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OPTIMAL DESIGN OF STATIC NORMALIZING PRECOMPENSATORS Marcos Vicente Moreira , Jo ao Carlos Basilio COPPE - Programa de Engenharia El´etrica Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brasil Emails: moreira@pee.coppe.ufrj.br, basilio@dee.ufrj.br Abstract Normalization is the ﬁrst step in the design of multivariable controllers within the Characteristic Locus Method, i.e. , if the plant transfer matrix is far from normal at the frequency band of interest, then a normalizing precompensator should be designed in order to make the precompensated plant approximately normal at this frequency band. This problem is addressed in this paper, where an optimal static normalizing precompensator is proposed. The design strategy is based on an optimization problem with no constraints on the precompensator structure, and whose solution can be obtained directly from an eigenvalue problem. Keywords Linear systems, Multivariable systems, Optimization, Frequency domain. 1 Introduction The Characteristic Locus Method (CLM) is an ef- fective tool for the design of multivariable con- trol systems, allowing conﬂicting design objec- tives, such as stability and performance, be ad- dressed by a single system representation: namely that, the frequency response of the eigenfunctions of the open-loop system (characteristic loci). The essence of the CLM (MacFarlane and Belletruti, 1973) is to construct a commutative controller, i.e. , a controller with the same eigen- vector and dual-eigenvector matrices (frames) as the plant transfer matrix and to manipulate the controller eigenfunctions so as to achieve closed- loop stability, satisfying the generalized Nyquist stability criterion (MacFarlane and Postlethwaite, 1977), and to satisfy performance requirements such as tracking, disturbance rejection and good transient response. However, it is well known (Doyle and Stein, 1981; Postlethwaite, 1982; Wilkinson, 1965) that for plants whose frequency responses are not nor- mal at a certain frequency band, the character- istic loci are very sensitive to perturbations at the plant input and output at these frequencies. Therefore, in order to deal with the characteristic locus sensitivity problem, the design of multivari- able controllers within the CLM, must be done in two stages (Basilio and Sahate, 2000; Moreira and Basilio, 2005): ﬁrst, precompensate the plant in order to approximately normalize it in the nec- essary frequency range and then, ii apply the CLM to the precompensated system. A precompensation scheme with the view to making the precompensated plant as normal as possible has recently been presented (Basilio and Sahate, 2000), where the design of a nor- malizing dynamic precompensator is proposed. The main disadvantages of the precompensation method proposed in Basilio and Sahate (2000) are: the characteristic locus sensitivity with re- spect to perturbations at the plant input are not considered in the formulation of the optimization problem, and ii an optimization problem is for- mulated and solved for a speciﬁc precompensator structure. With the view to considering the sensitivity of the characteristic loci with respect to pertur- bations at both the plant input and output, it is proposed in Moreira and Basilio (2005) an opti- mization problem based on a measure of the mis- alignment between the input and output principal directions of the plant. Although this precompen- sation scheme has led to insensitive characteris- tic loci for the precompensated plant when an ex- act commutative controller is considered, it also imposes constraints on the precompensator struc- ture. In this paper, the design of an optimal sta- tic precompensator with the view to making the precompensated plant approximately normal at the vicinity of the crossover frequency is pro- posed. This new method is based on an optimiza- tion problem which has no constraints on the pre- compensator structure. The solution to the op- timization problem is very simple and can be ex- pressed as an eigenvalue problem. Another advan- tage of this method is that it improves the char- acteristic locus sensitivity indices with respect to perturbations at both the plant input and output when an exact commutative controller (Moreira and Basilio, 2004; Moreira et al., 2005; Moreira et al., 2006) is considered. 2 Main concepts Let and be the transfer matrices of the plant and controller, respectively. Accord- ing to the generalized Nyquist stability criterion (MacFarlane and Postlethwaite, 1977), the feed- back system of Fig. 1 will be stable if and only if the net sum of anti-clockwise encirclements of the critical point 1 + , by the characteristic loci of , is equal to the number of unstable poles of and 1739 of 1744

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±° ²¯ Figure 1: Closed-loop feedback control system. In order to be able to use the generalized Nyquist stability criterion as a design tool, a con- troller such that ) = is sought. This condition is satisﬁed providing that and share the same eigenvector and dual eigenvector frames and, therefore, the eigenvalues of the product are equal to the product of the eigenvalues of and . Thus, to achieve closed-loop stability and to satisfy performance requirements, such as track- ing, disturbance rejection and good transient re- sponse, one can manipulate the eigenfunctions of the open-loop transfer matrix, choosing ade- quately the eigenfunctions of the controller providing that the eigenvectors of are those of . However, in some cases, the character- istic loci can be very sensitive to uncertainties in the plant transfer function and therefore, in these cases, the CLM is not a reliable tool for the design of a multivariable control system. In or- der to show this Doyle and Stein (1981) consider a stable matrix perturbation, , at any point in the conﬁguration of Fig. 1, and show that the closed-loop system remains stable if and only if j )] multiplied by j )] is less than , for all frequencies, where de- notes the