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International Journal of Control Vol. 79, No. 7, July 2006, 707–718 Diagonal dominance via eigenstructure assignment B. LABIBI*, H. J. MARQUEZ and T. CHEN Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada Received 21 October 2005; in final form 21 February 2006 This paper presents a new methodology for diagonal dominance of large-scale systems via eigenstructure assignment. For a given large-scale system in general form, an equivalent descriptor system in the input–output decentralized form is defined. Sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance attenuation via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating disturbances, closed-loop poles of the overall system are placed in a desirable region, by assigning the eigenstructure of each isolated subsystem appropriately. The presented algorithm alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. The designed controller is in the decentralized form and plays the role of pre-compensator as well. An illustrative example is given to show the effectiveness of the proposed method. 1. Introduction The classical techniques of frequency domain design for single-input–single-output systems have been general- ized and applied to multivariable feedback systems by Hawkins (1972), Rosenbrock (1974), MacFarlane and Kouvaritakis (1977) and Patel and Munro (1985). Rosenbrock’s diagonal dominance design method con- sists essentially of determining a multivariable pre- compensator matrix , constant or dynamic, such that the resulting forward path-transfer function is diagonal dominant. When this condition is fulfiled, a diagonal compensator matrix ) can be used to implement single-loop compensators as required to meet the overall design specifications (Chughtai and Munro 2004). Various attempts have been made to determine a pre-compensator such that the resulting system is diagonal dominant. In many cases, however, this is very difficult to achieve. Manual cut-and-try methods, which were the ones originally suggested, often fail to achieve diagonal dominance with reasonable effort (Maciejowski 1989). Hawkins (1972) has proposed a pseudo-diagonalization scheme which attempts to minimize the off-diagonal effects at specific frequencies using a constant compensator. In Hawkins (1972) it is assumed that inverse arrays are to be used, but his method can be applied equally well to direct arrays; the methodology gives a constrained minimization. By using Lagrange multiplier methods, this can be shown to lead to an eigenvalue problem, so the solution can be obtained easily (Patel and Munro 1982). Ford and Daly (1979) has extended this approach to dynamic compensators. Pesudo-diagonalization can be applied to either direct or inverse Nyquist arrays (Maciejowski 1989). A practical difficulty arises if a dynamic compensator is found for an inverse array: its inverse needs to be realizable (Maciejowski 1989). MacFarlane and Kouvaritakis (1977) has suggested the ALIGN algorithm for diagonal dominance achievement. However, all of these algorithms are designed to obtain dominance at a particular frequency which is normally selected by trial and error. The overall dominance, obtained by the compensated system, depends on the selected frequency. These algorithms can be modified for the multi-frequency case, by selecting certain frequencies and finding the most *Corresponding author. Email: labibi@ece.ualberta.ca (On leave from K. N. Toosi University of Technology, Tehran, Iran) International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170600644860

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appropriate pre-compensator for them (Patel and Munro 1982). In Nobakhti et al . (2003), genetic algorithms (GA) were used to achieve diagonal dom- inance. The method is powerful and gives good results. The algorithm presented in Nobakhti et al . (2003) alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. However, it inherits the problems associated with GAs in that it requires substantial computational effort and in general the designed controller is a centralized one. In Chughtai et al . (2004) a new method for determining a constant pre-compensator for reducing the effects of interactions in multivariable systems is presented. It is shown that linear matrix inequalities (LMIs) can be used in the design of pre- compensators to achieve diagonal dominance. The method is then modified to reduce the conservatism by the introduction of scaling as an LMI variable and a path-shaping matrix. The path-shaping matrix enables a designer to find an optimal pre-compensator to achieve a specific forward path structure. In this method frequency selection is removed and the LMIs formulated in this paper are computationally cheap compared to GAs. This method however, has its own problems, i.e., it gives a centralized controller and finding path-shaping matrix is found by trial and error. The feasibility of LMIs problem should also be considered. Decentralized control with Nyquist like methods can be very effective, if one can obtain the required degree of diagonal dominance fairly easily. Some attempts have been made to obtain a diagonal compensator to achieve diagonal dominance. Mees (1981) suggested that the diagonal dominance with a real, constant compensator may be obtained by choosing the elements of a constant diagonal compensator to be the elements of the Perron- Frobenius right eigenvectors of the properly approxi- mated matrix. Patel and Munro (1982) pointed out that dynamic compensation could be used for the original transfer matrix to obtain diagonal dominance. All of these, however, involve simultaneous dynamic input and output scaling which may be unacceptable. In fact, we must be careful about using output scaling as part of the strategy for achieving diagonal dominance. Roughly speaking, output scaling corresponds to inserting a post- compensator (that is, inserting a compensator between the outputs and variables being controlled). This is physically impossible, since the meaningful plant outputs cannot be affected by mathematical operations (Nwokah and Yau 1993). The first major drawback of all the existing decentralized control methods based on generalized diagonal dominance is the fact that a compensated open-loop diagonally dominant system in no way guarantees that the resulting closed-loop system is also diagonally dominant. On the other hand, closed- loop diagonal dominance is necessary for almost decoupled closed-loop response. More fundamentally, there is a need to investigate what level of open-loop dominance will guarantee a desired level of the closed- loop dominance. To do this, an appropriate measure of dominance must be defined. Decentralized design methods that address the above issue are therefore desirable (Nwokah and Yau 1993). In Labibi (2005) a new method for closed-loop diagonal dominance is proposed. In this methodology, designing a separate compensator to achieve diagonal dominance is not required. The designed controller consists of both compensator and control. The frequency selection step is removed, and in spite of the previous pieces of work, it gives closed-loop diagonal dominance. In Labibi (2005) in order to achieve sufficient conditions for overall closedl-loop diagonal dominance for a given large-scale system, interactions between subsystems are considered as external disturbances for each isolated subsystem; then, in order to attenuate the disturbances, the method proposed in Duan et al. (2000) is used. The method suggested in Duan et al . (2000) is based on parametric eigenstructure assignment which was introduced by Fahmy and O’Reilly (1983) and Roppenecker (1983) in the early 1980s. Parametric eigenstructure assignment provides, for a linear system, a parametrization of all the controllers which assign the desired closed-loop eigen- values and eigenvectors, and hence provides all the degrees of freedom available in the controller design (Fahmy and O’Reilly 1983, Roppenecker 1983). Many of the previous pieces of work in decentralized control, including Labibi (2005), are only applicable to systems which are in the input–output decentralized form. In this paper in order to overcome the above mentioned problem, the method in Labibi (2005) is extended to the general case where the system is not in the input output decentralized form. The proposed methodology can be applied to non-minimum phase, non-square and open-loop unstable systems and guarantees a closed- loop diagonally dominant system; this is achieved using a decentralized controller. In order to simplify the design procedure and to eliminate the interaction due to input output centralization, for a given large-scale system, a finite-spectrum-equivalent descriptor system in the input–output decentralized form is introduced. This new proposed representation converts a general system in which the matrices and are non-block diagonal, to an equivalent descriptor system in the input–output decentralized form, where and are block-diagonal and the interaction between subsystems is only due to non-block-diagonal state matrix, (Labibi 2001). The results of Labibi (2005) are then extended to systems in the general form. It is to be noted the descriptor system concept is a mathematical tool employed in the proof of the relevant theorems and by proving the theorems, 708 B. Labibi et al.

