Robust Neural Control Strategies For Discrete-Time Uncertain Nonlinear .
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016Robust Neural Control Strategies for Discrete-TimeUncertain Nonlinear SystemsImen ZaidiMohamed ChtourouMohamed DjemelElectrical EngineeringDepartmentNational School of Engineers,Sfax, TunisiaElectrical EngineeringDepartmentNational School of Engineers,Sfax, TunisiaElectrical EngineeringDepartmentNational School of Engineers,Sfax, TunisiaABSTRACTIn this paper, three neural control strategies are addressed to aclass of single input-single output (SISO) discrete-timenonlinear systems affected by parametric variations.According to the control scheme, in a first step, a direct neuralmodel (DNM) is developed to emulate the behavior of thesystem, then an inverse neural model (INM) is synthesizedusing specialized learning technique and cascaded to thesystem as a controller. The sliding mode backpropagationalgorithm (SM-BP), which presents in a previous studyrobustness and high speed learning, is adopted for the trainingof the neural models. However, in the presence of strongparametric variations, the synthesized (INM) showslimitations to present satisfactory tracking performances. Infact, in order to improve the control results, two neural controlstrategies such as hybrid control and neuro-sliding modecontrol are proposed in this work. A simulation example istreated to show the effectiveness of the proposed controlstrategiesKeywordsSISO Discrete-time uncertain nonlinear systems, neuralmodelling, sliding mode, backpropagation algorithm, INMcontrol, hybrid control, neuro-sliding mode control.1. INTRODUCTIONIn practice, a large number of systems are strongly nonlinearand uncertain. Thus, in recent years, several studies dealingwith modeling and control of uncertain nonlinear systemshave been developed [1-2-3]. The first step of the control ofan uncertain nonlinear system is to find a mathematical modelable to reproduce the dynamic of this system with a requiredaccuracy. However, conventional modeling methods haveshown limitations to approximate correctly nonlinear systemsaffected by parametric uncertainties. In fact, the last decadehas witnessed an ever increasing research in Non conventionalmodeling methods as fuzzy system [4-5] and neural networks[6] since they have been considered as positional solutions toovercome these difficulties of modeling owing to theiruniversal approximation property. Topalov and kaynakpresented in [7] a robust neural identification of roboticmanipulators using learning algorithm based on sliding modecontrol technique. In [8], a problem of identification andcontrol of uncertain nonlinear system was investigated basedon fuzzy neural networks. Reference [9] proposed an adaptiverobust control based on neural network approximation for aclass of uncertain strict-feedback discrete-time nonlinearsystems.Moreover, control techniques using classical controllerspresent performance indexes degradation in case of uncertainnonlinear system. Indeed, it is important to develop effectiverobust control techniques [10]-[13] to guarantee stability,robustness and satisfactory tracking performances. In manystudies, neural networks have been proven useful andeffective for controlling a wide class of uncertain nonlinearsystem. In fact, Tellez et al, proposed in [14] a neural inverseoptimal controller to achieve stabilization for discrete timeuncertain nonlinear systems. In [15] a new approach for thecalibration and the control of spark ignition engines using acombination of neural networks and sliding mode controltechnique was presented. Internal model control (IMC) is alsoconsidered as a robust control technique. Indeed, Alzohairyproposed in [16] a neural internal model control approach forthe tracking of unknown nonaffine nonlinear discrete timesystems subject to external disturbances.This paper suggests robust neural control strategies for a classof single input-single output (SISO) discrete-time uncertainnonlinear systems. Indeed, in a first step, a direct neuralmodel (DNM) is elaborated to reproduce the dynamic of thesystem, then, in a second step, an inverse neural model (INM)is developed. After satisfactory training, the synthesized(INM) is applied as a controller for the uncertain nonlinearsystem. The most popular algorithm for the training offeedforward