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QUATERNIONIC SYMMETRY OF LINEAR DYNAMICAL SYSTEMS
QUATERNIONIC SYMMETRY OF LINEAR DYNAMICAL SYSTEMS Ricardo Pereira and Paolo Vettori
Department of Mathematics, University of Aveiro email: {ricardopereira, pvettori}@ua.pt
Abstract: In this paper we study a particular kind of symmetry of linear dynamical systems, the quaternionic symmetry. After giving some basic notions about quaternions and behavioral systems, we introduce and characterize linear systems with quaternionic symmetry. Finally we propose a state-space realization algorithm c 2007 IFAC for input/output systems. Copyright Keywords: linear dynamical systems, symmetries, behavioral approach
1. INTRODUCTION General symmetries of linear dynamical systems have already been thoroughly investigated (see, e. g., De Concini and Fagnani (1993), Fagnani and Willems (1993), and Fagnani and Willems (1994)). This paper is devoted to the analysis of a particular kind of symmetry, the quaternionic symmetry (see Pereira and Vettori (2006)). Quaternions can be represented by noncommuting, squared real matrices of dimension four. On the contrary, real systems which exhibit quaternionic symmetry can be transformed into more compact quaternionic systems. In this paper we study and characterize these kind of real systems. Realization algorithms are proposed.
For any ν = a + bi + cj + dk ∈ H, we define its real part as Re ν = a, its imaginary part as Im ν = bi + cj + dk, its conjugate as ν = Re ν − Im a − bi − cj − dk, and its norm as |ν| = √ ν =√ νν = a2 + b2 + c2 + d2 . Given an p × q matrix with quaternionic entries M = (mhl ) ∈ Hp×q we define its conjugate as M = (mhl ) and its transpose as M > = (mlh ) ∈ Hq×p . Since each matrix M ∈ Hp×q may be uniquely written as M = M1 + M2 i + M3 j + M4 k, where M1 , M2 , M3 , M4 ∈ Rp×q we can define an injective linear map: Hp×q → R4p×4q such that
2. QUATERNIONS Let R and C denote the fields of real and of complex numbers respectively. The quaternion skewfield H is an associative but non-commutative algebra over R defined as the set H = {a + bi + cj + dk : a, b, c, d ∈ R} , where i, j, k are called imaginary units and satisfy i2 = j 2 = k 2 = ijk = −1.
M1 −M 2 M→ 7 MA = −M3 −M4
M2 M1 M4 −M3
M3 −M4 M1 M2
M4 M3 . −M2 M1
(1)
The matrix M A is called the real adjoint matrix of M . In general, any real matrix with the structure (1) is said to be a real adjoint matrix. We may as well define a bijective linear map: Hp×q → R4p×q such that
M1 −M2 M→ 7 MR = −M3 , −M4
(2)
which in particular maps quaternionic column vectors into real column vectors. Definition 1. A quaternionic polynomial r(s) is defined by r(s) =
N X
rl sl , rl ∈ H, N ∈ N.
l=0
Sum and product of polynomials are defined as in the commutative case with the additional rule (asn ) (bsm ) = absn+m , as if the indeterminate commuted with constant values. To simplify the notation, we omit the indeterminate and write r ∈ H[s] instead of r(s), if no ambiguity arises. The set of quaternionic polynomials is denoted by H[s] and Hp×q [s] is the set of p × q matrices with entries in H[s].
It is possible to treat discrete and continuous linear systems in a unified fashion by means of polynomial operators. Define, in the discrete-time case, the backward shift operator by (σ τ w)(t) = w(t + τ ), for any t, τ ∈ Z. Then the condiPN tion defining B in (3) is l=M Rl w(t + l) = PN l Rl σ w(t) = R(σ)w = 0, where R(s) = Pl=M N l p×q [s, s−1 ] is a quaternionic Laul=M Rl s ∈ H rent polynomial matrix (i.e., a polynomial with both positive and negative powers of s) acting on w as a linear difference operator. For the sake of simplicity we will suppose, without loss of generality, that R ∈ Hp×q [s]. Indeed, by definition (3), w ∈ B if and only if σ τ w ∈ B, i.e., R(σ)σ τ w = 0, for any t ∈ Z. So, if we take τ = −M , the behavior B can be equivalently defined by R(s)s−M , which is a polynomial matrix (see also (Pereira et al., 2005, Corollary 3.12)). PN Analogously, if R(s) = l=0 Rl sl ∈ Hp×q [s], the condition in (4) can be written in the operator PN dl d w(t) = l=0 Rl dt form R dt l w(t) = 0.
In the sequel, following (Willems, 1989), we introduce the concept of behavior.
Eventually, both in the discrete and in the continuous case, the behavior is the kernel of the operator R, B = ker R, where R(σ) is a difference d operator when T = Z and R( dt ) is a differential operator when T = R. The polynomial matrix R(s) is a kernel representation of B. Note that different representations may give rise to the same behavior. In particular ker R = ker U R for any unimodular matrix U (Pereira et al., 2005).
