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A C E GHJLN ST U X \] f u  yz|}~    L2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 $2$Jg-Vj= 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj $2$rYUncertainty in Control Problems 4^\Randa Herzallah Al-Balqa Applied University, Jordan 16th July, 2008$E7 -2 Plan of talk ^\OControl Problem (1)^\ Control :  decision for  optimal performance in stochastic  uncertain environment subject to (partially uncertain) observation. General form of nonlinear stochastic discrete time control system The admissible control sequence which minimizes some specified performance index need to be found Such laws may be based on forward model which may adapt in two ways: uncertain state (Kalman filter), uncertain parameters (adaptive control). In control the following N-stage performance index which is expressed in terms of the tracking error, is usually of interest $PcPDPPcPPPPPP  |     3 UncertaintyDefinition: An uncertainty is a state of having limited knowledge where it is impossible to exactly describe existing or future states. Uncertainty is due to: Data noise Sensor variability Model uncertainty Dual Control Concept: Controller must have knowledge about its environment as well as knowledge about its knowledge validity. Certaintyequivalence is not optimal under modelling error. \1qZ1, ^)QControl Problem (2)^\' Direct Inverse Control^\.1Direct Inverse Adaptive Control & Uncertainty (1)22(^\Incorporating uncertainty of forward and inverse models by modelling the conditional distributions of both of them, (Herzallah and Lowe, 2004) lpdqduBFB/Nonlinear SISO Example (1)^\A liquid system described by the following equation is considered V, is Gaussian random variable N(0.03259,0.2) The input-output neural network model of the plant is taken to be rCGFU0Nonlinear SISO Example (2)^\LThe inverse neural model of the plant was The desired output is given by ML1&Classical and Sampling Control Results''(^\B Classical Inverse Controller Proposed sampling approach CBB 1Direct Inverse Adaptive Control & Uncertainty (2)22(^\Bayes theorem is used to estimate the pdf of the inverse controller, (Herzallah & Lowe, 2004). Bayes Rule: The likelihood density function The prior distribution of control signals The evidence probability density function _pdZ dZZ!dZZ+dZZ*dZZZ_ z  * !The negative log of the Posterior""^\wThe optimal control law is then found by minimizing the negative log of the posterior. Optimal control signal XWHdPPqdPW- Nonlinear Example^\ /(Neural adaptive Control: Indirect Method))(^\)Neural Adaptive Control & Uncertainty (1)**(^\A dual adaptive control scheme which takes into consideration models uncertainty is proposed, Herzallah 2005, for stochastic nonlinear discrete systems which could be described by any of the four different classes of models defined by Narendra. The main results were based on systems described by following general form To incorporate parameters uncertainty, the following form of the performance index is suggested Since non-affine system, a closed form for the optimal control law could not be obtained. @pdZIdZdZbdZdZZdZdZ} 9 adZB,^ 29Architecture of the Proposed Indirect Adaptive Controller::(^\3Nonlinear SISO Example (1)^\kThe dynamic equation of the system is e(k+1) is assumed to be sampled from a Gaussian distribution N(0,0.001) The forward model of the plant is taken to be a Gaussian RBF network with 9 Gaussian basis functions described by the following input-output model Uncertainty of the forward model is taken to have the same form and inputs as that of the forward model>l&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN ST U)X \] f u  yz|}~      !"#$%&'()*+,-./0123456789:=>?@ACDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~    L2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 $2$Jg-Vj= 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj $2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~    L2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI $2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYUncertainty in Control Problems 4^\Randa Herzallah Al-Balqa Applied University, Jordan 30th November, 2008$I7," 2 Plan of talk ^\3 UncertaintyControl :  decision for  optimal performance in stochastic  uncertain environment subject to (partially uncertain) observation. Definition: An uncertainty is a state of having