Algorithm for Multiple Model Adaptive Control Based on Input-Output Plant Model

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume No Sofia Algorithm for Multiple Model Adaptive Control Based on Input-Output Plant Model sonyo Slavov Department of Automatics echnical University of Sofia 756 Sofia Bulgaria Abstract: An algorithm for multiple model adaptive control of a -variant plant in the presence of measurement noise is proposed. his algorithm controls the plant using a bank of PID controllers designed on the base of invariant input/output models. he control signal is formed as weighting sum of the control signals of local PID controllers. he main contribution of the paper is the obective function minimized to determine the weighting coefficients. he proposed algorithm minimizes the sum of the square general error between the model bank output and the plant output. An equation for on-line determination of the weighting coefficients is obtained. hey are determined by the current value of the general error covariance matrix. he main advantage of the algorithm is that the derived general error covariance matrix equation is the same as this in the recursive least square algorithm. hus most of the well known RLS modifications for the tracking variant parameters can be directly implemented. he algorithm performance is tested by simulation. Results with both SISO and MIMO variant plants are obtained. Keywords: Multiple model adaptive control input-output model PID controllers variant plants.. Introduction he control system design has to be often realized under apriori uncertainty of the process model parameters. On the other hand many processes are significantly 3

2 changing their parameters during their normal functioning. Consequently a general control system design problem is to provide efficient control of the processes with significant parameters changes. he Multiple Model Adaptive Control (MMAC) is one of the maor approaches for control under significant parameters uncertainty [- 5]. he main idea of MMAC is that the complex plant dynamics can be represented by a discrete finite set of simple local models with constant parameters. Each of them describes the dynamics for one or more regimes. hen a limited discrete set of simple local controllers tuned according to the corresponding simple model is designed. he control is formed as weighting sum of local controllers control signals. he weighting coefficients are determined on-line. Historically the MMAC arises with the necessity to use the set of linear controllers in a state space system under the conditions of apriori uncertainty in the plant dynamics. In order to estimate the corresponding state vector in the plant linearized in a certain operation mode linear Kalman filters are used [4]. he first idea for utilizing a set of linear Kalman filters is formed into the so called static multiple-model state space estimator [6]. Later a scheme of interactive multiplemodel estimator is proposed in [7] where a general state vector is calculated with different weight of each local filter operation according to apriori defined transition matrix. here are many different MMAC algorithms based on the state space controllers and Kalman filters [8] and there is a lack of such based on the inputoutput model. Very close to MMAC based on the input-output model is the Multiple-Model Adaptive Switching Control [9-]. he main idea is that each controller of the bank takes an independent action in the control system tuned according to the corresponding plant model at the corresponding regime. he online controller switching is based on the performance index evaluation of the bank of models and/or controllers [3 3]. his approach is suitable when the plant operating regimes are apriori known and/or well defined. he useful MMAC algorithm based on the linear input-output model and deadbeat controller is proposed in [ 4 5]. his algorithm does not use Kalman filters. It is especially suitable for control in case of low variance measurement noise. he weighting coefficients are determined based on the current value of the inverse output model error or the current value of the inverse exponential smoothing output model error. In this paper MMAC algorithm based on input-output models and PID controllers is proposed. Each model has the same structure but different values of the parameters. he MMAC algorithm is similar to the multiple model adaptive control and state estimation algorithm presented in [6]. he principal difference is that the algorithm suggested in [6] is based on the state space models. It uses the bank of linear Kalman filters and the corresponding bank of LQR controllers. he other significant difference is in the obective function minimized to determine the weighting coefficients. he proposed algorithm minimizes the sum of the square general error between the model bank output and the plant output whereas the algorithm described in [6] minimizes the trace of the general residual variance or the trace of the general innovation term variance. he general residual and the innovation term are a multiple model Kalman filter residual and an innovation term. he advantage of the algorithm proposed here is that the current values of the 4