neural network (FNN) is backpropagation (BP)algorithm [17]. However, this training method is notcompletely robust face to disturbance and parametervariations. The sliding mode backpropagation (SM-BP) [18][20] has been adopted in a specialized learning technique ofthe (INM). The training of both (DNM) and (INM) isaccomplished through this algorithm which has been provenas the best configuration in previous study [19]. In order toimprove the robustness and the tracking performance of theabove neural control strategy, in the presence of strongparametric variations, two control strategies are proposed inthis work such as: hybrid control and neuro-sliding modecontrol. Thus, a proportional-integral controller (PI) and asecond order neuro-sliding mode corrective controller areadded to operate with the synthesized (INM) for the case ofthe hybrid control and the neuro-sliding mode controlrespectively. The rest of paper is organized as follows.Section 2 introduces problem statement. Neural modeling ispresented in section 3. In section 4, the neural controlstrategies are described in order to develop a robust neuralcontroller for the discrete-time nonlinear affected by smalland strong parametric uncertainties. A simulation example istreated in section 5 to show the effectiveness of the proposedcontrol strategies. Finally, in section 6 conclusions are given.2. PROBLEM STATEMENTConsider the SISO uncertain nonlinear system described bythe following equation:y(k 1) y(k ),., y(k n 1), u(k ),., u(k m 1), p (1)23
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016y and u are, respectively, the output and the input of they k mu k Fsystem, n is the order of,is the order of, isan unknown nonlinear function to be estimated by a neuralnetwork and p is an uncertain parameters vector. In thiswork, an additive uncertainty is considered.p p0 prepresents the nominal parameters andvector affecting the system. p'(6)X 2mj k X1mj k X1mj k 1 (7)'is uncertain3. NEURAL MODELLING: DNMIn order to reproduce the dynamic of system (1) a DNM isused. Indeed, it estimates the output of the system through olddata of its inputs and outputs. Two approaches often discussedin the literature are the series parallel model and the parallelone [6]. In this work, we are interested in series parallelmodel.The block diagram of the DNM training process is presentedby Fig.1:u (k )ZSystem iDNMym (k 1) h 1,2 , Ncm(8)C 0Where h is the hidden node, 0 Hand:'X1mHh (k ) X1mj (k ).W jhm (k ). f Hm Rhm (k ) (9)mjh(10)represents the weight between the output node j and the'e Thus, the weights update equations based on SM-BP are givenby:m Wjhm k m .sgn S mj k . X1mj k .YHh k ym (k 1) Fˆ y(k ),., y(k n 1), u(k ),. Z(3)and F̂ denote respectively the output of the DNM and theestimate of F .The weights of the DNM are adjusted to minimize the costfunction defined by:1 m 2 e 2 (4)em y k 1 ym (k 1)is the error between the output of they (k 1)and the one of the DNM m.The learning algorithm adopted is this work is SM-BPalgorithm which combines gradient descent method andsliding mode theory [18-19-20]. In fact, the SM-BP equationsare presented by the following equations.For the node j from the output layer, sliding surface isdefined as [21]:With j 1mhi k m .sgn S k . X k . Ti k mHhm1Hh(5)(11)mWith: i 1,.,(n m 1)ymS mj (k ) X 2mj (k ) C0 . X1mj (k )mSHh k X 2mHh k C0H .X1mHh k fmhidden node h , H denotes the derivative of the hiddenRmactivation function, h is the global input of the hidden nodemh and N c represents the number of neurons in the hiddenlayer.mThe output of the DNM is given by the following equation:y k 1 Let the sliding surface for each node of the hidden layer besuch as:WalgorithmFig. 1: DNM training process, u(k m 1), p mWhere f denotes the derivative of the output activationVmfunction, j is the global input of the output node j .X 2mHh k X1mHh k X1mHh k 1 y(k 1)LearningsystemX1mj k y k ym k . f m V jm k (2)p0J The index m refers to DNM’s parameters, j is the outputC 0node and 0(12)h Tmis the output of the hidden node , i is thei Zminput of the input node , hi represents the weight between 0 0the hidden node h and the input node i , mand m.WhereYHhmAccording to Utkin [21], the condition for existence of slidingmode and system stability is defined by the followingequation:Sds 0dt(13)For discrete time, Sarpturk et al. [22] defined the equationS (k ) S (k 1)instead of equation (13) as the necessaryand sufficient condition to guarantee the sliding manifold. Thecomputing of the limits of[18-19]. mand mis presented in4. NEURAL CONTROLIn this section, the design of the neural controller for uncertainnonlinear system through different control strategies ispresented.24
