A Generalised Minimum Variance Controller for Time-Varying MIMO Linear Systems with Multiple Delays

WSEAS RANSACIONS on SYSEMS and CONROL A Generalised Minimum Variance Controller for ime-varying MIMO Linear Systems with Multiple Delays ZHENG LI School of Electrical, Computer and elecom. Eng. University of Wollongong Wollongong NSW 5 AUSRALIA zheng@elec.uow.edu.au CHRISIAN SCHMID Institute of Automation and Computer Control Ruhr-Universität Bochum D-448 Bochum GERMANY cs@atp.rub.de Abstract: - In this paper we study optimal control of time-varying multi-input multi-output stochastic systems and develop a generalised minimum variance controller. he system to be controlled is described using a multiinput multi-output time-varying autoregressive moving average model that has multiple delays between the output and input. he controller minimises the sum of output tracking error variances and squared current control variables. Key-Words: - MIMO systems, minimum variance control; multiple delays, time-varying systems. Introduction he multi-input multi-output (MIMO) generalised minimum variance controller (GMVC) minimises a generalised minimum variance cost functional for stochastic optimal control. It is based on a MIMO controlled autoregressive moving average (CARMA) model and extends the MIMO minimum variance controller (MVC) by adding quadratic control variables to the minimum variance cost functional for a generalised minimum variance cost functional. Although making the output tracking suboptimal the inclusion of the control variables in the cost functional improves the stability of the closed-loop control system by penalising large control actions. Consequently, large fluctuations in control variables are reduced and the controller is no longer restricted to systems with a stable inverse. he MIMO GMVC for linear time-invariant (LI) systems was developed by Koivo []. It extends the MIMO minimum variance controller of Borisson [] from minimum phase systems to nonminimum phase systems. his LI GMVC extends also the LI GMVC of Clarke and Gawthrop [3], [4] from single-input and single-output (SISO) case to MIMO LI systems. Both the SISO and MIMO LI GMVCs are very useful and have seen many applications in stochastic adaptive control. he basic difficulty in extending the SISO LI GMVC to MIMO LI GMVC is the noncommutativity of MIMO transfers functions. A pseudocommutation technique is used for the development of the MIMO LI GMVC for overcoming noncommutativity. his technique was developed by Wolovich [5] and was introduced to the design of stochastic optimal controllers by Borisson []. he pseudocommutation has seen many applications in analysis and design of MIMO LI systems. here are many industrial systems that have time-varying dynamics [6], [7]. In this paper we will develop a generalised minimum variance controller for MIMO linear time-varying (LV) systems. Noncommutativity is also an obstacle for extending a GMVC from SISO LI plants for SISO LV systems. A pseudocommutation technique was developed for SISO LV transfer operators in the previous work [8] on minimum variance prediction of SISO LV systems. Based on this technique the SISO LI GMVC was extended for SISO LV plants under a generalised cost functional that uses time-varying filters and weighting functions [9]. However, a pseudocommutation technique is unavailable for MIMO LV transfer operators for extending the SISO LV GMVC for MIMO LV plants. An LV GMVC was developed without using the pseudocommutation for an exponentially stable SISO LV CARMA model []. However, when the LV filters are involved in the cost functional the pseudocommutation cannot be avoided [], []. An MIMO LV GMVC [3] was recently developed for MIMO LV systems as an extension of the LV MIMO MVC [4], [5]. However, this LV GMVC can only be applied to the MIMO CARMA model with a uniform single delay and cannot be used for the MIMO plants that have different delays between the inputs and outputs. ISSN: 99-8763 6 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL It is very common that MIMO processes have multiple delays and cannot be controlled using a MIMO GMVC that has a uniform single delay. In this paper, we extend the LV GMVC for MIMO LV plants from the uniform