3 weighting coefficients are determined by the current value of the general error covariance matrix. As it can be later seen the derived general error covariance matrix equation is the same as this in the recursive least square algorithm (RLS). his means that most of the well known RLS modifications for the tracking variant parameters can be directly implemented in the suggested algorithm. he content of the paper is as follows. In Section the proposed MMAC algorithm is derived. In Section 3 the pseudo code of the MMAC algorithm is given. he results from the simulation of MMAC of both SISO and MIMO systems are presented in Section 4 and some conclusions are made in Section 5.. Multiple model adaptive control algorithm he block-diagram of the control system based on the proposed MMAC algorithm is shown in Fig.. r u μ u ξ y o u μ u q μ q y y q y e e e q Fig.. Block-diagram of the control system based on MMAC algorithm Let the controlled plant is variant and be described with the equation () yo ( s) = W ( s u( s) + ξ ( s) r m where yo ( s) R is a vector containing plant outputs u( s) R is a vector r containing plant inputs ξ ( s) R is a vector containing the measurement noises and W ( s is the transfer matrix. It has the form W( s W( s... W m ( s = W( s W( s... Wm ( s W ( s. M M M M Wr( s Wr ( s... Wrm ( s he elements of W ( s are given by 5

4 i i b ( s + b t s b t i ( ) i n ( ) i Wi ( s = i =... r =... m ni ni s + a ( s a t i n ( ) i where b ( b (... b ( i i n and a ( ) ( )... ( ) i t a t a t i i n are the transfer function i parameters. It is supposed that the parameters b and a l =... n are changing 6 n n according to known value intervals bl [ b min max ] i l b i l and a [ ] i l a i li min al i max. hen the complex plant dynamics can be approximated with a limited set of invariant models referred as local ones. Each local model contains a combination of bl i and a l i values of parameters into the known variation intervals. hese continuous- transfer functions are put into a discrete- form in order to design a set of discrete PID controllers. he set of local models forms the model bank in the structure scheme presented in Fig.. he description of i-th discrete- local model is given by () b W ( q) = i di q + a + b di d i q q y ( = W i b a i l i ( q) u( ni dn q i ni dn q i l i i =... r =... where b d b i d... b i dn and a i d a i d... a i dn are the local model parameters. For i each sample the combination of the local models is used to model the global plant behavior. A PID controller is designed for each local model. he set of PID controllers forms the controller bank in the scheme presented in Fig.. he control signal is obtained as weighting sum of the local controllers control signals (3) u( = μ u( + μu ( μ q uq ( where u i ( i =... q are the local controllers control signals and μi i =... q are the normalized weighting coefficients q (4) μ i =. i= he description of the -th digital PID controller is given by: u ( = u ( + u ( + u ( (5) p int d (6) up ( = K ( ) ( ) p d r k y k u ( = u ( k ) + b r( y( + b r( k ) y( k ) (7) [ ] [ ] int int int int (8) u ( = a u ( k ) + b [ c r( c r( k ) y( + y( k )] where d d bint = K p b int = int d d a d d + d = b N m = d N K p N r (k ) d d +

5 is the reference signal K p integral of the -th PID controller d the proportional gain of the -th PID controller int d derivative of the -th PID controller a constant of the -th first-order low pass filter the N sample d c weighting coefficients of the -th PID controller. he block named Supervisor determines on-line the current value of the weighting coefficients according to the proposed MMAC algorithm. In the algorithm a general model output ~ y is used. It is formed as weighting sum of the local models outputs (9) ~ y = μ y where () μ = [ μ ] μ... μq is a vector containing the normalized weighting coefficients and q () y = [ y y... y ] is a r q matrix containing the local model outputs y i i =... q. he equation (4) can be described in the form () μ = where = [... ]. 443 q he general output error e ~ is given by (3) e ~ = y ~ o y. aking into account equations (9) and () the general output error can be expressed as (4) e ~ = μ yo μ y = μ [ yo y ] = μ e where e = [ e e... eq ] is a q r matrix containing the errors i (5) ei = yo y i =... q between the plant output and the local model output. he main contribution of the paper is the obective function used for determination of the weighting coefficients. his function is defined as k (6) ~ J ( μ ) = ( ) ~ e i e ( i). i= From expressions (6) and (4) the following equation is obtained 7

