AP05_6.vp 1 Introduction Eigenstructure assignment is one of the basic techniques for designing linear control systems. The eigenstructure as- signment problem is the problem of assigning both a given self-conjugate set of eigenvalues and the corresponding eigenvectors. Assigning the eigenvalues allows one to alter the stability characteristics of the system, while assigning eigenvectors alters the transient response of the system. Ei- genstructure assignment via state feedback has developed the design methods for a wide class of linear systems under full-state feedback with the objective of stabilizing the control systems. The parametric solution of eigenstructure assign- ment for state feedback has been studied by many researchers [6–10]. Fahmy and Tantawy [7] and Fahmy and O’Reilly [8–9] have developed solutions to the eigenstructure assignment problem. A parametric characterization of the assignable eigenvalues and generalized eigenvectors is presented. Duan [6] presented two complete parametric approaches for eigen- structure assignment in linear systems via state feedback. This methodology is deeply utilized in this research work. This paper focuses on a special feedback using only state derivatives instead of full-state feedback. Therefore this feedback is called state-derivative feedback. The problem of arbitrary eigenstructure assignment using full-state derivative feedback naturally arises. The motivation for state deriva- tive feedback comes from controlled vibration suppression of mechanical systems. The main sensors of vibration are accel- erometers. From accelerations we can reconstruct velocities with reasonable accuracy, but not the displacements. There- fore the available signals for feedback are accelerations and velocities only, and these are exactly the derivatives of states of the mechanical systems, which are the velocities and displace- ments. Direct measurement of the state is difficult to achieve. One necessary condition for a control strategy to be imple- mentable is that it must use the available measured responses to determine the control action. All of the previous research in control has assumed that all of the states can be directly mea- sured (i.e., that there is full-state feedback). Many papers have been published on controlling this class of systems, (e.g. [12-17]) describing the acceleration feedback for controlled vibration suppression. However, the eigenstructure assignment approach for feedback gain deter- mination has not been used at all or has not been solved generally. Other papers dealing with acceleration feedback for mechanical systems are [18–19], but here the feed- back uses all states (positions, velocities) and accelerations additionally. Abdelaziz and Valasek [1–3] have recently presented an eigenvalue assignment technique via state-derivative feed- back for single-input and multi-input time-invariant linear systems. Eigenstructure assignment via state-derivative feed- back is introduced in [4-5]. In this paper, two complete parametric approaches for eigenstructure assignment in lin- ear systems via state-derivative feedback are proposed. Two complete parametric expressions for the closed-loop eigen- vector matrices and the feedback gains are established in terms of closed-loop eigenvalues and a group of parameter vectors. Both the closed-loop eigenvalues and this group of parameters can be properly chosen to produce a closed-loop system with some additional desired system specifications. The necessary and sufficient conditions for the existence of the eigenstructure assignment problem are described. The proposed controller is based on the measurement and feed- back of the state derivatives of the system. This work has successfully extended previous techniques by state feedback and has modified them to state-derivative feedback. Finally, numerical examples are included to demonstrate the effec- tiveness of this approach. The main contribution of this work is an efficient technique that solves the eigenstructure as- signment problem via state-derivative feedback systems. The procedure defined here represents a unique treatment for extending the eigenstructure assignment technique using the state-derivative feedback in the literature. This paper is organized as follows. In the next section, the problem formulation and the necessary and sufficient conditions for the existence of the eigenstructure assignment problem are described. Additionally, two complete