Feedback rectifiable pairs and stabilization of switched linear systems

We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.


Introduction
Switched systems constitute a special kind of hybrid dynamical systems that consist of a family of subsystems with a switching signal that orchestrates between them.Such systems appear for instance in modelling of mechanical systems, electric power systems, communication networks and intelligent control systems with logic-based controllers [Lib03,SG05,SG11].In particular the dynamics of the switched system may be discontinuous at the switching instances.As such, the dynamics of the overall switched system are much more complex and may exhibit very different dynamical behaviour as can be found in the constituent subsystems.
For more than two decades the stability of switched linear systems has been subject of inspiring research [LM99, DBPL00, SWM + 07, LA09] and monographs [Lib03,SG05,SG11].However, constructive algebraic conditions for the stability of switched systems and thereby also efficient design-procedures that refrain from numerical tools remain scarce.
In this paper we consider switched linear systems that switch autonomously in an arbitrary fashion, i.e. the switching is not restricted in neither time-domain nor statespace.This work is dedicated to the study of feedback control design using eigenstructure assignment.Such design method is well-established for linear time-invariant systems [Moo76,Rop90,LP98,RAS09] and has already seen some successful transfer to switched linear systems in terms of pole-placement design [WWS05,WWS09] and eigenstructure assignment approaches [BFK13,BF15,HB13,Hai16,WHR21].In [BFK13,BF15] subsystems that share a common right invariant eigenspace are considered and assign the left eigenspace of the closed-loop subsystems to achieve stability for arbitrary switching.In [HB13,Hai16] simultaneous triangularization of the subsystems is achieved for subsystems with input matrices with transverse subspaces.Both restrictions have been dropped in the approach presented in [WHR21], where a simultaneous eigenstructure assignment for two third-order subsystems is proposed.Thereby the closed-loop subsystems are brought simultaneously similar to upper triangular form which ensures the existence of a common quadratic Lyapunov function for the subsystems [MMK98,SN98] and hence, the closed-loop system is asymptotically stable for arbitrary switching [Lib03].
In this contribution we generalize the results from [WHR21] for switched systems with two subsystems of arbitrary order.The following section gives a formal problem statement in the context of the stabilization of switched linear systems.We recall the seminal result of Moore [Moo76] which gives necessary and sufficient conditions for the eigenstructure assignment problem for a single linear time-invariant system.Our approach exploits this result in order to rectify the eigenstructure of two subsystems.To this effect we introduce the notion of feedback rectifiable pairs, which states the existence of a feedback for every subsystem with the property that the closed-loop subsystems have the same eigenvectors; for details see Definition 3.1 below.It turns out that the eigenspaces of two subsystems may only be rectified if the assigned eigenvalues satisfy certain conditions, see Example 3.6.In Proposition 3.3 we formulate necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems.Essentially the intersection of two subspaces determined by the subsystems has to span the whole space in order to allow for an arbitrary assignment of the eigenvalues.This subspace can be described by a polynomial matrix as elaborated in Section 4. The rectifiability can be characterised via properties of a polynomial matrix in Theorem 4.5 and Theorem 5.3.Further, we propose a constructive procedure to calculate the polynomial matrix in Corollary 5.2.With this characterization at hand, the eigenvalues and eigenvectors for the closed-loop subsystems can be chosen and the respective feedback matrices that stabilize the switched system for arbitrary switching are readily obtained, see Algorithm 5.4.Some notation used: is the matrix that is obtained from C by deleting the i-th row and the j-th column.The complex conjugate of a complex number λ is denoted by λ.For a matrix A ∈ R n×n we denote by σ(A) ⊂ C its set of eigenvalues.