6 (7) J ( μ) = μ P ( μ where k (8) P ( = e( i) e ( i) i= is a q q matrix. After taking into account the normalization (4) the weighting coefficients are determined from (9) min L ( μ λ) 8 μ where () L ( μ λ) = J ( μ) + λ[ μ ] and λ is the Lagrange gain. he necessary conditions for the extremum of () are () L μ = L λ = where L μ is the gradient with respect to c and Lλ is the gradient with respect to λ. After differentiation of () according to () one obtains () P ( μ + λ = μ =. hus the vector μ can be obtained from (3) μ = P ( λ. After multiplying (3) to the left by one finds (4) μ = P ( λ. hus taking into account the normalization (4) the Lagrange gain can be expressed as (5) λ =. After substituting (5) into (3) the vector μ is determined from (6) μ =. he matrix P ( is inverse of the one defined by (8). he equation (8) can be expressed as (7) k P ( = e( i) e ( i) + e( e ( = P ( k ) + e( e (. i= It is important to note that the equation (7) is the same as the one for the inverse covariance matrix in the RLS algorithm. he determination of the weighting coefficients current values requires real computation of the matrix rather than of the matrix P (. hus it is necessary to derive a recursive equation for the

7 computation of P (. hen the matrix inverse lemma is useful [7]. After applying the matrix inverse lemma to equation (7) one obtains (8) = k ) k ) e( [ I r + e ( k ) e( ] e ( k ) where Ir is the unit matrix. he current value of μ is determined from equation (6) where the current value of P ( is determined from equation (8). As can be seen for the single output plant the equation (8) is the same as one for the covariance matrix in the RLS algorithm if the regressors are substituted by the error e (. It is well known that the determination of according to equation (8) makes the recursive algorithm insensitive to the plant parameters changes. here are many useful modifications of RLS that modify the covariance matrix equation in order to keep algorithm s sensitivity. he main advantage of the proposed MMAC algorithm is that most of these RLS modifications are applicable for equation (8). In this paper four well known modifications for covariance matrix determination are used. heir covariance matrix equations are presented in able [8]. able Name Covariance matrix equation cmin RLS with = k ) c regularization of max Pk ( ) k ) e( [ I r + e ( k ) e( ] e ( k ) + cmin I where c < λ ( ) < λ ( ) c ) RLS with dependent update of Pk ( ) RLS with directional forgetting(only for SISO system) RLS with exponential forgetting = k ) k ) e( [ I min min max max r ) = P + e ( k ) e( ] P ( = + ci if trace( ) cmin = ci if trace( ) cmax P ( = if c min < trace( ) < cmax e ( k ) = k ) k ) e( [ ε + e ( k ) e( ] e ( k ) λ ε( k ) = λ e ( k ) e( where λ can be chosen as in RLS with exponential forgetting algorithm = [ k ) k ) e( [ I λ 3. Pseudo code of MMAC algorithm + e ( k ) e( ] he pseudo code of the MMAC algorithm involves the following steps. r e ( k )] Step. Input initial conditions for the design process and choice of a limited model set. 9

8 his step is performed off-line and it includes: - Choice of the model number q. his is a very important task. MMAC offers an approach to model the complex plant dynamics by combination of simple local models. MMAC algorithm will ensure control system performance if the model set represents adequately the plant dynamics. On the other hand utilization of unnecessary large number of models and controllers cannot guarantee the control performance. here are no general rules for the local model choice. he solutions for the particular tasks based on the state space models can be found in [9 ]. he amount of the selected models is usually related to the operating condition at which the control system is expected to work. If the value intervals of the plant parameters changes are unknown then the parameters of the local models can be estimated by an identification procedure for the variant plant. - Choice of the sample - Choice of the weighting coefficients initial values. he initial values of the weighting coefficients are usually chosen as μ ( ) = =... q. q - uning of a limited set of local PID controllers. Each local PID controller is tuned off-line according to the corresponding local model in the model bank. - Choice of the initial value of the covariance matrix. he initial value of the covariance matrix has to be chosen in a similar manner as the one in the corresponding RLS algorithm. - Set the zero initial conditions for the local models and local controllers. Step. he output y i ( of each local model is determined from equations (). Step 3. he error e i of each local model is determined from equation (5) Step 4. he covariance matrix P ( is determined from one of the equations presented in able. c i =... q are evaluated according to Step 5. he weighting coefficients i equation (6). Step 6. he control signal of each local PID controller is determined from the equations (5). Step 7. he general control signal uk ( ) is calculated from equation (3). 4. Simulation results he performance of the proposed MMAC algorithm in presence of measurement noise is tested by several simulated experiments. For this purpose software working in MALAB and Simulink environment is developed. he MMAC algorithm performance is investigated in comparison with the control system based on the conventional PID controller tuned for an average plant model.