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the design will be done for isolated conventional subsystems. The paper is organized as follows. In 2, the problem of finding suitable decentralized dynamical controllers for the subsystems of a linear large-scale system is presented. In 3, the eigenstructure assignment method- ology proposed in Duan et al . (2000) is examined. In 4, a descriptor system for the large-scale system is defined, such that the equivalent system is in the input–output decentralized form. In 5 new sufficient conditions for closed-loop diagonal dominance are derived. It is shown that by assigning the closed-loop eigenstructure of each isolated subsystem appropriately, interactions between subsystems are attenuated and closed-loop poles are tried to be assigned in the desirable region. In 6an illustrative example is given to show the effectiveness of the proposed method. 2. The problem formulation Consider a large-scale system ), with the following state-space equations Ax Bu Cx where and Writing 11 ... ... ... NN 11 ... ... ... NN , and 11 ... ... ... NN ) is composed of linear time-invariant subsystems ), described by ii ii ij ij ii ij where ii ii , and ii . The terms ij ij and ij are due to interactions of the other subsystems. The objective in this paper is to design a local output feedback dynamical controller for each isolated subsystem ii ), described by ii ii ii such that the poles are assigned in a desirable region and at the same time effects of interactions of the other subsystems are minimized. By this, the decentralized controller diag g stabilizes the overall system and also provides desired dynamical characteristics for the overall system, if some sufficient conditions are satisfied. 3. Eigenstructure assignment In this section the method for eigenstructure assignment proposed in Duan et al . (2000) is investigated. Consider a system given by state space matrices ( ). Let ( ) be controllable and ( ) observable and the matrices and be full rank. A general output dynamical compensator, of order , for the system can be written in the following form 22 21 12 11 where is the compensator state vector and ij 1, 2, are four controller coefficient matrices of appropriate dimensions. Applying the dynamical com- pensator to the system gives the following closed-loop system Diagonal dominance via eigenstructure assignment 709

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where and BK 11 CBK 12 21 CK 22 The closed loop system matrix is assumed to be non- defective, where its Jordan form is a diagonal matrix which possess better robustness than defective ones (Wilkinson 1965). Since the system is controllable and observable and the matrices and are full rank, therefore there exist right coprime factorizations of the form sI sI where 2 2 2 and 2 are all polynomial matrices, and the pairs ), )) and ( ), )) are both right coprime. Lemma 1 (see Duan (1992) for proof ) Let be controllable and be observable and s 1, 2, ... be a group of self-conjugate complex numbers. Then (a) there exist matrices K ij 1, 2, and T such that BK 11 CBK 12 21 CK 22 10 11 diag 1, 2, ... 12 hold for a set of self conjugate complex numbers s 1, 2, ... if and only if there exist vectors f ij and g ij 0, 1, 0, 1, ... p satisfying the following constraints C1 ij il and g ij il if s C2 ij 1, 2, ... here ij is the Kronecker delta (b) when constraints C1 and C2 are met the matrix V is given by 13 where V 01 02 11 12 and the matrix T is given by 14 where T 01 02 and 11 12 ... The corresponding matrices ij are either given by 11 12 11 CV 21 22 21 CV 15 with 01 02 11 12 CV and CV Or by 11 21 11 12 22 21 16 with 01 02 11 12 and The parameters ij and ij 0, 1, 2, ... represent the degrees of freedom available in the compensator design for the system. The suggested methodology for eigenstructure assignment in this section will be used to assign eigenstructure of each isolated subsystem. 4. Systems in the general form Many practical systems do not satisfy the stringent conditions given in Lunze (1992) for transforming into 710 B. Labibi et al.

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the input–output decentralized form. In order to employ the method of Labibi (2005) for systems in the general form whose state space matrices and are not block-diagonal, the method for input–output decentral- ized transformation via the descriptor system represen- tation proposed in Labibi (2001) is used. Consider a system given by equation (1), in order to obtain an input–output decentralized form, all of the inputs and outputs of the system are defined as state variables. Then the augmented system has the following equations Bu 17 where ... ... ... ... ... ... ... ... and ... ... ... From the previous definitions, it can be seen the new input and output matrices, and are block diagonal and the results of Labibi (2005) can be applied to the new input–output decentralized system. The objective is to design a decentralized controller for each isolated subsystem given by ii ii ii ii 18 where ii ii ii ii ii ... ... ... ... ... ... ... 11 ... 11 11 ... ... ... ... NN ... NN NN ... Diagonal dominance via eigenstructure assignment 711

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and ii such that the eigenvalues of the overall closed-loop system are assigned as close as possible to the desirable locations. It is well known that a descriptor system has a more complicated structure and contains not only finite dynamical modes, but also infinite modes including infinite non-dynamical and dynamical modes. The infinite dynamical modes can generate undesired impulse behaviour. To guarantee the exis- tence and uniqueness of solution for a design system, it is always assumed that the system is regular (Cobb 1984). It is simple to show that the descriptor system in (17) is regular and impulse-free, and if the isolated subsystems given by (4) are controllable and observable, the descriptor sub- systems ii ii ii are strongly controllable, observable and sE sI 19 sE sI 20 where ), ), and are transfer functions of the original large-scale system, the equivalent descriptor system, the diagonal system composed of isolated subsystems given by (4), and the diagonal descriptor system composed of isolated subsystems given by (18), respectively (Labibi 2001). Therefore, control of the descriptor system results in control of the original system with the same controller. No assumption is made on the controll- ability and observability of the isolated subsystems. Since the large scale system given in (1) is observable, it is always possible to find a similarity transforma- tion to transform the system into the output decentralized form where matrix is block diagonal and ij 0 for (Lunze 1992). This transformation is based on the observability matrix of the original system given in (1) and is defined by 21 here, is the th row of the output matrix , and is the degree of the th subsystem. Then the isolated subsystems are observable and by appropriately decomposing the input matrix , it is possible to have controllable isolated subsystems. Hence, without loss of generality it is assumed the th isolated subsystem has the following equations or can be transformed to this case by using appropriate similar- ity transformation ii ii ij ij ii 22 Similarity transformations do not affect output feedback controllers. Therefore using a similarity transformation does not affect the diagonal dominance property. 5. Decentralized control via disturbance attenuation Consider the th isolated subsystem given by equation (4). In general the controller designed for each isolated subsystem is a dynamical controller. Assuming the th controller has state-space equations given by (6), it is simple to show that designing a dynamical output feedback controller with degree for the th subsystem can be reduced to design of a static controller for the augmented subsystem with the following state-space equations ii ii ii ii ii 23 where ii ii 00 ii ii and ii ii In order to avoid complexity in formulation, without loss of generality, in the rest of the paper it is assumed that in equation (22), the augmented matrices given in (23) are used. It can be shown that the overall closed-loop system has the 712 B. Labibi et al.