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 20164.1 Inverse Neural Model ControllerThe INM is trained to provide a control action that allows thebehavior of the uncertain system to be as close as possible tothe desired one. The specialized learning technique shown inFig.2 is considered in this work for training the INM [23].Thus, based on the DNM presented in section 3, which givesy ygood representation of the system mafter satisfactorytraining, the INM is trained.y d (k 1) ecy(k 1)𝑢(𝑘)INMDNMFig. 2 : Specialized method for INM trainingThe cost function to be minimized in the training step isexpressed as follows:21J c ec 2(14)ec y k 1 y d (k 1)is the error between the output of thedy(k 1)DNMand the desired one y (k 1) .Based on the SM-BP algorithm, updating rules for adjustingthe weights of the INM are expressed by the followingequations:jfrom the output layerS cj k X 2c j k C1. X1cj k Xc Z hic k c .sgn SHh k . X1cHh k .Ti c k C1 0 k [ y k 1 y k 1 ]. dmhand cAfter satisfactory training, the synthesized INM is simplycascaded with the plant as a neural controller of the uncertainnonlinear system [23] as given in Fig.3. Clearly, this approachrelies heavily on the fidelity of the inverse neural model usedas controller. For general purpose use serious questions ariseregarding the robustness of the INM controller. This lack ofrobustness can be attributed primarily to the absence offeedback. This problem can be overcome to some extent byusing on-line learning: the parameters of the inverse modelcan be adjusted on-line [24]. In this work, in order to improvethe performance of the developed INM controller, in presenceof strong parametric variations, other control strategies suchthat: hybrid control and neuro-sliding mode control areadopted.Z iy d (k 1) k . f [V k ]c'cjWithINM(16)y(k 1)System hm (k ) f m V jm k .W jhm k . f Hhm Rhm k .Z hm1 k '(22)can be found in reference [19].N 1mc(21)is the INM’s weight between the hidden node h and thei Ychinput node , Hh represents the output of the hidden nodec 0Tiirepresents the input of the input node of the INM , c 0 and cfurther information of the limits for the gain c(15)Where j is the output node,c1jThus, the weights update equations of the INM based on theSM-BP algorithm are given by equations (21) and (22):Z hic𝑍 𝑖Let’s the sliding surface for the nodebe defined as:'c Wjhc k c .sgn S cj k . X1cj k .YHh k Learning algorithmW jhcis the weight between the output node j and thefchidden node h of the INM, H is the derivative of the hiddenRcactivation function and h denotes the global input of thehhidden node .WhereZ' ih 1Fig. 3 : INM control structureX 2c j k X1cj k X1cj k 1 (17)4.2 Hybrid Controldenotes theVcderivative of the output activation function and j is thejglobal input of the output node .This approach has been proposed in [25-26]. It consists onoperating simultaneously a conventional controller and aconnectionist model to improve the control performance. Infact, the INM developed in section 4.1, trained using SM-BP,is used with a proportional-integral controller PI to generate acontrol action given as:For the node h from the hidden layer, the sliding surface isexpressed as follows:u(k ) uINM (k ) uPI (k )The indexcrefers to INM’s parameters,cSHh k X 2cHh k C1H .X1cHh k fc'(18)C 0Where h is the hidden node, 1HX1cHh k X1cj k .W jhc k . f Hc Rhc k (23)u (k )u (k )With INMand PIrepresent the output of the INMcontroller and the output of the PI controller respectively.uPI (k ) uPI (k 1) k p e(k ) ki e(k 1)(24)'Xc2 Hh k X k X k 1 c1Hhc1Hh(19)kp(20)e(k ) is the tracking error defined as:andkiare proportional and integral gains, respectively.25