9 Example he variant SISO plant is described by equation (). he transfer function is given by K( ( ( s + ) (9) W ( s = ( s + )(.7s + )(.8s + ) where K ( [ 5] and ( [.5 ]. he measurement noise ξ ( is zero mean white Gaussian noise with covariance D ξ =.3. It is supposed that the plant dynamics can be approximated by three local models. heir transfer functions are chosen as: (.5s + ) Model : W ( s) = ( s + )(.7s + )(.8s + ) 3.5(.75s + ) Model : W ( s) = ( s + )(.7s + )(.8s + ) 5( s + ) Model 3: W ( s) =. ( s + )(.7s + )(.8s + ) he sample is chosen as.5 s. During the simulation the reference signal and the parameters of transfer function (9) are varied as follows.5 for t < s t < 3 s 4 t < 5 s 6 t < 7 s 8 t < 9 s t < s r ( =.5 for t < s 3 t < 4 s 5 t < 6 s 7 t < 8 s 9 t < s.5 t < 8 s t < 8 s.75 8 t < 53 s 4 8 t < 53 s K ( = ( = t < 73 s t < 73 s 73 t < 93 s 5 73 t < s.6 93 t < s It is important to note that only in the range of -8 s the transfer function of the plant matches with some of the transfer functions in the model bank. he local PID controllers are tuned by optimization technique. he obective function used in the optimization procedure is given by I ise = ( r( y( ) dt. he following PID controller parameters are obtained: PID controller : Kp =.557 int =.9597 d =.9 c = b = N = PID controller : Kp =.397 int =.844 d =.9954 c = b = N = PID controller 3: Kp3 =.398 int =.89 =.99 c3 = b3 = N3 =. 3 d 3 he conventional PID controller is tuned for Model. In Fig. the output signals of the system based on MMAC algorithm with regularization of the

10 covariance matrix (denoted by ) and the system based on the conventional PID controller (denoted by PID ) are shown. he simulation of the MMAC algorithms are done for the following initial conditions algorithm: c min =.3 c max = P ( ) = I3 MMACCI algorithm: c min = c max = 4 c =. P ( ) =.I 3 MMACDF algorithm: λ =. 9 P ( ) =.I 3. For better visualization the output signals of the same systems within the range of 7- s are depicted in Fig. 3. In Figs. 4-5 the output signals of the system based on algorithm the system based on MMAC algorithm with a directional forgetting factor (denoted by MMACDF ) and the system based on MMAC algorithm with dependent updating of the covariance matrix (denoted by MMACCI ) are indicated. In Figs. 6-9 the control signals of the same systems as the ones shown in Figs. -5 are presented output PID.5 Output Fig.. Output signals of the control systems based on and PID controllers 3.5 output PID.5 output Fig. 3. Output signals of systems based on and PID controllers in the range 7- s

11 MMAC outputs MMACDF MMACCI output Fig. 4. Output signals of systems based on MMACDF and MMACCI controllers MMAC outputs MMACDF MMACCI. output Fig. 5. Outputs of systems based on MMACDF and MMACCI controllers in the range 7- s.5 control PID.5 control Fig. 6. Control signals of systems based on and PID controllers 3

12 ..8 control PID.6.4 control Fig. 7. Control signals of systems based on and PID controllers in the range 7- s It is seen from the figures that the performance of the systems based on all MMAC algorithms is better than this of the system based on a PID controller. he PID system response has large oscillations in the range of 7- s where the plant gain is higher than this used for the PID controller tuning. In the same range the systems based on MMAC algorithms kept their performance. he settling of the systems based on all MMAC is considerably smaller than this of the system based on a PID controller. Furthermore the PID system cannot work the reference in the of 8-9 s. he PID system performance is good in the range 8-53 where the plant model is close to the one used for PID controller tuning. he results in Figs. 4-5 show that the output signals of the systems based on and MMACDF have smaller oscillations than those of the system based on MMACCI. Furthermore the output response of MMACDF system is without overshoot in more cases. he MAACCI maximum deviation is greater than this of and MMACDF when the plant gain changes from up to 4. In order to characterize more precisely the dynamic behaviour of the control systems their maximal overshoot σ max in the range - s and the square mean error are computed. he square mean error e ise is determined as eise = ( r( y( ) dt. he computed performance indices are shown in able. able. Mean square error and maximal overshoot of the control systems Indices MMACCI MMACDF PID e ise σ max % he indices presented in able point out the advantages of the proposed MMAC algorithms. he maximal overshoot of MMACCI and MMACDF is approximately 9 s smaller than the corresponding value for PID. 4