2 Preliminaries and problem definition.

Problem definition
We consider the switched system that is composed by the constituent linear time-invariant (LTI) systems with A q ∈ R n×n , B q ∈ R n×p and q ∈ I = {1, 2}.The column rank of the input matrix B q is denoted by m q := rk B q ≤ p and without loss of generality we assume that These subsystems constitute the switched system where the switching signal s : R + → I is a piecewise constant function with discontinuities t j satisfying t j+1 − t j > τ > 0 for all j ∈ N and some τ > 0. Note that the system and input matrices of (2) switch simultaneously such that at any given time t one of the subsystems (1) is active.We consider the asymptotic stability of the origin of the homogeneous solution of (2) for arbitrary switching signals of the above characteristics.It is well known that asymptotic stability of the subsystems (1) does not imply asymptotic stability of the switched system for arbitrary switching.Instead the existence of a common Lyapunov function for the subsystems is necessary and sufficient for the asymptotic stability for arbitrary switching [LM99, DBPL00, SWM + 07].
This paper aims at the control synthesis problem for the switched system (2).Assuming the current value s(t) of the switching signal is known at all times we consider the feedback law with F s(t) ∈ R p×n .The resulting closed-loop control system again constitutes a switched linear system and takes the form The control problem considered in this contribution is to design feedback matrices F q ∈ R p×n , q ∈ I for a given switched system (2) such that the closed-loop system (4) is asymptotically stable for arbitrary switching.
Our approach to design stable switched systems is inspired by the seminal results for the stability of switched systems for arbitrary switching [MMK98,SN98].
Theorem 2.1.Given the switched system (2) with A q Hurwitz, q ∈ I.If there exists a T ∈ C n×n , regular, such that T −1 A q T is in upper triangluar form for all q ∈ I then the origin of (2) is asymptotically stable for arbitrary switching and there exists a common quadratic Lyapunov function.
Note that if two matrices commute, i.e.A 1 A 2 = A 2 A 1 holds, then they are simultaneously triangulizable.More precisely, see [DDG51], simultaneous triangularisation holds true if and only if for every polynomial p in two variables the matrix A special case for which Theorem 2.1 can be applied is when A 1 and A 2 share the same set of n linearly independent eigenvectors.
Since the above structural assumptions rarely hold for practical control applications, we aim to shape the eigenspaces of the individual subsystems via state feedback.In particular, we aim to find feedback matrices F q for a given switched system (2) such that the closed-loop system matrices in (4) have eigenvalues with negative real parts and are in upper triangular form.

Preliminary results
For our design of the switched controller (3) we shall exploit the seminal result by Moore, [Moo76], that characterizes the eigenvalue and eigenvector assignment for a single closedloop LTI system with (A, B).Consider N (λ) and M (λ) satisfying Using this notion we have the following, [Moo76].
Theorem 2.2.Given (A, B) ∈ R n×n × R n×p and the self-conjugate set Λ = {λ 1 , . . ., λ n } with distinct entries.Then there exists F ∈ R p×n such that (A + BF )v i = λ i v i , v i = 0, for all i ∈ {1, . . ., n} if and only if the following three conditions are satisfied: where v and λ denote the complex conjugate of v and λ, respectively.
Remark 2.3.In Theorem 2.2 the vectors v i are eigenvectors to the eigenvalues λ i and, as the entries of Λ are distinct, the v i are automatically linear independent.Clearly, Theorem 2.2 still holds under the assumption that the matrix A+BF is diagonalizable (that is, multiple eigenvalues are allowed but no generalized eigenvectors), see the remarks after Proposition 1 in [Moo76].Moreover, there exists also a version of Theorem 2.2 without any assumptions on the Jordan structure of the matrix A + BF , that is, arbitrary Jordan chains are allowed, see [KM77].
Remark 2.4.Recall that (A, B) is called controllable if rk[λI −A, B] = n holds for all λ ∈ C, which implies that the kernel in (5) is of dimension p for all λ ∈ C.However, controllability is not mentioned in Theorem 2.2.The equivalence of controllability and pole assignability implies that the three conditions in Theorem 2.2 cannot be satisfied if the uncontollable eigenvalues are not included in the selected set Λ of closed loop eigenvalues (i.e., the numbers λ i in Theorem 2.2), see [Moo76].