Definition 2. A behavior B is a set of functions, called trajectories, having the same domain T, called time set, and the same codomain W , i.e., B ⊆ W T = {w : T → W }
For anybehavior B we define the real behavior B R = wR : w ∈ B , where wR (t) = (w(t))R . B R is called the real form of B. Analogously to (Pereira et al., 2005, Proposition 3.5) the following proposition holds.
The definition of the real adjoint matrix of R(s) ∈ Hp×q [s], RA (s), is analogous to the constant case.
3. LINEAR DYNAMICAL SYSTEMS
In this paper, behaviors are solution sets of linear systems of quaternionic difference or differential equations. In other words, we will deal with discrete-time systems, where T = Z and ( ) N X q B= w:Z→H , Rl w(t + l) = 0, ∀t ∈ Z , l=M
(3) and with continuous-time systems, where T = R and ( ) N X q (l) B= w:R→H , Rl w (t) = 0, ∀t ∈ R .
Proposition 3. Let R(s) ∈ Hp×q [s]. Then (ker R)R = ker RA .
4. REPRESENTATIONS AND SYMMETRIES In this section we will introduce some definitions and notations regarding representations and symmetries which will be used in this paper (see also Serre (1977)). Let W be a vector space over the field K and GL(W ) be the group of isomorphisms on W .
l=0
(4) The systems are time-invariant, i.e., Rl ∈ Hp×q are constant matrices, and in the continuous case, where w(l) is the l-th order derivative of w, trajectories are supposed to be sufficiently smooth, otherwise equations have to be intended in a distributional sense (see Polderman and Willems (1997)).
Definition 4. Let G be a group and W a vector space. Then a representation of G on V is a group anti-homomorphism ρ : G → GL(W ), g 7→ ρg . Remark 5. Usually, representations are homomorphisms, i.e., ρgh = ρg ρh but this property is
not compatible with the the traditional notation which is used for quaternions. Hence, in this papers representations satisfy ρgh = ρh ρg , ∀g, h ∈ G. Definition 6. An element v ∈ W is G-symmetric if it is a fixed point of ρ, i.e., ρg v = v, ∀g ∈ G. If dim W = q < ∞, the representation ρ has finite degree q. In this case ρg ∈ Kq×q and is an invertible matrix for every g ∈ G. Example 7. A representation of the additive group G = R/Z on R2 is cos 2πg − sin 2πg ρg = . sin 2πg cos 2πg This is the group of rotations (by 2πg degrees) around the origin.
and only if it admits a full row rank kernel representation R(s) ∈ Kp×q [s] such that R(s)ρ = ρ0 R(s) where ρ0 is a subrepresentation of ρ.
5. QUATERNIONIC SYMMETRY Let G = {±1, ±i, ±j, ±k} be the quaternionic group and consider the representation ρ of G on R4 so defined 0 1 0 0 −1 0 0 0 ρ1 = I4 , ρi = 0 0 0 1 , 0 0 −1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 −1 ρj = −1 0 0 0 , ρk = 0 −1 0 0 . −1 0 0 0 0 1 0 0
Note that, for any v ∈ R2 with r = |v|, Cr = {ρg v : g ∈ G} is the circle with radius r and therefore the only symmetric point is v = 0 (or r = 0). However, by extending the action of ρ to subsets of R2 , we obtain a richer set of symmetric objects: circles, disks, rings etc. For example, ρg Cv = Cv , ∀g ∈ G, meaning that {ρg v : v ∈ Cr } = Cr .
It is easily checked that ρ is indeed a representation of G since
Most important is the case of symmetric subspaces of W since these are connected with the structure of the representation.
We may write just ρ instead of ρn when the value of n is clear from the context and no confusion arises.
Definition 8. Given a representation ρ on W , a subspace V ⊆ W is ρ-symmetric if ρg V ⊆ V, ∀ g ∈ G.
Definition 11. A behavior B is quaternionic-symmetric (q-symmetric) if it is ρ-symmetric (i.e., ρn symmetric for some n ∈ N).
In this case the restrictions of ρg to V are isomorphisms of V and thus ρ|V is itself a representation which is called a subrepresentation of ρ.
The following theorem states when a behavior is ρ-symmetric.
Finally, we consider the action of ρ on a trajectory w with values in W : (ρg w)(t) = ρw(t). In this case too it is useful to extend ρ to sets of trajectories, which leads us to the most important definition of this section. Definition 9. A behavior B is said to be ρsymmetric if ρg B = B, ∀ g ∈ G.
ρ2i = ρ2j = ρ2k = ρk ρj ρi = −ρ1 . In the sequel we will use the representation ρn = ρ ⊗ In of G on R4n and call it simply the quaternionic representation (A ⊗ B = [alh B] is the Kronecker product).