limited knowledge where it is impossible to exactly describe existing or future states. Uncertainty is due to: Data noise Sensor variability Model uncertainty #ZdZZ1qZZqZZ | n/OControl Problem (1)^\General form of nonlinear stochastic discrete time control system The admissible control sequence which minimizes some specified performance index need to be found In control the following N-stage performance index which is expressed in terms of the tracking error, is usually of interest Such laws may be based on forward model which may adapt in two ways: uncertain state (Kalman filter), uncertain parameters (adaptive control). BqPPPdPPPPC d ~     :QControl Problem (2)^\4 Dual ControlDual Control Concept: Controller must have knowledge about its environment as well as knowledge about its knowledge validity. Certainty equivalence is not optimal under modelling error. (  ' Direct Inverse Control^\.1Direct Inverse Adaptive Control & Uncertainty (1)22(^\Incorporating uncertainty of forward and inverse models by modelling the conditional distributions of both of them, (Herzallah and Lowe, 2004) lpdqduBFB/Nonlinear SISO Example (1)^\A liquid system described by the following equation is considered V, is Gaussian random variable N(0.03259,0.2) The input-output neural network model of the plant is taken to be rCGFU0Nonlinear SISO Example (2)^\LThe inverse neural model of the plant was The desired output is given by ML1&Classical and Sampling Control Results''(^\B Classical Inverse Controller Proposed sampling approach CBB 1Direct Inverse Adaptive Control & Uncertainty (2)22(^\Bayes theorem is used to estimate the pdf of the inverse controller, (Herzallah & Lowe, 2004). Bayes Rule: The likelihood density function The prior distribution of control signals The evidence probability density function _pdZ dZZ!dZZ+dZZ*dZZZ_ z  * !The negative log of the Posterior""^\wThe optimal control law is then found by minimizing the negative log of the posterior. Optimal control signal XWHdPPqdPW- Nonlinear Example^\ /(Neural adaptive Control: Indirect Method))(^\)Neural Adaptive Control & Uncertainty (1)**(^\A dual adaptive control scheme which takes into consideration models uncertainty is proposed, Herzallah 2005, for stochastic nonlinear discrete systems which could be described by any of the four different classes of models defined by Narendra. The main results were based on systems described by following general form To incorporate parameters uncertainty, the following form of the performance index is suggested Since non-affine system, a closed form for the optimal control law could not be obtained. @pdZIdZdZbdZdZZdZdZ} 9 adZB,^ 29Architecture of the Proposed Indirect Adaptive Controller::(^\3Nonlinear SISO Example (1)^\kThe dynamic equation of the system is e(k+1) is assumed to be sampled from a Gaussian distribution N(0,0.001) The forward model of the plant is taken to be a Gaussian RBF network with 9 Gaussian basis functions described by the following input-output model Uncertainty of the forward model is taken to have the same form and inputs as that of the forward model>l&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~    L2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI $2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYUncertainty in Control Problems 4^\Randa Herzallah Al-Balqa Applied University, Jordan 30th November, 2008$I712 Plan of talk ^\3 UncertaintyControl :  decision for  optimal performance in stochastic  uncertain environment subject to (partially uncertain) observation. Definition: An uncertainty is a state of having limited knowledge where it is impossible to exactly describe existing or future states. Uncertainty is due to: Data noise Sensor variability Model uncertainty #ZdZZ1qZZqZZ | n/OControl Problem (1)^\General form of nonlinear stochastic discrete time control system The admissible control sequence which minimizes some specified performance index need to be found In control the following N-stage performance index which is expressed in terms of the tracking error, is usually of interest Such laws may be based on forward model which may adapt in two ways: uncertain state (Kalman filter), uncertain parameters (adaptive control). BqPPPdPPPPC d ~     :QControl Problem (2)^\4 Dual ControlDual Control Concept: Controller must have knowledge about its environment as well as knowledge about its