13 he mean square error of the proposed algorithms is approximately 5% smaller than the corresponding value of the PID..5.5 MMAC controls MMACDF MMACCI control Fig. 8. Control signals of systems based on MMACDF and MMACCI controllers.8.6 MMAC controls MMACDF MMACCI.4 control Fig. 9. Control signals of systems based on MMACDF and MMACCI algorithms in the range of 7- s.4 weighting coefficients. μ μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on In Figs. - the weighting coefficients of the system based on MMAC algorithms are shown. As can be seen from the figures the value of coefficient μ for and MMACCI is close to in the range -8 s where the plant has the same transfer function as the one of Model. he weighting coefficient of 5

14 and MMACCI are reevaluated faster after each change of the parameters k % and/or % than the corresponding values of MMACDF. he value of coefficient μ 3 is close to in the range of 75- s where the plant parameters are close to the ones of Model 3. MMACCI weighting coefficients. μ.8 μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on MMACCI. MMACDF weighting coefficients μ.8 μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on MMACDF Example he variant two input two output plant is described by equation (). Its transfer matrix has the form W( s W ( s W ( s =. W( s W ( s he elements of W ( s are given by ~ ~ ~ ( kkk3 ( s + )( s + )( 3s + ) + ) k4 W ( s = ~ ( s + )( s + )( 3s + )( 4s + ) ~ ~ ~ kk3k4 kk W ( s = ~ W ( s = ~ ( s + )( 3s + )( 4s + ) ( s + )( s + ) ~ k W ( s = ~ ( s + ) 6

15 where k = k ~ [ 5] k 3 =. 5 k ~ [ 3] 4 = ~ [.5 ] 3 =. 7 and 4 =.7. he measurement noise ξ ( is zero mean white Gaussian noise with covariance D. ξ = I. It is supposed that the plant dynamics can be approximated with the help of three local models. heir parameters are chosen as: Model : k = k = k 3 =.5 k 4 = 3 = =.5 3 =.7 4 =.7 Model : k = k = 3.5 k 3 =.5 k 4 = = =.75 3 =.7 4 =.7 Model 3: k = k = 5 k 3 =.5 k 4 = 3 = = 3 =.7 4 =.7. wo PID controllers are tuned for each local model. he first one of them is based on a feedback from the first output and the second one is based on a feedback from the second output. he conventional PID controllers are tuned for a model with parameters as follows: k = k = 3 k 3 =.5 k 4 =.5 = =.5 3 =.7 4 =.7. he sample is chosen as.5 s. During simulation the reference signal and the parameters of the transfer matrix vary as follows:.5 for t < s t < 3 s 4 t < 5 s 6 t < 7 s 8 < 9 s < s ( t t r = r ( =.5 for t < s 3 t < 4 s 5 t < 6 s 7 t < 8 s 9 t < s t < 8 s t < 8 s.5 8 s s t < ~ t < ~.5 8 t < 53 s ~ k = k4 = =.8 8 t < 53 s t < 73 s 3 53 t < 73 s 53 s 5 73 s.7 73 s t < t < t < he local PID controllers are tuned by optimization technique. he obective function is given by I ise = ( r ( y( ) dt + ( r ( y ( ) dt. he following PID controller parameters are obtained: Model : PID K p =.6 bint =.89 ad =.6 bd =.467 PID K.69 b =.385 a =.3 b. 899 p = int d d = Model : PID K p =.7463 bint =.68 ad =.7 bd =. 4 PID K.689 b =.64 a =.6 b.87 p = int d d = Model 3: PID K p3 =.6 bint =.94 ad =.8 bd =.549 PID K p3 =.648 bint =.783 ad =.4 bd =.8. Conventional PID controllers: PID K p =.578 bint =.35 ad =.39 bd =.3 7