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following form where is the th reference input, and sE ii ii ii ii The outputs of the th subsystem of the overall closed-loop descriptor system are given by ii hi ii ii ii 25 where ii ii iN hi and ij 00 0 ij ij 00 0 Therefore, minimizing ii , such that ii < ii 1, 26 where is a positive scalar and kk is a norm of ( ), reduces interactions between the subsystems and the output of the th closed-loop descriptor subsystem is given by ii ii ii ii 1, ... 27 where sI ii ii ii which is exactly the output of the th closed-loop system given by equations in (4). This means that ii is an approximation to ii ii The residual of ii is the matrix ii The relative error in ii is the number (Stewart 1973) ii ii < 1, ... 28 Therefore by choosing small values for ’s, this error will be reduced. Considering the th output ii ii ii 29 it can be seen that minimizing the term ii minimizes the interactions between the subsystems. Thus, in equation (29) the states of the other descriptor subsystems can be considered as external disturbances for each isolated descriptor subsystem. Since 30 where ii sI ii ii ii ii sI ii ii ii ii ii sI ii ii ii ii ii sI ii ii sI ii ii ii sI ii ii and ii sI ii ii ii ii sI ii ii ii ii ii sI ii ii ii ii it is simple to show that ii ii ABi kk 31 11 11 12 ... 11 22 21 22 ... 22 NN NN ... NN 11 11 22 22 NN NN 24 Diagonal dominance via eigenstructure assignment 713

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where ABi ii ii ii ii iN iN Thus, to attenuate the effects of disturbance to the th output, the following index can be minimized: ii sI ci 32 Here is the th column of ABi and ci ii ii ii The th closed-loop subsystem is stabilized; therefore the equation ci ci 33 has a unique solution with respect to , and this solution is also symmetric positive semi-definite. Further, it follows that the following holds (Duan et al. 2000). ii sI ci trace ii ii 34 Using equations (10)–(12) for the th subsystem, and (33), we have 35 Assuming 36 or 37 where the matrix is also symmetric positive semi- definite, we have ii sI ci trace ii ii 38 Denoting ijl , equation (35) can then be decomposed as ij il ijl ij il 1,2, ... 39 where, ij is the th left eigenvector of the th closed-loop subsystem (Duan et al . 2000), thus ijl ij il ij il 1, ... 40 The algorithm for solving the decentralized control problem can be summarized as follows. Algorithm: (a) Find an appropriate similarity transformation to transform the system into the output decentralized form if it is not. (b) For the th subsystem, select degree of controller, 0. (c) For the th subsystem, solve for the polynomial matrices ), ), ), and ), satisfying the right coprime factorization equations (8) and (9). (d) Solve for the expression in constraints C1 and C2. (e) Solve for the expression of index according to equations (38) and (40). (f) Specify the desired closed-loop eigenvalue location regions, according to the closed-loop stability and performance requirements. (g) Solve the local optimization problem, min subject to constraints C1 and C2 with some numerical optimization algorithm. If is sufficiently less than ii , the sufficient conditions for closed-loop diagonal dominance given in (28) are satisfied, go to step(h); otherwise increment by 1 and go to step (e). (h) Solve il for the th subsystem according to equation (15) or (16). Remark 1: The suggested algorithm will eventually converge. According to pole assignment theory for linear systems, the closed-loop poles of the dynamical compensation system in (1) and (6) can be arbitrary assigned when the order of the dynamical compensator is sufficiently large. Under these condition, it can be easily reasoned that the optimization problem always has a solution. However, since the minimization is generally a non-convex non-linear programming prob- lem, an optimal solution cannot often be found (Duan et al . 2000). For seeking a solution (generally subopti- mal) to the minimization, the optimization toolbox of MATLAB or the software named general algebraic modelling system (GAMS) can be used. It is also possible by combining the genetic algorithms and the gradient-based optimization to provide a technique for solving the optimization problem. The gradient-based methods lack robustness over broad spectrum of optimization functions. The direct-search techniques, which are problem independent, have been proved as a 714 B. Labibi et al.

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panacea for the difficulties associated with the tradi- tional techniques. The genetic algorithms (GA) are different from normal search methods because they search from a population of points, not a single point and use probabilistic and not deterministic transition rules. But the convergence speed of the genetic algorithms is slower than the gradient-based optimiza- tion. Although the gradient-based optimization probably gives the local optimal solution, its conver- gence speed is faster. Thus, a numerical algorithm by combining the advantages of the genetic algorithms and the gradient-based optimization may be more useful to solve the optimization problems (Patton and Liu 1994). Remark 2: The proposed algorithm has this flexibility that increasing the degrees of sub-controllers results in the increase of free parameters in the optimization problem to achieve a higher degree of diagonal dominance and even to assign the closed-loop eigen- values in exact locations. Remark 3: The robustness problem may be consid- ered in the proposed method. The condition number of the eigenvector matrix of each isolated closed-loop subsystem can be considered as a kind of eigenvalue sensitivity (Wilkinson 1965). Hence, in order to have minimum eigenvalue sensitivity in addition to diag- onal dominance, it is possible to consider a multi- objective problem. The cost function can be considered as a weighted summation of cost function as given in (38) and the condition number of the modal matrix of the th isolated closed-loop sub- system. This condition number can be parameterized with the parameters given in constraints C1 and C2 (Duan 1992). Therefore, the optimization problem can be done on the same parameter space as the previous problem. 6. Illustrative example In this section an example is provided to show the effectiveness of the proposed method. Consider the three-input, three-output, five-state aircraft model AIRC (Maciejowski 1989), described by the following state-space matrices. 00 1 132 0 0538 1712 0 0 0705 00010 00 0485 0 8556 013 2909 0 1 0532 6859 00 0 12 1 0 00 0 419 0 665 575 0 0732 10000 01000 00100 000 000 000 The system is highly interacted and figure 1 shows the Nyquist array of the open-loop system. The aim is to achieve a bandwidth of about 15 rad/sec for each loop, with little interaction between outputs. The system can be decomposed into three subsystems. Since the third subsystem is unobservable at first based on observability matrix of the system, a similarity transformation given by 100 0 0 001 0 0 000 1 0 000 0 1 101 132 0 is applied to the system. The transformed system has the following state-space matrices: 01000 6859 0 2909 0 7764 0 0788 0705 0538 0914 0 00001 01 013 0 0485 1467 8556 000 575 0 0 0732 12 1 0 000 419 0 165 10000 00100 00010 000 000 000 The transformed system is composed of three subsystems, 11 01 6859 11 575 Diagonal dominance via eigenstructure assignment 715