Without loss of generality we shall assume in the derivation of our main results that all eigenvalues are controllable.For a discussion on uncontrollable modes see Section 6.
In any case the choice of N (λ) and M (λ) is unique up to multiplications with invertible matrices from the right.If λ ∈ C \ σ(A), then a possible choice of the matrices in (5) is the following where we have However, (6) is not valid for λ ∈ σ(A), consider e.g.A = B = 0, then adj(λI In what follows, we shortly recall the main ideas of the proof of Theorem 2.2.Assume that for some F ∈ R p×n and for {v 1 , . . ., v n } and {λ 1 , . . ., λ n } as in Theorem 2.2 the following holds Then this implies Therefore v i ∈ im N (λ i ) which shows the third condition in Theorem 2.2.The first two conditions follow directly from elementary matrix theory.
Conversely, if we have linearly independent vectors v i with v i = N (λ i )w i for w i ∈ R p , we can choose the feedback matrix F via which shows (A + BF )v i = λ i v i .It remains to show that F can be chosen to be a real matrix, for this we refer to [Moo76].

Feedback rectifiable pairs of matrices
Our goal is to design feedback matrices F q such that the closed-loop subsystems (4) are in upper triangular form and exhibit eigenvalues with negative real parts.In this section we investigate conditions for which two subsystems can be brought into upper triangular form using state-feedback.This motivates the following notion.
Note that, contrary to the situation in Theorem 2.2, multiple eigenvalues are allowed, see also Remark 2.3.
With this notion together with Theorem 2.1 we readily obtain the following result [WHR21].
Theorem 3.2.The switched system (2) is stabilizable for arbitrary switching via statefeedback (3) if the pairs (A 1 , B 1 ) and (A 2 , B 2 ) are feedback rectifiable over a set D that only contains elements with negative real parts.
In order to achieve recifiability together with stabilization a sufficient number of parameters have to be available for the control design [WHR21], which yields the following condition Furthermore let for some are feedback rectifiable over D, then (11) together with (9) implies Hence, we have span(( where we set Moreover, we have the following characterization of feedback rectifiability which generalizes the result for n = 3 from [WHR21, Theorem 4]. Proposition 3.3.Consider the pairs (A q , B q ) ∈ R n×n × R n×p , q = 1, 2. Then the following holds: (c) The pairs are feedback rectifiable over D ⊆ C×C if there exists {(λ 1 , µ 1 ), . . ., (λ n , µ n )} ⊆ D satisfying the following conditions: ) for all i = 1, . . ., n such that {v 1 , . . ., v n } is linearly independent; (ii) v i = v j holds whenever λ i = λ j , furthermore, in this case we can choose without restriction µ j = µ i .
Proof.(a) Assume that span(( Then the assumption implies that there exist α i,j ∈ R with i = 1, . . ., n, j = 1, . . ., k i as well as For the construction of feedback matrices F q ∈ R p×n , q = 1, 2, we can apply the construction proposed in [Moo76].By the linear independence of its columns, Since V −1 v i equals the i-th canonical unit vector the feedback matrices satisfy (11) as A similar calculation shows (A 2 + B 2 F 2 )v i = µ i v i which implies (11).The remaining statement of (a) and the statement in (b) follow from the reasoning (14) preceding this proposition, which is true for both real and complex numbers.Assertion (c) follows from Theorem 2.2 and Remark 2.3 applied to each system (A 1 , B 1 ) and (A 2 , B 2 ).
Remark 3.4.Note that Theorem 2.2 cannot be directly applied to prove the converse of Proposition 3.3 (b).If we construct a complex-valued basis {v 1 , . . ., v n } ⊆ C n as in the proof of Proposition 3.3 (a) then this might lead to set which does not contain pairs of complex conjugate vectors.
From Proposition 3.3 (b) we have that the pairs (A q , B q ) are not feedback rectifiable over D if the intersection N 1 (λ) ∩ N 2 (µ) is empty for all λ, µ ∈ D. Therefore we obtain with (6) the following result.
