Least Squares Based Self-Tuning Control Systems: Supplementary Notes

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1 Least Squares Based Self-Tuning Control Systems: Supplementary Notes S. Garatti Dip. di Elettronica ed Informazione Politecnico di Milano, piazza L. da Vinci 32, 2133, Milan, Italy. July 17, 212 These notes provides some complimentary material to the paper Least Squares Based Self-Tuning Control Systems by S. Bittanti and M. Campi. In particular, they provide smoothed versions of the paper proofs by providing some reasonings left to the reader and simplifying the arguments when possible. The notes presume the reader got through the paper at least up to page 351 or that the reader attended the classes given by myself. I used here the same notation adopted in class, which is slightly different from that of the paper. The main differences with the paper are now briefly explained to ease the reader. 1

2 - The delay operator z 1 is used in place of q 1 - The true system is indicated with S - The RLS estimate is indicated by ϑ t and the RLS algorithm is given by the following equations: ϑ t+1 = ϑ ] t + S(t + 1) 1 ϕ(t) [y(t + 1) ϕ(t) T ϑt S(t + 1) = S(t) + ϕ(t)ϕ(t) T ϑ = ϑ S() = S (here S(t) is equal to P (t) 1 in the paper). - The control law is as follows: C(ϑ) : A c (ϑ, z 1 )u(t) = B c (ϑ, z 1 )y(t) + C c (ϑ, z 1 )r(t), where r(t) is the reference signal and A c (ϑ, z 1 ) = 1 α 1 (ϑ)z 1 α nα (ϑ)z nα B c (ϑ, z 1 ) = β (ϑ) + β 1 (ϑ)z β nβ (ϑ)z n β C c (ϑ, z 1 ) = γ (ϑ) + γ 1 (ϑ)z γ nγ (ϑ)z nγ 1 RLS estimate convergence In this section we want to prove that the RLS estimate ϑ t converges for t to a finite value for any possible data record {..., y(1), u(1),..., y(t), u(t),...} 2

3 generated according to the true system, i.e. such that y(t + 1) = ϕ(t) T ϑ o. From the RLS equations: ϑ t+1 = ϑ ] t + S(t + 1) 1 ϕ(t) [y(t + 1) ϕ(t) T ϑt [recall that y(t + 1) = ϕ(t) T ϑ o ] ϑ t+1 ϑ o = ϑ [ t ϑ o + S(t + 1) 1 ϕ(t)ϕ(t) T ϑ o ϑ ] t. Calling ϑ t = ϑ t ϑ o we have that ϑ t+1 = ϑ t S(t + 1) 1 ϕ(t)ϕ(t) T ϑt S(t + 1) ϑ t+1 = S(t + 1) ϑ t ϕ(t)ϕ(t) T ϑt [note that S(t + 1) ϕ(t)ϕ(t) T = S(t)] S(t + 1) ϑ t+1 = S(t) ϑ t, (1) i.e. the quantity S(t) ϑ t is an invariant, and, recursively applying (1), S(t) ϑ t = S() ϑ = S (ϑ ϑ o ), t. We need to recall here the definition of positive (semi-)definite matrix. Definition 1 M, N are symmetric quadratic matrices. M means that M is positive definite, i.e. x T Mx >, x. M means that M is positive semi-definite, i.e. x T Mx, x. We will write M N and M N when M N and M N, respectively. Since S(t + 1) = S(t) + ϕ(t)ϕ(t) T we have that x T S(t + 1)x = x T S(t)x + x T ϕ(t)ϕ(t) T x = [note that x T ϕ(t) = ϕ(t) T x is a scalar] = x T S(t)x + ( x T ϕ(t) ) 2. 3