delay case for multiple delay systems. We consider the case where the delay can be described using a diagonal matrix such that each output can have a different delay to its inputs. he reminder of this paper is organised as the following. Section describes the MIMO LV CARMA model, the LV operators, the multiple time delays and the generalised minimum variance cost functional. Section 3 develops the MIMO LV GMVC, analyses its closed-loop stability and compares it with other LV GMVCs. Section 4 presents design and simulation examples followed by a conclusion in Section 5. Problem Formulation he MIMO LV plants to be controlled is described by the LV CARMA model A( kq, ) Y' Bkq (, ) Uk ( ) Ckq (, ) W', () where W' is a p vector of zero mean independent Gaussian processes that represent the stochastic disturbances. he variance of W' is time varying and uniformly bounded away from infinite. U and Y' are the plant input and output. hey are all p vectors. In the CARMA model, q is the onestep-advance operator and Akq (, ) I A( kq ) A( kq ) Bkq (, ) B B( kq ) B( kq ) Ckq (, ) I C( kq ) C( kq ) n n m m h h () are LV moving average operators (MAO's) with A i, B j and C r, i=,,..., n, j=,,..., m, r=,,..., h being pp matrices whose elements are uniformly bounded away from infinite. It is also assumed that the determinant of B is uniformly bounded away from zero. In the CARMA model the signal vectors Y' y( k d) y( k d) yp( k d ) p U u u up W' w( k d) w( k d) wp( k dp) (3) are used, where the superscript denotes matrix transpose, d i, i=,,, p, represent the multiple time delays between the plant inputs and outputs. he above equation shows that the CARMA model allows each output to have a different delay with respect to the inputs. Without losing generality it is assumed that d i d i+. Letting d d d p ( ) diag,,..., D q q q q (4) be a diagonal matrix for description of the multiple time delays in terms of the one-step-advance operator. he LV CARMA model can be expressed as Akq (, ) DqY ( ) B( kq, ) Uk ( ) Ckq (, ) DqWk ( ) ( ) where Yk ( ) y y yp W w w wp (5) (6) When all the delays are equal the CARMA model (5) reduces to the uniform single delay LV CARMA model in [3]. he inverse of an LV MAO is called an LV autoregressive operator (ARO) and is denoted as A - (k,q - ) whose stability is defined using the stability of the state transition matrix A A A3 An An I I ( k, k) I (7) where I and are identity and zero matrix of appropriate dimension. he LV ARO is exponentially stable if and only if there are constants C > and c > such that c( k s) ( ks, ) Ce (8) for all k s. he maximum of all possible c is called the rate of exponential stability for the LV ARO. When the LV ARO is exponentially stable it leads to the cancellation rules A( k, q ) A ( k, q ) X X A kq Akq Xk Xk k (, ) (, ) ( ) ( ) ( ) where satisfies (9) ISSN: 99-8763 6 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL Akq (, ). () It is a zero input solution to the above autoregressive equation and will decay to zero exponentially when A - (k,q - ) is exponentially stable. In this paper we assume that both A - (k,q - ) and C - (k,q - ) are exponentially stable and all the plant parameters in the MIMO LV CARMA model are known. he reference vector for the LV GMVC to follow has the form S' s( k d) s( k d) sp( k dp) DqSk ( ) ( ) () Given the uniformly bounded reference vector the generalised minimum variance control objective is to minimise the cost functional J k E Y k S k P k Y k S k ( ) {[ '( ) '( )] ( )[ '( ) '( )] Uk ( ) RkUk ( ) ( ) Datak ( )} () using a sequence of control vectors, U, U(k-), U(k-),. In the control objective () Data is the set of input and output data up to and including time k, both P and R are uniformly positive definite matrix that are uniformly bounded away from infinite. In comparison with the LI GMVC the assumption on exponential stability of A - (k,q - ) is additional and the others are simple and natural extensions of those from the LI case for LV plants. In addition, the degrees of the LV MAO's in the LV CARMA model are time varying because the time varying matrices A i, B j and C r, i=,,..., n, j=,,..., m, r=,,..., h, are allowed to become zero. Noting (3) we know that this objective includes the generalised