paramet- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 19 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 6/2005 A Complete Parametric Solutions of Eigenstructure Assignment by State-Derivative Feedback for Linear Control Systems T. H. S. Abdelaziz, M. Valášek In this paper we introduce a complete parametric approach for solving the problem of eigenstructure assignment via state-derivative feed- back for linear systems. This problem is always solvable for any controllable systems iff the open-loop system matrix is nonsingular. In this work, two parametric solutions to the feedback gain matrix are introduced that describe the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve robust- ness of the closed-loop system. Accordingly, the sensitivity of the assigned eigenvalues to perturbations in the system and gain matrix is mini- mized. Numerical examples are included to show the effectiveness of the proposed approach. Keywords: eigenstructure assignment, state-derivative feedback, linear control systems, feedback stabilization, parametrization. ric solutions to the eigenstructure assignment problem via state-derivative feedback are presented. In section 3, illustra- tive examples are presented. Finally, conclusions are discussed in section 4. 2 Eigenstructure assignment by state-derivative feedback for time-invariant systems In this section, we present two complete parametric ap- proaches for solving the eigenstructure assignment problem via state-derivative feedback for linear time-invariant systems. 2.1 Eigenstructure assignment problem formulation Consider a linear, time-invariant, completely controllable system �( ) ( ) ( ), ( )x Ax Bu x xt t t t� � �0 0 , (1) where �( )x t n�R , x( )t n�R and u( )t m�R are the state-deriva- tive, the state and the control input vectors, respectively, ( )m n� , while A � �R n n and B � �Rn m are the system and control gain matrices, respectively. The fundamental assumption imposed on the system is that the system is com- pletely controllable and matrix B has a full column rank m. The objective is to stabilize the system by means of a linear feedback that enforces the desired characteristic behavior for the states. The problem is to find the state-derivative feedback control law u K x( ) �( )t t� � , (2) which assigns prescribed closed-loop eigenvalues and corre- sponding eigenvectors that stabilize the system and achieve the desired performance. Here, the first derivative vector of state-space �( )x t is utilized instead of the vector of state-space x( )t . Then, the closed-loop system dynamics becomes �( ) ( ) ( )x I BK A xt tn� � �1 , (3) where In is the n×n identity matrix. In what follows, we as- sume that (In+ BK) has a full rank in order that the closed- -loop system is well defined. The closed-loop characteristic polynomial is given by � det ( )� � � ���I I BK An n 1 0. (4) Let �� � � � � �� �i i i s s n, , , , ,C 1 1� be a set of desired self-conjugate eigenvalues, where s is the number of distinct eigenvalues, and denote the algebraic and geometric multi- plicity of the ith eigenvalue �i by mi and qi, respectively, ( )1 � �q mi i . The length of qi chains of generalized eigen- vectors with �i are denoted by pij, ( j � 1, …, qi). Then in the Jordan canonical form of the closed-loop matrix, there are qi blocks associated with the ith eigenvalue �i of orders pij. It is satisfying that p nijj q i s i � �� 11 . In this work, we restrict ourselves by mi � qi. This means that the multiple eigenvalues are not split; they are placed in one Jordan block. The partial multiplicities are not placed. Therefore, the Rosenbrock’s inequalities are fulfilled. The right eigenvector and generalized eigenvectors of the closed-loop matrix with �i are denoted by vij k n�C , i � 1, …, s, j � 1, …, qi, k � 1, …, pij. According to the definition of the right eigenvector and generalized eigenvectors for multiple eigenvalues, then � �( )I BK A In i n ijk ijk� � �� �1 1� v v , vij0 � 0, �i j k, , . (5) This equation demonstrates the relation of assignable right generalized eigenvectors with the associated eigenvalue. The notations are defined as � V V V� � �1, ,� s n nC , V V Vi i iq n m i i� � �� � �� � �1, ,� C , Vij ij ij p n pij ij� � �� � �� � � v v1 , ,� C , where Vi contains all right eigenvectors and generalized ei- genvectors associated with the eigenvalue �i, and det( )V � 0. Then, the eigenstructure assignment problem for system (1) via state-derivative