maximum singular value of a matrix and is the return ratio matrix for the point where the uncertainty is considered . Since the CLM is based on the eigenvalues of the open-loop trans- fer matrix [or, equivalently, and not on its singular values, then it is shown in Doyle and Stein (1981) that the CLM can only be eﬀective from the robustness point of view only if the plant transfer matrix is normal. The deﬁnition of normal matrix is as follows. Deﬁnition 1 A matrix is normal if it commutes with its conjugate transpose, i.e. GG In order to deal with the sensitivity problem of the characteristic loci within the CLM, it is pro- posed in Basilio and Sahate (2000), the design of a controller in two stages. The ﬁrst stage is the design of a dynamic normalizing precom- pensator, i.e. , a precompensator which ap- proximately normalizes the precompensated plant j j at the frequency range of interest. The second stage is the design of a controller, , that commutes with and also In general, it is more common to consider the uncer- tainty at the plant input or at the plant output, which corresponds to ) = or ) = , re- spectively. internally stabilizes the feedback system. With the view to deﬁning approximately normal matri- ces, it is proposed in Basilio and Sahate (2000) the following measure of the deviation from nor- mality: ) = GG (1) where denotes the Frobenius norm, which, for a matrix is deﬁned as: tr (2) where tr denotes the trace of a matrix. Other ways of measuring the approximate normality of a complex matrix is presented in Hung and Mac- Farlane (1982). One of these measures is based on the following theorem. Theorem 1 A matrix is normal if and only if it has an orthonormal set of eigen- vectors. Proof: See Horn and Johnson (1990). Theorem 1 leads to the following deﬁnition of approximately normal matrices. Deﬁnition 2 A matrix is said to be an approximately normal matrix if ) = 1 + where denotes condition number, is the eigenvector matrix of , and deﬁnes the degree of normality of The condition number of the eigenvector ma- trix of j j and j j can be used to provide an upper bound for a measure of the robustness of the closed-loop system with respect to perturbations at the plant output and input, respectively. This will be done with the help of the following result. Lemma 2 (Small Gain Theorem) Deﬁne ) = [ )] (3) as the plant transfer matrix with multiplica- tive perturbation at the plant output, where j )] , and is a non-negative real function. Then, stabilizes if and only if stabilizes and j j ) [ j j )] (4) for all Proof: See Vidyasagar (1985). It is well known that singular values are upper bounds for the moduli of the eigenvalues of a ma- trix, and thus, a necessary condition for robust stability of the closed-loop system, given in terms of the eigenvalues of the open-loop system at each frequency, can be given as: max j j 1 + j j (5) 1740 of 1744

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In order to obtain now a suﬃcient condition, the following result is needed. Lemma 3 Let be a square matrix of order and suppose that , where , is a spectral decomposition of . Then, max }| (6) Proof: Notice that = ¯ } max }| (7) Lemmas 2 and 3 provide a suﬃcient condition for the robustness of the closed-loop system, as follows. Theorem 4 If stabilizes and max j j 1 + j j )[1 + )] (8) for each , where ) = j )] with j denoting the eigenvector matrix of j j , then stabilizes , given by Eq. (3). Proof: See Moreira (2006) Remark 1 Similar results to those given in the- orem 4 and inequality (5) can be obtained con- sidering a multiplicative perturbation at the plant input, i.e. ) = )[ )] . In this case the upper bounds are based on the eigenvalues of j j and on the condition number of the eigenvector matrix of j j Lemma 2, theorem 4 and inequality (5) pro- vide all the necessary information for the design of reliable commutative controllers, as far as normal- ity of the open-loop transfer matrix is concerned. Notice that: At high frequencies j j , and thus j j )] = 1 ,...,m Therefore, conditions (5) and (8), become, re- spectively: ² < and ² < (1 + , ² (9) which are satisﬁed for arbitrarily large values of . This implies that there is no need for normalization at high frequencies. ii At low frequencies, due to integral action, j j becomes inﬁnite and thus, ac- cording to inequality (4), becomes a necessary and suﬃcient condition for the robustness of the closed-loop system. This implies that the suﬃcient condition given by inequality (8) becomes [1+ )] and thus, normalization is not a crucial re- quirement for low sensitivity of the charac- teristic loci at low frequencies. iii At the frequencies near the crossover fre- quency , condition (8) is also necessary when i.e. , if j j becomes approximately normal. The discussion above allows us to conclude that it suﬃces to use a static precompensator, in- stead of a dynamic normalizing precompensator as designed in Basilio and Sahate (2000) and Moreira and Basilio (2005). As pointed out in the introduction, normaliza- tion should be understood as the ﬁrst step in the design of commutative controllers, i.e. , if j is not normal at the frequency band near the crossover frequency, then a static normalizing pre- compensator will be designed in order to make j approximately normal in this frequency band. After that, a controller that com- mutes with must be designed. In the end, the controller to be implemented will be given as: ) = (10) This paper deals with the design of an opti- mal static precompensator that makes both j j and j j approxi- mately normal at the frequency range of interest. 3 A normalizing precompensator 3.1 Problem formulation In this paper, it is assumed multiplicative pertur- bations to model uncertainties in the plant trans- fer matrix. This choice of representation is prefer- able over an additive representation (Doyle and Stein, 1981), since the same uncertainty represen- tation can be applied to both the plant and the compensated system . As a conse- quence, robustness of the closed-loop system may be considered only at the ﬁnal stage of the con- troller design, namely, the design of the commu- tative controller With the view to considering the normaliza- tion of both and it is in- troduced in Hung and MacFarlane (1982) the so called reversed-frame-normalizing-controllers (RFNC), where it can be seen that both and are normal if and only if the singular-vector frames of are those of taken in reversed order, as stated in the following lemma. Lemma 5 Suppose and are both of rank and let (11) 1741 of 1744