Theorem 12. A behavior is q-symmetric if and only if it is the real form of a quaternionic behavior. PROOF. “If” part. If B is the real form of a quaternionic behavior then there exists a full row rank matrix R ∈ Hp×q [s] such that B = ker RA , according to Proposition 3. By direct calculation we have that ρ0g RA = RA ρg , ∀g ∈ G
In (Fagnani and Willems, 1993) the following characterization of symmetric systems was given in terms of their kernel representation.
where ρ0l is a subrepresentation of ρ.
Theorem 10. Given a representation of G on W = Kq , the behavior B ⊆ W T is ρ-symmetric if
“Only if” part. Let R ∈ Rp×4q be a full row rank kernel representation of B and partition
Therefore, the behavior B is q-symmetric by Theorem 10.
R = [M1 |M2 |M3 |M4 ] ∈ Rp×4q [s]. Then let M = M1 + iM2 + jM3 + kM4 ∈ Hp×q and note, as Formula (1) shows, that the first block-row of M A is R. Therefore, ker M A ⊆ ker R = B. On the other hand, consider w ∈ ker R partitioned like R in order to obtain w1 w2 [M1 |M2 |M3 |M4 ] (5) w3 = 0. w4 Since B is ρ-symmetric we have that ρB = B and therefore the trajectories w2 w3 w4 −w1 −w4 w3 ρi w = w4 , ρj w = −w1 , ρk w = −w2 (6) −w3 w2 −w1 also belong to B. By the first equality of (6) we have that (5) is equivalent to w2 −w1 [M1 |M2 |M3 |M4 ] w4 = 0, −w3 which in turn is equivalent to w1 w2 [−M2 |M1 | − M4 |M3 ] w3 = 0. w4 Analogously we have also that w1 w2 [−M3 |M4 |M1 | − M2 ] =0 w3 w4
(3) Premultiply M by a unimodular matrix U ∈ ˜ M ˜ full Hp×p [s] such that U M = , with M 0 row rank. ˜ A is full row rank and B = (4) The matrix M A ˜ . ker M In order to apply Algorithm 1, a question arises naturally: is it possible to characterize q-symmetric behaviors by analyzing their kernel representation? The answer is rather simple. Let B be a qsymmetric behavior with full row rank kernel representation R ∈ Rp×n . By Theorem 12, B = ker M A thus there exist a unimodular U such that R = U M A . This means that R and M A share the same Smith Form and so the following result is a consequence of Pereira et al. (2005, Theorem 4.10) which gives a characterization of the Smith Form of complex adjoint matrices. Theorem 13. Let ∆ be the Smith Form of R. Then the behavior B = ker R is q-symmetric if and only if ∆ = diag(δ1 , δ1 , δ10 , δ10 , . . . , δn , δn , δn0 , δn0 , 0, . . . , 0) where δ1 |δ10 | · · · |δn |δn0 and δl , δl0 are monic polynomials with exactly the same real zeros, for every l = 1, . . . , n. 6. SYMMETRIC INPUT/OUTPUT SYSTEMS In Polderman and Willems (1997, section 3.3) behavioral input/output (i/o) systems are introduced and deeply analyzed. In this section, after giving their definition, we study i/o system from the viewpoint of quaternionic symmetry and finally give a realization algorithm.
and w1 w2 [−M4 | − M3 |M2 |M1 ] w3 = 0. w4 This means that, looking again at Formula (1), w ∈ ker M A , hence B ⊆ ker M A . We conclude that B = ker M A = (ker M )R . Given a q-symmetric behavior B, the proof of the previous theorem suggests a way to find a full row rank real adjoint kernel representation of B. This leads to the following algorithm. Algorithm 1. (1) Let B = ker R be q-symmetric, where R = [M1 |M2 |M3 |M4 ] ∈ Rp×4q [s]. (2) Construct M = M1 + M2 i + M3 j + M4 k ∈ Hp×q [s].
Definition 14. Let B ⊆ {[ uy ] : T → Rp+m }. Then u is an input variable and y is an output variable of B if (1) u is free in B: ∀u ∈ (Rm ) , ∃y ∈ (Rp ) such that [ uy ] ∈ B; (2) once u is fixed, no component of y is free in {y : [ uy ] ∈ B}. T
T
The notation Bi/o is used to denote behaviors which satisfy Definition 14. In general, B is an i/o behavior if the components of its trajectories w can be partitioned into input and output variables, i.e., a permutation of coordinates transforms it into a behavior Bi/o . A partition of any kernel representation R = P | −Q of an i/o behavior is naturally induced, which is made explicit by the i/o representation Bi/o = [ uy ] : T → Rp+m : P y = Qu . (7)
We assume that det P 6= 0 and that the transfer matrix P −1 Q of the behavior (7) is a proper rational matrix (Willems, 1989). Definition 15. A dynamical system defined by the equation P y = Qu, (8) p×p p×m where P ∈ R [s] and Q ∈ R [s], is a (proper) i/o system, with behavior Bi/o defined by equation (7), if P admits a rational inverse and its transfer matrix P −1 Q ∈ Rp×m (s) is proper.