feedback can be stated as follows: Eigenstructure assignment problem: Given the real pair (A, B) and the desired self-conjugate set �� �1, ,� n �C , find the real state-derivative feedback gain matrix K � �Rm n that will make the closed-loop matrix ( )I BK An � �1 have admissable eigenvalues and the associ- ated set of right eigenvector and generalized eigenvector matrix V. The necessary and sufficient conditions that ensure solv- ability of the eigenstructure assignment problem via state- -derivative feedback are presented in the following lemma. Lemma 1: The eigenstructure assignment problem for the real pair (A, B) is solvable for any arbitrary nonzero, self-conjugate, closed-loop poles, if (A, B) is completely controllable, that is � rank B AB A B, , ,� n n� �1 , or � rank �I A Bn n� �, , � �� C , and A is nonsingular. Proof: From this condition, the closed-loop matrix must be defined. This means the matrix ( )I BKn � is of full rank and det( )I BKn � � 0. Then, from (5) it can easily be rewritten as ( )I BK AV Vn � � �1 � (6) where � � �Cn n is in Jordan canonical form with the desired eigenvalues on the diagonal. Then AV V BKV� �� � (7) which can be written as I BK AV Vn � � � � � 1 1. (8) Then, det( ) det( ) det( ) det( )I BK AV V An � � � � � � � � � 1 1 1 0. (9) Since V must be nonsingular, then det( )A � 0 and det( )� � 0. Therefore, matrices A and � should be of full rank in order for the closed-loop matrix to be defined. � Thus, the necessary and sufficient conditions for existence of the solution to the eigenstructure assignment problem via state-derivative feedback is that the system is completely con- 20 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 6/2005 Czech Technical University in Prague trollable and all eigenvalues of the original system are non- zero (A has full rank). We remark that the requirement that matrix � is diagonal, together with the invertibility of V, ensures that the closed-loop system is non-defective. Non-de- fective systems are desirable because the poles of such systems are less sensitive to system parameter perturbation [10]. Based on the above necessary and sufficient conditions, two parametric forms are derived for the state-derivative feedback gain matrix K that assigns the desired closed-loop poles. 2.2 Eigenstructure assignment The main work now is to find a parametric solution to the state-derivative feedback gain matrix K that assigns the de- sired closed-loop eigenvalues and associated eigenvectors. Equation (5) can be rewritten as A ij k i n ij k n ij kv v v� � � � �� ( ) ( )I BK I BK 1, vij 0 � 0, �i j k, , . (10) Let the auxiliary vectors w vij k ij k m� �K C , i s� 1, ,� , j qi� 1, ,� , k pij� 1, ,� , (11) be introduced. The set of w ij k is defined in a similar manner to the set of vij k as follows, W W W� � � � � � � � � 1 , ,� s m nC , W W Wi i iq m m i i� � � � � � � � � 1, ,� C , Wij ij ij p m pij ij� � �� � �� � � w w1 , ,� C . This leads to ( ) ,� �i n ij k i ij k ij k ij k ij ij I A B B 0 0 � � � � � � � � �v w v w v w 1 1 0 0, , �i j k, , . , (12) The above equation can be equivalently written in the fol- lowing compact matrix form, � � � �i n i ij k ij k n ij k ij kI A B I B� � � � � � � � � � � � � � � � � � �, , v w v w 1 1 � � � � � , , , , , .v wij ij i j k 0 00 0 (13) Finally, the parametric equation to the right eigenvector and generalized eigenvectors can be expressed as � � � � � � � � i n i ij k n ij k ij k ij k ij k I A B I B� � � � � � � � � � � � �, , , , 1 v w ij i j k 0 � �0, , , . (14) Then, parameter vectors � ij k n m� �C are chosen arbitrary under the condition that the columns of matrix V are linearly independent. A parametric solution to the eigenstructure assignment problem via state-derivative feedback is derived from (11) as K WV� �1 (15) where � � V V V W W W� �1 1 1 1( ), , ( ) , ( ), , ( )� � � �� �s s s s . Then the feedback gain matrix is parametrized directly in terms of the eigenstructure of the closed-loop system, which can be selected to ensure robustness by exploiting freedom of these parameters. There exists a real feedback gain matrix K if and only if the following three conditions are satisfied: 1. The assigned eigenvalues are symmetric with respect to the real axis. 2. The right generalized eigenvectors �vijk n i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � � are linearly independent and for complex-conjugate poles, � �i i2 1� then v vi j k i j k 2 1 � . 