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be a singular value decomposition of , where Σ = diag ,i = 1 ,...,m . Then GK and KG are both normal if and only if (12) for some nonsingular diagonal matrix Proof: See Hung and MacFarlane (1982). According to lemma 5, the characteristic loci are at their least sensitivity to small perturba- tions at the plant input and output if and only if the controller has the structure given by Eq. (12). However, in this paper, the controller is deﬁned by Eq. (10), which implies that must satisfy some conditions such that and be both normal, as stated below. Theorem 6 Suppose and are both of rank and let the singular value decompo- sition of be given by Eq. (11). In addition let , where commutes exactly with GK . Then, GK and KG are both normal matri- ces if and only if (13) for some nonsingular diagonal matrix Proof: See Moreira and Basilio (2005). According to theorem 6, at a given fre- quency , to achieve the exact normality of j j and j j one should choose any nonsingular diagonal matrix and obtain according to Eq. (13). However, in this paper, it is considered the computation of a normalizing static precompensator, which implies that must be a real matrix. It can be veriﬁed that, since j j , then, only for special cases it is possible to obtain real. In Hung and MacFarlane (1982), the design of a static precompensator with the structure given by Eq. (13) is considered in two cases where it is pos- sible to obtain real precompensators, namely that, at d.c. frequency ( = 0 ) and at very high fre- quencies ( ). In these situations j j . However, the need for normal- ization is more critical at frequencies where the characteristic loci are close to the critical point 1 + i.e. , at intermediate frequencies, as ex- plained in section 2. Although exact normalities of j j and j j at intermediate frequencies are usually not achievable, it is more realistic to aim at approximate normality. The following results relates the approximate normality of j j and j j to the approximate normality of j and j , respectively. Theorem 7 Let be given by Eq. (10) and let be a given frequency. Then j j is approximately normal if j is approximately normal. Proof: The proof is straightforward and will be omitted. Theorem 8 Under the same assumptions of the- orem 7, and assuming that all eigenvalues of j are distinct, then j j is also approximately normal if j is approxi- mately normal. Proof: Let a singular value decomposition of j be given by Eq. (11) and deﬁne j ) = j j (14) for a nonsingular matrix . From Eqs. (11) and (14), it follows that: GK UMY MY (15) Suppose now that has the following spectral decomposition: (16) where is a diagonal matrix, is an eigen- vector matrix of and . Then, ac- cording to Eqs. (15) and (16), GK can be written as: GK Y W (17) Since, by assumption, commutes exactly with GK , it has the same eigenvector matrix as GK and, therefore, a spectral decomposition of can be given as: Y W (18) where is a diagonal matrix formed with the eigenvalues of . Thus, according to Eqs. (14), (18) and (11), KG can be written as: KG UMW (19) Notice that, according to Eq. (16), Σ = , and thus, Eq. (19), turns out to be: KG UMW (20) Thus, according to Eq. (20), for every commuta- tive controller , an eigenvector matrix of KG is given by UMW . In particular, for , a spectral decomposition of is given by: UMW (21) Supposing that all eigenvalues of are distinct, then the eigenvector matrix of is equal to UMW scaled to have the least condition num- ber as possible (Braatz and Morari, 1994). There- fore, if the eigenvector matrix of has condi- tion number approximately equal to one, i.e. , if is approximately normal, then KG is also approximately normal. Theorems 6, 7 and 8 lead to the formulation of the following optimization problem with the view 1742 of 1744

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to ﬁnding a real precompensator that approx- imately normalizes j and j at a given frequency min (22) subject to be a nonsingular diago- nal matrix, and , where j and j are obtained, according to Eq. (11), from a singular value decomposition of j 3.2 Solution to the optimization problem Deﬁne the cost function of problem (22) as: Φ) = (23) Then, applying the deﬁnition of Frobenius norm, given by Eq. (2), in Eq. (23), and after some simple algebraic manipulations, one obtains: Φ)= tr Re tr tr ( Φ) (24) where Re denotes the real part of a complex matrix. Let ), ), and ) be the real (imaginary) parts of , and , respec- tively. Thus: j ) = j ) + jU j j ) = j ) + jY j Φ( j ) = j ) + j (25) Using Eqs. (25), then it is possible to write Eq. (24) as follows: Φ) = tr tr ( Φ) (26) where Re tr [( jU jY )( )] (27) After some simple algebraic manipulations then Eq. (27) can be written as: tr ) + tr tr ) + tr (28) Notice that each term of the right-hand side of Eq. (28) has the same structure given by tr Φ) , where and are equal to the real or imaginary parts of and , respec- tively. Deﬁne, now, the vector as the column vector formed with the diagonal entries of and let = 1 ,...,m , denote the columns of and form the vector ... k . Thus, denot- ing the columns of by , for = 1 ,...,m and the elements of by ij , then it can be veriﬁed that: tr Φ) = (29) where 11 12 ... 