minimum variance control cost functional in [3] as a special case by introducing multiple time delays for the plant outputs. A k q C k q D q F k q (, ) (, ) ( ) (, ) (4) A ( kq, ) Gkq (, ) where dp dp d p Fkq (, ) F( kq ) F( kq ) F ( kq ) (5) is the quotient and G(k,q - ) is the remainder. he maximum power of q in G(k,q - ) is zero. Substituting (4) into (3) we have DqYk ( ) ( ) A ( kq, ) Bkq (, ) Uk ( ) FkqWk (, ) ( ) A ( kq, ) Gkq (, ) Wk ( ) he LV CARMA model can be rewritten as DqYk ( ) ( ) FkqWk (, ) ( ) A ( kq, ) Bkq (, ) Uk ( ) A ( kq, ) Gkq (, ) Wk ( ) (6) (7) Letting k being the current time we have the plant output and future noise on the left hand side of the above equation because both LV MAO s, D(q) and F(k,q), have positive power of at least one in the one-step-advance operator q. On the right hand side we have the current and past plant input and noise because A(k,q - ) is monic and the maximum powers of both G(k,q - ) and B(k,q - ) are zero. As a result, if the noise up to and including the current time k can be estimated we can use (7) as a predictor for the plant output because the future noise is of zero mean. Letting Y ˆ ' Y' F( k, q) W (8) be the output prediction of the CARMA model and noting (7) and (4) we have 3 Controller Left dividing (5) using A(k,q - ) we have DqYk ( ) ( ) A ( kq, ) Bkq (, ) Uk ( ) A ( kq, ) Ckq (, ) DqWk ( ) ( ) (3) ˆ '( ) (, ) (, Y k A k q B k q ) U ( k ) A kq Gkq Wk (, ) (, ) ( ) (9) Substituting this predictor into the generalised minimum variance cost functional we have the following GMVC. In order to divide the noise term on the far right hand side of the above equation into past and future component left dividing C(k,q - )D(q) by A(k,q - ) gives GMVC heorem If the LV AROs A - (k,q - ) and C - (k,q - ) are exponentially stable, the LV GMVC for the MIMO LV CARMA model is given by ISSN: 99-8763 63 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL ˆ ( ) ( ) (, ) (, ) ( ) ( ) Wk Dq C kq Akq DqYk Bkq (, ) Uk ( ) U k k q A k q D q S k Gkq (, ) Wk ˆ ( ) ( ) (, ) (, ) ( ) ( ) kq (, ) Bkq (, ) Akq (, ) P ( kb ) ( krk ) ( ) where W ˆ ( k ) is the estimate of W. () () () Proof Substituting the minimum variance prediction (9) into the cost functional () we have J [ Yˆ' S'] P[ Yˆ' S'] U R U E F k q W k P k F k q W k Data k [ (, ) ( )] ( )[ (, ) ( )] ( ) From (), (), and (8) it can be verified that (3) Yˆ '( k ) Y '( k ) B. (4) Uk ( ) Uk ( ) hus Jk ( ) B ˆ ( kpk ) ( )[ Y' S'] Uk ( ) RkUk ( ) ( ) and (5) Jk ( ) B ( kpkb ) ( ) Rk ( ). (6) U Because the second derivative is uniformly positive definite there exists optimal control U such that the generalised minimum variance cost functional () is minimised. From (5) we know that the optimal control U can be determined from P B R U S' Yˆ '. (7) It follows from (9) that P B R U S' A ( k, q )[ B( k, q ) U Gkq (, ) Wk ( )] (8) Solving for the control variable we have [ Akq (, ) P ( kb ) Rk ( ) Bkq (, )] Uk ( ) Akq (, ) S' Gkq (, ) W (9) In order to use this equation to determine the optimal control we need the estimate of the noise W. he estimator () can be rewritten as follows Ckq DqWk ˆ Akq DqYk (, ) ( ) ( ) (, ) ( ) ( ) Bkq (, ) Uk ( ) (3) Comparing the above equation with the CARMA model (5) we have Ckq (, ) DqWk ( ) ( ) where, (3) W k W k Wˆ k D q W k ( ) ( ) ( ) ( ) '( ) (3) is the estimation error. Equation (3) shows that the error will decay exponentially due to exponential stability of C - (k,q - ). Replacing the noise term in (9) using its estimate obtained by () and noting (4) we have (), which can be further expressed as k q U k A k q D q S k (, ) ( ) (, ) ( ) ( ) Gkq (, ) Wk ( ) Gkq (, ) Wk ( ) (33) Noting (), (3), (3) and (33) we have the closedloop system of the LV GMVC as follows Ckq (, ) W ' Gkq (, ) Dq ( ) kq (, ) Uk ( ) B( kq, ) Akq (, ) Y' (34) W Gkq (, ) Akq (, ) Dq ( ) Sk ( ) Ckq (, ) Dq ( ) Remarks: a) he left most matrix in the above equation is the LV ARO matrix for the closed-loop system. It is a lower triangle matrix. Its stability is determined by the three diagonal elements, A - (k,q - ), C - (k,q - ) and - (k,q - ). he exponential stability of B - (k,q - ) is not necessary for stability of the closed-loop system because it is not a diagonal element unless the weighting