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016e(k ) y d (k ) y(k )(25)dWith y (k ) represents the desired output and y (k ) is theactual system output.y (k 1)uC (k )2is the control action provided by the INM andrepresents the second order sliding mode correctivecontrol.S (k ) e(k ) e(k ) uINM (k )For the system defined by equation (1), the following slidingsurface is selected:INM dWherey(k 1)u (k )ClassicaSystelme(k 1)ControllerFig. 4 : Hybrid control structure(28)e(k ) y d (k ) y(k ) e(k ) e(k ) e(k 1)(29)y d (k ) Withand donate respectively the desired output anda positive constant that determines the slope of the slidingsurface.4.3 Neuro-sliding mode controlSliding-mode has been widely used to control uncertainnonlinear systems. In fact many studies have been proposedtowards finding a controller that guarantees robustness andsatisfactory tracking performances [27-28-29].In general, the sliding mode control law based on Lyapunovstability theory is given by:u(k ) ueq (k ) uc (k )uINM (k )INMy d (k 1) SecondOrder SMCy(k 1)Systemuc (k )2Corrective control(26)u (k )u (k )Where eqis the equivalent control law and cis thecorrective term added to ensure robustness.The classical SMC suffers mainly from two disadvantages.The first one is the high frequency oscillations of thecontroller output, termed “chattering”. The second is that acomplete knowledge of the plant dynamics is needed in thecomputation of the equivalent control [30]. In the literature,many works adopt the neuro-sliding mode control as astructure to solve these problems [31-32-33]. In fact, twoparallel neural networks are used to realize the equivalentcontrol and the corrective control as in Fig.5.NN1y d (k 1) NN 2y(k 1) SystemWith 0,1 (30)to ensure the stability of (k ).uc (k ) uc (k 1) K sign( (k ))22is a constant andsign . (31)is the signum function definedas:uc (k )NN1 and NN2 are two neural networks used, respectively, toestimate the equivalent control and to generate the correctiveu (k )control to estimate the chattering effects. The sum of equ (k )and cforms the control signal to be applied to thecontrolled system.In this work, the adopted neuro-sliding control structure isshown in Fig.6. The INM as presented in section 4.1 is usedwith a second order sliding mode corrective control togenerate the control signal to be applied to the uncertainsystem.The control action is computed as follows:2 (k ) S (k ) S (k 1)KFig. 5 : Neuro-sliding mode control structureu(k ) uINM (k ) uC (k )In case of second order sliding mode control, the slidingsurface and the sliding mode corrective term are given byequations (30) and (31), respectively [34]:The associated control action is given by the followingequation:ueq (k ) Fig. 6 : The adopted neuro-sliding mode control structure 1 sign (k ) 0 1 (32)5. SIMULATION RESULTSIn this section, the different proposed neural control strategiesare evaluated through a numerical example described by arecurrent nonlinear equation inspired from [6]:Consider the nonlinear uncertain system given by equation(33) which is a modified version of the one presented in [6]:y(k 1) a(k ) y (k ) c(k )u 3 (k )1 b( k ) y 2 ( k )(33)u (k )and y (k ) indicate respectively the inputkand the output of the system at the instant .The variables(27) (k ) 0 (k ) 0 (k ) 026
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016cand are bonded uncertaina(k ) 0.25;1.75 b(k ) 0.5;1.5 , case 1c(k ) 0.95;1.05 c(k ) case 2c(k ) 0.9;1.1 c(k ) 0.8;1.2 case 3a, bparametersas:- case 1 -105a, bandc0can1.40204060100k10ymcase 11.580- case 2 (3)(2)1.5ymyAssume that the variations of the parametersbe given by the following figure:(1)suchycase 2case 31.215110.500.5200 0100 k1000.8k 200 00100k200020406080100k- case 3-Fig. 7 : Variation of a (1), variation of b (2), and variationof c (3)10ymy5.1 Neural Modeling: DNM5According to the control structure, a DNM has to bedeveloped to emulate the behavior of the uncertain nonlinearsystem (33). The proposed DNM has two inputs u (k ) andy (k ) , ym (k 1) is the output of this model. The input is a 0 , 2 signal with amplitude distributed over the interval . Inorder to ensure compromise between the quality of modelingand the time of convergence, the choice of the DNM’sparameters has been done after several simulations. In fact,Nm 5neural model parameters are chosen such that: cand atotal of training sets N 200. In this work, the SM-BPalgorithm considers fixed learning rates. Thus severalsimulation results were carried out in order to find the bestvalues such that the sliding surface be smooth enough,avoiding thus chattering problems. The deduction of the boundaries of m and m is not shown here, nevertheless, thevalues chosen are within these boundaries. Fig. 