 2 (t)+µ2 (t)+o(ξ,µ) 3, ξ 2 (t +1)=µ(t)+µ 2 (t)+o(ξ,µ) 3 Solutions In this system, n=2 and coefficients are [ ] 1 F 1 =, 1

16 54 W-L Jin F 2 =, G =, [ 1 h = 1 ] Following Algorithm 53, we find the Brunovsky form, and the corresponding transformations Thus, this system is linearized x 1 (t +1)=x 2 (t)+o(x, ν) 3, x 2 (t +1)=ν(t)+O(x, ν) 3, ξ 1 (t) =x 1 (t)+2x 2 1(t)+x 2 2(t), ξ 2 (t) =x 2 (t) x 2 1 (t)+x2 2 (t), µ(t) =ν(t) x 2 2(t) 6 Conclusions In this paper, we proposed a method for computing Brunovsky forms of both continuous and discrete quadratic control systems, which are linearly controllable with a single input Our approach is constructive in nature and computes Brunovsky forms explicitly in a three-step manner, with the introduction of linear operators L and X i (i =,,): (i) we first derived relationships between the coefficients of original systems, those of the quadratically equivalent systems, and the corresponding transformation matrices; (ii) after simplifying these relationships, we obtained a necessary condition for all equivalent systems; (iii) from the necessary conditions, we finally computed the Brunovsky forms and the corresponding transformations However, the linear operators L and X i and, therefore, the Brunovsky forms are fundamentally different for continuous and discrete systems For a continuous quadratic system, which has at most n(n +1) 2 /2 nonlinear terms, there are two types of Brunovsky forms with n(n 1)/2 nonlinear terms each; for a discrete quadratic system, there is only one type, and the numbers of nonlinear terms of a general system and its Brunovsky form are n(n+1) 2 /2+n and n(n+1)/2, respectively For continuous systems, we used the same transformations as in [3] and found that the inclusion of transformation vector r yields multiple solutions of a type of Brunovsky forms Therefore, we suggested to set r = in order to maintain the uniqueness That is, under transformations defined by (18) for both continuous and discrete systems, the Brunovsky forms and the corresponding quadratic transformations always exist and are uniquely determined by the original systems In this sense, our study can be viewed as a constructive proof of the existence and uniqueness of Brunovsky forms Finally, the method proposed in this paper can be extended for studying quadratic control systems, not linearly controllable or with multiple input In addition, the method could be useful for applications involving analysis and control of quadratic systems

17 Quadratic Brunovsky Forms 55 REFERENCES [1] P Brunovsky A classification of linear controllable systems Kybernetika, 3: , 197 [2] T Kailath Linear Systems Prentice-Hall, Englewood Cliffs, New Jersey, 198 [3] W Kang and A J Krener Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems SIAM Journal on Control and Optimization, 3: , 1992