knowledge validity. Certainty equivalence is not optimal under modelling error. (  ' Direct Inverse Control^\.1Direct Inverse Adaptive Control & Uncertainty (1)22(^\Incorporating uncertainty of forward and inverse models by modelling the conditional distributions of both of them, (Herzallah and Lowe, 2004) lpdqduBFB/Nonlinear SISO Example (1)^\A liquid system described by the following equation is considered V, is Gaussian random variable N(0.03259,0.2) The input-output neural network model of the plant is taken to be rCGFU0Nonlinear SISO Example (2)^\LThe inverse neural model of the plant was The desired output is given by ML1&Classical and Sampling Control Results''(^\B Classical Inverse Controller Proposed sampling approach CBB 1Direct Inverse Adaptive Control & Uncertainty (2)22(^\Bayes theorem is used to estimate the pdf of the inverse controller, (Herzallah & Lowe, 2004). Bayes Rule: The likelihood density function The prior distribution of control signals The evidence probability density function _pdZ dZZ!dZZ+dZZ*dZZZ_ z  * !The negative log of the Posterior""^\wThe optimal control law is then found by minimizing the negative log of the posterior. Optimal control signal XWHdPPqdPW- Nonlinear Example^\ /(Neural adaptive Control: Indirect Method))(^\)Neural Adaptive Control & Uncertainty (1)**(^\A dual adaptive control scheme which takes into consideration models uncertainty is proposed, Herzallah 2005, for stochastic nonlinear discrete systems which could be described by any of the four different classes of models defined by Narendra. The main results were based on systems described by following general form To incorporate parameters uncertainty, the following form of the performance index is suggested Since non-affine system, a closed form for the optimal control law could not be obtained. @pdZIdZdZbdZdZZdZdZ} 9 adZB,^ 29Architecture of the Proposed Indirect Adaptive Controller::(^\3Nonlinear SISO Example (1)^\kThe dynamic equation of the system is e(k+1) is assumed to be sampled from a Gaussian distribution N(0,0.001) The forward model of the plant is taken to be a Gaussian RBF network with 9 Gaussian basis functions described by the following input-output model Uncertainty of the forward model is taken to have the same form and inputs as that of the forward model>l&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\                          ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | }  V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~    L2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI $2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI $2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$168=;}f 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] $2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models u                         ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  ncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] $2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-List of Papers^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\ sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] $2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN STU,X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y 2$[mvwCm 2$@ZBl4 9DH 2$D֥oKRP 2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] $2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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Definition: An uncertainty is a state of having limited knowledge where it is impossible to exactly describe existing or future states. Uncertainty is due to: Data noise Sensor variability Model uncertainty Parameters uncertainty ZbZZIqZZqZZ  ; n/OControl Problem (1)^\General form of nonlinear stochastic discrete time control system The admissible control sequence which minimizes some specified performance index need to be found In control the following N-stage performance index which is expressed in terms of the tracking error, is usually of interest Such laws may be based on forward model which may adapt in two ways: uncertain state (Kalman filter), uncertain parameters (adaptive control). BqPPPdPPPPC d ~     :QControl Problem (2)^\4 Dual ControlDual Control Concept: Controller must have knowledge about its environment as well as knowledge about its knowledge validity. Two important properties: Direction: maintain small tracking error, with due consideration given to the uncertainty. Probing: elicit further information to reduce further parameter uncertainty. Certainty equivalence is not optimal under modelling error. Increasing number of cautious adaptive controllers in industrial applications is necessary. PPP " P " P<PP]Pl  ' Direct Inverse Control^\.1Direct Inverse Adaptive Control & Uncertainty (1)22(^\Incorporating uncertainty of forward and inverse models by modelling the conditional distributions of both of them, (Herzallah and Lowe, 2004) ppdqduBFB /Nonlinear SISO Example (1)^\A liquid system described by the following equation is considered V, is Gaussian random variable N(0.03259,0.2) The input-output neural network model of the plant is taken to be rCGFU0Nonlinear SISO Example (2)^\LThe inverse neural model of the plant was The desired output is given by ML1&Classical and Sampling Control Results''(^\B Classical Inverse Controller Proposed sampling approach CBB 1Direct Inverse Adaptive Control & Uncertainty (2)22(^\Bayes theorem is used to estimate the pdf of the inverse controller, (Herzallah & Lowe, 2004). Bayes Rule: The likelihood density function The prior distribution of control signals The evidence probability density function _pdZ dZZ!dZZ+dZZ*dZZZ_ z  * !The negative log of the Posterior""^\wThe optimal control law is then found by minimizing the negative log of the posterior. Optimal control signal XWHdPPqdPW- Nonlinear Example^\ /(Neural adaptive Control: Indirect Method))(^\)Neural Adaptive Control & Uncertainty (1)**(^\A dual adaptive control scheme which takes into consideration models uncertainty is proposed, Herzallah 2005, for stochastic nonlinear discrete systems which could be described by any of the four different classes of models defined by Narendra. The main results were based on systems described by following general form To incorporate parameters uncertainty, the following form of the performance index is suggested Since non-affine system, a closed form for the optimal control law could not be obtained. @pdZIdZdZbdZdZZdZdZ} 9 adZB,^ 29Architecture of the Proposed Indirect Adaptive Controller::(^\3Nonlinear SISO Example (1)^\kThe dynamic equation of the system is e(k+1) is assumed to be sampled from a Gaussian distribution N(0,0.001) The forward model of the plant is taken to be a Gaussian RBF network with 9 Gaussian basis functions described by the following input-output model Uncertainty of the forward model is taken to have the same form and inputs as that of the forward model>l&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN S TU.X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y $$$2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$䎝S^\NS 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. 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A C E GHJLN S TU.X \] f u  yz|}~     2$"~ .>Į\2$<3Bݱ x.xi`X2$so̓5Z2$K#ijY`v,\2$ A)s 9_2$M@ @p!3a2$y,3 G@3Ed2$sY9OpHDo PG~2$* 2$Kh2x X/cA 2$vR*iC2[oTD 2$8Ari 8Y]F 2$qLJH#G~I 2$Cg%[X':/ 2$?q?(/3YD0y $$$2$!t ^y21 2$LvFI 2$ Rm^!O 2$}X8* #Z 2$ͱlq/< P|FLA} 2$3vMwHLWԂ 2$kb ʤ+~! 2$#~a!ox sy*F$ 2$y YYwp( 2$5'5|}?~$* 2$_ la^i'(}kA 2$ܨF <:>̛.(/"C 2$L\L~[ɓ|Jr 2$ڱYrT "ss 2$j؟UW_uo)+lYu 2$vl03/9 ,8v 2$US p6' 2$CN s:-; ] 2$䎝S^\NS 2$hFd*y{nk 2$Xgh茓;)߷ 2$Esс)[t  2$Yn'CQ(-Dj 2$"m)ĝ-y#cK 2$rYl&=BKNonlinear SISO Example (2)^\The inverse model of the plant is taken to be a Gaussian RBF network with 7 Gaussian basis functions described by the following model The desired output is specified as 5Certainty equivalence Results^\"Certainty equivalence, r=-1, l = 0#A6Cautious Control Results^\Cautious, r=0, l = 1&A7Dual Control Results^\Dual control, r=-0.6, l = 0.8&A&)Neural Adaptive Control & Uncertainty (2)**(^\Bayes theorem is used to estimate the posterior distribution of the inverse controller (Herzallah & Lowe, 2008) The objective in control is to make the output of the system follow a predetermined desired output. Forward model of the system pqdddqddqdpd$S 'Bayes theorem (1)(vs tWe propose Bayes theorem to estimate the pdf of the inverse controller The likelihood density function L ddG , I(Bayes theorem (2)vs  WThe prior distribution of control signals Posterior distribution of control signals>*dd-dXW )Optimal control law(kMaximum of the posterior minimum of the expected value of the negative logarithm of the posterior lk+Forms of control problems(1)7Affine class of discrete