21 22 ... ... mm (30) Therefore, deﬁning the vectors and as the column vectors formed with the diagonal entries of and , respectively, and forming the vector , then Eq. (24) can be rewritten as: , ) = A (31) where RR II RI IR (32) with the matrices RR II IR and RI formed in accordance with Eq. (30), where the ﬁrst and second indexes represent, respectively, the real (R) or imaginary (I) part of or From Eq. (31), it can be seen that if one ﬁxes the values of the elements of vector , then the cost function , describes a paraboloid with minimum given by: min ) = (33) where A , and, therefore, the problem of min- imizing , , given by Eq. (31), is equivalent to the following optimization problem: min (34) subject to be such that is a nonsingular ma- trix. Remark 2 Notice that the cost function of prob- lem (34) is equivalent to the norm deﬁned in Eq. (23). Therefore, the matrix is, in gen- eral, positive deﬁnite (it can also be positive semi- deﬁnite). Since must be a nonsingular matrix, then the Euclidean-norm of the vector must be dif- ferent from zero. Thus, suppose that = 1 where denotes the Euclidean-norm. Then, using the Rayleigh’s principle (Horn and John- son, 1990) and remark 2, it can be seen that the minimum value for problem (34) is achieved for , where denotes the eigenvector associ- ated with the smallest eigenvalue, , of the ma- trix . Moreover, the minimum value of the cost function of problem (34) is given by Remark 3 In general, the eigenvector associ- ated with the smallest eigenvalue of leads to a matrix nonsingular. However, if the matrix associated with is singular, then it is always possible to make a linear combination of the eigenvectors of , in order to obtain a vector , such that and obtained from , be nonsingular. Thus, must be normalized leading to a cost value min ) = , where 1743 of 1744

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4 Examples Consider the transfer matrix of an automotive gas turbine (Nobakhti and Munro, 2004) ) = ij )] , where 11 ) = 806 + 0 264 + 1 15 + 0 202 12 ) = 15 42 + 12 + 13 + 2 36 21 ) = 95 + 2 12 + 0 49 + 9 15 + 9 39 + 1 62 22 ) = 14 + 25 + 9 35 + 20 + 116 + 111 + 18 and suppose, as speciﬁed in Nobakhti and Munro (2004), that the closed-loop bandwidth be equal to 1 rad/s. The condition number of the eigenvector ma- trix of j is shown in Fig. 2 (solid line), where it can be seen that it is larger than one at the de- sired bandwidth frequency (it is approximately 2.6 at 0.4 rad/s). As a consequence, it is necessary to design a normalizing precompensator. Let 1) = 1)Σ( 1) 1) be a singular value decomposition of 1) , where 1) and 1) are given as: 1)= 5612 + 0 8181 1101 0606 0454 1172 8857 + 0 4470 1)= 5469 8372 8200 1689 5356 + 0 1103 Forming the matrix , according to Eq. (32): 3069 0 0921 0 4474 0507 5984 0 0523 5760 0446 0248 0 7415 0641 0 3742 0174 0 5237 0 1037 0 1417 and computing now the smallest eigenvalue of , and the eigenvector associated with , one obtains: = 0 0001 and 4750 6514 0 5238 0 2751 Therefore, the optimal normalizing precompen- sator is given by: 4262 5642 5642 4262 (35) The condition numbers of the eigenvector matrices of j and j are also both equal for all frequencies, and, in Fig. 2, it can be seen that the condition number of the eigenvector matrix of j (dash-dotted line), is equal to one at 1 rad/s as desired, and also approximately equal to one at the vicinity of 1 rad/s. Therefore, the CLM can now be eﬀectively applied to the precompen- sated plant Acknowledgement This work has been partially supported by the Brazilian Research Council (CNPq). 10 −3 10 −2 10 −1 10 10 10 10 (rad/s) Condition numbers Figure 2: Condition numbers of the eigenvec- tor matrices of j (solid line), and j (dash-dotted line) for example 2. References Basilio, J. C. and Sahate, J. A. (2000). A normalizing precompensator for the design of eﬀective and reli- able commutative controllers, International Journal of Control 73 : 1280–1297. Braatz, R. D. and Morari, M. (1994). Minimizing the euclidian condition number, SIAM J. Control and Optimization 32 : 1763–1768. Doyle, J. C. and Stein, G. (1981). Multivariable feed- back design: Concepts for a classic/modern syn- thesis, IEEE - Transactions on Automatic Control 26 : 4–16. Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis Cambridge University Press. Hung, Y. S. and MacFarlane, A. G. J. (1982). Multivariable Feedback: A Quasi-Classical Approach , Springer- Verlag. MacFarlane, A. G. J. and Belletruti, J. J. (1973). The char- acteristic locus design method, Automatica : 575 588. MacFarlane, A. G. J. and Postlethwaite, I. (1977). The gen- eralized nyquist stability criterion and multivariable root loci, International Journal of Control 25 : 81 127. Moreira, M. V. (2006). Uma contribui¸c˜ao por meio de pr´e- compensa¸c˜ao normalizante e parametriza¸c˜ao para o efetivo projeto de controladores utilizando o m´etodo do lugar caracter´ıstico , D.Sc. Thesis, UFRJ/COPPE - Programa de Engenharia El´etrica. Moreira, M. V. and Basilio, J. C. (2004). Controladores comutativos estabilizantes para plantas inst´aveis, XV Congresso Brasileiro de Autom´atica , Gramado: RS. Moreira, M. V. and Basilio, J. C. (2005). Design of normal- izing precompensators via alignment of output-input principal directions, 44th IEEE Conference on Deci- sion and Control and European Control Conference Sevilla : Spain, pp. 2170–2175. Moreira, M. V., Basilio, J. C. and Kouvaritakis, B. (2005). Rational stabilizing commutative controllers for un- stable plant, 44th IEEE Conference on Decision and Control and European Control Conference , Sevilla : Spain, pp. 7864–7869. Moreira, M. V., Basilio, J. C. and Kouvaritakis, B. (2006). Rational stabilising commutative controllers: para- meterisation and characterisation of degrees of free- dom, International Journal of Control (accepted for publication) Nobakhti, A. and Munro, N. (2004). A new method for sin- gular value loop shaping in design of multiple-channel controllers, IEEE-Transactions on Automatic Con- trol 49 : 249–253. Postlethwaite, I. (1982). Sensitivity of the characteristic gain loci, Automatica 18 : 709–712. Vidyasagar, M. (1985). Control System Synthesis: A Fac- torization Approach , MIT Press. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Prob- lem , Clarendon Press, Oxford. 1744 of 1744

