NEURAL NETWORK-BASED MRAC CONTROL OF DYNAMIC NONLINEAR SYSTEMS

Transcription

1 Int J Appl Math Comput Sci 6 Vol 6 No 9 NEURAL NETWORK-BASED MRAC CONTROL OF DYNAMIC NONLINEAR SYSTEMS GHANIA DEBBACHE ABDELHAK BENNIA NOUREDDINE GOLÉA Electrical Engineering Institute Oum El-Bouaghi University 4 Oum El-Bouaghi Algeria {gdebbachenour_golea}@yahoofr Electronic Department Constantine University 5 Constantine Algeria abdelhakbennia@lapostenet This paper presents direct model reference adaptive control for a class of nonlinear systems with unknown nonlinearities The model following conditions are assured by using adaptive neural networks as the nonlinear state feedback controller Both full state information and observer-based schemes are investigated All the signals in the closed loop are guaranteed to be bounded and the system state is proven to converge to a small neighborhood of the reference model state It is also shown that stability conditions can be formulated as linear matrix inequalities LMI) that can be solved using efficient software algorithms The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters Simulation results are presented to show the effectiveness of the approach Keywords: neural networks reference model nonlinear systems adaptive control observer stability LMI Introduction Nonlinear control design can be viewed as a nonlinear function approximation problem see eg Billings et al 99; Narendra and Parthasarathy 99)) On the other hand neural networks have been proved to be universal approximators for nonlinear systems Cybenko 989; Cotter 99; Ferrari and Stengel 5; Park and Sandberg 99) Neural networks are capable of learning and reconstructing complex nonlinear mappings and have been widely studied by control engineers in the identification analysis and design of control systems A large number of control structures have been proposed including direct inverse control Plett ) model reference control Narendra and Parthasarathy 99; Patino and Liu ) sliding mode control Zhihong et al 998) internal model control Rivals and Personnaz ) feedback linearization Yesildirek and Lewis 995) backstepping Zhang et al ) indirect adaptive control Chang and Yen 5; Narendra and Parthasarathy 99; Neidhoefer et al ; Poznyak et al 999; Patino and Liu ; Seshagiri and Khalil ; Yu and Annaswamy 997; Zhang et al ) and direct adaptive control Ge et al 999; Huang et al 6; Sanner and Slotine 99; Spooner and Passino 996; Zhihong et al 998) The principal types of neural networks used for control problems are the multilayer perceptron MLP) neural networks with sigmoidal units and radial basis function RBF) neural networks Model reference adaptive control Landau 979) is a technique well established in the framework of linear systems In the direct model reference control approach the parameters of the linear controller are adapted directly either by gradient- or stability-based approaches to drive the plant output to follow a desired reference model This structure can be extended by utilizing the nonlinear function approximation capability of feedforward neural networks such as the multilayer perceptron MLP) to form a more general flexible control law Narendra and Parthasarathy 99; Patino and Liu ; Zhihong et al 998) Such a network could be trained on-line within the MRAC structure to facilitate the generation of a nonlinear control law that may be necessary to drive the output of the plant to follow the desired linear reference model In this paper we propose a new neural-networkbased model reference adaptive control scheme for a class of nonlinear systems Unlike in the cited works neural networks are not directly used to learn the system nonlinearities but they are used to adaptively realize like in linear MRAC the model following conditions The idea behind this approach is that activation functions play the role of a continuous scheduler of local linear feedback