4 Hence, x T (S(t + 1) S(t))x = ( x T ϕ(t) ) 2, for all x, i.e. S(t + 1) S(t). Recursively reasoning, the following relationship is obtained: S(t + 1) S(t) S(1) S, where the last relation is because the user-chosen initialization of the RLS algorithm is such that S is positive definite. Since S(t), t, the inverse of S(t) exists for all t and moreover S(t) 1, t (this is a basic property of positive definite matrices). Hence, we can write ϑ t = S(t) 1 S ϑ and it is clear that if S(t) 1 converges then ϑ t does as well. Another well known property of positive definite matrices is that M N N 1 M 1, leading to the following relationship which is equivalent to S 1 S(t) 1 S(t + 1) 1, x T S 1 x x T S(t) 1 x x T S(t + 1) 1 x >, x. (2) Indicate with a i,j (t) the generic element of S(t) 1, that is a 1,1 (t) a 1,n (t) S(t) 1 = a n,1 (t) a n,n (t) 4

5 By taking x = e 1 = 1. in (2) we have (x T S(t) 1 x indeed selects the element a 1,1 (t) of S(t) 1 ): a 1,1 () a 1,1 (t) a 1,1 (t + 1) >, i.e., as t increases, the a 1,1 (t) s form a decreasing sequence which is bounded from below by. Therefore, a 1,1 (t) must converge i.e. a 1,1 (t) t ā 1,1. Similarly, letting x equal to e 2 = 1.,, e n 1 =. 1, e n =. 1, it can be shown that all the elements on the diagonal a i,i (t) must converge. Finally, letting x = e i + e j, i j we have a i,i (t) + a j,j (t) + 2a i,j (t) a i,i (t + 1) + a j,j (t + 1) + 2a i,j (t + 1) >, that is, a i,i (t)+a j,j (t)+2a i,j (t) must converge. Yet, because a i,i (t) and a j,j (t) converge, we deduce that a i,j (t) converges too. 5

6 Thus, altogether, we have proved that S(t) 1 S 1, t so that ϑ t = S(t) 1 S ϑ S 1 S ϑ, t, i.e. ϑ t converges to the finite value S 1 S ϑ which will be indicated by ϑ. Since ϑ t = ϑ t ϑ o, the convergence of ϑ t to a finite value is easily obtained from the convergence of ϑ t : ϑ t ϑ := ϑ o + ϑ, t. Clearly, if S(t) 1, then ϑ t and ϑ t ϑ o. Yet, we cannot rely on S(t) 1 in the context of adaptive control. 2 Characterization of ϑ For a given realization of {..., y(1), u(1),..., y(t), u(t),...} we can define the unexcitation subspace as E = { } ϑ R n : lim ϑ T S(t)ϑ < +, t i.e. E is the space of directions where the information does not diverge (ϑ T S(t)ϑ can be interpreted as the projection of S(t) on the direction of ϑ). Note that since S(t + 1) S(t) for all t, we have that ϑ T S(t + 1)ϑ ϑ T S(t)ϑ, i.e. ϑ T S(t)ϑ is an increasing sequence of scalars. Hence, lim t ϑ T S(t)ϑ exists and it can be either a finite value or +. 6

7 The excitation subspace E is the orthogonal complement of E, i.e. E = E = { λ R n : λ T ϑ =, ϑ E }. Note that every ϑ R n can be decomposed as ϑ = ϑ u + ϑ e, where ϑ u is the orthogonal projection of ϑ on the unexcitation subspace E, while ϑ e is the projection on E. Clearly, ϑ u ϑ e, i.e. (ϑ u ) T ϑ e =. First, we want to prove that ϑ always belongs to the unexcitation subspace. Let v(t) = ϑ T t S(t) ϑ t. v(t) is of course a scalar, and, since S(t), v(t) >, for all t. We want to understand what kind of sequence is formed by v(t) as t is let increase. We have that v(t + 1) = ϑ T t+1s(t + 1) ϑ t+1 = ϑ T t+1s(t + 1)S(t + 1) 1 S(t + 1) ϑ t+1 = [recall that S(t + 1) ϑ t+1 = S(t) ϑ t ] = ϑ T t S(t)S(t + 1) 1 S(t) ϑ t. From the matrix inversion lemma we have that S(t + 1) 1 = S(t) 1 S(t) 1 ϕ(t)ϕ(t) T S(t) ϕ(t) T S(t) 1 ϕ(t), 7