8 illustratesthe behavior of the DNM on a test data set for the parametricvariations given by Fig.7.Table 1. DNM training parametersCase 1a(k ) 0.25 , 1.75 b(k ) 0.5 , 1.5 c(k ) 0.95 , 1.05 m 0.48, m 4.6C CH 10Case 3a(k ) 0.25 , 1.75 a(k ) 0.25 , 1.75 b(k ) 0.5 , 1.5 b(k ) 0.5 , 1.5 c(k ) 0.9 , 1.1 c(k ) 0.8 , 1.2 m 9, m 5 m 6.5, m 1C CH 5C CH 3.520406080100kFig. 8 : Evolution of the system output y (k ) , the DNMy (k )output mfor the validation set for different cases ofa, bvariation of parametersand cAccording to the simulation results given by Fig.8, the DNMpresents acceptable accuracy for the representation of thedynamic of system (33) for small variation of the parameter c c 0.95,1.05 , c 0.9,1.1 . However, for more important c 0.8,1.2 the DNM is less accurate.variation5.2 Neural ControlThe design of the robust neural controller for uncertainnonlinear systems through three control strategies is presentedin this section. Case 20INM controllerAfter satisfactory training of the DNM, it is used to train theINM. The input vector of the INM is composed by the desireddoutput y (k 1) and the output of the neural model y (k ) , onehidden layer with five hidden neuronsNcc 5and u (k ) asoutput. The training set is composed by N 200. INMtraining parameters are given by table 2.Table 2. INM training parametersCase 2Case 3a(k ) 0.25 , 1.75 Case 1a(k ) 0.25 , 1.75 a(k ) 0.25 , 1.75 b(k ) 0.5 , 1.5 b(k ) 0.5 , 1.5 b(k ) 0.5 , 1.5 c(k ) 0.95 , 1.05 c(k ) 0.9 , 1.1 c(k ) 0.8 , 1.2 c 3, c 9.8 c 2, c 13 c 3, c 15C CH 1C CH 2C CH 527
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016After a satisfactory training, the synthesis INM is placed incascade with the plant to be controlled, thus it is used as aneural controller for the uncertain nonlinear system. Theevolution of the system output for the parametric variationsgiven by Fig. 7, the desired output and different control signalare illustrated by Fig.9.2yd(3)1.51.21.51.11110.90.5- case 1-8(2)(1)20uy1.80500.80.5k 100 0k 100 05050k 10061.641.4Fig. 10 : Variation of a (1), variation of b (2) andvariation c (3)1.2210020406080100k0.8020406080100k- 6080k100- case 3ydy2u1.861.64 Hybrid controluAccording to the hybrid control structure, a proportionalintegral controller is used to operate simultaneously with theINM developed in the previous section. The proportional andk 0.1k 0.01the integral gains are chosen such as: pand i.The evolution of the system output, the desired one and thecontrol signal are shown in Fig.11.The performance result of the hybrid control is compared withthe INM one.1.4y,yd1.28(1)desired output1020406080k1000.8INM control6020406080k1.5100u0(2)2Hy brid Control2Fig. 9 : Evolution of the system output y (k ) , the desireddoutput y (k ) and the control signal u (k ) for differentcases of the variation of parametersa, band c1100 y (k ) y (k ) 100d120.50To show the robustness performances of the synthesized INMcontroller against parametric variations given by Fig.7, theerror between the desired output and the system one iscomputed as follows:E 402040k 60801000050k100Fig. 11: Evolution of the system output y (k ) controlled byhybrid control strategy and INM controller, the desireddoutput y (k ) (1) and control signal u (k ) associated to thehybrid approach (2)2k 1(34)Table 3. Comparative resultsFig.11 shows that the performance of the hybrid controlstrategy is satisfactory when the system to be controlled isaffected by parametric variations given by Fig.10. Parametric variationsECase 10.0038Case 20.0114Case 30.2012Neuro-sliding mode controlFor this proposed neuro-sliding mode control approach, asecond order sliding mode corrective controller is used tooperate with the INM synthesized previously in order tocompensate the effect of the parametric variations on thesystem to be controlled.The simulations parameters are chosen as:It is noted from Fig.9 and table 3 that the INM controller isnot able to present satisfactory tracking performances for theparametric variations of case 3 where: a(k ) 0.25 ;1.75 , b(k ) 0.5 ;1.5 and c(k ) 0.8 ;1.2 .Thus it is recommended to propose others control strategies inorder to improve control results of the system (33) affected bystrong parametric uncertainties.Assume that:c(k ) 0.8;1.2 a(k ) 0.25;1.75 , b(k ) 0.5;1.5 in the following parts.and 2.5, 0.2, K 0.03, 0.47In order to avoid chattering problems, the sign(.) function isreplaced by the following function: (k ) sat ( (k )) sign( (k ) (k ) 1 (k ) 1 (35)Fig.12. illustrates the evolution of the system output, thedesired one and the control signal.28