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11 10 11 0, 22 0538, 22 1, 22 1, 22 0, 33 01 1467 8556 33 665 33 10 33 The interaction matrices are given by AB 00000 2909 0 7764 0 0788 0 0 0732 and AB 0705 0914 0 12 0 For the transformed system, ii 1, 1, 2, 3. Minimizing the appropriately defined cost func- tions for the isolated subsystems, 0.01274, 0.0014, and 0.0119 are achieved. Hence, conditions (28) are satisfied and the overall closed- loop system is diagonal dominant. The designed decentralized controller 30 7897 12 81349 00 010 0 00 344 9785 41 9423 is applied to the original system given in equation (1). The closed-loop eigenvalues of each isolated subsystem are assigned in the desirable region to have 15 rad/sec bandwidth and the closed-loop overall system is stabilized. Figure 2 shows the Nyquist array of the closed-loop system. It is clear that the closed loop system is diagonal dominant. Figure 3 shows the Bode diagram of the closed-loop transfer function, and shows that the bandwidth requirement has been achieved. 7. Conclusion This paper has introduced new sufficient conditions for closed-loop diagonal dominance achievement of multi- variable systems. In order to simplify the design 500 0 500 500 500 REAL IMAG 200 0 200 200 100 100 REAL IMAG 100 0 100 100 100 200 REAL IMAG 100 0 100 50 50 100 REAL IMAG 20 0 20 20 20 40 REAL IMAG 10 0 10 20 10 10 REAL IMAG 100 0 100 100 100 REAL IMAG 50 0 50 20 20 REAL IMAG 20 0 20 40 20 20 REAL IMAG Figure 1. The Nyquist array of the open-loop system. 716 B. Labibi et al.

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1012 0.5 0.5 REAL IMAG 0.2 0 0.2 0.1 0.05 0.05 0.1 REAL IMAG 0.5 0 0.5 0.2 0.1 0.1 0.2 REAL IMAG 1012 0.5 0.5 REAL IMAG 0.2 0 0.2 0.1 0.1 0.2 0.3 REAL IMAG 0.02 0 0.02 0.01 0.01 0.02 REAL IMAG 10 1 2 1.5 0.5 0.5 REAL IMAG 0.5 0 0.5 0.5 0.5 REAL IMAG 0.01 0 0.01 0.02 15 10 x 103 REAL IMAG Figure 2. The Nyquist array of the closed-loop system. 100 80 60 40 20 From: In(1) To: Out(1) 100 80 60 40 20 To: Out(2) 10 10 10 100 80 60 40 20 To: Out(3) From: In(2) 10 10 10 From: In(3) 10 10 10 Bode Diagram Frequenc (rad/sec) Magnitude (dB) Figure 3. The Bode diagram of the closed-loop system. Diagonal dominance via eigenstructure assignment 717

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procedure and to eliminate the interactions due to input output centralization, for a given multivariable system, an equivalent descriptor system in the input–output decentralized form is defined. Then for the equivalent descriptor system in the input–output decentralized form, sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance rejection via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating the disturbances, closed-loop poles of the overall system are assigned in the desirable region, by assigning eigenstructure of each isolated subsystem appropriately. The designed controller achieves closed- loop diagonal dominance for all of frequencies in the desired band-width and removes frequency selection. The proposed algorithm has the flexibility that by increasing the degrees of sub-controllers, we have increased, number of free parameters in the optimization problem to achieve a higher degree of diagonal dominance and to assign the closed-loop eigenvalues in exact locations. The robust- ness problem can be considered by simply solving an appropriately defined multi-objective optimization problem. Acknowledgements This research is supported by the Natural Sciences and Engineering Research Council of Canada, and Syncrude Canada Ltd. References S.S. Chughtai and N. Munro, ‘‘Diagonal dominance using LMIs’’, IEE Proceedings Part D, Control Theory and Applications , 151(2), pp. 225–233, 2004. D. Cobb, ‘‘Controllability, observability and duality in singular system’’, IEE Trans. Automat. Control , AC-29, pp. 1076–1082, 1984. G.R. Duan, ‘‘Simple algorithm for robust pole assignment in linear output feedback’’, IEE Proceedings Part D, Control Theory and Applications , 139(5), pp. 465–469, 1992. G.R. Duan, G.W. Irwin and G.P. Liu, ‘‘Disturbance attenuation in linear systems via dynamical compensators: a parametric eigenstructure assignment approach’’, IEE Proceedings Part D, Control Theory and Applications , 146(2), pp. 129–136, 2000. M.M. Fahmy and J. O’Reilly, ‘‘Eigenstructure assignment in linear multivariable systems – a parametric solution’’, IEEE Trans. Automatic Contr , AC-28, pp. 990–994, 1983. M.P. Ford and K.C. Daly, ‘‘Dominance improvement by pseudo- decoupling’’, IEE Proceedings Part D, Control Theory and Applications , 126, pp. 1316–1320, 1979. D.J. Hawkins, ‘‘Pseudo-diagonalisation and the inverse Nyquist array method’’, IEE Proceedings Part D, Control Theory and Applications 119(3), pp. 337–342, 1972. B. Labibi, ‘‘Stability and robustness in decentralized control of large scale systems’’. PhD thesis, University of Tehran, Tehran, Iran (2001). B. Labibi, ‘‘Decentralized control of large scale systems via disturbance attenuation’’, in The 16th IFAC World Congress 2005. J. Lunze, Feedback Control of Large-Scale Systems , New York: Prentice Hall, 1992. A. MacFarlane and B. Kouvaritakis, ‘‘A design technique for linear multivariable feedback systems’’, International Journal of Control 25, pp. 837–874, 1977. J. Maciejowski, Multivariable Feedback Design , Wakingham, England: Addison Wesley, 1989. A.I. Mees, ‘‘Achieving diagonal dominance’’, Systems and Control Letters , 1, pp. 155–158, 1981. N. Munro, ‘‘Recent extensions to the inverse Nyquist array methd’’, in The Proc. 24th IEE Conf. on Decision and Control Miami FL, USA, 1985, pp. 1852–1857. A. Nobakhti, N. Munro and B. Porter, ‘‘Evolutionary achievement of diagonal dominance in linear multivariable plants’’, Electronics Letters , 39, pp. 165–166, 2003. O.D. Nwokah and C.H. Yau, ‘‘Quantitative feedback design of decentralized control systems’’, Journal of Dynamic Systems, Measurement, and Control , 115, pp. 452–464, 1993. R.V. Patel and N. Munro, Multivariable Systems Theory and Design Oxford, UK: Pergamon Press, 1982. R.J. Patton and G.P. Liu, ‘‘Robust control design via eigenstructure assignment, genetic algorithms and gradient-based optimization’’, IEE Proceedings Part D, Control Theory and Applications , 141, pp. 202–208, 1994. G. Roppenecker, ‘‘Entwurf von Ausgangsrueckfuehrungen mit Hilfe der invarianten Prametervectoren’’, Regelungstechnik , 31, pp. 125–131, 1983. H.H. Rosenbrock, Computer-Aided Control System Design , London: Academic Press, 1974. G.W. Stewart, Introduction to Matrix Computations , New York: Academic Press, 1973. J.H. Wilkinson, The Algebraic Eigenvalue Problem , Oxford: Clarendon Press, 1965. 718 B. Labibi et al.