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OPTIMAL DESIGN OF STATIC NORMALIZING PRECOMPENSATORS Marcos Vicente Moreira , Jo ao Carlos Basilio COPPE - Programa de Engenharia El´etrica Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brasil Emails: moreira@pee.coppe.ufrj.br, basilio@dee.ufrj.br Abstract Normalization is the ﬁrst step in the design of multivariable controllers within the Characteristic Locus Method, i.e. , if the plant transfer matrix is far from normal at the frequency band of interest, then a normalizing precompensator should be designed in order to make the precompensated plant approximately normal at this frequency band. This problem is addressed in this paper, where an optimal static normalizing precompensator is proposed. The design strategy is based on an optimization problem with no constraints on the precompensator structure, and whose solution can be obtained directly from an eigenvalue problem. Keywords Linear systems, Multivariable systems, Optimization, Frequency domain. 1 Introduction The Characteristic Locus Method (CLM) is an ef- fective tool for the design of multivariable con- trol systems, allowing conﬂicting design objec- tives, such as stability and performance, be ad- dressed by a single system representation: namely that, the frequency response of the eigenfunctions of the open-loop system (characteristic loci). The essence of the CLM (MacFarlane and Belletruti, 1973) is to construct a commutative controller, i.e. , a controller with the same eigen- vector and dual-eigenvector matrices (frames) as the plant transfer matrix and to manipulate the controller eigenfunctions so as to achieve closed- loop stability, satisfying the generalized Nyquist stability criterion (MacFarlane and Postlethwaite, 1977), and to satisfy performance requirements such as tracking, disturbance rejection and good transient response. However, it is well known (Doyle and Stein, 1981; Postlethwaite, 1982; Wilkinson, 1965) that for plants whose frequency responses are not nor- mal at a certain frequency band, the character- istic loci are very sensitive to perturbations at the plant input and output at these frequencies. Therefore, in order to deal with the characteristic locus sensitivity problem, the design of multivari- able controllers within the CLM, must be done in two stages (Basilio and Sahate, 2000; Moreira and Basilio, 2005): ﬁrst, precompensate the plant in order to approximately normalize it in the nec- essary frequency range and then, ii apply the CLM to the precompensated system. A precompensation scheme with the view to making the precompensated plant as normal as possible has recently been presented (Basilio and Sahate, 2000), where the design of a nor- malizing dynamic precompensator is proposed. The main disadvantages of the precompensation method proposed in Basilio and Sahate (2000) are: the characteristic locus sensitivity with re- spect to perturbations at the plant input are not considered in the formulation of the optimization problem, and ii an optimization problem is for- mulated and solved for a speciﬁc precompensator structure. With the view to considering the sensitivity of the characteristic loci with respect to pertur- bations at both the plant input and output, it is proposed in Moreira and Basilio (2005) an opti- mization problem based on a measure of the mis- alignment between the input and output principal directions of the plant. Although this precompen- sation scheme has led to insensitive characteris- tic loci for the precompensated plant when an ex- act commutative controller is considered, it also imposes constraints on the precompensator struc- ture. In this paper, the design of an optimal sta- tic precompensator with the view to making the precompensated plant approximately normal at the vicinity of the crossover frequency is pro- posed. This new method is based on an optimiza- tion problem which has no constraints on the pre- compensator structure. The solution to the op- timization problem is very simple and can be ex- pressed as an eigenvalue problem. Another advan- tage of this method is that it improves the char- acteristic locus sensitivity indices with respect to perturbations at both the plant input and output when an exact commutative controller (Moreira and Basilio, 2004; Moreira et al., 2005; Moreira et al., 2006) is considered. 2 Main concepts Let and be the transfer matrices of the plant and controller, respectively. Accord- ing to the generalized Nyquist stability criterion (MacFarlane and Postlethwaite, 1977), the feed- back system of Fig. 1 will be stable if and only if the net sum of anti-clockwise encirclements of the critical point 1 + , by the characteristic loci of , is equal to the number of unstable poles of and 1739 of 1744

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±° ²¯ Figure 1: Closed-loop feedback control system. In order to be able to use the generalized Nyquist stability criterion as a design tool, a con- troller such that ) = is sought. This condition is satisﬁed providing that and share the same eigenvector and dual eigenvector frames and, therefore, the eigenvalues of the product are equal to the product of the eigenvalues of and . Thus, to achieve closed-loop stability and to satisfy performance requirements, such as track- ing, disturbance rejection and good transient re- sponse, one can manipulate the eigenfunctions of the open-loop transfer matrix, choosing ade- quately the eigenfunctions of the controller providing that the eigenvectors of are those of . However, in some cases, the character- istic loci can be very sensitive to uncertainties in the plant transfer function and therefore, in these cases, the CLM is not a reliable tool for the design of a multivariable control system. In or- der to show this Doyle and Stein (1981) consider a stable matrix perturbation, , at any point in the conﬁguration of Fig. 1, and show that the closed-loop system remains stable if and only if j )] multiplied by j )] is less than , for all frequencies, where de- notes the maximum singular value of