In the following low-dimensional example we consider feedback rectifiability with respect to different underlying sets D ⊂ C × C.
For λ, µ ∈ C \ {−1, 1} we obtain the upper parts of the kernels according to (6) and ( 7) Hence we have co-linearity of N 1 (λ) and N 2 (µ) for µ = −λ −1 .It can be verified using ( 18) and ( 19) that the intersection for all λ ∈ C \ {−1, 1, 0} is given by which yields linearly independent vectors.Hence, whenever the set Example 3.6 shows that rectified eigenstructures may only be achieved with some additional condition for the eigenvalues of the closed-loop systems, e.g.
and therefore it is not possible to choose both closed-loop eigenvalues, λ i and µ i , to have negative real-parts.Hence the stabilization of the switched system with rectified eigenstructure may not be possible.

Characterization of feedback rectifiability
In this section, we apply Proposition 3.3 to develop conditions for which two pairs of matrices (A 1 , B 1 ) and (A 2 , B 2 ) are feedback rectifiable.To this end we derive characterizations such that (N 1 ∩N 2 )(D) contains a basis of R n or C n .We compute the intersection of im N 1 (λ) and im N 2 (µ) along the lines from [POH99,Yan97].The following lemma is a slight generalization (see also Remark 4.2).
Proposition 4.1.Let F be an arbitrary field and consider the subspaces K = im K and L = im L for some K ∈ F n×r1 and L ∈ F n×r2 .Then there exists an invertible R ∈ F n×n satisfying and the intersection of the subspaces is given by Consider r 3 := rk L 2 .If r 3 = 0 then the following holds Moreover, if r 3 = r 2 > 0 then K ∩ L = {0} holds.If 0 < r 3 < r 2 holds then there exist invertible R ∈ F n×n and V ∈ F r2×r2 so that we can further decompose (20) as follows r3) .Furthermore, the following holds Proof.Applying Gaussian elimination there exist an invertible matrix R and matrices K 1 , L 1 , and L 2 such that (20) holds.As K 1 has full row-rank it is surjective.Therefore the following holds This together with the invertibility of R yields which shows (21).If r 3 = rk L 2 = 0 holds, then ker L 2 = R r2 and therefore (21) implies (22).If r 3 > 0 then applying Gaussian elimination to the rows of L 2 , we obtain for some invertible W ∈ F (n−n1)×(n−n1) and some invertible V ∈ F r2×r2 with rk L 21 = rk L 2 = r 3 .Here L 21 ∈ F r3×r3 is chosen to be square and therefore invertible.
Defining R := This together with a repetition of the steps in (24) proves (23).
Remark 4.2.In [POH99,Yan97] the invertible transformation R in Proposition 4.1 is chosen in such a way that [RK, RL] is in reduced row echelon form.This is a particular staircase form resulting from Gaussian elimination where the leading entries in each row are equal to one.The columns corresponding to such leading entries are unit vectors, i.e. all of their other entries are equal to zero.In particular, we have L 11 = 0 and L 21 = I r3 in (23).Hence, if R is chosen in such a way that [ RK, RL] is in reduced row echelon form then the intersection (23) is given by In the following, we apply Proposition 4.1 to compute the intersection of the parameter dependent subspaces im N 1 (λ) and im N 2 (µ) given by ( 6) and (7).To this end, we consider the quotient field R(λ, µ) of the ring R[λ, µ] of polynomials p(λ, µ) = I i=0 J j=0 p ij λ i µ j with real-valued coefficients p ij ∈ R, see e.g.[Lan13, Section IV. §7], i.e. the quotient field R(λ, µ) simply contains all rational functions in two variables with real-valued coefficients.Recall that a rational matrix-valued function Q(λ, µ) ∈ R(λ, µ) n×m is said to have a pole at (λ, µ) if one of its entries has a pole at (λ, µ).We shall denote the set of poles by P(Q).By applying Proposition 4.1 to F = R(λ, µ) in combination with (6) and (7) we obtain the following result.