79 No 7 July 2006 707718 Diagonal dominance via eigenstructure assignment B LABIBI H J MARQUEZ and T CHEN Department of Electrical and Computer Engineering University of Alberta Edmonton Canada Received 21 October 2005 in final form 21 F ID: 23691

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International Journal of Control Vol. 79, No. 7, July 2006, 707–718 Diagonal dominance via eigenstructure assignment B. LABIBI*, H. J. MARQUEZ and T. CHEN Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada Received 21 October 2005; in final form 21 February 2006 This paper presents a new methodology for diagonal dominance of large-scale systems via eigenstructure assignment. For a given large-scale system in general form, an equivalent descriptor system in the input–output decentralized form is defined. Sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance attenuation via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating disturbances, closed-loop poles of the overall system are placed in a desirable region, by assigning the eigenstructure of each isolated subsystem appropriately. The presented algorithm alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. The designed controller is in the decentralized form and plays the role of pre-compensator as well. An illustrative example is given to show the effectiveness of the proposed method. 1. Introduction The classical techniques of frequency domain design for single-input–single-output systems have been general- ized and applied to multivariable feedback systems by Hawkins (1972), Rosenbrock (1974), MacFarlane and Kouvaritakis (1977) and Patel and Munro (1985). Rosenbrock’s diagonal dominance design method con- sists essentially of determining a multivariable pre- compensator matrix , constant or dynamic, such that the resulting forward path-transfer function is diagonal dominant. When this condition is fulfiled, a diagonal compensator matrix ) can be used to implement single-loop compensators as required to meet the overall design specifications (Chughtai and Munro 2004). Various attempts have been made to determine a pre-compensator such that the resulting system is diagonal dominant. In many cases, however, this is very difficult to achieve. Manual cut-and-try methods, which were the ones originally suggested, often fail to achieve diagonal dominance with reasonable effort (Maciejowski 1989). Hawkins (1972) has proposed a pseudo-diagonalization scheme which attempts to minimize the off-diagonal effects at specific frequencies using a constant compensator. In Hawkins (1972) it is assumed that inverse arrays are to be used, but his method can be applied equally well to direct arrays; the methodology gives a constrained minimization. By using Lagrange multiplier methods, this can be shown to lead to an eigenvalue problem, so the solution can be obtained easily (Patel and Munro 1982). Ford and Daly (1979) has extended this approach to dynamic compensators. Pesudo-diagonalization can be applied to either direct or inverse Nyquist arrays (Maciejowski 1989). A practical difficulty arises if a dynamic compensator is found for an inverse array: its inverse needs to be realizable (Maciejowski 1989). MacFarlane and Kouvaritakis (1977) has suggested the ALIGN algorithm for diagonal dominance achievement. However, all of these algorithms are designed to obtain dominance at a particular frequency which is normally selected by trial and error. The overall dominance, obtained by the compensated system, depends on the selected frequency. These algorithms can be modified for the multi-frequency case, by selecting certain frequencies and finding the most *Corresponding author. Email: labibi@ece.ualberta.ca (On leave from K. N. Toosi University of Technology, Tehran, Iran) International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170600644860

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appropriate pre-compensator for them (Patel and Munro 1982). In Nobakhti et al . (2003), genetic algorithms (GA) were used to achieve diagonal dom- inance. The method is powerful and gives good results. The algorithm presented in Nobakhti et al . (2003) alleviates the necessity of choosing a suitable frequency in designing a pre-compensator, as required by previous methods. However, it inherits the problems associated with GAs in that it requires substantial computational effort and in general the designed controller is a centralized one. In Chughtai et al . (2004) a new method for determining a constant pre-compensator for reducing the effects of interactions in multivariable systems is presented. It is shown that linear matrix inequalities (LMIs) can be used in the design of pre- compensators to achieve diagonal dominance. The method is then modified to reduce the conservatism by the introduction of scaling as an LMI variable and a path-shaping matrix. The path-shaping matrix enables a designer to find an optimal pre-compensator to achieve a specific forward path structure. In this method frequency selection is removed and the LMIs formulated in this paper are computationally cheap compared to GAs. This method however, has its own problems, i.e., it gives a centralized controller and finding path-shaping matrix is found by trial and error. The feasibility of LMIs problem should also be considered. Decentralized control with Nyquist like methods can be very effective, if one can obtain the required degree of diagonal dominance fairly easily. Some attempts have been made to obtain a diagonal compensator to achieve diagonal dominance. Mees (1981) suggested that the diagonal dominance with a real, constant compensator may be obtained by choosing the elements of a constant diagonal compensator to be the elements of the Perron- Frobenius right eigenvectors of the properly approxi- mated matrix. Patel and Munro (1982) pointed out that dynamic compensation could be used for the original transfer matrix to obtain diagonal dominance. All of these, however, involve simultaneous dynamic input and output scaling which may be unacceptable. In fact, we must be careful about using output scaling as part of the strategy for achieving diagonal dominance. Roughly speaking, output scaling corresponds to inserting a post- compensator (that is, inserting a compensator between the outputs and variables being controlled). This is physically impossible, since the meaningful plant outputs cannot be affected by mathematical operations (Nwokah and Yau 1993). The first major drawback of all the existing decentralized control methods based on generalized diagonal dominance is the fact that a compensated open-loop diagonally dominant system in no way guarantees that the resulting closed-loop system is also diagonally dominant. On the other hand, closed- loop diagonal dominance is necessary for almost decoupled closed-loop response. More fundamentally, there is a need to investigate what level of open-loop dominance will guarantee a desired level of the closed- loop dominance. To do this, an appropriate measure of dominance must be defined. Decentralized design methods that address the above issue are therefore desirable (Nwokah and Yau 1993). In Labibi (2005) a new method for closed-loop diagonal dominance is proposed. In this methodology, designing a separate compensator to achieve diagonal dominance is not required. The designed controller consists of both compensator and control. The frequency selection step is removed, and in spite of the previous pieces of work, it gives closed-loop diagonal dominance. In Labibi (2005) in order to achieve sufficient conditions for overall closedl-loop diagonal dominance for a given large-scale system, interactions between subsystems are considered as external disturbances for each isolated subsystem; then, in order to attenuate the disturbances, the method proposed in Duan et al. (2000) is used. The method suggested in Duan et al . (2000) is based on parametric eigenstructure assignment which was introduced by Fahmy and O’Reilly (1983) and Roppenecker (1983) in the early 1980s. Parametric eigenstructure assignment provides, for a linear system, a parametrization of all the controllers which assign the desired closed-loop eigen- values and eigenvectors, and hence provides all the degrees of freedom available in the controller design (Fahmy and O’Reilly 1983, Roppenecker 1983). Many of the previous pieces of work in decentralized control, including Labibi (2005), are only applicable to systems which are in the input–output decentralized form. In this paper in order to overcome the above mentioned problem, the method in Labibi (2005) is extended to the general case where the system is not in the input output decentralized form. The proposed methodology can be applied to non-minimum phase, non-square and open-loop unstable systems and guarantees a closed- loop diagonally dominant system; this is achieved using a decentralized controller. In order to simplify the design procedure and to eliminate the interaction due to input output centralization, for a given large-scale system, a finite-spectrum-equivalent descriptor system in the input–output decentralized form is introduced. This new proposed representation converts a general system in which the matrices and are non-block diagonal, to an equivalent descriptor system in the input–output decentralized form, where and are block-diagonal and the interaction between subsystems is only due to non-block-diagonal state matrix, (Labibi 2001). The results of Labibi (2005) are then extended to systems in the general form. It is to be noted the descriptor system concept is a mathematical tool employed in the proof of the relevant theorems and by proving the theorems, 708 B. Labibi et al.