2 such that at each operating point there exists a controller that achieves the model following objective Both full state information and observer-based approaches are developed The stability in both cases is established using Lyapunov tools It is shown that this approach is robust against external disturbances and the tracking and observation errors are bounded Compared with the previously developed direct adaptive approaches this adaptive scheme requires few assumptions and its implementation is much simpler ie neither supervisory nor switching control is needed to ensure stability and no upper bound on the high-frequency gain derivative is required Simulation results for an unstable inverted pendulum show the capabilities of the proposed approach for both tracking a reference model and estimating the system states The rest of the paper is organized as follows: Section introduces the neural networks used and states the nonlinear systems direct model reference adaptive control problem Section develops the full state information approach based on neural network state feedback Section 4 develops the observer-based approach using an adaptive observer to estimate the system states Section 5 presents simulation results and Section 6 concludes the paper Preliminaries The following notation will be used throughout the paper: The symbol y denotes the usual Euclidean norm of a vector y In case y is a scalar y denotes its absolute value If A is a matrix then A denotes the Frobenius matrix norm defined as A = a ij =tr A T A ij where tr denotes the trace of the matrix Neural Networks The general function of a one hidden layer feedforward network can be described as a weighted combination of N activation functions y = N ϕ i xθ i ) w i ) i= Here the input vector x and ϕ i ) represent the i-th activation function with its parameter vector θ i ) connected to the output by the weight w i The number of the input and output layers coincides with the dimension of the input vector and the output information number respectively The activation functions are chosen following the neural network at hand For example the MLP uses generally the sigmoid function defined as ϕ i xθ i )= +exp a i x) ) or the logistic function defined by G Debbache et al ϕ i xθ i )= exp a i x) +exp a i x) ) On the other hand RBF networks use Gaussian functions defined as ) ϕ i xθ i )=exp x c i σi 4) Since the above neural networks will be trained on-line to realise a control task in order to reduce computation load we will assume that the parameters of the activation functions θ i are fixed ie their number and shapes are fixed a priori The only adjustable parameters are the weights w i Then ) is rewritten in the compact form where and y = φ x) w 5) φ x) = ϕ x) ϕ N x) w T = w w N From approximation theory it is known that the modeling error can be reduced arbitrarily by increasing the number N ie the number of linear independent activation functions in the network That is a smooth function f x) x Ω x R n can be written as f x) =φ x) w + ɛ x) 6) where ɛ x) is the approximation error inherent in the network and w is an optimal weight vector Following the universal approximation results for the neural network 5) there exist a finite set of w and a constant ɛ such that 6) holds with ɛ x) <ɛ Problem Statement Consider the class of continuous-time nonlinear systems given by x = Ax + b fx)+gx)u 7) y = c T x 8) where x T = x x x n R n is the state vector u y R are the system input and output respectively and f x) g x) are smooth nonlinear unknown functions Furthermore A = n ) I n n n n

3 Neural network-based MRAC control of dynamic nonlinear systems b = n c = n where I n is the n n identity matrix The nonlinear system 7) 8) satisfies the following assumptions: P The high-frequency gain g x) is strictly positive and bounded by <gx) <g for some known constant g in the relevant control space Ω x The negative case can also be handled P The nonlinear function fx) can be decomposed as f x) = a x) x + a x) with a x) = a x) a x) a n x) R n as an unknown smooth nonlinear vector function and a x) as a bounded nonlinear function Neuro-State Feedback Design The neural adaptive controller to be designed consists of three neural components see Fig ) such that u = φ x) K x + φ x) k + φ x) k r ) where φ x) R N φ x) R N and φ x) R N are the vectors of activation functions in the three networks of dimensions N N and N respectively K R N n k R N and k R N are adjustable parameters of the three networks The first and third terms in ) are a nonlinear replica of the standard reference model feedback control The second term is added to account for the nonlinearity a x) As can be seen from Fig the first term in ) is realized by a single multi-output n outputs) network and the others are formed by single output networks Note that P is a usual assumption in adaptive control and P specifies the discussed class of nonlinear systems Note that from a practical point of view a large class of real plants satisfies Assumption P eg robot manipulators electrical drives hydraulic rigs power plants and turbofan engines etc The reference model is defined by the following stable LTI state equation: x m = A m x m + bb m r 9) y m = c T x m ) where x m R n is the state vector y m is the reference model output r is a bounded reference input A m is given by A m = n ) I n a m n n with a m R n and b m > Substracting 7) and 8) from 9) and ) respectively and using P yield the following tracking error dynamic: e = A m e + b a m + ax)) x + b m r gx)u a x) ) e y = c T e ) where e = x m x and e y = y m y are the state and output tracking errors respectively The control problem can be stated as follows: Design the control input u such that the reference model following is achieved under the condition that all involved signals in the closed loop remain bounded Fig Neural state feedback structure Now introducing ) in ) yields ) e = A m e + b a m + ax)+gx)φ x) K x ) a x)+gx)φ x) k ) gx)φ x) k b m r 4) According to the universal approximation results there exist optimal parameters K k and k defined as { } K =arg min K Ω k =arg min k Ω k =arg min k Ω am sup + ax)+gx)φ x)k x Ω x { } a sup x)+gx)φ x)k x Ω x { } gx)φ sup x)k b m x Ω x where Ω Ω Ω and Ω x are constraint sets for K k k and x respectively Define the minimum approxima-