8 and substituting we obtain that v(t + 1) = ϑ T t S(t) ϑ t ϑ T t ϕ(t)ϕ(t) T ϑt 1 + ϕ(t) T S(t) 1 ϕ(t) = [note that ϑ T t ϕ(t) = ϕ(t) T ϑt is a scalar] = v(t) ( ϑ T t ϕ(t)) ϕ(t) T S(t) 1 ϕ(t). Hence, v(t + 1) is equal to v(t) minus a term which is positive (the numerator is a square, while the denominator is positive in view of the positive definiteness of S(t) 1 ). This means that v(t) is decreasing with t: This proves that v() v(t) v(t + 1) >. lim v(t) = lim ϑ T t S(t) ϑ t v() < +. t t Unfortunately, this latter statement does not permit us to draw the final conclusion from the convergence of ϑ t to ϑ only, see Appendix A. Yet, it is not required to exploit the particular structure of ϑ t, since it suffices to consider the monotonicity of S(t) to finalize the proof. Since S(τ) S(t) for every t and τ t, we have that ϑ τ S(t) ϑ τ ϑ τ S(τ) ϑ τ, t, τ t. The right-hand-side of this inequality is equal to v(τ) which is smaller than or equal to v(), so that ϑ τ S(t) ϑ τ v(), t, τ t. Since this inequality holds for all τ t, it applies also to the limit for τ, that is (remember that ϑ τ converges to ϑ ): ϑ S(t) ϑ = lim τ ϑτ S(t) ϑ τ v(), t. 8

9 Since this other inequality holds for all t we eventually have lim ϑ S(t) ϑ v() < +, t showing that ϑ belongs to unexcitation subspace E according to its very definition. In virtue of this result, it holds that ϑ = ϑ u since, belonging ϑ to E, ϑ e =. Thanks to the convergence theorem, we then have that ϑ e t. Recalling that ϑ = ϑ o + ϑ, we also have ϑ e = (ϑ o ) e + ϑ e = (ϑ o ) e and ϑ e t (ϑ o ) e, i.e. the RLS estimate converges to the true parameter value along the direction of the excitation subspace only. Clearly if the excitation subspace were the whole R n, then we would have ϑ t ϑ o. Yet, since the input sequence is determined by the adaptive control scheme it is not possible to a-priori know what is the excitation subspace in the current system operation condition. As for ϑ u and ϑ u nothing can be said, they depend on the u, y signals realization. Yet, if we consider the regression vector ϕ(t) and its projections on E and E, i.e. ϕ(t) = ϕ u (t) + ϕ e (t), we can prove that ϕ u (t) gets smaller and smaller till to as t increases. In other words, the information about the real system carried by the regression vector ϕ(t) vanishes along the directions of the unexcitation subspace. We now prove that ϕ u (t) as t. Consider lim t ϑ T [ t i=1 ϕu (i 1)ϕ u (i 1) T ] ϑ. The following chain of 9

10 equalities/inequalities holds true. [ ] ϕ u (i 1)ϕ u (i 1) T ϑ lim t ϑt i=1 = lim t (ϑ e + ϑ u ) T [ ] ϕ u (i 1)ϕ u (i 1) T (ϑ e + ϑ u ) i=1 = [ϕ u (t) ϑ e since they belong to E and E, respectively] [ = lim t (ϑ u ) T i=1 ϕ u (i 1)ϕ u (i 1) T ] = [ϕ(t) = ϕ u (t) + ϕ e (t) and ϑ u ϕ e (t)] [ = lim t (ϑ u ) T < lim t (ϑ u ) T [ i=1 S + = lim t (ϑ u ) T S(t)ϑ u < +. ϕ(i 1)ϕ(i 1) T ] ϑ u ϑ u ] ϕ(i 1)ϕ(i 1) T i=1 ϑ u Let ϕ u 1(t) ϕ ϕ u (t) = u 2(t). ϕ u n(t) 1