Since ker P = ker Q = {0} it is clear that By and Bu are q-symmetric. By definition, the behavior Bi/o is q-symmetric if and only if ker R = ker P | −Q is ρ⊕ρ-symmetric and this does not T happen: if w = [ uy ] = 1 6 4 1 | 2 5 4 1 then T w ∈ Bi/o but ρi ⊕ρi w = 6 −1 1 4 | 5 −2 1 4 6∈ Bi/o .
6.1 Realization algorithm We aim now to answer the question: when is an i/o system symmetric? This leads to the following definition. Definition 16. Consider the behavior Bi/o defined as in (7), with input u ∈ R4m and output y ∈ R4p . Then Bi/o is q-symmetric if it is ρp ⊕ ρm symmetric, i.e., p
w ∈ B ⇐⇒
A characterization of q-symmetric i/o behaviors which is based on the Smith-McMillan Form of the transfer matrix P −1 Q can be given. In this paper we only show a simple necessary condition. Proposition 17. If the behavior Bi/o defined in (7) is q-symmetric then the behaviors ker P and ker Q are q-symmetric too. PROOF. Observe that the i/o behavior (7) has full row rank kernel representation R = P | −Q and therefore, by Theorem 10 there exists a subrepresentation ρ0 such that ρ0 P | −Q = P | −Q ρp ⊕ ρm . This equation can be divided into and ρ0 Q = Qρm
Note that even if Q is not full row rank, the symmetry condition is met since for every w ∈ ker Q, also ρm w ∈ ker Q. Indeed, Qw = 0 ⇒ ρ0 Qw = 0 ⇒ Qρm w = 0. However, note that the reciprocal of Proposition 17 is not true as can be seen in the following example. Let 00 0 0 1 0 01
and
1 1 Q= 0 0
0 1 0 0
0 0 1 0
x0 = Ax + Bu y = Cx + Du
in the continuous case, where [ uy ] is an adequate i/o partition of the trajectory w. Given a behavior B = ker R, there exist standard realization algorithms. However, more efficient ones can be given, which are appropriate for behaviors that exhibit quaternionic symmetry. Indeed, as we saw, if B is q-symmetric, it admits a kernel representation R with a special structure. Therefore, matrices (A, B, C, D) having a special structure too, could be constructed. In this case, the realization algorithm just computes a reduced number of entries which provide all the necessary information to completely determine the remaining ones. As an example, a simplified algorithm is proposed below. Algorithm 2.
thus giving the desired result.
Example 18. 1 0 1 1 P = 0 0 1 0
in the discrete case or
m
ρ ⊕ ρ Bi/o = Bi/o .
ρ0 P = P ρp
Consider a behavior B = ker R, with R ∈ Rp×q [s]. The realization problem consists in obtaining matrices (A, B, C, D) and a (state) vector x such that σx = Ax + Bu w ∈ B ⇐⇒ y = Cx + Du
−1 0 . 0 2
(1) Let B be a q-symmetric behavior. By AlgoeA , rithm 1 we may suppose that B = ker R e is a full row rank quaternionic mawhere R e trix. We also assume h that iker R is an i/o e = Pe| − Q e with the usual behavior, i.e., R h i y partition w e= e . e u To simplify the algorithm, Pnwe suppose that the p × p matrix Pe(s) = l=0 Pel sl has a full rank highest coefficient, Pen = I. (2) By definition of i/o behavior we know that Pn e l e Q(s) = l=0 Q ls . (3) With these matrices a quaternionic statespace representation is given by the four e B, e C, e D) e which are defined as matrices (A, follows:
0 ··· .. I . e= A . .. 0 e = 0 ··· C
e en Q0 − Pe0 Q 0 −Pe0 .. .. .. . . . B e = .. . .. 0 . I −Pen−1 e n−1 − Pen−1 Q en Q e =Q en 0 I D
(4) In the end, the real state-space system is eA , B = B eA, C = C eA , given by matrices A = A A e . and D = D Note that the relation between h i w and y and u is y not immediate as for w e= e since e u R R R ye y ye ye w=w eR = but = R 6= . u e u u e u e
ACKNOWLEDGEMENTS This work was supported in part by Portuguese Science Foundation (FCT–Funda¸c˜ ao para a Ciˆencia e Tecnologia) through the Unidade de Investiga¸c˜ ao Matem´ atica e Aplica¸c˜ oes of University of Aveiro, Portugal.