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016The performance result of neuro-sliding control is comparedwith the INM ones.(1)y ,y d2.5Neuro-sliding controldesired output2INM control61.5u4120.50020406080k 1000050k 100Fig. 12 : Evolution of the system output y (k ) controlled byneuro-sliding control strategy and INM controller, theddesired output y (k ) (1) and control signal u (k )associated to theneuro-sliding mode approach (2)It is noted from the simulation results given by Fig.12 that theadding of the second order neuro-sliding mode controller tothe INM has improved the system tracking performances. Theperformances of the different control strategies presentedabove are recapitulated in table 4.Table 4. Comparative resultsControl StrategyEINM control0.2012Hybrid control0.0062Neuro-sliding mode control0.0057pp. 8925-[6] K.S. Narendra, K. Parthasarathy “Identification andcontrol of dynamical systems using neural networks”IEEE Transaction on Neural Networks, vol. 1, no.1, pp.4–27, 1990.(2)8Systems with Applications, vol.36, no. 5,8931, 2009.6. CONCLUSIONTree neural control strategies of a class of SISO nonlineardiscrete time system affected by parametric variations wereproposed in this paper. The dynamic of this system wasapproximated by a DNM, in a first step, then based on thespecialized learning technique, an INM was synthesized. TheSM-BP algorithm was the used training algorithm adopted inthis work. In order to improve the tracking performances ofthe INM controller in case of important parametric variations,hybrid control and neuro-sliding mode control strategies wereproposed. A simulation example was employed to illustratethe effectiveness of the proposed control strategies. As futurework, others neural control strategies will be studied.7. REFERENCES[7] A.V. Topalov, O. Kaynak “Robust neural identificationof robotic manipulators using discrete time adaptivesliding mode learning” In Proceeding of InternationalFederation of Automatic Control World Congress, vol.38, no.1, pp. 336-341, 2005.[8] X-H. Ji “Fuzzy neural network control and identificationfor uncertain nonlinear systems” Chinese Control andDecision Conference, pp.4237-4242, 2010[9] X, Wang, T.Li, C.L.Philip Chen, B. Lin “Adaptive robustcontrol based on single neural network approximation fora class of uncertain strict-feedback discrete-timenonlinear systems” Neurocomputing, vol. 138, pp. 325331, 2014.[10] Z. Wang, D. W. C. Ho, Y. Liu, X. Liu “Robust controlfor a class of nonlinear discrete delay stochastic systemswith missing measurements” Automatica, vol.45, no. 3,pp. 684-691, 2009.[11] S. Sam Ge, J. Wang “Robust adaptive tracking for timevarying uncertain nonlinear systems with unknowncontrol coefficients” IEEE Transaction on AutomaticControl, vol. 48, no.8, pp. 1463-1469, 2003.[12] Z.Yan,J.Wang“Robustmodelpredictive control of nonlinear systems with unmodeleddynamics and bounded uncertainties based on neuralnetworks” IEEE Transactions on Neural Networks andLearning system, vol. 25, no. 3, pp. 457-469, 2014.[13] H. Abid, M. Chtourou, A. Toumi “Robust fuzzy slidingmode controller for discrete nonlinear systems”International Journal of Computers, Communications &Control vol. 3, no. 1, pp. 6-20, 2008.[14] F O. Tellez, E N. Sanchez, R G. Hernandez, J A. RuzHernandez, J L. Rullan-Lara “Neural inverse optimalcontrol for discrete time uncertain nonlinear systemsstabilization” TheInternational Joint Conferenceon Neural Networks (IJCNN), pp. 1-6, 2012.[1] Y-W. Chen, J-B. Yang, C-C. Pan, D-L. Xu, Z-J. Zhou“Identification of uncertain nonlinear systems:Constructing belief rule-based models” KnowledgeBased System, vol. 73, pp. 124-133, 2015.[15] T. Huang “Nonlinear torque and air-to-fuel ratio controlof spark ignition engines using neuro-sliding modetechniques” International Journal of Neural Systems, vol.21, no. 3, pp. 213–224, 2011.Y. Lin, Y. Shi, R. Burton “Modeling andRobust Discrete-Time Sliding-Mode Control Design fora Fluid Power Electrohydraulic Actuator (EHA) System”IEEE/ASME Transactions on Mechatronics, vol. 18, no.1, pp. 1-10, 2013[16] T.A.A. Alzohairy. “Neural internal model control fortracking unknown nonaffine nonlinear discrete-timesystems under external disturbances” InternationalJournal of Computer Applications, vol. 40, no.6, pp. 1926, 2012.