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the design will be done for isolated conventional subsystems. The paper is organized as follows. In 2, the problem of finding suitable decentralized dynamical controllers for the subsystems of a linear large-scale system is presented. In 3, the eigenstructure assignment method- ology proposed in Duan et al . (2000) is examined. In 4, a descriptor system for the large-scale system is defined, such that the equivalent system is in the input–output decentralized form. In 5 new sufficient conditions for closed-loop diagonal dominance are derived. It is shown that by assigning the closed-loop eigenstructure of each isolated subsystem appropriately, interactions between subsystems are attenuated and closed-loop poles are tried to be assigned in the desirable region. In 6an illustrative example is given to show the effectiveness of the proposed method. 2. The problem formulation Consider a large-scale system ), with the following state-space equations Ax Bu Cx where and Writing 11 ... ... ... NN 11 ... ... ... NN , and 11 ... ... ... NN ) is composed of linear time-invariant subsystems ), described by ii ii ij ij ii ij where ii ii , and ii . The terms ij ij and ij are due to interactions of the other subsystems. The objective in this paper is to design a local output feedback dynamical controller for each isolated subsystem ii ), described by ii ii ii such that the poles are assigned in a desirable region and at the same time effects of interactions of the other subsystems are minimized. By this, the decentralized controller diag g stabilizes the overall system and also provides desired dynamical characteristics for the overall system, if some sufficient conditions are satisfied. 3. Eigenstructure assignment In this section the method for eigenstructure assignment proposed in Duan et al . (2000) is investigated. Consider a system given by state space matrices ( ). Let ( ) be controllable and ( ) observable and the matrices and be full rank. A general output dynamical compensator, of order , for the system can be written in the following form 22 21 12 11 where is the compensator state vector and ij 1, 2, are four controller coefficient matrices of appropriate dimensions. Applying the dynamical com- pensator to the system gives the following closed-loop system Diagonal dominance via eigenstructure assignment 709

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where and BK 11 CBK 12 21 CK 22 The closed loop system matrix is assumed to be non- defective, where its Jordan form is a diagonal matrix which possess better robustness than defective ones (Wilkinson 1965). Since the system is controllable and observable and the matrices and are full rank, therefore there exist right coprime factorizations of the form sI sI where 2 2 2 and 2 are all polynomial matrices, and the pairs ), )) and ( ), )) are both right coprime. Lemma 1 (see Duan (1992) for proof ) Let be controllable and be observable and s 1, 2, ... be a group of self-conjugate complex numbers. Then (a) there exist matrices K ij 1, 2, and T such that BK 11 CBK 12 21 CK 22 10 11 diag 1, 2, ... 12 hold for a set of self conjugate complex numbers s 1, 2, ... if and only if there exist vectors f ij and g ij 0, 1, 0, 1, ... p satisfying the following constraints C1 ij il and g ij il if s C2 ij 1, 2, ... here ij is the Kronecker delta (b) when constraints C1 and C2 are met the matrix V is given by 13 where V 01 02 11 12 and the matrix T is given by 14 where T 01 02 and 11 12 ... The corresponding matrices ij are either given by 11 12 11 CV 21 22 21 CV 15 with 01 02 11 12 CV and CV Or by 11 21 11 12 22 21 16 with 01 02 11 12 and The parameters ij and ij 0, 1, 2, ... represent the degrees of freedom available in the compensator design for the system. The suggested methodology for eigenstructure assignment in this section will be used to assign eigenstructure of each isolated subsystem. 4. Systems in the general form Many practical systems do not satisfy the stringent conditions given in Lunze (1992) for transforming into 710 B. Labibi et al.

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the input–output decentralized form. In order to employ the method of Labibi (2005) for systems in the general form whose state space matrices and are not block-diagonal, the method for input–output decentral- ized transformation via the descriptor system represen- tation proposed in Labibi (2001) is used. Consider a system given by equation (1), in order to obtain an input–output decentralized form, all of the inputs and outputs of the system are defined as state variables. Then the augmented system has the following equations Bu 17 where ... ... ... ... ... ... ... ... and ... ... ... From the previous definitions, it can be seen the new input and output matrices, and are block diagonal and the results of Labibi (2005) can be applied to the new input–output decentralized system. The objective is to design a decentralized controller for each isolated subsystem given by ii ii ii ii 18 where ii ii ii ii ii ... ... ... ... ... ... ... 11 ... 11 11 ... ... ... ... NN ... NN NN ... Diagonal dominance via eigenstructure assignment 711

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and ii such that the eigenvalues of the overall closed-loop system are assigned as close as possible to the desirable locations. It is well known that a descriptor system has a more complicated structure and contains not only finite dynamical modes, but also infinite modes including infinite non-dynamical and dynamical modes. The infinite dynamical modes can generate undesired impulse behaviour. To guarantee the exis- tence and uniqueness of solution for a design system, it is always assumed that the system is regular (Cobb 1984). It is simple to show that the descriptor system in (17) is regular and impulse-free, and if the isolated subsystems given by (4) are controllable and observable, the descriptor sub- systems ii ii ii are strongly controllable, observable and sE sI 19 sE sI 20 where ), ), and are transfer functions of the original large-scale system, the equivalent descriptor system, the diagonal system composed of isolated subsystems given by (4), and the diagonal descriptor system composed of isolated subsystems given by (18), respectively (Labibi 2001). Therefore, control of the descriptor system results in control of the original system with the same controller. No assumption is made on the controll- ability and observability of the isolated subsystems. Since the large scale system given in (1) is observable, it is always possible to find a similarity transforma- tion to transform the system into the output decentralized form where matrix is block diagonal and ij 0 for (Lunze 1992). This transformation is based on the observability matrix of the original system given in (1) and is defined by 21 here, is the th row of the output matrix , and is the degree of the th subsystem. Then the isolated subsystems are observable and by appropriately decomposing the input matrix , it is possible to have controllable isolated subsystems. Hence, without loss of generality it is assumed the th isolated subsystem has the following equations or can be transformed to this case by using appropriate similar- ity transformation ii ii ij ij ii 22 Similarity transformations do not affect output feedback controllers. Therefore using a similarity transformation does not affect the diagonal dominance property. 5. Decentralized control via disturbance attenuation Consider the th isolated subsystem given by equation (4). In general the controller designed for each isolated subsystem is a dynamical controller. Assuming the th controller has state-space equations given by (6), it is simple to show that designing a dynamical output feedback controller with degree for the th subsystem can be reduced to design of a static controller for the augmented subsystem with the following state-space equations ii ii ii ii ii 23 where ii ii 00 ii ii and ii ii In order to avoid complexity in formulation, without loss of generality, in the rest of the paper it is assumed that in equation (22), the augmented matrices given in (23) are used. It can be shown that the overall closed-loop system has the 712 B. Labibi et al.