REFERENCES De Concini, Corrado and Fabio Fagnani (1993). Symmetries of differential behaviors and finite group actions on free modules over a polynomial ring. Math. Control Signals Systems 6(4), 307–321. Fagnani, Fabio and Jan C. Willems (1993). Representations of symmetric linear dynamical systems. SIAM J. Control Optim. 31(5), 1267– 1293. Fagnani, Fabio and Jan C. Willems (1994). Symmetries of differential systems. In: Differential Equations, Dynamical Systems, and Control Science. pp. 491–504. Marcel Dekker Inc.. New York. Lecture Notes in Pure and Appl. Math., 152. Pereira, Ricardo and Paolo Vettori (2006). Stability of quaternionic linear systems. IEEE Trans. Automat. Control 51(3), 518–523. Pereira, Ricardo, Paula Rocha and Paolo Vettori (2005). Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140. Polderman, Jan Willem and Jan C. Willems (1997). Introduction to Mathematical Systems Theory: A Behavioral Approach. Vol. 26 of Texts in Applied Mathematics. SpringerVerlag. Berlin. Serre, Jean-Paul (1977). Linear Representations of Finite Groups. Springer-Verlag. Berlin.
Willems, Jan C. (1989). Models for dynamics. In: Dynamics Reported (U. Kirchgraber and H. O. Walther, Eds.). Vol. 2. pp. 171–269. John Wiley & Sons Ltd.. Chichester.
Department of Mathematics, University of Aveiro email: {ricardopereira, pvettori}@ua.pt
Abstract: In this paper we study a particular kind of symmetry of linear dynamical systems, the quaternionic symmetry. After giving some basic notions about quaternions and behavioral systems, we introduce and characterize linear systems with quaternionic symmetry. Finally we propose a state-space realization algorithm c 2007 IFAC for input/output systems. Copyright Keywords: linear dynamical systems, symmetries, behavioral approach
1. INTRODUCTION General symmetries of linear dynamical systems have already been thoroughly investigated (see, e. g., De Concini and Fagnani (1993), Fagnani and Willems (1993), and Fagnani and Willems (1994)). This paper is devoted to the analysis of a particular kind of symmetry, the quaternionic symmetry (see Pereira and Vettori (2006)). Quaternions can be represented by noncommuting, squared real matrices of dimension four. On the contrary, real systems which exhibit quaternionic symmetry can be transformed into more compact quaternionic systems. In this paper we study and characterize these kind of real systems. Realization algorithms are proposed.
For any ν = a + bi + cj + dk ∈ H, we define its real part as Re ν = a, its imaginary part as Im ν = bi + cj + dk, its conjugate as ν = Re ν − Im a − bi − cj − dk, and its norm as |ν| = √ ν =√ νν = a2 + b2 + c2 + d2 . Given an p × q matrix with quaternionic entries M = (mhl ) ∈ Hp×q we define its conjugate as M = (mhl ) and its transpose as M > = (mlh ) ∈ Hq×p . Since each matrix M ∈ Hp×q may be uniquely written as M = M1 + M2 i + M3 j + M4 k, where M1 , M2 , M3 , M4 ∈ Rp×q we can define an injective linear map: Hp×q → R4p×4q such that
2. QUATERNIONS Let R and C denote the fields of real and of complex numbers respectively. The quaternion skewfield H is an associative but non-commutative algebra over R defined as the set H = {a + bi + cj + dk : a, b, c, d ∈ R} , where i, j, k are called imaginary units and satisfy i2 = j 2 = k 2 = ijk = −1.
M1 −M 2 M→ 7 MA = −M3 −M4
M2 M1 M4 −M3
M3 −M4 M1 M2
M4 M3 . −M2 M1
(1)
The matrix M A is called the real adjoint matrix of M . In general, any real matrix with the structure (1) is said to be a real adjoint matrix. We may as well define a bijective linear map: Hp×q → R4p×q such that
M1 −M2 M→ 7 MR = −M3 , −M4
(2)
which in particular maps quaternionic column vectors into real column vectors. Definition 1. A quaternionic polynomial r(s) is defined by r(s) =
N X
rl sl , rl ∈ H, N ∈ N.
l=0
Sum and product of polynomials are defined as in the commutative case with the additional rule (asn ) (bsm ) = absn+m , as if the indeterminate commuted with constant values. To simplify the notation, we omit the indeterminate and write r ∈ H[s] instead of r(s), if no ambiguity arises. The set of quaternionic polynomials is denoted by H[s] and Hp×q [s] is the set of p × q matrices with entries in H[s].
It is possible to treat discrete and continuous linear systems in a unified fashion by means of polynomial operators. Define, in the discrete-time case, the backward shift operator by (σ τ w)(t) = w(t + τ ), for any t, τ ∈ Z. Then the condiPN tion defining B in (3) is l=M Rl w(t + l) = PN l Rl σ w(t) = R(σ)w = 0, where R(s) = Pl=M N l p×q [s, s−1 ] is a quaternionic Laul=M Rl s ∈ H rent polynomial matrix (i.e., a polynomial with both positive and negative powers of s) acting on w as a linear difference operator. For the sake of simplicity we will suppose, without loss of generality, that R ∈ Hp×q [s]. Indeed, by definition (3), w ∈ B if and only if σ τ w ∈ B, i.e., R(σ)σ τ w = 0, for any t ∈ Z. So, if we take τ = −M , the behavior B can be equivalently defined by R(s)s−M , which is a polynomial matrix (see also (Pereira et al., 2005, Corollary 3.12)). PN Analogously, if R(s) = l=0 Rl sl ∈ Hp×q [s], the condition in (4) can be written in the operator PN dl d w(t) = l=0 Rl dt form R dt l w(t) = 0.