a matrix and is the return ratio matrix for the point where the uncertainty is considered . Since the CLM is based on the eigenvalues of the open-loop trans- fer matrix [or, equivalently, and not on its singular values, then it is shown in Doyle and Stein (1981) that the CLM can only be eﬀective from the robustness point of view only if the plant transfer matrix is normal. The deﬁnition of normal matrix is as follows. Deﬁnition 1 A matrix is normal if it commutes with its conjugate transpose, i.e. GG In order to deal with the sensitivity problem of the characteristic loci within the CLM, it is pro- posed in Basilio and Sahate (2000), the design of a controller in two stages. The ﬁrst stage is the design of a dynamic normalizing precom- pensator, i.e. , a precompensator which ap- proximately normalizes the precompensated plant j j at the frequency range of interest. The second stage is the design of a controller, , that commutes with and also In general, it is more common to consider the uncer- tainty at the plant input or at the plant output, which corresponds to ) = or ) = , re- spectively. internally stabilizes the feedback system. With the view to deﬁning approximately normal matri- ces, it is proposed in Basilio and Sahate (2000) the following measure of the deviation from nor- mality: ) = GG (1) where denotes the Frobenius norm, which, for a matrix is deﬁned as: tr (2) where tr denotes the trace of a matrix. Other ways of measuring the approximate normality of a complex matrix is presented in Hung and Mac- Farlane (1982). One of these measures is based on the following theorem. Theorem 1 A matrix is normal if and only if it has an orthonormal set of eigen- vectors. Proof: See Horn and Johnson (1990). Theorem 1 leads to the following deﬁnition of approximately normal matrices. Deﬁnition 2 A matrix is said to be an approximately normal matrix if ) = 1 + where denotes condition number, is the eigenvector matrix of , and deﬁnes the degree of normality of The condition number of the eigenvector ma- trix of j j and j j can be used to provide an upper bound for a measure of the robustness of the closed-loop system with respect to perturbations at the plant output and input, respectively. This will be done with the help of the following result. Lemma 2 (Small Gain Theorem) Deﬁne ) = [ )] (3) as the plant transfer matrix with multiplica- tive perturbation at the plant output, where j )] , and is a non-negative real function. Then, stabilizes if and only if stabilizes and j j ) [ j j )] (4) for all Proof: See Vidyasagar (1985). It is well known that singular values are upper bounds for the moduli of the eigenvalues of a ma- trix, and thus, a necessary condition for robust stability of the closed-loop system, given in terms of the eigenvalues of the open-loop system at each frequency, can be given as: max j j 1 + j j (5) 1740 of 1744

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In order to obtain now a suﬃcient condition, the following result is needed. Lemma 3 Let be a square matrix of order and suppose that , where , is a spectral decomposition of . Then, max }| (6) Proof: Notice that = ¯ } max }| (7) Lemmas 2 and 3 provide a suﬃcient condition for the robustness of the closed-loop system, as follows. Theorem 4 If stabilizes and max j j 1 + j j )[1 + )] (8) for each , where ) = j )] with j denoting the eigenvector matrix of j j , then stabilizes , given by Eq. (3). Proof: See Moreira (2006) Remark 1 Similar results to those given in the- orem 4 and inequality (5) can be obtained con- sidering a multiplicative perturbation at the plant input, i.e. ) = )[ )] . In this case the upper bounds are based on the eigenvalues of j j and on the condition number of the eigenvector matrix of j j Lemma 2, theorem 4 and inequality (5) pro- vide all the necessary information for the design of reliable commutative controllers, as far as normal- ity of the open-loop transfer matrix is concerned. Notice that: At high frequencies j j , and thus j j )] = 1 ,...,m Therefore, conditions (5) and (8), become, re- spectively: ² < and ² < (1 + , ² (9) which are satisﬁed for arbitrarily large values of . This implies that there is no need for normalization at high frequencies. ii At low frequencies, due to integral action, j j becomes inﬁnite and thus, ac- cording to inequality (4), becomes a necessary and suﬃcient condition for the robustness of the closed-loop system. This implies that the suﬃcient condition given by inequality (8) becomes [1+ )] and thus, normalization is not a crucial re- quirement for low sensitivity of the charac- teristic loci at low frequencies. iii At the frequencies near the crossover fre- quency , condition (8) is also necessary when i.e. , if j j becomes approximately normal. The discussion above allows us to conclude that it suﬃces to use a static precompensator, in- stead of a dynamic normalizing precompensator as designed in Basilio and Sahate (2000) and Moreira and Basilio (2005). As pointed out in the introduction, normaliza- tion should be understood as the ﬁrst step in the design of commutative controllers, i.e. , if j is not normal at the frequency band near the crossover frequency, then a static normalizing pre- compensator will be designed in order to make j approximately normal in this frequency band. After that, a controller that com- mutes with must be designed. In the end, the controller to be implemented will be given as: ) = (10) This paper deals with the design of an opti- mal static precompensator that makes both j j and j j approxi- mately normal at the frequency range of interest. 3 A normalizing precompensator 3.1 Problem formulation In this paper, it is assumed multiplicative pertur- bations to model uncertainties in the plant trans- fer matrix. This choice of representation is prefer- able over an additive representation (Doyle and Stein, 1981), since the same uncertainty represen- tation can be applied to both the plant and the compensated system . As a conse- quence, robustness of the closed-loop system may be considered only at the ﬁnal stage of the con- troller design, namely, the design of the commu- tative controller With the view to considering the normaliza- tion of both and it is in- troduced in Hung and MacFarlane (1982) the so called reversed-frame-normalizing-controllers (RFNC), where it can be seen that both and are normal if and only if the singular-vector frames of are those of taken in reversed order, as stated in the following lemma. Lemma 5 Suppose and are both of rank and let (11) 1741 of 1744