3. There exists a set of vectors �w ijk m i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � � , satisfying (13) and w wi j k i j k 2 1 � for � �i i2 1� . The parametric formula for the state-derivative feedback gain matrix K that assigns the desired closed-loop poles sys- tem is now derived. In the following, we obtain the more gen- eral parametric solutions of vij k and w ij k in (13). Two complete parametric forms are introduced and a new procedure is de- rived, which yields a parametric expression for K involving free parameter vectors. 2.3 A parametrization approach for eigenstructure assignment The aim now is to find a parametric solution to the eigen- structure assignment problem via state-derivative feedback. We remark that the development of parametric solutions to this problem is useful in that one can then think of solving other important variations of the problem, such as the robust eigenstructure assignment problem, by exploiting freedom of these parameters. The relation demonstrating the assignable right generalized eigenvectors with the eigenvalues is (13). Definition 1: A square polynomial matrix P(�) is called a uni- modular matrix if its determinant is a nonzero constant. Definition 2: polynomial matrix P(�) is a unimodular matrix if and only if P(�) equals the product of some finite number of elementary row (or column) transformation matrices. It is well known that the matrix pair (A, B) is controllable if and only if � rank C� �I A Bn n� � � �, , . Due to the controllability of (A, B), there exist unimodular matrices P( )� � �C n n and Q( ) ( ) ( )� � � � �C n m n m satisfying the following equation: � � P I A B Q 0 I( ) , ( ) ,� � �n n� � . (16) Partition the polynomial matrix Q(�) into the following form Q Q Q Q Q Q Q ( ) ( ) ( ) ( ) ( ) ( ) ( ) � � � � � � � � � � �� � � �� � � � 1 2 11 12 21 22 �� � � �� with Q1( ) ( ) � � � �C n n m , Q2( ) ( ) � � � �C m n m , Q11( )� � �Cn m, Q12( )� � �Cn n, Q21( )� � �C m m and Q22( )� � �C m n. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 21 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 6/2005 Then, converting (16) to the following form, � � P I A B Q Q 0 I Q Q ( ) , ( ) ~ ( ) , , ~ ( ) � � � � � � n n� � � � � � � � � � � 1 2 2 2with ( ) . � � � � �� � � �� (17) Now, the following theorem gives a parametric solution to the eigenstructure assignment problem via state-derivative feedback. The parametric solutions of vij k and w ij k in (13) can now be given. Theorem 1: Let the matrix pair (A, B) be controllable, where matrix A � �R n n is nonsingular and matrix B � �R n m has a full column rank m. Then all solutions of (13), vij k and w ij k , are given by: v w v ij k ij k ij k i ij k f� � � � � � � � � � � � � � � � � � Q Q P 1 2 ( ) ~ ( ) ( ) � � � � �� �� � � � � � � � � � � 1 1 0 0 B 0 0 w v w ij k ij ij . , , (18) Or, equivalently, as v v w vij k i ij k i i ij k ij k ijf� � � � �Q Q P B11 12 1 1 0( ) ( ) ( )( ),� � � � 0 and � �w v w wijk i i ij k i i ij k ij kf� � �� � 1 21 22 1 1 � � � �Q Q P B( ) ( ) ( )( ) , ij 0 � 0 i s� 1, ,� , j qi� 1, ,� , k pij� 1, ,� , where P( )� and Q( )� are unimodular matrices satisfying (16), and fij k m�C are arbitrarily free parameter vectors satisfying the following two contraints: det( )V 0� and f fi j k i j k 2 1 � if � �i i2 1� . Proof: First, we need to show that the set of vectors satisfying (13) and the set of vectors given by (18) are equal. Then, using (13) and (18), we have � � � � � � � � i n i ij k ij k i n i i i I A B I A B Q Q � � � � � � � � � � �, , ( ) ~ ( ) v w 1 2 � � � � � � � � � � � � � � � � � � � � � � � � fij k i ij k ij k i P B P 0 ( ) ( ) , � � v w1 1 1 � � � I P B B n ij k i ij k ij k ij k ij k f ( )� � � � � � � � � � � � � � � � � � v w v w 1 1 1 1 � � � � � � � � � � � � � � � �I B 0 0 n ij k ij k ij ij i j k , , , , , , . v w v w 1 1 0 0 (19) Therefore, the vectors given by (18) satisfy (13). Now, we show that vectors vij k and w ij k (i � 1, …, s, j � 1, …, qi, k � 1, …, pij) satisfying (13) can be expressed in the form of (18). From (17) we can obtain � � P I A B 0 I Q Q ( ) , , ( ) ~ ( ) , , ,� � � � � i i n i n i i i� � � � � � � � � � � � 1 2 1 1 � s. (20) Then � � P I A B 0 I Q Q ( ) , , ( ) ~ ( ) � � � � � i i n i ij k ij k n i i � � � � � � � � � � �v w 1 2� � � � � � � � � � � � � � � �1 v w ij k ij k (21) and � � � P B 0 I 0 ( ) , , , � i ij k ij k n ij k ij k ij f � � � � � � � � � � � � � �v w e v w 1 1 0 ij i j k 0 � �0, , , , , (22) where f ij k ij k i i ij k ij ke v w � � � � � � � � � � � � � � � � � � � � Q Q 1 2 1 ( ) ~ ( ) � � � � � � � � �, , ,i j k . (23) Then from (22) we obtain � �e v w v wijk i ijk ijk ij ij i j k� � � � � �� �P B 0 0( ) , , , , ,� 1 1 0 0 . (24) Substituting (24) into (23) we obtain (18). � Assuming zero initial conditions and applying the Laplace transformation to (1), we obtain X I A BU G U( ) ( ) ( ) ( ) ( )� � � � �� � ��n 1 . (25) Then the behaviour of our linear system is described by a rational matrix function ( )�I A Bn � �1 of size n×m of a com- plex variable �. The input-state transfer function of the system can be factorized as G I A B N D( ) ( ) ( ) ( )� � � �� � �� �n 1 1 , (26) where N( )� � �C n m and D( )� � �C m m are right coprime poly- nomial matrices in �. The above equation can be written as ( ) ( ) ~ ( )� � � �I A N BDn � � � 0, with ~ ( ) ( ) D D � � � � � . (27) Now, the following theorem gives a parametric solution to the eigenstructure assignment problem via state-derivative feedback. Theorem 2: Let the matrix pair (A, B) be controllable, where matrix A � �R n n is nonsingular and matrix B � �Rn m has a full column rank m. Then all solutions of (13), vij k and w ij k , are given by: v w ij k ij k i i ij k if � � � � � � � � � � � �� � � �� � N D N( ) ~ ( ) ( ) ~ � � � �d d D N D ( ) ( ) ! ( ) ~ ( ) � � � � i ij k k k i i f k � � �� � � �� � � � � � � � 1 1 1 1 1 � d d � �� � � �� fij 1. (28) Or, equivalently, as � �v vij k i ij k l l i ij k l l k l ijf l f� � �� � � N N 0( ) ! ( ) ,� � � 1 1 0d d , and � �w wijk i ijk l l i ij k l l k l ijf l f� � �� � � ~( ) ! ~ ( ) ,D D 0� � � 1 1 0d d , i s� 1, ,� , j qi� 1, ,� , k pij� 1, ,� , 22 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 6/2005 Czech Technical University in Prague where N( )� and ~ ( )D � are polynomial matrices satisfying (27) and fij k m�C are arbitrarily free parameter vectors satisfying the following two contraints: det( )V 0� and f fi j k i j k 2 1 � if � �i i2 1� . Proof: We need only to show that the set of vectors given by (28) satisfies (13). Then take differential of order l on both sides of (27), and we obtain ( ) ( ) ( ) ~ ( ) � � � � � � � � I A N N B D B n l l l l l l l l l � � � � � � � d d d d d d d 1 1 1 1 0d� � l � � ~ ( ) .D (29) Substituting � by �i and postmultiplying by a vector 1 1 l fij k ! � on both sides of (29) gives � � � �( ) ! ( ) ! ~ ( )� � � � � �i n l l i ij k i l l i ij k l f l fI A N B D� �� � 1 11d d d d � � 1 1 1 1 1 1 1 1 1 1 � � � � � � � � � �( )! ( ) ( )! ~ l f l l l i ij k l l d d d d� � � N B D� �( ) .�i ijkf �1 (30 ) Summing up all the equations in (30) and using (28) we obtain ( ) , , � �i n ij k i ij k ij k ij k ij ij I A B B 0 0 � � � � � � � � �v w v w v w 1 1 0 0 , , , ,�i j k (31) where the assignable chains of right eigenvector generalized eigenvectors associated with the assigned eigenvalue �i are given by � �vij k i ij k i ij k k k f f k � � � � � � � � N N N ( ) ( ) ( ) ! ( � � � � d d d d 1 1 1 1 1 � � �� i ij ijf .) , 1 0v � 0 (32) Similarity, the gain-eigenvector product � �w ijk i ijk i ijk k k f f k � � � � � � � � ~ ( ) ~ ( ) ( ) ! D D� � � � d d d d 1 1 1 1 1 � � �~( ) ,D 0� i ij ijf .1 0w � (33) Summing up, all the equations (28) hold. � Then, Theorems 1 and 2 give two complete and explicit parametric solutions with the complete and explicit freedom of eigenstructure assignment via state-derivative feedback. These solutions are expressed by the eigenvalues and a group of free parameter vectors, fij k. By specially choosing the free parameter vectors in (18) and (28), solutions with desired properties can be obtained. Remark 1: It should be noted that for the case of distinct eigenvalues (mi � qi � 1, s � n). Then, the computations of vij k and w ij k , taking the simple form, are given by: v w i i i i if1 1 1 1 1 2 1 1� � � � � � � � � � � � � � � � � � � � � � � � Q Q 0 ( ) ~ ( ) � � � or (34) v w i i i i if i n 1 1 1 1 1 1 1 � � � � � � � � � � � �� � � �� � N D ( ) ~ ( ) , , , . � � � Remark 2: For the single-input system (m � 1), the parameter vectors f ij k reduce to scalars and, accordingly, the solutions of (18) and (28) are the same (unique), regardless of the choice of f ij k. This leads to the well-known result that solution K in this case is unique [1, 2]. The general expressions for the closed-loop eigenvectors and the feedback gains can immediately be written out as soon as the two polynomial matrix reductions (16) and (26) are carried out. These reductions can be completed