In the sequel, following (Willems, 1989), we introduce the concept of behavior.
Eventually, both in the discrete and in the continuous case, the behavior is the kernel of the operator R, B = ker R, where R(σ) is a difference d operator when T = Z and R( dt ) is a differential operator when T = R. The polynomial matrix R(s) is a kernel representation of B. Note that different representations may give rise to the same behavior. In particular ker R = ker U R for any unimodular matrix U (Pereira et al., 2005).
Definition 2. A behavior B is a set of functions, called trajectories, having the same domain T, called time set, and the same codomain W , i.e., B ⊆ W T = {w : T → W }
For anybehavior B we define the real behavior B R = wR : w ∈ B , where wR (t) = (w(t))R . B R is called the real form of B. Analogously to (Pereira et al., 2005, Proposition 3.5) the following proposition holds.
The definition of the real adjoint matrix of R(s) ∈ Hp×q [s], RA (s), is analogous to the constant case.
3. LINEAR DYNAMICAL SYSTEMS
In this paper, behaviors are solution sets of linear systems of quaternionic difference or differential equations. In other words, we will deal with discrete-time systems, where T = Z and ( ) N X q B= w:Z→H , Rl w(t + l) = 0, ∀t ∈ Z , l=M
(3) and with continuous-time systems, where T = R and ( ) N X q (l) B= w:R→H , Rl w (t) = 0, ∀t ∈ R .
Proposition 3. Let R(s) ∈ Hp×q [s]. Then (ker R)R = ker RA .
4. REPRESENTATIONS AND SYMMETRIES In this section we will introduce some definitions and notations regarding representations and symmetries which will be used in this paper (see also Serre (1977)). Let W be a vector space over the field K and GL(W ) be the group of isomorphisms on W .
l=0
(4) The systems are time-invariant, i.e., Rl ∈ Hp×q are constant matrices, and in the continuous case, where w(l) is the l-th order derivative of w, trajectories are supposed to be sufficiently smooth, otherwise equations have to be intended in a distributional sense (see Polderman and Willems (1997)).
Definition 4. Let G be a group and W a vector space. Then a representation of G on V is a group anti-homomorphism ρ : G → GL(W ), g 7→ ρg . Remark 5. Usually, representations are homomorphisms, i.e., ρgh = ρg ρh but this property is
not compatible with the the traditional notation which is used for quaternions. Hence, in this papers representations satisfy ρgh = ρh ρg , ∀g, h ∈ G. Definition 6. An element v ∈ W is G-symmetric if it is a fixed point of ρ, i.e., ρg v = v, ∀g ∈ G. If dim W = q < ∞, the representation ρ has finite degree q. In this case ρg ∈ Kq×q and is an invertible matrix for every g ∈ G. Example 7. A representation of the additive group G = R/Z on R2 is cos 2πg − sin 2πg ρg = . sin 2πg cos 2πg This is the group of rotations (by 2πg degrees) around the origin.
and only if it admits a full row rank kernel representation R(s) ∈ Kp×q [s] such that R(s)ρ = ρ0 R(s) where ρ0 is a subrepresentation of ρ.
5. QUATERNIONIC SYMMETRY Let G = {±1, ±i, ±j, ±k} be the quaternionic group and consider the representation ρ of G on R4 so defined 0 1 0 0 −1 0 0 0 ρ1 = I4 , ρi = 0 0 0 1 , 0 0 −1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 −1 ρj = −1 0 0 0 , ρk = 0 −1 0 0 . −1 0 0 0 0 1 0 0
Note that, for any v ∈ R2 with r = |v|, Cr = {ρg v : g ∈ G} is the circle with radius r and therefore the only symmetric point is v = 0 (or r = 0). However, by extending the action of ρ to subsets of R2 , we obtain a richer set of symmetric objects: circles, disks, rings etc. For example, ρg Cv = Cv , ∀g ∈ G, meaning that {ρg v : v ∈ Cr } = Cr .
It is easily checked that ρ is indeed a representation of G since
Most important is the case of symmetric subspaces of W since these are connected with the structure of the representation.
We may write just ρ instead of ρn when the value of n is clear from the context and no confusion arises.
Definition 8. Given a representation ρ on W , a subspace V ⊆ W is ρ-symmetric if ρg V ⊆ V, ∀ g ∈ G.
Definition 11. A behavior B is quaternionic-symmetric (q-symmetric) if it is ρ-symmetric (i.e., ρn symmetric for some n ∈ N).
In this case the restrictions of ρg to V are isomorphisms of V and thus ρ|V is itself a representation which is called a subrepresentation of ρ.
The following theorem states when a behavior is ρ-symmetric.