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be a singular value decomposition of , where Σ = diag ,i = 1 ,...,m . Then GK and KG are both normal if and only if (12) for some nonsingular diagonal matrix Proof: See Hung and MacFarlane (1982). According to lemma 5, the characteristic loci are at their least sensitivity to small perturba- tions at the plant input and output if and only if the controller has the structure given by Eq. (12). However, in this paper, the controller is deﬁned by Eq. (10), which implies that must satisfy some conditions such that and be both normal, as stated below. Theorem 6 Suppose and are both of rank and let the singular value decompo- sition of be given by Eq. (11). In addition let , where commutes exactly with GK . Then, GK and KG are both normal matri- ces if and only if (13) for some nonsingular diagonal matrix Proof: See Moreira and Basilio (2005). According to theorem 6, at a given fre- quency , to achieve the exact normality of j j and j j one should choose any nonsingular diagonal matrix and obtain according to Eq. (13). However, in this paper, it is considered the computation of a normalizing static precompensator, which implies that must be a real matrix. It can be veriﬁed that, since j j , then, only for special cases it is possible to obtain real. In Hung and MacFarlane (1982), the design of a static precompensator with the structure given by Eq. (13) is considered in two cases where it is pos- sible to obtain real precompensators, namely that, at d.c. frequency ( = 0 ) and at very high fre- quencies ( ). In these situations j j . However, the need for normal- ization is more critical at frequencies where the characteristic loci are close to the critical point 1 + i.e. , at intermediate frequencies, as ex- plained in section 2. Although exact normalities of j j and j j at intermediate frequencies are usually not achievable, it is more realistic to aim at approximate normality. The following results relates the approximate normality of j j and j j to the approximate normality of j and j , respectively. Theorem 7 Let be given by Eq. (10) and let be a given frequency. Then j j is approximately normal if j is approximately normal. Proof: The proof is straightforward and will be omitted. Theorem 8 Under the same assumptions of the- orem 7, and assuming that all eigenvalues of j are distinct, then j j is also approximately normal if j is approxi- mately normal. Proof: Let a singular value decomposition of j be given by Eq. (11) and deﬁne j ) = j j (14) for a nonsingular matrix . From Eqs. (11) and (14), it follows that: GK UMY MY (15) Suppose now that has the following spectral decomposition: (16) where is a diagonal matrix, is an eigen- vector matrix of and . Then, ac- cording to Eqs. (15) and (16), GK can be written as: GK Y W (17) Since, by assumption, commutes exactly with GK , it has the same eigenvector matrix as GK and, therefore, a spectral decomposition of can be given as: Y W (18) where is a diagonal matrix formed with the eigenvalues of . Thus, according to Eqs. (14), (18) and (11), KG can be written as: KG UMW (19) Notice that, according to Eq. (16), Σ = , and thus, Eq. (19), turns out to be: KG UMW (20) Thus, according to Eq. (20), for every commuta- tive controller , an eigenvector matrix of KG is given by UMW . In particular, for , a spectral decomposition of is given by: UMW (21) Supposing that all eigenvalues of are distinct, then the eigenvector matrix of is equal to UMW scaled to have the least condition num- ber as possible (Braatz and Morari, 1994). There- fore, if the eigenvector matrix of has condi- tion number approximately equal to one, i.e. , if is approximately normal, then KG is also approximately normal. Theorems 6, 7 and 8 lead to the formulation of the following optimization problem with the view 1742 of 1744

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to ﬁnding a real precompensator that approx- imately normalizes j and j at a given frequency min (22) subject to be a nonsingular diago- nal matrix, and , where j and j are obtained, according to Eq. (11), from a singular value decomposition of j 3.2 Solution to the optimization problem Deﬁne the cost function of problem (22) as: Φ) = (23) Then, applying the deﬁnition of Frobenius norm, given by Eq. (2), in Eq. (23), and after some simple algebraic manipulations, one obtains: Φ)= tr Re tr tr ( Φ) (24) where Re denotes the real part of a complex matrix. Let ), ), and ) be the real (imaginary) parts of , and , respec- tively. Thus: j ) = j ) + jU j j ) = j ) + jY j Φ( j ) = j ) + j (25) Using Eqs. (25), then it is possible to write Eq. (24) as follows: Φ) = tr tr ( Φ) (26) where Re tr [( jU jY )( )] (27) After some simple algebraic manipulations then Eq. (27) can be written as: tr ) + tr tr ) + tr (28) Notice that each term of the right-hand side of Eq. (28) has the same structure given by tr Φ) , where and are equal to the real or imaginary parts of and , respec- tively. Deﬁne, now, the vector as the column vector formed with the diagonal entries of and let = 1 ,...,m , denote the columns of and form the vector ... k . Thus, denot- ing the columns of by , for = 1 ,...,m and the elements of by ij , then it can be veriﬁed that: tr Φ) = (29) where 11 12 ... 