11 Taking ϑ = e 1 we have [ lim t et 1 i=1 ϕ u (i 1)ϕ u (i 1) T ] e 1 = lim t This implies that ϕ u 1(t). = lim t = lim t < +. e T 1 ϕ u (i 1)ϕ u (i 1) T e 1 i=1 (e 1 ϕ u (i 1)) 2 i=1 (ϕ u 1(i 1)) 2 Letting ϑ = e 2, e 3,..., e n we also obtain ϕ u i (t), i = 2, 3,..., n, showing that ϕ u (t). i=1 3 BIBO stability and self-optimality of the self-tuning adaptive scheme Throughout, ϑ t has to be intended as the RLS estimate obtained based on the measurements of u and y (input and output of the true system) up to time t. When not explicitly required, the equations for ϑ t will be omitted. Let s start from the equations of the self-tuning control scheme: Σ(ϑ o, ϑ y(t) = [1 A(ϑ o, z 1 )]y(t) + B(ϑ o, z 1 )u(t d) t ) : u(t) = [1 A c ( ϑ t, z 1 )]u(t) + B c ( ϑ t, z 1 )y(t) + r (t) where to ease the notation we have put r (t) = C c ( ϑ t, z 1 )r(t) = γ ( ϑ t )r(t) + γ 1 ( ϑ t )r(t 1) + + γ nγ ( ϑ t )r(t n γ ). 11

12 Since ϑ t ϑ and by continuity, it holds that γ i ( ϑ t ) γ i ( ϑ ), so that if r(t) is bounded then r (t) is bounded too 1. Hence, BIBO stability and self-optimality can be proved with reference to r (t) instead of r(t). The first equation of Σ(ϑ o, ϑ t ) can be also written as y(t) = ϕ(t 1) T ϑ o = [ ϕ(t 1) T ϑ o ϑ t + ϑ ] t = ϕ(t 1) T ϑt ϕ(t 1) T ϑt = [1 A( ϑ t, z 1 )]y(t) + B( ϑ t, z 1 )u(t d) + e(t), where we have defined e(t) = ϕ(t 1) T ϑt. Hence, the self-tuning scheme Σ(ϑ o, ϑ t )can be re-written as y(t) = 1 A(ϑo, z 1 ) B(ϑ o, z 1 )z d y(t) + e(t) (3a) u(t) 1 A c ( ϑ t, z 1 ) B c ( ϑ t, z 1 ) u(t) r (t) e(t) = ϕ(t 1) T ϑt (3b) In (3), the first (2 inputs/2 outputs) equation is exactly the same equation of the so-called imaginary system Σ( ϑ t, ϑ t ), with an additional input e(t). The second equation then reveals that e(t), however, is not an exogenous input as it is calculated based on the values taken by the signals u and y. Hence, 1 Indeed, let g i = sup t γ i ( ϑ t ). Since ϑ t converges, g i exists and is finite. Let also R such that r(t) R t. Then r (t) (g + g g nγ )R < +. 12

13 altogether these equations reveal the fundamental fact that the self-tuning scheme is nothing but the imaginary system in a feedback configuration as pictorially represented in Figure 1. This suggest the following steps to com- Figure 1: The self-tuning controller scheme is the imaginary system feedback connected with the perturbation system. plete the proof about the BIBO-stability and self-optimality of Σ(ϑ o, ϑ t ). 1. We first consider e(t) = and show that the imaginary system Σ( ϑ t, ϑ t ) (which is a linear time-variant system) is asymptotically stable for ϑ t generated by the RLS algorithm. 2. Thanks to the convergence property of the RLS estimate and the asymptotic stability of Σ( ϑ t, ϑ t ) we will show that u and y in (3) must be bounded whenever r is bounded (BIBO stability of Σ(ϑ o, ϑ t )). 3. Thanks to the boundedness of u and y and the convergence property of 13