Finally, we consider the action of ρ on a trajectory w with values in W : (ρg w)(t) = ρw(t). In this case too it is useful to extend ρ to sets of trajectories, which leads us to the most important definition of this section. Definition 9. A behavior B is said to be ρsymmetric if ρg B = B, ∀ g ∈ G.
ρ2i = ρ2j = ρ2k = ρk ρj ρi = −ρ1 . In the sequel we will use the representation ρn = ρ ⊗ In of G on R4n and call it simply the quaternionic representation (A ⊗ B = [alh B] is the Kronecker product).
Theorem 12. A behavior is q-symmetric if and only if it is the real form of a quaternionic behavior. PROOF. “If” part. If B is the real form of a quaternionic behavior then there exists a full row rank matrix R ∈ Hp×q [s] such that B = ker RA , according to Proposition 3. By direct calculation we have that ρ0g RA = RA ρg , ∀g ∈ G
In (Fagnani and Willems, 1993) the following characterization of symmetric systems was given in terms of their kernel representation.
where ρ0l is a subrepresentation of ρ.
Theorem 10. Given a representation of G on W = Kq , the behavior B ⊆ W T is ρ-symmetric if
“Only if” part. Let R ∈ Rp×4q be a full row rank kernel representation of B and partition
Therefore, the behavior B is q-symmetric by Theorem 10.
R = [M1 |M2 |M3 |M4 ] ∈ Rp×4q [s]. Then let M = M1 + iM2 + jM3 + kM4 ∈ Hp×q and note, as Formula (1) shows, that the first block-row of M A is R. Therefore, ker M A ⊆ ker R = B. On the other hand, consider w ∈ ker R partitioned like R in order to obtain w1 w2 [M1 |M2 |M3 |M4 ] (5) w3 = 0. w4 Since B is ρ-symmetric we have that ρB = B and therefore the trajectories w2 w3 w4 −w1 −w4 w3 ρi w = w4 , ρj w = −w1 , ρk w = −w2 (6) −w3 w2 −w1 also belong to B. By the first equality of (6) we have that (5) is equivalent to w2 −w1 [M1 |M2 |M3 |M4 ] w4 = 0, −w3 which in turn is equivalent to w1 w2 [−M2 |M1 | − M4 |M3 ] w3 = 0. w4 Analogously we have also that w1 w2 [−M3 |M4 |M1 | − M2 ] =0 w3 w4
(3) Premultiply M by a unimodular matrix U ∈ ˜ M ˜ full Hp×p [s] such that U M = , with M 0 row rank. ˜ A is full row rank and B = (4) The matrix M A ˜ . ker M In order to apply Algorithm 1, a question arises naturally: is it possible to characterize q-symmetric behaviors by analyzing their kernel representation? The answer is rather simple. Let B be a qsymmetric behavior with full row rank kernel representation R ∈ Rp×n . By Theorem 12, B = ker M A thus there exist a unimodular U such that R = U M A . This means that R and M A share the same Smith Form and so the following result is a consequence of Pereira et al. (2005, Theorem 4.10) which gives a characterization of the Smith Form of complex adjoint matrices. Theorem 13. Let ∆ be the Smith Form of R. Then the behavior B = ker R is q-symmetric if and only if ∆ = diag(δ1 , δ1 , δ10 , δ10 , . . . , δn , δn , δn0 , δn0 , 0, . . . , 0) where δ1 |δ10 | · · · |δn |δn0 and δl , δl0 are monic polynomials with exactly the same real zeros, for every l = 1, . . . , n. 6. SYMMETRIC INPUT/OUTPUT SYSTEMS In Polderman and Willems (1997, section 3.3) behavioral input/output (i/o) systems are introduced and deeply analyzed. In this section, after giving their definition, we study i/o system from the viewpoint of quaternionic symmetry and finally give a realization algorithm.
and w1 w2 [−M4 | − M3 |M2 |M1 ] w3 = 0. w4 This means that, looking again at Formula (1), w ∈ ker M A , hence B ⊆ ker M A . We conclude that B = ker M A = (ker M )R . Given a q-symmetric behavior B, the proof of the previous theorem suggests a way to find a full row rank real adjoint kernel representation of B. This leads to the following algorithm. Algorithm 1. (1) Let B = ker R be q-symmetric, where R = [M1 |M2 |M3 |M4 ] ∈ Rp×4q [s]. (2) Construct M = M1 + M2 i + M3 j + M4 k ∈ Hp×q [s].
Definition 14. Let B ⊆ {[ uy ] : T → Rp+m }. Then u is an input variable and y is an output variable of B if (1) u is free in B: ∀u ∈ (Rm ) , ∃y ∈ (Rp ) such that [ uy ] ∈ B; (2) once u is fixed, no component of y is free in {y : [ uy ] ∈ B}. T
T
The notation Bi/o is used to denote behaviors which satisfy Definition 14. In general, B is an i/o behavior if the components of its trajectories w can be partitioned into input and output variables, i.e., a permutation of coordinates transforms it into a behavior Bi/o . A partition of any kernel representation R = P | −Q of an i/o behavior is naturally induced, which is made explicit by the i/o representation Bi/o = [ uy ] : T → Rp+m : P y = Qu . (7)
We assume that det P 6= 0 and that the transfer matrix P −1 Q of the behavior (7) is a proper rational matrix (Willems, 1989). Definition 15. A dynamical system defined by the equation P y = Qu, (8) p×p p×m where P ∈ R [s] and Q ∈ R [s], is a (proper) i/o system, with behavior Bi/o defined by equation (7), if P admits a rational inverse and its transfer matrix P −1 Q ∈ Rp×m (s) is proper.