4 tion errors a m + ax)+gx)φ x)k = ε x) 5) a x)+gx)φ x) k = ε x) 6) gx)φ x)k b m = ε x) 7) where ε x) R n and ε x) ε x) R are the minimum approximation errors achieved by the three neural networks in ) with the optimal parameters K k and k The properties 5) 7) can be viewed as a nonlinear version of the linear model following conditions where the activation functions play the role of scheduling parameters ie for every sample x t) the activation functions φ x) to φ x) quantify the contribution of the linear feedback to the controller output The adaptive neural controller ) satisfies the following assumptions: C The activation functions in φ x) are such that φ x) <ϕ where ϕ is a known positive constant and φ x) k > C Suitable upper bounds Ω = {K K <κ } Ω = {k k <κ } and Ω = {k k <κ } with κ κ κ > are given C The approximation errors are upper bounded by ε x) ɛ ε x) ɛ and ε x) ɛ for some positive constants ɛ ɛ and ɛ Assumption C indicates that the activation functions in φ x) should be chosen as smooth functions with a bounded rate of change Note that φ x) may not necessarily be globally bounded but it will have a constant bound within Ω x due to the continuity of φ x) Assumption C is made to prevent any drift of the neural controller parameters Assumption C follows from the universal approximation property of neural networks Hence using 5) 7) in 4) we obtain e = A m e + b gx)φ x) K x + gx)φ x) k + gx)φ x) k r ɛ x) x ɛ x) ɛ x) r 8) where K = K K k = k k and k = k k are the estimation errors of the neural networks parameters G Debbache et al Further exploiting the property 7) 8) can be rewritten as e = A m e + b m φ x)k b φ x) K x + φ x) k + φ x) k r + φ x)k b α e + ξ 9) with ξ = α x m + α + α r α = ε x) φ x)k ε x) φ x) K α = ε x) φ x) k ε x) φ x)k α = ε x) φ x)k Combining ) and 9) yields the following transfer function: b m e y = G s) φ x)k φ x) K x + φ x) k + φ x) k r + G s) φ x)k α e + ξ ) where G s) =c T si n A m b and s is the Laplace operator Define the polynomial Hs) =s n + β s n + + β n s + β n such that H s) is a proper stable transfer function Then dividing and multiplying ) by Hs) weget b m e y = G s) φ x)k φ f x) K x + φ f x) k + φ f x) k r + G s) φ x)k αf e + ξ f ) with G s) = H s) G s) α f = H s)α ξ f = H s)ξ and φ if x) =H s)φ i x) for i = Hence ) can be rewritten in the following state space form: b m e = A m e + φ x)k b c φ f x) K x + φ f x) k + φ f x) k r + φ x)k b c αf e + ξ f ) e y = c T e ) where b T c = β β n R n