matrix for the control variable R is zero. hus, the stable in- ISSN: 99-8763 64 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL vertability condition of the LV MVC [8] is removed. b) Among the three diagonal elements both A - (k,q - ) and C - (k,q - ) are exponentially stable. herefore, the closed-loop system is exponentially stable if - (k,q - ) is exponentially stable. From () it is known that stability of - (k,q - ) depends on the choice of P and R. Because A - (k,q - ) is exponentially stable we can choose P B R I with being a very large positive number in order for the stability of A - (k,q - ) to dominate the stability of - (k,q - ). c) he equations (34) and () show that stability of the closed-loop system is independent on the time delay of the system because all the three diagonal elements of the LV ARO for the closed-loop system do not include the delay operator D(q). his agrees with the stability results for the LV MVCs and LI GMVCs. d) he estimator () is introduced for the estimation of the current process noise from the plant input and output. he estimation error depends on the initial conditions used in the estimator. When the initial conditions are accurate the estimator will produce accurate estimate right from the beginning for all the time and the performance of the LV GMVC will be optimal all the time. When the initial condition is inaccurate there will be an estimation error that decays exponentially to zero according to equation (3). As a result, the performance of the LV GMVC will converge exponentially to the optimal performance specified by (). he rate of the exponential convergence is faster than or equal to the rate of the exponential stability of the LV ARO C - (k,q - ) as is shown by the error equation (3). e) his GMVC includes the LV GMVC for MIMO CARMA models with a uniform single delay [3] as a special case. When all diagonal elements in the delay operator D(q) are equal the generalised minimum variance control cost functional () becomes the cost functional for the LV MIMO CARMA models with a uniform single delay and the LV GMVC becomes the LV MIMO GMVC of [3]. f) his GMVC includes also the LV MVC for MIMO CARMA models with multiple delays as a special case. When the time-varying input weighting matrix R is set to zero the generalised minimum variance cost functional () becomes the minimum variance cost functional of [4] and the LV GMVC becomes the LV MVC for MIMO LV systems with multiple delays. g) his GMVC also includes our LV GMVC for SISO LV systems [] as a special case. When the LV CARMA model () reduces to an LV SISO system the cost functional () reduces to that for the SISO LV CARMA models of [] and the MIMO LV GMVC reduces to the SISO LV GMVC. However, it cannot include our flexible LV GMVC [] for LV SISO CARMA models as a special case. ime-varying filters for the plant input and output were used in the flexible GMVC cost functional in []. hey were dealt with using the pseudocommutation for the development of the flexible SISO LV GMVC. However, the pseudocommutation is unavailable for MIMO LV systems. 4 Examples We present the design and simulation of our LV GMVC using a two-input and two-output LV plant described by the CARMA model y ( k ) y Ak ( ) ( ) ( ) y k y k u u ( k ) Bk ( ) ( ) ( ) u k u k w ( k ) w Ck ( ) ( ) ( ) w k w k (35) where both noises are independent Gaussian processes with zero mean and unit variance. Specifying the delays using the one-step-advance operator we have D q ( ) diag( q, q ) (36) he system can then be represented using the LV CARMA model (5) where Akq (, ) I Akq ( ) B( kq, ) IBkq ( ) Ckq (, ) ICkq ( ) (37) Example In this example we study the performance of our LV GMVC using constant weighting functions. hey have the forms ISSN: 99-8763 65 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL 6 Pk ( ).7 Rk ( ).3 he time-varying parameter matrices are.7sin(. k).cos(.3 k) Ak ( ).7sign[cos(. )] k..5sin(.5 k) Bk ( ).sign[cos(.6 )] k k Ck ( ) k 5.4sin(.k.).