Proof.If α ∈ C n \ {0} fulfills the assumption in (a) then α is orthogonal to sp G (P ).Hence dim sp G (P ) < n.We continue with the proof of (b).If dim sp G (P ) < n then there exists α ∈ C n such that α ⊤ P (λ, µ) = 0 holds for all (λ, µ) ∈ G. Since G is dense in C × C, for all (λ, µ) ∈ G there exists a sequence (λ l , µ l )) l∈N in G converging to (λ, µ).Then the continuity of P implies for all (λ, µ) ∈ G Below we present the first main result.
If the pair is feedback rectifiable over Ω R then Proposition 3.3 (a) and (30) lead to

As shown above, this implies (31).
If we combine Theorem 4.5 with the fact that feedback rectifiability over Ω R also implies feedback rectifiability over the larger set Ω, and, by applying Proposition 3.3 (b) we obtain the following necessary and sufficient condition.
Corollary 4.6.Let A q ∈ R n×n , B q ∈ R n×mq , q = 1, 2 and let Ω by given by Proposition 4.3 and assume that P ∈ R[λ, µ] n×r is as in (29).Then the pairs (A q , B q ) are feedback rectifiable over Ω if and only if (31) holds.
Note in particular, if (31) does not hold for the set Ω R , then the pairs are also not feedback rectifiable over any extension Θ ⊃ Ω R .

Construction of the feedback rectification
In this section we consider pairs of matrices for which holds for some points (λ, µ) ∈ C × C. We obtain an explicit formula for the intersection N 1 (λ) ∩ N 2 (µ) which allows us to specify a set D for rectification and to formulate a constructive algorithm for the design of the feedback matrices.
In particular the consider pairs of matrices satisfying the following assumption.
Assumption 5.1.Let A q ∈ R n×n , B q ∈ R n×mq , q = 1, 2 and let Θ ⊂ C × C. We assume that condition (33) holds for all (λ, µ) ∈ Θ. Define Without loss of generality we can reorder the columns of B 1 and B 2 such that N (λ, µ) is invertible for all (λ, µ) ∈ Θ, if condition (33) holds.The matrix N (λ, µ) in ( 34) is used in the following corollary to compute the intersection of im N 1 (λ) and im N 2 (µ) based on Remark 4.2.
we have L 12 = 0 and the following holds Proof.For (λ, µ) ∈ Θ for which Assumption 5.1 holds, we obtain the following estimate As a consequence, there exist nonzero x, y ∈ R p fulfilling N 1 (λ)x = −N 2 (µ)y.Since rk B 1 = p = m 1 holds, we have that N 1 (λ)x = 0 holds which proves (a).
To prove (b), we use the definition of the inverse and Assumption 5.1 which implies Hence, the intersection is given by Note that in Corollary 5.2 the point (λ, µ) ∈ C × C is fixed whereas in Proposition 4.3 the intersection of N 1 (λ) and N 2 (µ) is determined as a function of λ and µ on a dense set Ω which excludes λ ∈ σ(A 1 ) and µ ∈ σ(A 2 ).Hence, the set Θ may contain eigenvalues of A q if N q is not constructed using the adjugate as in (6).
Next, we use Proposition 3.3 (c) to obtain a sufficient condition for feedback rectifiability over Θ ⊂ C × C.
With the previous results and the construction of the intersection (35) at hand we obtain the following procedure for the design of rectifying feedback.
Algorithm 5.4.If the pairs (A 1 , B 1 ) and (A 2 , B 2 ) are feedback rectifiable and satisfy Assumption 5.1 then the feedback matrices F q satisfying equation (11) can be constructed as follows.
Step 4 Choose the eigenvalues (λ i , µ i ) ∈ Θ for both closed loop subsystems.
Step 6 The parameter vectors w 1i ∈ R p in equation ( 16) are obtained by Step 7 The parameter vectors w 2i for the second system can be found by solving Step 8 Calculate the feedback matrices F 1 and F 2 as in (17).