[3] A.aydi, M. Djemal, M. Chtourou “Robust poleassignement for control uncertain nonlinear discrete-timesystems” 12th International Multi-Conference onSystems, Signals & Devices, pp. 1-5, 2015.[17] D.E. Rumelhart, G.E. Hinton, R.J. Williams “Learninginternal representation by error propagation. Paralleldistributed processing: explorations in the microstructureof cognition, MIT Press Cambridge, MA, vol.1, pp. 318362, 1986.[2][4] L.A. Zadeh “Fuzzy Sets” Information and control, vol.8, pp. 338-353, 1965[5] C.C. Chuang, J-T. Jeng, C-W. Tao “Hybrid robustapproach for TSK fuzzy modeling with outliers” Expert[18] G.G. Parma, B.R. Menezes, A.P. Barga “Improvingbackpropagation with sliding mode control” Proceedingsof the Vth Brazilian Symposium on Neural Networks.29
International Journal of Computer Applications (0975 – 8887)Volume 143 – No.10, June 2016Belo Horizonte, Brazil: IEEE Computer Society Press,pp. 8-13, 1998[19] Zaidi, M, Chtourou, M, Djemel “Robust neural control ofdiscrete time uncertain nonlinear systems using slidingmode backpropagation training algorithm” submitted toInternational Journal of Automation and Computing.[20] G. Parma, B.R. Menezes, A.P. Barga “Neural networkslearning with sliding mode control: the sliding modebackpropagation algorithm” International Journal ofNeural Networks, vol. 9, pp.187-193, 1999[21] V.I Utkin. “Sliding modes and their application invariable structure systems” MIR, Moscow, 1978.[22] S.Z. Sarpturk, Y. Istefanopulos, O. Kaynak “On thestability of discrete time sliding mode control system”IEEE Transaction on Automatic Control, vol. 32, no. 10,pp. 930-932, 1987[23] D. Psaltis, A.Sideris, A.A. Yamamura “A multilayeredneural network controller” IEEE Control SystemsMagazine, pp. 17-21, 1988.[28] Y. Lin, Y. Shi, R. Burton “Modeling and robust discretetime sliding-mode control design for a fluid powerelectrohydraulic actuator (EHA) System” IEEE/ASMETransactions on Mechatronics, vol.18, no.1, pp. 1-10,February 2013.[29] D.M. Tuan, Z. Man, C. Zhang, J. Jin, H. Wang “Robustsliding mode learning control for uncertain discrete-timemulti-input multi-output systems” IET Control Theoryand Applications, vol. 8, no. 12, pp. 1045-1053, 2014.[30] F.Huang, Y. Jing, G.M. Dimirovski “ Sliding modefeedback control for uncertain discrete-time MarkovJump systems” Proceedings of the 18th World CongressThe International Federation of Automatic Control(IFAC), vol. 4, no. 1, pp. 2419-2423, 2011.[31] V.I Utkin. “Sliding mode in Control and Optimization”.Springer- Verlag, 1981.[32] M. Ertugrul, O. Kaynak “Neuro sliding mode control ofrobotic manipulators”. Mechatronics, vol.
affected by parametric uncertainties. In fact, the last decade has witnessed an ever increasing research in Non conventional modeling methods as fuzzy system [4-5] and neural networks [6] since they have been considered as positional solutions to overcome these difficulties of modeling owing to their universal approximation property.
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
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Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
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H Control 12. Model and Controller Reduction 13. Robust Control by Convex Optimization 14. LMIs in Robust Control 15. Robust Pole Placement 16. Parametric uncertainty References: Feedback Control Theory by Doyle, Francis and Tannenbaum (on the website of the course) Essentials of Robust Control by Kemin Zhou with Doyle, Prentice-Hall .File Size: 1MB
Neuroblast: an immature neuron. Neuroepithelium: a single layer of rapidly dividing neural stem cells situated adjacent to the lumen of the neural tube (ventricular zone). Neuropore: open portions of the neural tube. The unclosed cephalic and caudal parts of the neural tube are called anterior (cranial) and posterior (caudal) neuropores .
A growing success of Artificial Neural Networks in the research field of Autonomous Driving, such as the ALVINN (Autonomous Land Vehicle in a Neural . From CMU, the ALVINN [6] (autonomous land vehicle in a neural . fluidity of neural networks permits 3.2.a portion of the neural network to be transplanted through Transfer Learning [12], and .