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following form where is the th reference input, and sE ii ii ii ii The outputs of the th subsystem of the overall closed-loop descriptor system are given by ii hi ii ii ii 25 where ii ii iN hi and ij 00 0 ij ij 00 0 Therefore, minimizing ii , such that ii < ii 1, 26 where is a positive scalar and kk is a norm of ( ), reduces interactions between the subsystems and the output of the th closed-loop descriptor subsystem is given by ii ii ii ii 1, ... 27 where sI ii ii ii which is exactly the output of the th closed-loop system given by equations in (4). This means that ii is an approximation to ii ii The residual of ii is the matrix ii The relative error in ii is the number (Stewart 1973) ii ii < 1, ... 28 Therefore by choosing small values for ’s, this error will be reduced. Considering the th output ii ii ii 29 it can be seen that minimizing the term ii minimizes the interactions between the subsystems. Thus, in equation (29) the states of the other descriptor subsystems can be considered as external disturbances for each isolated descriptor subsystem. Since 30 where ii sI ii ii ii ii sI ii ii ii ii ii sI ii ii ii ii ii sI ii ii sI ii ii ii sI ii ii and ii sI ii ii ii ii sI ii ii ii ii ii sI ii ii ii ii it is simple to show that ii ii ABi kk 31 11 11 12 ... 11 22 21 22 ... 22 NN NN ... NN 11 11 22 22 NN NN 24 Diagonal dominance via eigenstructure assignment 713

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where ABi ii ii ii ii iN iN Thus, to attenuate the effects of disturbance to the th output, the following index can be minimized: ii sI ci 32 Here is the th column of ABi and ci ii ii ii The th closed-loop subsystem is stabilized; therefore the equation ci ci 33 has a unique solution with respect to , and this solution is also symmetric positive semi-definite. Further, it follows that the following holds (Duan et al. 2000). ii sI ci trace ii ii 34 Using equations (10)–(12) for the th subsystem, and (33), we have 35 Assuming 36 or 37 where the matrix is also symmetric positive semi- definite, we have ii sI ci trace ii ii 38 Denoting ijl , equation (35) can then be decomposed as ij il ijl ij il 1,2, ... 39 where, ij is the th left eigenvector of the th closed-loop subsystem (Duan et al . 2000), thus ijl ij il ij il 1, ... 40 The algorithm for solving the decentralized control problem can be summarized as follows. Algorithm: (a) Find an appropriate similarity transformation to transform the system into the output decentralized form if it is not. (b) For the th subsystem, select degree of controller, 0. (c) For the th subsystem, solve for the polynomial matrices ), ), ), and ), satisfying the right coprime factorization equations (8) and (9). (d) Solve for the expression in constraints C1 and C2. (e) Solve for the expression of index according to equations (38) and (40). (f) Specify the desired closed-loop eigenvalue location regions, according to the closed-loop stability and performance requirements. (g) Solve the local optimization problem, min subject to constraints C1 and C2 with some numerical optimization algorithm. If is sufficiently less than ii , the sufficient conditions for closed-loop diagonal dominance given in (28) are satisfied, go to step(h); otherwise increment by 1 and go to step (e). (h) Solve il for the th subsystem according to equation (15) or (16). Remark 1: The suggested algorithm will eventually converge. According to pole assignment theory for linear systems, the closed-loop poles of the dynamical compensation system in (1) and (6) can be arbitrary assigned when the order of the dynamical compensator is sufficiently large. Under these condition, it can be easily reasoned that the optimization problem always has a solution. However, since the minimization is generally a non-convex non-linear programming prob- lem, an optimal solution cannot often be found (Duan et al . 2000). For seeking a solution (generally subopti- mal) to the minimization, the optimization toolbox of MATLAB or the software named general algebraic modelling system (GAMS) can be used. It is also possible by combining the genetic algorithms and the gradient-based optimization to provide a technique for solving the optimization problem. The gradient-based methods lack robustness over broad spectrum of optimization functions. The direct-search techniques, which are problem independent, have been proved as a 714 B. Labibi et al.

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panacea for the difficulties associated with the tradi- tional techniques. The genetic algorithms (GA) are different from normal search methods because they search from a population of points, not a single point and use probabilistic and not deterministic transition rules. But the convergence speed of the genetic algorithms is slower than the gradient-based optimiza- tion. Although the gradient-based optimization probably gives the local optimal solution, its conver- gence speed is faster. Thus, a numerical algorithm by combining the advantages of the genetic algorithms and the gradient-based optimization may be more useful to solve the optimization problems (Patton and Liu 1994). Remark 2: The proposed algorithm has this flexibility that increasing the degrees of sub-controllers results in the increase of free parameters in the optimization problem to achieve a higher degree of diagonal dominance and even to assign the closed-loop eigen- values in exact locations. Remark 3: The robustness problem may be consid- ered in the proposed method. The condition number of the eigenvector matrix of each isolated closed-loop subsystem can be considered as a kind of eigenvalue sensitivity (Wilkinson 1965). Hence, in order to have minimum eigenvalue sensitivity in addition to diag- onal dominance, it is possible to consider a multi- objective problem. The cost function can be considered as a weighted summation of cost function as given in (38) and the condition number of the modal matrix of the th isolated closed-loop sub- system. This condition number can be parameterized with the parameters given in constraints C1 and C2 (Duan 1992). Therefore, the optimization problem can be done on the same parameter space as the previous problem. 6. Illustrative example In this section an example is provided to show the effectiveness of the proposed method. Consider the three-input, three-output, five-state aircraft model AIRC (Maciejowski 1989), described by the following state-space matrices. 00 1 132 0 0538 1712 0 0 0705 00010 00 0485 0 8556 013 2909 0 1 0532 6859 00 0 12 1 0 00 0 419 0 665 575 0 0732 10000 01000 00100 000 000 000 The system is highly interacted and figure 1 shows the Nyquist array of the open-loop system. The aim is to achieve a bandwidth of about 15 rad/sec for each loop, with little interaction between outputs. The system can be decomposed into three subsystems. Since the third subsystem is unobservable at first based on observability matrix of the system, a similarity transformation given by 100 0 0 001 0 0 000 1 0 000 0 1 101 132 0 is applied to the system. The transformed system has the following state-space matrices: 01000 6859 0 2909 0 7764 0 0788 0705 0538 0914 0 00001 01 013 0 0485 1467 8556 000 575 0 0 0732 12 1 0 000 419 0 165 10000 00100 00010 000 000 000 The transformed system is composed of three subsystems, 11 01 6859 11 575 Diagonal dominance via eigenstructure assignment 715

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11 10 11 0, 22 0538, 22 1, 22 1, 22 0, 33 01 1467 8556 33 665 33 10 33 The interaction matrices are given by AB 00000 2909 0 7764 0 0788 0 0 0732 and AB 0705 0914 0 12 0 For the transformed system, ii 1, 1, 2, 3. Minimizing the appropriately defined cost func- tions for the isolated subsystems, 0.01274, 0.0014, and 0.0119 are achieved. Hence, conditions (28) are satisfied and the overall closed- loop system is diagonal dominant. The designed decentralized controller 30 7897 12 81349 00 010 0 00 344 9785 41 9423 is applied to the original system given in equation (1). The closed-loop eigenvalues of each isolated subsystem are assigned in the desirable region to have 15 rad/sec bandwidth and the closed-loop overall system is stabilized. Figure 2 shows the Nyquist array of the closed-loop system. It is clear that the closed loop system is diagonal dominant. Figure 3 shows the Bode diagram of the closed-loop transfer function, and shows that the bandwidth requirement has been achieved. 7. Conclusion This paper has introduced new sufficient conditions for closed-loop diagonal dominance achievement of multi- variable systems. In order to simplify the design 500 0 500 500 500 REAL IMAG 200 0 200 200 100 100 REAL IMAG 100 0 100 100 100 200 REAL IMAG 100 0 100 50 50 100 REAL IMAG 20 0 20 20 20 40 REAL IMAG 10 0 10 20 10 10 REAL IMAG 100 0 100 100 100 REAL IMAG 50 0 50 20 20 REAL IMAG 20 0 20 40 20 20 REAL IMAG Figure 1. The Nyquist array of the open-loop system. 716 B. Labibi et al.

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1012 0.5 0.5 REAL IMAG 0.2 0 0.2 0.1 0.05 0.05 0.1 REAL IMAG 0.5 0 0.5 0.2 0.1 0.1 0.2 REAL IMAG 1012 0.5 0.5 REAL IMAG 0.2 0 0.2 0.1 0.1 0.2 0.3 REAL IMAG 0.02 0 0.02 0.01 0.01 0.02 REAL IMAG 10 1 2 1.5 0.5 0.5 REAL IMAG 0.5 0 0.5 0.5 0.5 REAL IMAG 0.01 0 0.01 0.02 15 10 x 103 REAL IMAG Figure 2. The Nyquist array of the closed-loop system. 100 80 60 40 20 From: In(1) To: Out(1) 100 80 60 40 20 To: Out(2) 10 10 10 100 80 60 40 20 To: Out(3) From: In(2) 10 10 10 From: In(3) 10 10 10 Bode Diagram Frequenc (rad/sec) Magnitude (dB) Figure 3. The Bode diagram of the closed-loop system. Diagonal dominance via eigenstructure assignment 717

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procedure and to eliminate the interactions due to input output centralization, for a given multivariable system, an equivalent descriptor system in the input–output decentralized form is defined. Then for the equivalent descriptor system in the input–output decentralized form, sufficient conditions for diagonal dominance of the closed-loop system are introduced. These conditions are in terms of the isolated subsystems. Based on them, interactions between subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used for disturbance rejection via dynamical output compensators based on complete parametric eigenstructure assignment. By attenuating the disturbances, closed-loop poles of the overall system are assigned in the desirable region, by assigning eigenstructure of each isolated subsystem appropriately. The designed controller achieves closed- loop diagonal dominance for all of frequencies in the desired band-width and removes frequency selection. The proposed algorithm has the flexibility that by increasing the degrees of sub-controllers, we have increased, number of free parameters in the optimization problem to achieve a higher degree of diagonal dominance and to assign the closed-loop eigenvalues in exact locations. The robust- ness problem can be considered by simply solving an appropriately defined multi-objective optimization problem. Acknowledgements This research is supported by the Natural Sciences and Engineering Research Council of Canada, and Syncrude Canada Ltd. References S.S. Chughtai and N. Munro, ‘‘Diagonal dominance using LMIs’’, IEE Proceedings Part D, Control Theory and Applications , 151(2), pp. 225–233, 2004. D. Cobb, ‘‘Controllability, observability and duality in singular system’’, IEE Trans. Automat. Control , AC-29, pp. 1076–1082, 1984. G.R. Duan, ‘‘Simple algorithm for robust pole assignment in linear output feedback’’, IEE Proceedings Part D, Control Theory and Applications , 139(5), pp. 465–469, 1992. G.R. Duan, G.W. Irwin and G.P. Liu, ‘‘Disturbance attenuation in linear systems via dynamical compensators: a parametric eigenstructure assignment approach’’, IEE Proceedings Part D, Control Theory and Applications , 146(2), pp. 129–136, 2000. M.M. Fahmy and J. O’Reilly, ‘‘Eigenstructure assignment in linear multivariable systems – a parametric solution’’, IEEE Trans. Automatic Contr , AC-28, pp. 990–994, 1983. M.P. Ford and K.C. Daly, ‘‘Dominance improvement by pseudo- decoupling’’, IEE Proceedings Part D, Control Theory and Applications , 126, pp. 1316–1320, 1979. D.J. Hawkins, ‘‘Pseudo-diagonalisation and the inverse Nyquist array method’’, IEE Proceedings Part D, Control Theory and Applications 119(3), pp. 337–342, 1972. B. Labibi, ‘‘Stability and robustness in decentralized control of large scale systems’’. PhD thesis, University of Tehran, Tehran, Iran (2001). B. Labibi, ‘‘Decentralized control of large scale systems via disturbance attenuation’’, in The 16th IFAC World Congress 2005. J. Lunze, Feedback Control of Large-Scale Systems , New York: Prentice Hall, 1992. A. MacFarlane and B. Kouvaritakis, ‘‘A design technique for linear multivariable feedback systems’’, International Journal of Control 25, pp. 837–874, 1977. J. Maciejowski, Multivariable Feedback Design , Wakingham, England: Addison Wesley, 1989. A.I. Mees, ‘‘Achieving diagonal dominance’’, Systems and Control Letters , 1, pp. 155–158, 1981. N. Munro, ‘‘Recent extensions to the inverse Nyquist array methd’’, in The Proc. 24th IEE Conf. on Decision and Control Miami FL, USA, 1985, pp. 1852–1857. A. Nobakhti, N. Munro and B. Porter, ‘‘Evolutionary achievement of diagonal dominance in linear multivariable plants’’, Electronics Letters , 39, pp. 165–166, 2003. O.D. Nwokah and C.H. Yau, ‘‘Quantitative feedback design of decentralized control systems’’, Journal of Dynamic Systems, Measurement, and Control , 115, pp. 452–464, 1993. R.V. Patel and N. Munro, Multivariable Systems Theory and Design Oxford, UK: Pergamon Press, 1982. R.J. Patton and G.P. Liu, ‘‘Robust control design via eigenstructure assignment, genetic algorithms and gradient-based optimization’’, IEE Proceedings Part D, Control Theory and Applications , 141, pp. 202–208, 1994. G. Roppenecker, ‘‘Entwurf von Ausgangsrueckfuehrungen mit Hilfe der invarianten Prametervectoren’’, Regelungstechnik , 31, pp. 125–131, 1983. H.H. Rosenbrock, Computer-Aided Control System Design , London: Academic Press, 1974. G.W. Stewart, Introduction to Matrix Computations , New York: Academic Press, 1973. J.H. Wilkinson, The Algebraic Eigenvalue Problem , Oxford: Clarendon Press, 1965. 718 B. Labibi et al.

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