The following example illustrates the use of the results obtained in this section.We consider the stabilization of a fourth-order switched system with three inputs and two modes.In order to calculate the intersection we resort to the row reduced echelon form utilized in Corollary 5.2.The polynomial matrix representation (29) is used to verify the recifiability of the two pairs via Theorem 5.3.The procedure given by Algorithm 5.4 is then applied to choose the eigenvectors and obtain the feedback matrices of the control law (3).
Note, that this representation is only valid for λ, µ = 0.
where ⋆ denotes the repetition of the second last column.
Identifying the upper m 1 = 3 entries of the last two columns in the above matrix with L 12 in (35) we obtain the intersection by Note, that the latter is a representation of Q(λ, µ) in (26).Accordingly the set of poles P(Q) consists of the roots of the polynomial (2λ − µ) which are excluded from the set Θ. Note that a different representation of the kernels or computation of their intersection may possibly yield a different set for feedback rectifiability.
The polynomial matrix in (28),(29), respectively, is given by with The matrix C 03 C 04 C 12 C 13 C 14 has full rank.Thus by Theorem 4.5, the two pairs are feedback rectifiable over Since Ω R ⊂ Θ we have feedback rectifiablity over Θ.It follows immediately, that the switched system (2) is stabilizable via rectification, c.f. Theorem 3.2.Note that the obtained set Ω R may not contain all pairs in R × R for which the rectifiability condition (11) can be satisfied, see Example 5.6.Next we design the control law (3) that stabilizes the switched system.Therefore we constructed the feedback matrices F q from equation (11) using Algorithm 5.4.
We choose the eigenvalues of the closed loop system (4) as λ 1 = −3, λ 2 = −1, λ 3 = −2, λ 4 = −4 for mode q = 1 and µ 1 = −1, µ 2 = −3, µ 3 = −2 and µ 4 = −4 for mode q = 2.The assignable eigenvectors v i can be chosen from im P (λ i , µ i ).Substituting (λ i , µ i ) into (38) yields Note that due to the dimensions of the system the eigenvector for each pair (λ i , µ i ) is uniquely defined.For higher-order system with larger number of inputs the intersection space for each pair of eigenvalues may be increased, such that P (λ i , µ i ) in (38) has several columns.In such case the eigenvector associated with a pair of eigenvalues can be chosen from a set parameterized by the vector c i ∈ dom P (λ i , µ i ).
Using this vector we determine the parameter vectors w 1i in (16) for subsystem q = 1 By solving v i = N 2 (λ 2 )w 2i for the second system we obtain the parameter vectors Note that the last two columns of N 2 (µ) are linearly dependent and therefore the last column does not provide any degree of freedom.As a result, the last entry of each w 2i is calculated to zero.
Finally, the feedback gains (10) for each subsystem are calculated to providing the closed loop system matrices A q + B q F q , q = 1, 2, with the eigenvalues and eigenvectors chosen above.Note again that the last column of the input matrix B 2 does not provide any further degrees of freedom.This results in the zero row of the feedback matrix F 2 .In fact, F 2 includes a standard input transformation which leads to an input matrix with full rank.
Example 5.6.The following example demonstrates that even though the system is not feedback rectifiable by constructing the intersection by (35) of Corollary 5.2 it might be feedback rectifiable over another set D p .
Consider the systems (A 1 , B 1 ) and (A 2 , B 2 ) given by Choosing the approach of Corollary 5.2, the intersection can be computed with the transformation matrix with Following the Algorithm 5.4 we obtain the intersection im N In this case Q(λ, µ) coincides with P (λ, µ) in (28).Obviously Condition (31) of Theorem 4.5 is not satisfied and thus it is not possible to select n pairs (λ i , µ i ) to generate n linearly independent eigenvectors from the obtained intersection.Now we consider the kernel with the following restriction for the eigenvalues to be chosen: µ i = λ i .We consider the new set Note that N (λ, µ) in (39) cannot be used to compute the intersection for (λ, µ) ∈ D p .But we can compute the intersection im N 1 (λ) ∩ im N 2 (µ) via the reduced row echelon form of .