( k ).6.5( e )sin(.6 k) (38) (39).67 ( k, q ) B( k, q ) A( k, q ).3 (4) and Gkq (, ) Ckq (, ) Dq ( ) Akq (, ) Fkq (, ). (43) he reference inputs are square waves. Fig. and Fig. show that the LV GMVC is able to drive the MIMO CARMA model to follow both square wave references. However, there are steady-state tracking errors in both outputs. his is typical for LV GMVC as a result of the tradeoff between the output tracking performance and stability by introducing a penalizing term of control variables into the minimum variance cost functional. he plant parameters are shown in Fig. 3 to Fig. 5. Equations (37) and (39) show that B - (k,q - ) is exponentially unstable because B is a triangular matrix and the absolute value of its first diagonal element is uniformly greater than unit. However, both A - (k,q - ) and C - (k,q - ) are exponentially stable because they are upper triangular and the absolute values of their diagonal elements are uniformly less than unit. Applying the LV GMVC the estimator () has the following form wˆ c( k ) wˆ( k ) c( k ) wˆ( k ) y a( k ) y( k ) a( k ) y( k ) u( k ) b( k ) u( k 3) b( k ) u ( k 3) (4) wˆ ˆ ˆ c( k ) w( k ) c( k ) w y a( k ) y( k ) a( k ) y u( k ) b( k ) u( k ) b ( k ) u ( k ) where a ij are the elements of A on the ith row and jth column. he same applies to B and C. he controller () has the form u s ( k ) ( k, q ) A( k, q ) u s( k ) wˆ Gkq (, ) wˆ (4) 5 4 3 y s - - -3-4 -5 3 time step k 7 8 9 Fig. Example first output. 5 y s 5-5 - -5 3 time step k 7 8 9 Fig. Example second output. where ISSN: 99-8763 66 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL.8.6.4. a a a a oscillatory than the second as shown in Fig. and Fig.. his is partly due to the stably uninvertible nature of the system as is characterised by the top left element in the LV MAO B(k,q - ). It has the form b ( k, q ).q (44) -. -.4 -.6 -.8 3 time step k 7 8 9 Fig. 3 Example plant parameters of A..5.5.5 -.5 - -.5 - b b b b -.5 3 time step k 7 8 9 Fig. 4 Example plant parameters of B..8.6.4. -. -.4 c c c c -.6 3 time step k 7 8 9 Fig. 5 Example plant parameters of C. Fig. 6 shows the two control variables produced by the LV GMVC. While the second control variable is quite smooth the first fluctuates a lot over a wide range. As a result, the first output is also more his LV property is corresponding to the nonminimum phase characteristics in LI case. When the coefficient of the second term in the above equation is changed from. to.9 the LV CARMA model becomes stably invertible. he plant outputs and control variables of this modified example are shown in Fig. 7 to Fig. 9. 5 5-5 u u - 3 time step k 7 8 9 Fig. 6 Example control variables. 5 4 3 y s - - -3-4 -5 3 time step k 7 8 9 Fig. 7 Modified example first output. Fig. 9 shows that the oscillation in the first control variable is reduced from a range of -9 to 4 in Fig. 6 to that of -5 to 7 in Fig. 9 when the system changes from stably uninvertible to stably invertible. Consequently, the oscillation in the first plant output is also reduced significantly as shown ISSN: 99-8763 67 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL in Fig. 7. However, there are very little changes in the second plant output and second control variable. he reasons can be seen from the triangular form of the equations (37) and (39). hey show that the second equation in the LV CARMA model (35) is independent of the first one. 5 5-5 - y s -5 3 time step k 7 8 9 Fig. 8 Modified example second output..5cos(.3 k).cos(. k) Ak ( ).3sign[cos(.5 k)]..5sin(.5 k)cos(.5 k) Bk ( ).sign[cos(.5 k)] k.( k ).3.6( e )sin(.5 k) Ck ( ) k 5.7sin(.3k.3) he weighting matrices are chosen as ( k 65) Pk ( ).5 ( 65) k.7 Rk ( ).5.49 ( 35) k where t () t t (45) (46) (47) 8 6 4 - -4 u u -6 3 time step k 7 8 9 Fig. 9 Modified example control variables. Example he first example shows that the output of the control system can be quite oscillatory due to the stably uninvertible nature of the system. he choice of the weighting functions can also influence the tradeoffs between the tracking performance and stability. In this example we study the affects of time-varying weighting functions. he structure of the system remains the same as in the first example and the plant parameters are the following. is the step function. he nonzero weightings are shown in Fig. and the plant output, input and parameters are shown in Fig. to Fig. 6. Fig. shows oscillatory tracking in the plant first output. he weighting function P is doubled and P is halved on k=65 