by a series of simple elementary matrix transformations [6]. There are two methods for computing the polynomial matrices. The first is the Smith canonical form, which ex- ploits the fact that for a controllable pair (A, B) the matrix ( ,�I A Bn � ) maintains full rank for all values of �. The Smith canonical form constructs two unimodular matrices P(�) and Q(�) that diagonalized a given polynomial matrix as (16). Subject to the controllability of (A, B) the augmented matrix G I I A B I � �� � �� � � �� � n n n m � 0 , (35) can be changed into the form of H P 0 I 0 Q � � � �� � � �� ( ) ( ) � � n . (36) By applying a series of row elementary transformations within the upper n rows and a series of column elementary transformations within the last n � m columns, the matrices P(�) and Q(�) in the final transformed matrix H are uni- modular and automatically satisfying (16). Consequently, we can partition Q(�) to find N(�) and D(�) as Q N D ( ) ( ) ( ) � � � � � � � � �� � � ��. (37) The second approach uses the matrix fraction description (MFD). If all the elements of the matrix are proper rational polynomials, then the matrix may be factored as N(�) D�1(�). The elements of ( )�I A Bn � �1 are rational polynomials. Thus ( )� � �I A B N Dn � � � �1 1( ) ( ). The convenient solution can be found by inspection N I A B( )� �� � �( )n 1 and D I( )� � n , (38) or N I A B( ) adj� �� �( )n and D I A I( )� �� �det( )n n , (39) where adj(.) and det(.) represent, respectively, the adjoint and the determinant of matrix (.). The Smith canonical form and MFD approaches both require symbolic manipulation to perform the Smith decom- position or matrix inversion. This presents no difficulty when working by hand or using a symbolic package such as Maple. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 23 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 6/2005 Based on the discussion and analysis above, an algorithm for solving the eigenstructure assignment problem via state-derivative feedback can be given as follows: Algorithm Input Controllable real pair (A, B), where matrix A � �Rn n is nonsingular, and a set of n self-conjugate complex numbers {�1, …, �n}. Step 1 Construct the right coprime matrix polynomials N(�) and D(�) by applying the method presented in this section. Step 2 Choose arbitrary parameter vectors f ij k m�C (i � 1, …, s, j � 1, ..., qi, k � 1, ..., pij) in such a way that f fi j k i j k 2 1 � implies � �i i2 1� . Step 3 Calculate the right eigenvectors vij k n�C (i � 1, …, s, j � 1, ..., qi, k � 1, ..., pij) using (28). If the eigenvectors matrix V is singular, then return to Step 2 and select different parameters fij k, until V is nonsingular. Step 4 Compute the gain-eigenvectors w ij k m�C (i � 1, …, s, j � 1, ..., qi, k � 1, ..., pij) using (28), and construct matrix W. Step 5 Compute the real derivative feedback gain matrix us- ing, K � WV �1. From the above results we can observe that the system poles can always be assigned by a state-derivative feedback controller for any controllable system if and only if the open- -loop system matrix A is nonsingular. In the case of single-in- put, m � 1, there only at most one solution. In the case of multi-input, 1 < m � n, the solution is generally non-unique, and extra conditions must be imposed to specify a solution. The extra freedom can be used to give the closed-loop system other desirable properties. The extra freedom can be used in different ways, for example to decrease the norm of the feedback gain matrix or to improve the condition of the eigenvectors of the closed-loop matrix. Additionally, it in- creases the robustness of the closed-loop system against the system parameter perturbation. This issue becomes very im- portant when the system model is not sufficiently precise or the system is subject to parameter uncertainty. Then the feedback gain matrix is parameterized directly in terms of the eigenstructure of the closed-loop system, which can be selected to ensure robustness by exploiting freedom of these parameters. Eigenstructure assignment is a very flexible technique. It provides access to all the available design freedom. The draw- back of eigenstructure assignment is that it has no inherent mechanism for insuring robustness and can assign a robust solution as easily as a catastrophically unrobust solution. Then the optimization techniques are used with the objective of finding the optimum design vectors f ij k so that the closed- -loop system is robust to parameter variations. 