Eigenstructure Assignment Method and Its Applications to the Constrained Problem

A partial eigenstructure assignment method that keeps the open-loop stable eigenvalues and the corresponding eigenspace unchanged is presented. This method generalizes a large class of systems previous methods and can be applied to solve the constrained control problem for linear invariant continuous-time systems. Besides, it can be also applied to make a total eigenstructure assignment. Indeed, the problem of finding a stabilizing regulator matrix gain taking into account the asymmetrical control constraints is transformed to a Sylvester equation resolution. Examples are given to illustrate the obtained results.


Introduction
Eigenstructure assignment method plays a capital role in control theory of linear systems.The state feedback control law is used in this end, leading to change eigenvalues or eigenvectors of the open loop to desired ones in the closed loop.This method is usually realized in order to perform optimal or stabilizing control laws ([1]- [9]).These articles and the references therein constitute a comprehensive summary and an important bibliography on the control of linear systems with input saturation.Indeed, authors have used different concepts leading to many methods for eigenstructure assignment control laws to regulate linear systems with input saturation.
Throughout this paper, we will be interested in continuous time systems of the form ( ) ( ) ( ).
x t Ax t Bu t The matrices A and B are real and constant: x t ∈  represents the state vector of the system and ( ) m u t ∈  is the control vector.We suppose that the spectrum of the matrix A contains n m − desirable or stable eigenvalues 1 , , and m undesirable or unstable eig- envalues 1 , , m λ λ


. We also suppose that the pair ( ) A B is stabilizable.Presence of undesirable eigenvalues makes System (1) unstable and, in [4] or in [10], methods to overcome the instability of System (1), keeping unchanged the open-loop stable eigenvalues and the corresponding eigenspace s  and replacing the remaining undesirable eigenvalues by other chosen values, were given.But, in these methods, additional conditions on System (1) should be satisfied.
In this paper, we try to get rid of these additional conditions on System (1).First, we should give some outlines on the methods described in [4] or in [10].In [4], the method, called the inverse procedure, consists of giving a matrix with some desirable or stable spectrum and then computing, when possible, a full rank feedback matrix such that: ( ) and the kernel of F is the stable subspace.In this case, the spectrum of the matrix A BF + is stable since it is constituted by the desirable eigenvalues of A and the chosen spectrum of H.In other words, the eigenvalues of H will replace, in the closed-loop, the undesirable eigenvalues of A. With the change of variables ( ) ( ) which allows one to focus on the unstable part of this system.The inverse procedure described in [4] ensures the existence of a matrix F that satisfies the conditions mentioned above and gives a way to compute it under the following conditions: (a) The matrix H is diagonalizable; that is, there exist linearly independent eigenvectors 1 , , m θ θ The endomorphism induced by the matrix A on the stable subspace is diagonalizable; that is, there exist linearly independent eigenvectors 1 , , The spectrum of A and the spectrum of H are disjoint.(d) The matrix ( ) ( ) Under these conditions and according to the method described in [4], the matrix F is unique and is given by , , ,0 , ,0 where 0 i is the null vector of n  .In [10], the condition b) is kept and a new assumption should be fulfilled: (d').the matrix is invertible.The method described in [10] consists of computing a matrix V such that its rows span the orthogonal of the stable subspace by taking, for example the first m rows of 1 N − and then, for a stable matrix Then, according to the method described in [10], the feedback matrix F is given by F KV = where K is the inverse matrix of the solution of the Sylvester equation , and In this paper, we generalize the method described in [10] to a more general class of systems for which the condition d') is not necessary.In fact, no additional condition is needed to deal with System (1).As in [10], the feedback matrix F is always given by KV where K is an invertible matrix of m m ×  such that 1 X K − = is the solution of the Sylvester Equation (5).
The methods described in [10] or in [4] are, in fact, partial pole placement methods in which the desirable eigenvalues of the matrix A are kept in the closed-loop.But, it may happen that these desirable eigenvalues are close to the imaginary axis which causes a slow convergence rate to the origin.To overcome this problem, a total pole placement is needed.The technique of augmentation (see [5] and [6]) allows one to perform a total pole placement when the matrix B needs not be of full rank.This is possible with the inverse procedure [4], but not with the method described in [10] since, under the condition 1, the matrix B is of full rank.
So, from one hand, the method that we present constitutes a generalization of the two methods described in [10] or in [4] without any other additional condition on the systems and, from another hand, allows one to make, if necessary, a total pole placement by the use of the augmentation technique.
The paper is organized as follows: In Section 1, definitions, notations and some known facts are presented to be used in the sequel.The main results are presented in Section 2 together with an illustrative example.Some particular cases are presented in Section 3. Section 4 is devoted to the total eigenstructure problem with the illustrative example of the double integrator.• If M is a p q × real matrix, for some * , p q ∈  , its transpose, the q p × real matrix, will be denoted by T M .• When we view n  as an euclidean space, the usual inner product ( )

Preliminaries
where ( ) ( )  , its orthogonal will be denoted by S ⊥ , that is, will denote the coefficient of M corresponding to the ( ) • For real matrices (or real vectors) M and N , we say that M is less than or equal to N if every component of M is less than or equal to the corresponding component of N .We then write  spanned by the columns of M and also the set of vectors of the form MX with Re λ is the real part of λ .