Since ker P = ker Q = {0} it is clear that By and Bu are q-symmetric. By definition, the behavior Bi/o is q-symmetric if and only if ker R = ker P | −Q is ρ⊕ρ-symmetric and this does not T happen: if w = [ uy ] = 1 6 4 1 | 2 5 4 1 then T w ∈ Bi/o but ρi ⊕ρi w = 6 −1 1 4 | 5 −2 1 4 6∈ Bi/o .
6.1 Realization algorithm We aim now to answer the question: when is an i/o system symmetric? This leads to the following definition. Definition 16. Consider the behavior Bi/o defined as in (7), with input u ∈ R4m and output y ∈ R4p . Then Bi/o is q-symmetric if it is ρp ⊕ ρm symmetric, i.e., p
w ∈ B ⇐⇒
A characterization of q-symmetric i/o behaviors which is based on the Smith-McMillan Form of the transfer matrix P −1 Q can be given. In this paper we only show a simple necessary condition. Proposition 17. If the behavior Bi/o defined in (7) is q-symmetric then the behaviors ker P and ker Q are q-symmetric too. PROOF. Observe that the i/o behavior (7) has full row rank kernel representation R = P | −Q and therefore, by Theorem 10 there exists a subrepresentation ρ0 such that ρ0 P | −Q = P | −Q ρp ⊕ ρm . This equation can be divided into and ρ0 Q = Qρm
Note that even if Q is not full row rank, the symmetry condition is met since for every w ∈ ker Q, also ρm w ∈ ker Q. Indeed, Qw = 0 ⇒ ρ0 Qw = 0 ⇒ Qρm w = 0. However, note that the reciprocal of Proposition 17 is not true as can be seen in the following example. Let 00 0 0 1 0 01
and
1 1 Q= 0 0
0 1 0 0
0 0 1 0
x0 = Ax + Bu y = Cx + Du
in the continuous case, where [ uy ] is an adequate i/o partition of the trajectory w. Given a behavior B = ker R, there exist standard realization algorithms. However, more efficient ones can be given, which are appropriate for behaviors that exhibit quaternionic symmetry. Indeed, as we saw, if B is q-symmetric, it admits a kernel representation R with a special structure. Therefore, matrices (A, B, C, D) having a special structure too, could be constructed. In this case, the realization algorithm just computes a reduced number of entries which provide all the necessary information to completely determine the remaining ones. As an example, a simplified algorithm is proposed below. Algorithm 2.
thus giving the desired result.
Example 18. 1 0 1 1 P = 0 0 1 0
in the discrete case or
m
ρ ⊕ ρ Bi/o = Bi/o .
ρ0 P = P ρp
Consider a behavior B = ker R, with R ∈ Rp×q [s]. The realization problem consists in obtaining matrices (A, B, C, D) and a (state) vector x such that σx = Ax + Bu w ∈ B ⇐⇒ y = Cx + Du
−1 0 . 0 2
(1) Let B be a q-symmetric behavior. By AlgoeA , rithm 1 we may suppose that B = ker R e is a full row rank quaternionic mawhere R e trix. We also assume h that iker R is an i/o e = Pe| − Q e with the usual behavior, i.e., R h i y partition w e= e . e u To simplify the algorithm, Pnwe suppose that the p × p matrix Pe(s) = l=0 Pel sl has a full rank highest coefficient, Pen = I. (2) By definition of i/o behavior we know that Pn e l e Q(s) = l=0 Q ls . (3) With these matrices a quaternionic statespace representation is given by the four e B, e C, e D) e which are defined as matrices (A, follows:
0 ··· .. I . e= A . .. 0 e = 0 ··· C
e en Q0 − Pe0 Q 0 −Pe0 .. .. .. . . . B e = .. . .. 0 . I −Pen−1 e n−1 − Pen−1 Q en Q e =Q en 0 I D
(4) In the end, the real state-space system is eA , B = B eA, C = C eA , given by matrices A = A A e . and D = D Note that the relation between h i w and y and u is y not immediate as for w e= e since e u R R R ye y ye ye w=w eR = but = R 6= . u e u u e u e
ACKNOWLEDGEMENTS This work was supported in part by Portuguese Science Foundation (FCT–Funda¸c˜ ao para a Ciˆencia e Tecnologia) through the Unidade de Investiga¸c˜ ao Matem´ atica e Aplica¸c˜ oes of University of Aveiro, Portugal.
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