14 the RLS estimate we can easily show that e(t) always. But then, by (3), we have that the output of Σ(ϑ o, ϑ t ) behaves as the output of Σ( ϑ t, ϑ t ) additionally fed by a signal which goes to zero. Since Σ( ϑ t, ϑ t ) is linear and asymptotically stable, this output will tend to the output of Σ( ϑ t, ϑ t ) when e(t) =, i.e. Σ(ϑ o, ϑ t ) asymptotically behaves like Σ( ϑ t, ϑ t ). Since ϑ t ϑ and by a continuity property, we have that the output of Σ( ϑ t, ϑ t ) will tend to the output of Σ( ϑ, ϑ ), so that we eventually have that Σ(ϑ o, ϑ t ) asymptotically behaves like Σ( ϑ, ϑ ). To ease the notation is better to work with state-space representations rather than I/O representation. Let then ξ(t + 1) = F ( ϑ t )ξ(t) + G( ϑ t )w(t) z(t) = H( ϑ t )ξ(t) + M( ϑ t )w(t) be a state space representation of (3a) where z(t) denotes the 2-dimensional signal y(t) and w(t) the signal e(t). Moreover, for simplicity we will u(t) r (t) write F (t), G(t), H(t), and M(t) in place of F ( ϑ t ), G( ϑ t ), H( ϑ t ), and M( ϑ t ). As is clear, ξ(t + 1) = F (t)ξ(t) is the autonomous part of the imaginary system Σ( ϑ t, ϑ t ), while (4) ξ(t + 1) = F ( )ξ(t) is the autonomous part of the asymptotic imaginary system. 14

15 3.1 Stability issues Being Σ( ϑ t, ϑ t ) and Σ( ϑ, ϑ ) linear, the stability depends on their autonomous parts only. Let consider ξ(t + 1) = F ( )ξ(t) first. As is clear since ϑ Ξ and, by definition of Ξ, we have that the matrix F ( ) is Hurwitz 2 so that ξ(t + 1) = F ( )ξ(t) (and hence Σ( ϑ, ϑ )) is asymptotically stable. This in particular means that c 1 and ρ (, 1) such that F ( ) t c 1 ρ t and ξ(t) c 1 ρ t ξ(). Turn now to the imaginary system ξ(t + 1) = F (t)ξ(t). In this case, since ϑ t Ξ, t, we still have that each matrix F (t) at various time instants is Hurwitz, but unfortunately in this time-varying case, this not not enough to guarantee that the system ξ(t + 1) = F (t)ξ(t) is asymptotically stable as shown by the counterexample given in Appendix B. On the other hand, we also have that ϑ t ϑ so that, thanks to continuity, we have that F (t) F ( ). This property, together with the fact that F (t) is Hurwitz t and that F ( ) is Hurwitz, guarantee indeed the sought result, i.e. that ξ(t + 1) = F (t)ξ(t) is asymptotically stable. This result is taken here for granted, while the interested reader may find a proof in the books on time-varying linear systems. Before moving to the next step, we need to remark that in view of the asymptotic stability c 2 and ν (, 1) such that t i=1 F (i 1) c 2ν t so that ξ(t) = t i=1 F (i 1)ξ() t i=1 F (i 1) ξ() c 2ν t ξ(). 2 That is, all the eigenvalues are strictly inside the unit circle in the complex domain 15