Some Notes on the Stable-Unstable Subspaces
1. To System (1), we associate: (a) The two polynomials ( ) ( ) ( ) In case of m n = , polynomial Q is just the constant polynomial 1.
(b) The two subspaces of n  ; ( ) ( ) In case of m n = , the subspace s  is just the trivial subspace  and s  are complementary subspaces of n  .Moreover, we have ( ) ( )

Im and Im
and also Im ker and Im ker .

Since pair ( )
, A B is stabilizable, we have ( )

Some Notes on Sylvester Equation
Many problems in analysis and control theory can be solved using the well known of Sylvester equation.This equation is widely studied or used in the literature ([10]- [15]).Since Sylvester equation plays a central role in the development of this work, we shall recall conditions under which it has a unique solution.A Sylvester equation is any equation of the form where M is a p p × real or complex matrix, N is a q q × real or complex matrix and C is a p q × real or complex matrix while matrix X stands for an unknown p q × real or complex matrix.The following well known result gives a sufficient condition for the existence and uniqueness of a solution of Sylvester Equation (6).
Theorem 1 If spectrums of matrices M and N are disjoint, ( ) ( ) , then Sylvester Equation (6) has a unique solution.

System (1) with Constraints on the Control
Consider System (1) with the assumption that the control u is constrained to be in the region where min u and max u are positive vectors in m  .Note that the region D is a non symmetrical polyhedral set as is generally the case in practical situations.Let us first consider the unconstrained case where the regulator problem for System (1) consists in realizing a feedback law as where with full rank m .In this case, System (1) becomes The stability of the closed loop System (8) is obtained if, and only if, ( ) for all eigenvalues λ of the matrix A BF + .In the constrained case, the approach proposed in ([4]- [6]) consists of giving conditions allowing the choice of a stabilizing controller (7) in such a way that the state is constrained to evolve in a specified region of n  defined by Note that the domain In fact, when m n < , the sub- space ker F has dimension 0 n m − > and is a subset of this domain.Suppose now that there is a matrix Hence, by letting z Fx = , we get and then ( ) ( ) ( ) . We would get ( ) for all 0 t ≥ whenever ( ) We say that D is positively invariant with respect to the motion of System (12).More generally, we give the following definition of positive invariance.

Definition 1
A nonempty subset P of m  is said to be positively invariant with respect to the motion of System (12) if, for every initial state ( ) 0 z in P , the motion ( ) z t remains in P for every 0 t ≥ .The following theorem gives necessary and sufficient conditions for domain D to be positively invariant with respect to the motion of System (12).

Theorem 2 ([6])
The domain D is positively invariant with respect to the motion of System (12) if, and only if, where 2m  Till now, we have supposed the existence of a matrix H that satisfies Equation (11).The following result, which does not take into account the constrained problem, gives necessary and sufficient conditions for its existence. Theorem

Main Results
At first, we compute, and fix in the sequel, some matrix and its spectrum is { } Let f be the endomorphism of n  canonically associated to matrix T A , that is, T : .
).So, one can define the endomorphism g induced by f on s ⊥  .Matrix T V , when identified with its column vectors, can be seen as a basis of s ⊥  .In this basis, if we denote by L the matrix of g, we will have Let now T L Λ = .By transposition of formula (15), we get VA V = Λ .Since g is induced by f on ( )  which is also the spectrum of Λ. Formula ( 14) derives from the fact that V is of full rank m and shows the uniqueness of the matrix Λ.
Since this equality holds for all ) ( ) = is an invertible solution to Sylvester Equation ( 16).The only if part: Suppose now that Sylvester Equation ( 16) has an invertible solution X and let 1 K X − = .Matrix F = KV satisfies ker ker s F V = =  because matrix K is invertible and then is of full rank m.We also have The control vector is submitted to the constraint λ λ = = and 3 1 λ = − and pair ( ) A is stabilizable.We get first the matrix Note that the rows of V are orthogonal to the eigenvector of A associated to the stable eigenvalue 3 λ of A. We, then, choose the matrix This matrix is not diagonalizable and its eigenvalues are ( ) Then, we solve the equation ( 15) and use the inverse of its solution to get the feedback matrix Finally, 1 − is the unique eigenvalue of the matrix A BF + .Figure 1 shows that, starting from two different and admissible controls, the corresponding trajectories, in the control space, converge to the origin without saturations.

Case of Single Input Linear Systems
We discuss the particular case of single input linear systems, that is, when 1 m = .As described in the general case, we start by computing matrix V which is in case of 1 m = a row vector that is orthogonal to s  and is easy to get.Matrix Λ is only the real 1 0 λ ≥ ; the unique undesirable eigenvalue of A .If we choose H to be a negative real number, Theorem 5 ensures that all matrices F (row vectors) are of the form KV where 1 K − is a nonzero real solution to the simple "Sylvester equation" ( ) This equation has a nonzero solution if, and only if, the real number VB is nonzero.That is, VA V λ = shows that V is a left eigenvector of A associated to the undesirable eigenvalue 1 λ and then for every k ∈  .But pair ( ) ker Im .
If we suppose that 0 VB = , then we should have . This shows that 0 V = which is not true.As a consequence, we have the following result.
Proposition 3 For a given left eigenvector V of A associated to the unique eigenvalue 1 λ and for a ne- gative real number h , we have 0 VB ≠ and matrix F given by The spectrum of

Case Where the Matrix VB Is Nonsingular
We have seen in the last paragraph that, when 1 m = , a necessary and sufficient condition for Sylvester Equation (15) to have nonsingular solution (nonzero real number in fact) is that the real VB is nonzero.We also have seen that this last condition; 0 VB ≠ , is equivalent to the fact that pair ( ) A B is stabilizable.In case of 2 m ≥ , it may happen that pair ( ) , A B is stabilizable but matrix VB is singular as will show the following example.

Example 2
Consider System (1) with λ λ = = and 3 1 λ = − .The last desirable eigenvalue 3 λ is associated to the ei- genvector [ ] T 0 1 0 of A that spans the subspace s  .Matrix V is then and is of full rank 3.This shows that pair ( ) A B is stabilizable since it is even controllable.The case of matrix VB is nonsingular can be seen as a general case of 0 VB ≠ when 1 m = .So, it deserves a special study that will be the aim in the sequel of this paragraph.In the following theorem, we give a necessary and sufficient condition for matrix VB to be nonsingular.This shows that U = 0 since VB is nonsingular and, then, X = 0.The only if part: From the fact that VX = 0 for any vector X in s  ; that is  .Moreover, from the fact that ( ) since V is linear.This shows that the rank of VB is m and that it is nonsingular. The following theorem gives another method to get a partial pole assignment under the assumption that VB is nonsingular.

Theorem 7
Suppose that matrix VB is nonsingular and let matrix , , , , , This shows that ( ) is nonsingular and Λ is given ( )

A Total Eigenstructure Problem
Note that, when m and n are equal, matrix V described in the last section is now any invertible matrix of n n ×  .That is why, we suppose n V =  .Then matrix Λ is simply A and Sylvester Equation (15) becomes where H is a real n n × matrix with desirable spectrum.So, if the unique solution X of Equation ( 20) is invertible, then the spectrum of A BF + and the one of H are equal, where matrix F is 1 X − .Suppose now that m n < .The technique of augmentation (see [5] and [6]) consists of augmenting the matrix B by adding zeros in order to get a new matrix 1 n n B × ∈  .One should also complete the control vector u by fictive real numbers to get 1 n u ∈  .We also replace the assumption: "the pair ( ) , A B is stabilizable" by "the pair ( ) for which 1

Conclusion
A method for partial or total eigenstructure assignment problem was presented and examples to illustrate the method were given.The method uses Sylvester equation to find the feedback matrix F when some matrix H is given with a desirable spectrum that will replace all the undesirable eigenvalues of the initial matrix A of System (1) in the closed-loop.This method generalizes the one proposed in [10] without additional conditions on Sys- tem (5) and allows us to deal with the problem of asymmetrical constraints on the control vector.Examples to show its importance are presented.
n ≤ ≤ .The vector ( ) n . • Matrices of the form p λ are called scalar matrices for λ ∈  and * p ∈  .

FUT
=  .Since dimension of s  is n − m, rank formula shows that F is of full rank m.Let now mVV is invertible, so U = 0 and this shows that matrix T FV is invertible.Clearly, ( )

F
=  .From Proposition 2, there is an invertible matrix m m K × ∈  such that F = KV.Since VA = ΛV , we get

Figure 1 .
Figure 1.Trajectories of the system ( ) ( ) u t Hu t =  from two dif-

The feedback matrix 1 FFigure 1
, which represents the first row of F , is then but in the state space, Figure2plots two trajectories in the state space starting from two different and admissible initial states and, thanks to the asymptotic stability of the system and the invariance positive property, shows the state convergence to the origin without leaving the domain imposed by the constraints.

Figure 2 .
Figure 2. Trajectories of the system ( ) () ( )1 x t A BF x t = +  from (12) that ker F is positively invariant with respect to the motion of System(12)is the same as ker F is stable by matrix A. If the constrained problem is taken into account, the following theorem gives necessary and sufficient conditions for positive invariance of the domain of states 3 ([4]) F ker is positively invariant with respect to the motion of System (12) if, and only if, there is a matrix m m H × ∈  such that Equation (11) is satisfied.c H U ≤  Because we are focusing on the partial assignment problem, eigenvalues of matrix H should be desirable, that is, matrix H should be Hurwitz.The undesirable eigenvalues 1 , , m The if part: Since VB is nonsingular, matrix B is of full rank m.To complete the proof of the if part, we shall