3 Illustrative examples In this section, numerical examples are used to illustrate the feasibility and effectiveness of the proposed eigenstruc- ture assignment technique via state-derivative feedback. Example 1: Consider a controllable, time-invariant, multi-input linear system, �( ) ( ) ( )x x ut t t� � � � � � � � � � � � � � � � � � � � � � 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 . A pair of right matrix polynomials N(�) and D(�) satisfying (25) and ~ ( )D � can be found as: N( )� �� � � � � � � � � � � � 1 0 0 0 1 , D( )� � � � � � �� � �� � � �� 1 1 12 and ~ ( )D � � � � � � � � � � � �� � � �� 1 1 1 1 . In the following, we consider the assignment of three dif- ferent cases: Case 1: The desired closed-loop eigenvalues are selected as {�1, �2 and �3}. Then, the closed-loop eigenvector matrix V, and the cor- responding matrix W, can be written as � V N N N� ( ) , ( ) , ( )� � �1 111 2 211 3 311f f f and � W D D D� ~( ) , ~( ) , ~( )� � �1 111 2 211 3 311f f f . Specially choosing � f f11 1 31 1 1 0� � , T and � f21 1 0 1� , T. Then V � � � � � � � � � � � � � � 1 0 1 1 0 3 0 1 0 and W � � �� � �� � � �� 1 15 1 3 1 0 5 3 . . . Finally the state-derivative gain matrix is K WV� � � � � � � � � �� � � �� �1 4 3 1 3 15 0 1 0 5 . . . Case 2: The desired closed-loop poles are {�2 and �3 � i}. Choosing � f11 1 0 1� , Tand � f f21 1 31 1 1 0� � , T. Then V � � � � � � � � � � � � � � � � 0 1 1 0 3 3 1 0 0 i i and W � � � � � � � � � �� � � �� 15 0 3 0 3 0 5 3 3 . . . . i i i i . The gain matrix is K WV� � � � � � � � � �� � � �� �1 0 6 01 15 0 1 0 5 . . . . . Case 3: The desired eigenvalues are {�1, �1 and �3}. Then � �V N N N N� �� �� � �� ( ) , ( ) ( ) , ( )� � � � �1 11 1 1 11 1 1 11 2 2 21 1f f f f d d and � �W D D D D� �� � ~ ( ) , ~ ( ) ~ ( ) , ~ ( )� � � � �1 11 1 1 11 1 1 11 2 2 21 1f f f f d d� � �� . Choosing � f f11 1 21 1 1 0� � , T and � f11 2 0 1� , T. 24 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 6/2005 Czech Technical University in Prague We have V � � � � � � � � � � � � � � 1 0 1 2 0 3 0 1 0 and W � � �� � �� � � �� 0 5 15 1 3 2 0 5 3 . . . . Therefore K WV� � � � � � � � � �� � � �� �1 0 8333 01667 15 0 1 0 5 . . . . . Example 2: Consider a controllable, multi-input, linear system �( ) . . . . ( )x xt t� � � � � � � � � � � � � � � � � � 2 5 0 5 0 0 5 2 5 2 0 2 2 1 0 0 0 0 1 � � � � � � � u( )t . A pair of right matrix polynomials N( )� and D( )� can be obtained as: N( )� � � � �� � � � � � � � � � 2 5 4 1 0 0 1 and D( ) ( . ) . � � � � � � � � � � � � � � � � � � � 2 2 5 0 5 4 10 2 2 2 . In the following, we consider the assignment of three dif- ferent cases: Case 1: The desired closed-loop eigenvalues are selected as {–1, –2 and –3}. Specially choosing � f f11 1 31 1 1 0� � , T and � f21 1 0 1� , T. Then V � � �� � � � � � � � � � 3 4 1 1 0 1 0 1 0 and W � � � � � � �� � � �� 4 1 0 2 0 0 6667. . The gain matrix is K WV� � � � � � � �� � � �� �1 1 1 3 0 3334 1 13334. . . Case 2: The desired closed-loop eigenvalues are {–2 and –3� i}. Choosing � f11 1 0 1� , T and � f f21 1 31 1 1 0� � , T. Then V � � � � � �� � � � � � � � � � 4 1 2 1 2 0 1 1 1 0 0 i i , W � � � � � � � � � � � � �� � � �� 1 0 4 0 8 0 4 0 8 0 0 6 0 2 0 6 0 2 . . . . . . . . i i i i . The gain matrix K WV� � � � � � � � � � �� � � �� �1 0 4 0 8 2 6 01 0 7 0 4 . . . . . . . Case 3: The desired closed-loop eigenvalues are {–2, –2 and –3}. Choosing � f f11 1 21 1 1 0� � , T and � f11 2 0 1� , T. Then V � � �� � � � � � � � � � 1 4 1 1 0 1 0 1 0 and W � � � � � � �� � � �� 0 1 0 1 0 0 6667. . Therefore K WV� � � � � � � � �� � � �� �1 0 0 1 01667 0 8333 0 6667. . . . 4 Conclusions In this paper, two complete parametric approaches for solving the eigenstructure assignment problem via state-de- rivative feedback are proposed. The necessary conditions to ensure solvability are that the system is completely controlla- ble and the open-loop system matrix is nonsingular. The main result of this work is an efficient computational algo- rithm for solving the eigenstructure assignment problem of a linear system via state-derivative feedback. This parametric solution describes the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. The extra degrees of freedom on the choice of feedback gains are exploited to further improve the closed-loop robustness against perturba- tion. The main contribution of the present work is a compact parametric expression for the