16 3.2 BIBO stability of Σ(ϑ o, ϑ t ) In this section, we want to prove that if r (t) M, t (i.e. r (t) is a bounded input), then the ouput of the self-tuning scheme z(t) keeps bounded too. Based on (4), we have that (Lagrange formula): [ ] t 1 t 1 t 1 z(t) = H(t) F (i)ξ() + H(t) F (i)g(τ) w(τ) + M(t)w(t). i= τ= i=τ+1 We have then t 1 t 1 z(t) = H(t) F (i)ξ() + H(t) i= + M(t) w(t) [ bounding k 1 = max t k 2 = max t 1 i=τ+1 τ= t 1 i=τ+1 F (i) with νt τ 1 and letting t 1 H(t) F (i)ξ(), { max t,τ i= k 1 + k 2 ν t τ w(τ). τ= H(t) G(τ) ν } ], max M(t) t F (i) G(τ) w(τ) + Moreover, recalling that w(τ) = e(τ) e(τ) and since r (τ) r (τ) = e(τ) 2 + r (τ) 2 e(τ) + r (τ), we have that: z(t) k 1 + k 2 ν t τ e(τ) + k 2 τ= ν t τ r (τ), τ= 16

17 which in turn, by noting that e(τ) = ϕ(τ 1) T ϑτ = ϕ u (τ 1) T ϑu τ + ϕ e (τ 1) T ϑe τ ϕ u (τ 1) T ϑu τ + ϕ e (τ 1) T ϑe τ, can be split as follows z(t) k 1 + k 2 ν t τ r (τ) + k 2 ν t τ ϕ u (τ 1) T ϑu τ + τ= τ= +k 2 ν t τ ϕ e (τ 1) T ϑe τ. (5) τ= Since r (t) M, t, we have that ν t τ r (τ) M τ= ν t τ M τ= τ= ν t τ = M 1 ν, and hence the second term in the sum keeps bounded. Similarly, since, as we proved previously, ϕ u (τ 1) T ϑu τ so that ϕ u (τ 1) T ϑu τ M, t, the third term keeps bounded too. Bounding the first three terms in (5) with a unique constant, we have that z(t) k 3 + k 2 ν t τ ϕ e (τ 1) T ϑe τ τ= [by means of the Cauchy-Schwarz inequality] k 3 + k 2 ν t τ ϕ e (τ 1) ϑ e τ. (6) τ= 17

18 Now, let consider the regression vector ϕ(t). We have that y(t) y(t 1) z(t). z(t 1) ϕ(t) = y(t n) z(t) + z(t 1) + + z(t q),. u(t + 1 d) z(t q). u(t + 1 d m) where q = max{n, d+m 1}. Using repeatedly, for each term in the previous sum, the bound in (6), we obtain ϕ(t) k 4 + k 5 ν t τ ϕ e (τ 1) ϑ e τ τ= k 4 + max i t 1 ϕe (i) k 5 ν t τ ϑ e τ τ= [ ϕ e (i) ϕ e (i) + ϕ u (i) = ϕ(i) ] k 4 + max 5 ν t τ ϑ e τ. i t 1 (7) τ= Suppose now that ϕ(t) does not keep bounded. Since, as we proved in a previous section, ϑ e τ, it turns out ν t τ ϑ e τ, τ= when t. Indeed, t τ= νt τ ϑ e τ is nothing but the motion of an asymptotically stable system (x(t + 1) = νx(t) + ϑ e τ ) with vanishing input. 18

19 Hence, a t can be founded such that, simultaneously: k 5 max ϕ(i) 2k 4 i t ν t τ ϑ e τ 1 2, t t τ= (since ϕ(t) is not bounded) (since the convergence to zero of this term). Plugging these two properties in (7), we have that (note that max i t ϕ(i) 2k 4 k max i t): Moreover, ϕ( t + 1) k 4 + max ϕ(i) k 5 i t 1 2 max i t τ= ν t τ ϑ e τ ϕ(i) max ϕ(i) i t max ϕ(i). (8) i t ϕ( t + 2) k 4 + max ϕ(i) k 5 i t+1 t+1 ν t+1 τ ϑ e τ τ= 1 2 max ϕ(i) + 1 i t 2 max ϕ(i) i t+1 [ thanks to (8) ] 1 2 max i t max ϕ(i). i t ϕ(i) max ϕ(i) i t Proceeding recursively for t + 3, t + 4,... it can be shown that ϕ(t) max ϕ(i), t > t, i t contradicting the assumption that ϕ(t) was unbounded. Hence, eventually, we proved that ϕ(t) must keep bounded, and this implies that z(t) is bounded, i.e. Σ(ϑ o, ϑ t ) is BIBO stable. 19

20 3.3 Final derivations We have that e(t) = ϕ(t 1) T ϑt ϕ e (t 1) T ϑe t + ϕ u (t 1) T ϑu t. Since and ϕ u (t 1) T ϑu t ϕ u (t 1) T ϑu t leq ϕ e (t 1) ϑ e t (because ϕ e (t 1) keeps bounded and ϑ e t ), we obtain that e(t) when t. In conclusion, getting back to Figure 1, since Σ(ϑ o, ϑ t ) is equal to Σ( ϑ t, ϑ t ) additionally fed by e(t) which goes to zero and since Σ( ϑ t, ϑ t ) is linear and asymptotically stable, the output of Σ(ϑ o, ϑ t ) will tend to the output of Σ( ϑ t, ϑ t ), and by a continuity property, this latter will tend to the output of Σ( ϑ, ϑ ). This means that i.e. Σ(ϑ o, ϑ t ) is self-optimal. lim t y(t) y i (t) =, A Appendix: a counterexample Consider S(t) = 1 + t2 t 3, ϑ = 1, and ϑ t = S(t) 1 S() ϑ. t t 4 2

21 We have that ϑ t = 1+t 4 t 3 1+t 2 +t 4 1+t 2 +t 4 t 3 1+t 2 1+t 2 +t 4 1+t 2 +t 4 Simple calculations show that ϑ T t S(t) ϑ t = 1 = 1+t 4 1+t 2 +t 4 t 3 1+t 2 +t 4 t 1 + t4 1 + t 2 + t 4 t 1, 1 = ϑ. while ϑ T S(t) ϑ = 1 + t 2 which clearly tends to + as t increases. Note however that S(t) is not monotonically increasing, i.e. it is not true that S(t + 1) S(t) (hence this S(t) cannot be generated according to the rule S(t + 1) = S(t) + ϕ(t)ϕ(t) T ). This is clear even considering S(2) S(1) = = whose determinant is equal to = 4 (a positive semi-definite matrix has determinant always greater than or equal to ). B Appendix: another counterexample Consider the 2-dimensional system ξ(t + 1) = F (t)ξ(t), where the matrix F (t) obeys the following law: F 1 = if ξ 1 (t)ξ 2 (t) F (t) = F 2 = else

22 The systems ξ(t + 1) = F 1 ξ(t) and ξ(t + 1) = F 2 ξ(t) are both asymptotically stable and their free motions is depicted in Figure 2 (first two sub-figures). As it appears they correspond to convergent spirals elongated on the x-axis Figure 2: Motion associated to F 1 (first sub-figure), to F 2 (second sub-figure), and to F (t) (third sub-figure). and on the y-axis respectively. When F 1 and F 2 are combined together in F (t), however, it happens that we switch from one dynamics to the other exactly when we reach the point of maximum elongation according to the current dynamics. In other words, switching is such that it is not given enough time to the dynamics of F 1 or F 2 to bring the state close the origin. Rather, the state is driven further and further away, and an unstable behavior is obtained. This can be appreciate again in Figure 2 (third sub-figure) where the motion of ξ(t + 1) = F (t)ξ(t) is depicted. Perhaps, it is worth remarking that the unstable behavior is obtained proper because of switching. F (t) continues to oscillate between F 1 and F 2 and F (t) does not converge to any matrix when t. 22