in order to divert more emphasis on the tracking accuracy of the first plant output in the LV GMVC. However, this causes large oscillation in the first control variable as shown in Fig. 3 and the tracking performance in the first plant output does not improve after the weighting changes. his illustrates the difficulty in accurate tracking for stably uninvertible systems. Fig. shows a significant tracking error in the second plant output in the early stage. his error jumps at about the same time as the time-varying plant parameters a. he weighting function R is reduced to. on k=35 in order to allow large control actions to reduce this error and the performance improvement is significant. After k=65 the tracking accuracy degraded a little due to the changes made in the output weighting matrix P. ISSN: 99-8763 68 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL 5 4 3 P P R R 8 6 4 - u u -4-3 time step k 7 8 9 Fig. Example nonzero elements of the weighting functions. 5 4 3 - - -3-4 y s -5 3 time step k 7 8 9 Fig. Example first output. 5 5-5 y s - 3 time step k 7 8 9 Fig. Example second output. -6 3 time step k 7 8 9 Fig. 3 Example control variables..5 a.4 a.3 a. a. -. -. -.3 -.4 -.5 3 time step k 7 8 9 Fig. 4 Example plant parameters of A..5 b b.5 b b.5 -.5 - -.5 - -.5 3 time step k 7 8 9 Fig. 5 Example plant parameters of B. ISSN: 99-8763 69 Issue 7, Volume 5, July

WSEAS RANSACIONS on SYSEMS and CONROL.8.6.4. -. -.4 -.6 c c c c -.8 3 time step k 7 8 9 Fig. 6 Example plant parameters of C. 5 Conclusion Generalised minimum variance control has been studied for MIMO LV systems with multiple delays and an LV GMVC has been suggested. It extends the previous MIMO LV GMVC [3] from a uniform single delay case for multiple delay LV CARMA models. It extends also our previous LV MVC for MIMO systems [4], [5] with multiple delays by adding a penalised term for the control vector to the minimum variance cost functional and, thus, removes the stable invertability condition for closed-loop stability. References: [] Koivo, H. N., A multivariable self-tuning controller, Automatica, Vol. 6, 98, pp. 35-366. [] Borisson, U., Self-tuning regulators for a class of multivariable systems. Automatica, Vol. 5, 979, pp. 9-5. [3] Clarke, D. W. and P. J. Gawthrop, Self-tuning controller, Proc. IEE Part. D, Vol., 975, pp. 99-934. [4] Clarke, D. W. and P. J. Gawthrop, Self-tuning control, Proc. IEE Part. D, Vol. 6, 979, pp. 633-64. [5] Wolovich, W. A. Linear Multivariable Systems. Springer-Verlag, New York, 974. [6] Yang, A. M., J. P. Wu, W. X. Zhang and X. H. Kan, Research on Asynchronous Motor Vector Control System based on Rotor Parameters ime-varying, WSEAS RANSACIONS on SYSEMS, Vol. 7, 8, pp. 384-393. [7] Shahriari, K, S. arasiewicz and O. Adrot, Linear ime-varying Systems: heory and Identification of Model Parameters, WSEAS RANS- ACIONS on SYSEMS, Vol. 7, 8, pp. 445-454. [8] Li, Z., R.J. Evans and B. Wittenmark, Minimum variance prediction of linear time-varying systems. Automatica, vol. 33, no. 4, 997, pp. 67-68. [9] Li, Z. and R. J. Evans, Generalised minimum variance control of linear time-varying systems, IEE Proc.-Control heory Appl., Vol. 49,, pp. -6. [] Li, Z and C. Schmid, A generalised minimum variance controller for time-varying systems, Proc. of the 6 th IFAC World Congress, Prague, Czech, July 5, paperh-e3-p/. [] Li, Z. and C. Schmid, A Flexible ime- Varying Generalised Minimum Variance Controller, Proceedings of the 9th WSEAS International Conference on SYSEMS (ICS 5), Vouliagmeni, Athens, Greece, July, 5, paper 497-733. [] Li, Z. and C. Schmid, A ime-varying Generalised Minimum Variance Controller, WSEAS RANSACIONS ON SYSEMS, No. 5, Vol. 4, 5, pp. 55-5 [3] Li, Z., A Generalised Minimum Variance MIMO ime-varying Controller, Proc. of International Conference on Modelling, Identification and Control (Mon-4, A5-43), Shanghai, China, June 9 - July, 8. [4] Li, Z. and C. Schmid, A minimum variance controller for MIMO time-varying systems. IASME ransactions, Vol., 4, pp. 4-44. [5] Li, Z. and C. Schmid, A minimum variance controller for MIMO time-varying systems. Proc. of 4 IASME Int. Conf. on MECHAN- ICS and MECHARONICS, Udine, Italy, March 4, paper 483-55. ISSN: 99-8763 6 Issue 7, Volume 5, July