feedback controller gain matrix explicitly characterized by a set of free parameter vectors. The principle benefits of the explicit characterization of para- metric class of feedback controllers lie in the ability to directly accommodate various different design criteria. References [1] Abdelaziz. T. H. S., Valášek, M.: “Pole-Placement for SISO Linear Systems by State-Derivative Feedback.” IEE Proceeding Part D: Control Theory & Applications, 151, 4, 377–385, 2004. [2] Abdelaziz, T. H. S., Valášek, M.: “A Direct Algorithm for Pole Placement by State-Derivative Feedback for Single-Input Linear Systems.” Acta Polytechnica, Vol. 43 (2003), No. 6, p. 52–60. [3] Abdelaziz, T. H. S., Valášek, M.: “A Direct Algorithm for Pole Placement by State-Derivative Feedback for Multi- -Input Linear Systems – Nonsingular Case.” (Accepted in Kybernetika, 2004). [4] Abdelaziz, T. H. S., Valášek, M.: “Eigenstructure Assign- ment by State-Derivative and Partial Output-Derivative Feedback for Linear Time-Invariant Control Systems.” Acta Polytechnica, Vol. 44 (2004), No. 4, p. 54–60. [5] Abdelaziz, T. H. S., Valášek, M.: “Parametric Solutions of Eigenstructure Assignment by State-Derivative Feed- back for Linear Control Systems.” Proceedings of Inter- action and Feedbacks 2003, UT AV CR, Praha, 2003, p. 5–12. [6] Duan, G. R: “Solutions of the Matrix Equation AV+BW =VF and Their Application to Eigenstructure Assign- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 6/2005 ment In Linear Systems.” IEEE Transactions on Auto- matic Control, Vol. 38 (1993), No. 2. p. 276–280. [7] Fahmy, M. M., Tantawy, H. S.: “Eigenstructure Assign- ment via Linear State Feedback Control.” International Journal of Control, Vol. 40 (1984), No. 1, p. 161–178. [8] Fahmy, M. M., O’Reilly, J.: “Eigenstructure Assignment in Linear Multivariable Systems: a Parametric Solution.” IEEE Transactions on Automatic Control, Vol. 28 (1983), No. 10, p. 990–994. [9] Fahmy, M. M., O’Reilly, J.: “On Eigenstructure Assign- ment in Linear Multivariable Systems.” IEEE Trans- actions on Automatic Control, Vol. 27 (1982), No. 3, p. 690–693. [10] Clarke, T., Griffin, S. J., Ensor, J.: “A Polynomial Ap- proach to Eigenstructure Assignment Using Projection with Eigenvalue Trade-Off.” International Journal of Con- trol, Vol. 76 (2003), No. 4, p. 403–423. [11] Kautsky, J., Nichols, N. K., Van Dooren, P.: “Robust Pole Assignment in Linear State Feedback.” International Journal of Control, Vol. 41 (1985), p. 1129–1155. [12] Preumont, A., Loix, N., Malaise, D., Lecrenier, O.: “Ac- tive Damping of Optical Test Benches with Acceleration Feedback.” Machine Vibration, Vol. 2 (1993), p. 119–124. [13] Preumont, A.: Vibration Control of Active Structures, Kluwer, 1998. [14] Bayon de Noyer, M. P., Hanagud, S. V.: “Single Actuator and Multi-Mode Acceleration Feedback Control.” Adap- tive Structures and Material Systems, ASME, AD, Vol. 54 (1997), p. 227–235. [15] Bayon de Noyer, M. P., Hanagud, S. V.: “A Comparison of H2 Optimized Design and Cross-Over Point De- sign for Acceleration Feedback Control.” Proceedings of 39th AIAA/ASME/ ASCE/AHS, Structures, Structural Dynamics and Materials Conference, Vol. 4 (1998), p. 3250–3258. [16] Olgac, N., Elmali, H., Hosek, M., Renzulli, M.:“ Active Vibration Control of Distributed Systems Using Delayed Resonator with Acceleration Feedback.” Transactions of ASME Journal of Dynamic Systems, Measurement and Con- trol, Vol. 119 (1997), p. 380. [17] Kejval, J., Sika, Z., Valášek, M.: “Active Vibration Sup- pression of a Machine.” Proceedings of Interaction and Feedbacks 2000, UT AV CR, Praha, 2000, p. 75–80. [18] Deur, J., Peric, N.: A Comparative Study of Servosystems with Acceleration Feedback.” Proceedings of the 35th IEEE Industry Applications Conference, Roma, Italy, Vol. 2 (2000), p. 1533–1540. [19] Ellis, G.: “Cures for Mechanical Resonance in Industrial Servo Systems.” Proceedings of PCIM 2001 Conference, Nuremberg, 2001. Doc. Taha H. S. Abdelaziz e-mail: tahahelmy@yahoo.com Department of Mechanical Engineering Faculty of Engineering Helwan University 1 Sherif Street, Helwan, Cairo, Egypt Doc. Ing. Michael Valášek, DrSc. e-mail: valasek@fsik.cvut.cz Department of Mechanics Czech Technical University in Prague Faculty of Mechanical Engineering Karlovo nám. 13 121 35 Praha 2, Czech Republic 26 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 6/2005 Czech Technical University in Prague << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice