ABSTRACT

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.

Keywords:Sylvester Equation, Linear Continuous-Time Systems, Constrained Control

1. 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

(1)

The matrices A and B are real and constant: and with. The vector represents the state vector of the system and is the control vector. We suppose that the spectrum of the matrix A contains desirable or stable eigenvalues and undesirable or unstable eigenvalues. We also suppose that the pair 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 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:

(2)

and the kernel of is the stable subspace. In this case, the spectrum of the matrix 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 in the initial system, we get

(3)

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:

• The matrix H is diagonalizable; that is, there exist linearly independent eigenvectors of H associated to some stable eigenvalues.

• The endomorphism induced by the matrix A on the stable subspace is diagonalizable; that is, there exist linearly independent eigenvectors of A in the subspace.

• The spectrum of A and the spectrum of H are disjoint.

• The matrix

is invertible.

Under these conditions and according to the method described in [4] , the matrix is unique and is given by

(4)

where is the null vector of.

In [10] , the condition b) is kept and a new assumption should be fulfilled:

1. the matrix is invertible.

The method described in [10] consists of computing a matrix such that its rows span the orthogonal of the stable subspace by taking, for example the first rows of and then, for a stable matrix. Then, according to the method described in [10] , the feedback matrix is given by where is the inverse matrix of the solution of the Sylvester equation

(5)

and is a matrix such that.

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 such that 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.

2. Preliminaries

• 2.1. Notations and Definitions

• is the identity matrix of for.

• Matrices of the form are called scalar matrices for and.

• If is a real matrix, for some, its transpose, the real matrix, will be denoted by.

• When we view as an euclidean space, the usual inner product is considered, that is,

where and are two vectors of.

• If is a nonempty subset of, its orthogonal will be denoted by, that is•

• For a real number, we define

•

• If is a real matrix, for some, will denote the coefficient of corresponding to the row-column.

• If is a real matrix, for some, is the real matrix defined by

where and

• For real matrices (or real vectors) and, we say that is less than or equal to if every component of is less than or equal to the corresponding component of. We then write.

• If is a real matrix, for some, 

(a) is the kernel of: the subspace of of vectors such that.

(b) is the image of: the subspace of spanned by the columns of and also the set of vectors of the form with.

• If is a real matrix, for some, denotes its spectrum in the field of complex numbers.

• If is a complex number then is the real part of.

2.2. Some Notes on the Stable-Unstable Subspaces

1. To System (1), we associate:

(a) The two polynomials

that are factors of the characteristic polynomial of.

In case of, polynomial Q is just the constant polynomial 1.

(b) The two subspaces of; and.

In case of, the subspace is just the trivial subspace.

Remark 1

1. Since and are coprime, and are complementary subspaces of. Moreover, we have

and also

2. Since pair is stabilizable, we have

2.3. 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

(6)

where M is a real or complex matrix, N is a real or complex matrix and C is a real or complex matrix while matrix X stands for an unknown 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.

2.4. System (1) with Constraints on the Control

Consider System (1) with the assumption that the control is constrained to be in the region of defined by

where and are positive vectors in. Note that the region 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

(7)

where is chosen in with full rank. In this case, System (1) becomes

(8)

The stability of the closed loop System (8) is obtained if, and only if,

(9)

for all eigenvalues of the matrix. 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 defined by

(10)

Note that the domain is bounded only in case of. In fact, when, the subspace has dimension and is a subset of this domain. Suppose now that there is a matrix such that

(11)

Hence, by letting, we get

(12)

and then. We would get for all 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 of is said to be positively invariant with respect to the motion of System (12) if, for every initial state in, the motion remains in for every.

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,

(13)

where is the real vector

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 3 ([4])

is positively invariant with respect to the motion of System (12) if, and only if, there is a matrix such that Equation (11) is satisfied.

Note that is positively invariant with respect to the motion of System (12) is the same as 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.

Theorem 4 ([6])

The domain is positively invariant with respect to System (8) if, and only if, there is a matrix such that Equation (11) is satisfied and

3. Main Results

At first, we compute, and fix in the sequel, some matrix such that its rows span the subspace. Since the dimension of the subspace is, the matrix is of full rank.

Proposition 1 There is a unique matrix such that. This matrix is given by

(14)

and its spectrum is.

wang#title3_4:spwang#title3_4:spProof.

Let be the endomorphism of canonically associated to matrix, that is,

The subspace is stable under f (that is, f(x) belongs to whenever x is in). So, one can define the endomorphism g induced by f on. Matrix, when identified with its column vectors, can be seen as a basis of. In this basis, if we denote by L the matrix of g, we will have

(15)

Let now. By transposition of formula (15), we get. Since g is induced by f on, its spectrum is 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.                                ,

Proposition 2

For any matrix such that, there is an invertible matrix such that and is of full rank.

Proof.

Let such that. Since dimension of is n − m, rank formula shows that F is of full rank m. Let now such that. Then, and then. But matrix

is invertible, so U = 0 and this shows that matrix is invertible. Clearly, is invertible. For, we have

Since, we have and then

. So

Since this equality holds for all, we have.                                      Theorem 5 Let be a set of complex numbers stable under complex conjugation such that and with spectrum. Then, there is matrix of full rank such that and if, and only if, Sylvester equation

(16)

has a unique invertible solution such that.

wang#title3_4:spwang#title3_4:spProof.

The if part: Let be of full rank matrix such that and. From Proposition 2, there is an invertible matrix such that F = KV. Since VA = ΛV , we get

Then 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. Matrix F = KV satisfies because matrix K is invertible and then is of full rank m. We also have

,

Remark 2

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 of A must all be different from those of H. So, assumption in Theorem 5 above is necessary.

Example 1 Consider the linear time-invariant multivariable system described by

with

The control vector is submitted to the constraint such that,

Eigenvalues of A are and and pair 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 of A. We, then, choose the matrix

This matrix is not diagonalizable and its eigenvalues are. Besides, inequality (13) is satisfied. We use the formula to get

Then, we solve the equation (15) and use the inverse of its solution to get the feedback matrix

Finally, is the unique eigenvalue of the matrix.

Figure 1 shows that, starting from two different and admissible controls, the corresponding trajectories, in the control space, converge to the origin without saturations.

3. Particular Cases

3.1. Case of Single Input Linear Systems

We discuss the particular case of single input linear systems, that is, when. As described in the general case, we start by computing matrix V which is in case of a row vector that is orthogonal to and is easy to get. Matrix is only the real; the unique undesirable eigenvalue of. 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 is a nonzero real solution to the simple “Sylvester equation”

This equation has a nonzero solution if, and only if, the real number is nonzero. That is,

(17)

Formula shows that is a left eigenvector of associated to the undesirable eigenvalue and then for every. But pair is stabilizable, so

If we suppose that, then we should have

and then. We also have and. This shows that which is not true.

As a consequence, we have the following result.

Proposition 3 For a given left eigenvector of associated to the unique eigenvalue and for a negative real number, we have and matrix given by

(18)

The spectrum of is then.

3.2. Case Where the Matrix VB Is Nonsingular

We have seen in the last paragraph that, when, a necessary and sufficient condition for Sylvester Equation (15) to have nonsingular solution (nonzero real number in fact) is that the real is nonzero. We also have seen that this last condition;, is equivalent to the fact that pair is stabilizable. In case of, it may happen that pair is stabilizable but matrix is singular as will show the following example.

Example 2

Consider System (1) with

Eigenvalues of A are and. The last desirable eigenvalue is associated to the eigenvector of that spans the subspace. Matrix V is then

Matrix is singular since it is

Controllability matrix is given by

and is of full rank 3. This shows that pair is stabilizable since it is even controllable.

The case of matrix is nonsingular can be seen as a general case of when. 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 to be nonsingular.

Theorem 6

Matrix is nonsingular if, and only if, and are complementary subspaces of; that is,.

Proof.

The if part: Since VB is nonsingular, matrix B is of full rank m. To complete the proof of the if part, we shall show that since the dimension of the subspace is n − m. Recall that VX = 0 for any vector X in. If now X is both in and Im(B), then there is such that X = BU and then

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; that is, V is of full rank m; that is. Moreover, from the fact that, we get

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 is nonsingular.

Theorem 7

Suppose that matrix is nonsingular and let matrix be such that

, and is nonsingular. Then,

where.

wang#title3_4:spwang#title3_4:spProof.

Let and. Then,

This shows that is a nonsingular solution to Sylvester Equation (16) and, by Theorem 5, that matrix satisfies

                            ,

Example 3

Consider System (1) with

Eigenvalues of A are and. The last desirable eigenvalue is associated to the eigenvector of that spans the subspace. Matrix is then

Matrix is nonsingular and matrix is given by

Now choose that are the eigenvalues of the matrix. Then,

4. 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. That is why, we suppose. Then matrix is simply and Sylvester Equation (15) becomes

(19)

where H is a real matrix with desirable spectrum. So, if the unique solution X of Equation (20) is invertible, then the spectrum of and the one of H are equal, where matrix F is. Suppose now that. The technique of augmentation (see [5] and [6] ) consists of augmenting the matrix by adding zeros in order to get a new matrix. One should also complete the control vector by fictive real numbers to get. We also replace the assumption: “the pair is stabilizable” by “the pair is controllable.” System (1) which becomes

(20)

does not change. The matrix is now a real matrix with some desirable spectrum which does not contain any eigenvalue of matrix. If the unique solution of Sylvester equation

(21)

is invertible, we let. The matrices and have the same spectrum. If is the matrix of the first rows of, then assigns the spectrum of to.

Example 4

Consider the problem of stabilization of the following double integrator system

(22)

with

subject to the constraint. The matrix and the vector are respectively augmented to

such that where and are fictive positive real numbers. Let, for example, and to get the domain in the Example 1. We choose the matrix

for which the eigenvalues are and the inequality (13) is satisfied. Then, we solve the Sylvester equation (20) and get

The feedback matrix, which represents the first row of, is then

The spectrum of is then.

As in Figure 1 but in the state space, Figure 2 plots 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.

5. 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 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.

References