Thus we obtain As the C i above satisfy (31), the given pairs are feedback rectifiable over D p .In fact, (11) can be satisfied choosing any four distinct pairs (λ i , µ i ) ∈ D p and thus the switched system is indeed stabilizable via feedback rectification.Since dim im P (λ, λ) = 3, the intersection provides a three-dimensional space of each λ to choose the eigenvector from.For λ 1 = µ 1 = −1 the corresponding eigenvectors can be chosen as a linear combination of columns in

A note on uncontrollable modes
The procedure proposed in Section 4 can be applied to controllable as well as uncontrollable modes of the subsystems (A q , B q ).Of course the uncontrollable eigenvalues have to be eigenvalues of the closed loop system A q +B q F q and therefore cannot be re-assigned.However the corresponding eigenvector is not affected and can be chosen within the limits of Theorem 5.3.In fact, the dimension of the kernel from which the eigenvector for the single subsystem can be chosen increases, as can be seen as follows.
Consider (A 1 , B 1 ) with the unique uncontrollable eigenvalue λ u such that where ν(λ u ) denotes the geometric multiplicity of λ u , i.e. the dimension of the eigenspace associated with the so-called uncontrollable mode λ u .Then dim ker λ u I − A 1 B 1 = p + ν(λ u ).
Note that the dimension of the kernel at λ = λ u is larger than the kernel for all other λ ∈ C. Therefore we describe the upper part of the kernel associated with the uncontrollable mode using N u 1 (λ u ) ∈ R n×p+ν(λu) whereas for all λ = λ u it is described by N 1 (λ) ∈ R n×p .This increase of dimension, however, does not cause any further restrictions on the rectifiablility of the pairs (A 1 , B 1 ), (A 2 , B 2 ) as the intersection of the spaces in Proposition 3.3 increases as the following example shows.For the controllable eigenvalues we have Observe, that both subsystems have a zero row in the kernel matrices N q corresponding to the uncontrollable mode and thus their intersection will not span the first component in R 3 .For the uncontrollable modes λ u = µ u = −1 we have As both, N u 1 (λ u ) and N u 2 (µ u ), span R 3 their intersection still spans R 3 .Since the intersection of the controllable eigenvalues lacks the first component, we need to pair the uncontrollable eigenvalues λ u and µ u and assign an eigenvector with non-zero first component.

Conclusion
In this contribution we generalize results for switched systems of size at most three to subsystems of arbitrary order.Our approach discusses how to rectify the eigenstructure of two subsystems.To this effect we introduce the notion of feedback rectifiable pairs, which states the existence of a feedback for every subsystem with the property that the closed-loop subsystems have the same eigenvectors.It turns out that the eigenspaces of two subsystems may only be rectified if the assigned eigenvalues satisfy certain conditions.The formulated necessary and sufficient conditions essentially require that the intersection of two subspaces determined by the subsystems has to span the whole space.Moreover, we show that this subspace can be described by a polynomial matrix.We propose a constructive procedure to calculate this polynomial matrix.With this characterization at hand, the eigenvalues and eigenvectors for the closed-loop subsystems can be chosen and the respective feedback matrices that stabilize the switched system for arbitrary switching are readily obtained.The proposed methodology is not restricted to two subsystems, but for a larger number of subsystems, i.e. q ≥ 3, the complexity explodes.
e. rk λI − A B = n for all λ ∈ C, then t he kernel can be described by matrices of dimension N (λ) ∈ R n×p and M (λ) ∈ R p×p with rk N (λ) = rk B = m and rk M (λ) = p.For uncontrollable eigenvalues the kernel increases (see Section 6).

Example 6. 1 .
Consider the System with uncontrollable but stable eigenvalue λ u = µ u = −1 in each mode with