16 PID K p =.68 bint =.8 ad =.53 bd =.4. he simulation of MMAC algorithm operation is done according to the initial conditions algorithm: c min =. 3 c max = P ( ) = I3 MMACCI algorithm: c min =. c max = c = P ( ) = I3 MMACEF algorithm: λ =. 98 P ( ) = I3. In Figs. 3-4 the output signals of the system based on MMAC algorithm with regularization of the covariance matrix (denoted by ) and the system based on the conventional PID controller (denoted by PID ) are shown. In Figs. 5-6 the output signals of the system based on algorithm the system based on MMAC algorithm with dependent updating of the covariance matrix (denoted by MMACCI ) and the system based on MMAC algorithm with exponential forgetting (denoted by MMACEF ) are depicted. In Figs. 7-8 the control signals of the same systems as the ones shown in Figs 3-6 are presented. In Fig. 9 the square error in the range - s is presented. he control systems maximal overshoots the square errors and settling s are shown in ables output PID. output Fig. 3. First output of the control systems based on and PID controllers.8.6 output PID.4. output Fig. 4. Second output of the control systems based on and PID controllers 8

17 .8 MMAC output.6.4. output MMACCI. MMACEF Fig. 5. First output of the control systems based on MMACI and MMACEF MMAC output MMACCI MMACEF. output Fig. 6. Second output of the control systems based on MMACI and MMACEF.8 MMAC control.6.4. control MMACCI. MMACEF PID Fig. 7. First control signal of the systems based on MMACI MMACEF and PID It is seen from the figures that the performance of the systems based on all MMAC algorithms is better than this of the system based on a PID controller. he systems based on all MMAC algorithms have a step response with sufficiently small overshoot (except MMACEF in the range of 3-4 s) and a settling in the range of - s. 9

18 .8 MMAC control.6.4. control MMACCI. MMACEF PID Fig. 8. Second control signal of the systems based on MMACI MMACEF and PID MMAC square error square eroror MMACCI MMACEF PID Fig. 9. Square error of the systems based on MMACI MMACEF and PID in the range - s able 3. Overshoot of the control systems ime range MMACCI MMACEF PID able 4. Settling of the control systems ime range MMACCI MMACEF PID able 5. Square error of the control systems ime range MMACCI MMACEF PID

19 he indices presented in ables 3-5 point out the advantages of the proposed MMAC algorithms. he overshoot of MMACCI and MMACEF is smaller than the corresponding value for a PID. In almost all ranges the overshoot of is smaller than the corresponding value for MMACCI and MMACEF and considerably smaller than the corresponding value of PID. As can be seen from the results presented in Fig. 9 and able 5 the square error of the proposed algorithms is approximately 5 % smaller than the corresponding value of PID in all ranges. he settling of MMACCI and MMACEF is 5- % smaller than the corresponding value of PID. In Figs. - the weighting coefficients of the system based on MMAC algorithms are shown. It is seen that no value of the weighting coefficients is converging to in the range - s. his is due to the fact that the plant parameters do not coincide with the corresponding values of the models in the model bank. Nevertheless the performance of the control system based on MMAC algorithms is kept. he weighting coefficient of and MMACCI are reevaluated faster after each change of the parameters than the corresponding values of MMACEF. weighting coefficients μ.8.6 μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on MMACCI weighting coefficients μ.8.6 μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on MMACCI 3

20 . MMACEF weighting coefficients.8 μ μ μ 3 weighting coefficients Fig.. Weighting coefficients of systems based on 5. Conclusion In this paper a Multiple Model Adaptive Control (MMAC) algorithm for control of a -variant plant in the presence of measurement noise is proposed. his algorithm controls the plant using a bank of PID controllers designed on the base of invariant input-output models. he control signal is formed as weighting sum of the control signals of local PID controllers. he main contribution of the paper is the obective function minimized to determine the weighting coefficients. he proposed algorithm minimizes the sum of the square general error between the model bank output and the plant output. he equation for on-line determination of the weighting coefficients is obtained. hey are determined by the current value of the general error covariance matrix. he main advantage of the algorithm is that the derived general error covariance matrix equation is the same as this in the recursive least square algorithm (RLS). hus most of the well known RLS modifications for the tracking -variant parameters can be directly implemented in the suggested algorithms. Four well known RLS modifications (RLS with regularization RLS with dependent updating RLS with directional forgetting and RLS with exponential forgetting) are implemented. he algorithm performance is tested by simulation. For this aim software in Matlab/Simulink environment is developed. Simulation experiments with both SISO and MIMO variant plants are carried out. Comparison between the control systems based on the developed MMAC algorithms and the control system based on a conventional PID controller tuned for average plant model is performed. he results show the advantages of MMAC algorithms over the conventional PID. In more of the ranges the evaluated performance indices are significantly smaller for the systems based on MMAC algorithms than the corresponding values for the system based on a single PID controller. 3

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