21 22 ... ... mm (30) Therefore, deﬁning the vectors and as the column vectors formed with the diagonal entries of and , respectively, and forming the vector , then Eq. (24) can be rewritten as: , ) = A (31) where RR II RI IR (32) with the matrices RR II IR and RI formed in accordance with Eq. (30), where the ﬁrst and second indexes represent, respectively, the real (R) or imaginary (I) part of or From Eq. (31), it can be seen that if one ﬁxes the values of the elements of vector , then the cost function , describes a paraboloid with minimum given by: min ) = (33) where A , and, therefore, the problem of min- imizing , , given by Eq. (31), is equivalent to the following optimization problem: min (34) subject to be such that is a nonsingular ma- trix. Remark 2 Notice that the cost function of prob- lem (34) is equivalent to the norm deﬁned in Eq. (23). Therefore, the matrix is, in gen- eral, positive deﬁnite (it can also be positive semi- deﬁnite). Since must be a nonsingular matrix, then the Euclidean-norm of the vector must be dif- ferent from zero. Thus, suppose that = 1 where denotes the Euclidean-norm. Then, using the Rayleigh’s principle (Horn and John- son, 1990) and remark 2, it can be seen that the minimum value for problem (34) is achieved for , where denotes the eigenvector associ- ated with the smallest eigenvalue, , of the ma- trix . Moreover, the minimum value of the cost function of problem (34) is given by Remark 3 In general, the eigenvector associ- ated with the smallest eigenvalue of leads to a matrix nonsingular. However, if the matrix associated with is singular, then it is always possible to make a linear combination of the eigenvectors of , in order to obtain a vector , such that and obtained from , be nonsingular. Thus, must be normalized leading to a cost value min ) = , where 1743 of 1744

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4 Examples Consider the transfer matrix of an automotive gas turbine (Nobakhti and Munro, 2004) ) = ij )] , where 11 ) = 806 + 0 264 + 1 15 + 0 202 12 ) = 15 42 + 12 + 13 + 2 36 21 ) = 95 + 2 12 + 0 49 + 9 15 + 9 39 + 1 62 22 ) = 14 + 25 + 9 35 + 20 + 116 + 111 + 18 and suppose, as speciﬁed in Nobakhti and Munro (2004), that the closed-loop bandwidth be equal to 1 rad/s. The condition number of the eigenvector ma- trix of j is shown in Fig. 2 (solid line), where it can be seen that it is larger than one at the de- sired bandwidth frequency (it is approximately 2.6 at 0.4 rad/s). As a consequence, it is necessary to design a normalizing precompensator. Let 1) = 1)Σ( 1) 1) be a singular value decomposition of 1) , where 1) and 1) are given as: 1)= 5612 + 0 8181 1101 0606 0454 1172 8857 + 0 4470 1)= 5469 8372 8200 1689 5356 + 0 1103 Forming the matrix , according to Eq. (32): 3069 0 0921 0 4474 0507 5984 0 0523 5760 0446 0248 0 7415 0641 0 3742 0174 0 5237 0 1037 0 1417 and computing now the smallest eigenvalue of , and the eigenvector associated with , one obtains: = 0 0001 and 4750 6514 0 5238 0 2751 Therefore, the optimal normalizing precompen- sator is given by: 4262 5642 5642 4262 (35) The condition numbers of the eigenvector matrices of j and j are also both equal for all frequencies, and, in Fig. 2, it can be seen that the condition number of the eigenvector matrix of j (dash-dotted line), is equal to one at 1 rad/s as desired, and also approximately equal to one at the vicinity of 1 rad/s. Therefore, the CLM can now be eﬀectively applied to the precompen- sated plant Acknowledgement This work has been partially supported by the Brazilian Research Council (CNPq). 10 −3 10 −2 10 −1 10 10 10 10 (rad/s) Condition numbers Figure 2: Condition numbers of the eigenvec- tor matrices of j (solid line), and j (dash-dotted line) for example 2. References Basilio, J. C. and Sahate, J. A. (2000). A normalizing precompensator for the design of eﬀective and reli- able commutative controllers, International Journal of Control 73 : 1280–1297. Braatz, R. D. and Morari, M. (1994). Minimizing the euclidian condition number, SIAM J. Control and Optimization 32 : 1763–1768. Doyle, J. C. and Stein, G. (1981). Multivariable feed- back design: Concepts for a classic/modern syn- thesis, IEEE - Transactions on Automatic Control 26 : 4–16. Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis Cambridge University Press. Hung, Y. S. and MacFarlane, A. G. J. (1982). Multivariable Feedback: A Quasi-Classical Approach , Springer- Verlag. MacFarlane, A. G. J. and Belletruti, J. J. (1973). The char- acteristic locus design method, Automatica : 575 588. MacFarlane, A. G. J. and Postlethwaite, I. (1977). The gen- eralized nyquist stability criterion and multivariable root loci, International Journal of Control 25 : 81 127. Moreira, M. V. (2006). Uma contribui¸c˜ao por meio de pr´e- compensa¸c˜ao normalizante e parametriza¸c˜ao para o efetivo projeto de controladores utilizando o m´etodo do lugar caracter´ıstico , D.Sc. Thesis, UFRJ/COPPE - Programa de Engenharia El´etrica. Moreira, M. V. and Basilio, J. C. (2004). Controladores comutativos estabilizantes para plantas inst´aveis, XV Congresso Brasileiro de Autom´atica , Gramado: RS. Moreira, M. V. and Basilio, J. C. (2005). Design of normal- izing precompensators via alignment of output-input principal directions, 44th IEEE Conference on Deci- sion and Control and European Control Conference Sevilla : Spain, pp. 2170–2175. Moreira, M. V., Basilio, J. C. and Kouvaritakis, B. (2005). Rational stabilizing commutative controllers for un- stable plant, 44th IEEE Conference on Decision and Control and European Control Conference , Sevilla : Spain, pp. 7864–7869. Moreira, M. V., Basilio, J. C. and Kouvaritakis, B. (2006). Rational stabilising commutative controllers: para- meterisation and characterisation of degrees of free- dom, International Journal of Control (accepted for publication) Nobakhti, A. and Munro, N. (2004). A new method for sin- gular value loop shaping in design of multiple-channel controllers, IEEE-Transactions on Automatic Con- trol 49 : 249–253. Postlethwaite, I. (1982). Sensitivity of the characteristic gain loci, Automatica 18 : 709–712. Vidyasagar, M. (1985). Control System Synthesis: A Fac- torization Approach , MIT Press. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Prob- lem , Clarendon Press, Oxford. 1744 of 1744

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