time systems, where 46pp- ,Forms of control problems(2)[General nonlinear input-output discrete time systems, where No closed form @App[*ExampleSDynamic Programming^\C is the current state, g is the goal state The optimal control strategy from state c to g can be obtained by comparing the minimum cost of all the admissible decisions Adaptive Critic^\HThe cost function of the adaptive critic neural network is usually defined as: The forward model noise is usually assumed input independent when the adaptive critic is used. Defining a new variable called l[x(k)] as the derivative of the cost function with respect to the state x(k) at time k The optimality equation is OHdHdbBCBCBBBB,D/Y!Critic Architecture^\V ;Incorporating Uncertainty in Adaptive Critic Control Scheme<<(^\Adaptive critic network with forward model input-dependent noise is proposed, Herzallah 2005. The cost function of the adaptive critic network with input dependent noise is shown to be given by BHdN n]"Main Results for Linear System^\QFor systems described by the following general form And assuming input-independent noise, Werbos has shown that the true value of the cost function J satisfying Bellman equation is given by For input-dependent noise described by the following equation We show, Herzallah 2005, that the true value of the cost function J is given byFRZv  $,\ >d$Example^\Mixture Density Network(^\A Mixture density network for estimating general probability density functions of the inverse controller, (Herzallah & Lowe, 2004). It is applied to ill-posed control problems in which the solution to the inverse controller is not unique. The output from the mixture density network then can be selected in different ways Take the most probable output value corresponding to the most probable branch Take the most likely output value pddrdlSN#k B$ 0Mixture density network dealing with uncertainty11(^\tSampling from a mixture of Gaussians is then proposed in order to incorporate inverse and forward models uncertaintyuHdu8Nonlinear MIMO Example (1)^\Unstable nonlinear third order dynamic MIMO The outputs of the unknown plant are assumed to be given by Where the state dynamics are driven by two input control variables throughBhKiK?Nonlinear MIMO Example (2)^\The reference signals are taken to be Two separate forward NN models are used The inverse model is a MDN given by a MLP (7 hidden units) and two kernel functions (8 outputs) T')G9Most Probable Control Results^\:Most Likely Control Results ^\;!Sampling Approach Control Results""^\%  Conclusions ^\sRecently developed neural network control methods for incorporating models uncertainty have been presented. Methods of incorporating models uncertainty are also compared to the conventional control methods. It has been demonstrated through examples that a significant improvement could be obtained by incorporating knowledge of uncertainty in the control algorithms. VlHdZIdZHdZIdZZt-Papers of Interests^\sR. Herzallah and D. Lowe. A Bayesian Perspective on Stochastic Neuro Control. Volume 19, issue 5, 2008. R. Herzallah. Adaptive Critic Methods for Stochastic Systems with Input-Dependent Noise. Automatica. Volume 43, number 8, pages 1355-1362, 2007. R. Herzallah. Dual Version of Neural Adaptive Controller for Stochastic Nonlinear Systems with Functional Uncertainty. Journal of Control and Intelligent Systems. Volume 35, number 4, 2007. R. Herzallah and D. Lowe. Distribution Modelling of Nonlinear Inverse Controller Under a Bayesian Framework. IEEE Transactions on Neural Networks. Volume 18, issue 1, pages 107-114, 2007.tZt   3  '   M   1          /mopqv w x y z {|}~ !#$%&()*+,-./012} 0  $(  r  S 5@T  r  S 6@   H  0޽h ? 33___PPT10i./wn+D='  = @B +r^V:+Y1.  2 S8th.-@"Verdana-. 2 U?November, 2008.-rs Slide Titles,Root EntrydO)@-S<@PicturesGZ Current UserJ)SummaryInformation(                            ! " # $ % & ' ( ) * + , - 1 / 0 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { } ~  .                           ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s u v w x y z { | } ~   B  !"#$%&'()*K,-./0123456789:;<=>?@ABCDEFGHIMLNTOPQRS      !"#$%&'()*+,-./0123456789:=>?@ACDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~ely Control Results "Sampling Approach Control Results ConclusionsPapers of Interests  Fonts Used_CYDARDARr-.-@"Verdana-. 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