5 Neural network-based MRAC control of dynamic nonlinear systems Since Gs) is Hurwitz and the pairs {A m b} and {A m c} are controllable and observable respectively Hs) can always be chosen such that G s) is strictly positive real SPR) It follows that for a given Q> there exists P = P T > such that the following equations are satisfied Landau 979): A T mp + PA m = Q 4) b T c P = c T 5) The parameters of the neural controller are updated using the following projection algorithm Sastry and Bodson 989): K = γ b m e y φ T f x)xt ) I γ b m e y tr K T φ T x) + K f xt K κ 6) k = γ b m e y φ T x) I k k T φ T f γ b m e x) f y k 7) k = γ b m e y φ T x)r I k k T f γ b m e φt x) r f y k 8) where K <κ or if K = κ I = and e y tr K T φ Tf x) xt if K = κ and e y tr K T φ Tf x) xt > { k <κ or if I = k = κ and e y φ f x) k if k = κ and e y φ f x) k > { k <κ or if I = k = κ and e y φ f x) k r if k = κ and e y φ f x) k r> and γ γ γ > are design parameters The update laws 6) 8) ensure that K κ k κ and k κ t if the initial values of the parameters are set properly To prove that K κ let V K = K = trkt K Then V K = d K =tr dt K T K If I K = we have either K <κ or K = κ and V K = γ b m e y tr K T φ T f x)xt ie we always have K κ If I K =wehave K = κ and V K = γ b m e y tr K T φ T f x)xt tr K T φtf x) xt ) ) + K K κ = γ b m e y tr K T φ T f x)xt tr K T φ x) xt Tf ) + K tr K T K ) κ = γ b m e y tr K T φ x)xt Tf ) + K tr K T K ) 9) κ Since e y trk T φ T f x)xt > and K = κ we get V K which implies that K κ Therefore we get K κ t a similar analysis can be used to show that k κ and k κ t Now consider the Lyapunov function V = φ x) k e T Pe+ tr KT γ K + kt γ k + kt γ k ) where for a given Q > P is the solution of 4) and 5) The differentiation of ) along the trajectory of ) yields V = φ x) k et Qe + φ x) k et Pe +b m e T Pb c φ f x) K x+φ f x) k +φ f x) k r + e T Pb c α f e + e T Pb c ξ f + tr K + kt k γ γ + kt k γ ) K T Further using the facts that K = K k = k and k = k we can arrange ) as V = φ x)k et Qe + φ x)k et Pe + e T Pb c α f e + e T Pb c ξ f

6 4 + tr KT γ γ b m e y φ Tf x)xt ) K + kt γ b m e y φ Tf γ x) ) k + kt γ b m e y φ Tf γ x)r ) k ) Then introducing the update laws 6) 8) in ) yields V = φ x)k et Qe + φ x)k et Pe + e T Pb c α f e + e T Pb c ξ f + I tr K T b m e y tr K T φ x) xt Tf ) + K K κ b m e y k k T + I φt x) f kt k b m e y k k T + I φt x) r f kt k ) Further ) can be arranged as V = φ x)k et Qe + φ x)k et Pe + e T Pb c α f e + e T Pb c ξ f + I b m e y tr KT K tr K T φ x) xt Tf + K κ ) + I kt k b m e y k T φt x) f k + I kt k b m e y k T φ T x) f k r 4) We now prove that the last three terms in 4) are always nonpositive If I = I = I = the result is trivial If I = then K = κ and e y trk T φ T f x) xt > On the other hand we have tr KT K = tr K T K tr K T K tr KT K 5) G Debbache et al Since trk T K κ trkt K =κ and tr K T K wegettr K T K which means that the fourth term in 4) is always nonpositive The same analysis can be used to show that the last two terms in 4) are also nonpositive Using the above result 4) can be rewritten as V φ x)k et Qe + φ x)k et Pe + e T Pb c α f e + e T Pb c ξ f 6) Further using the fact that e T Pb c ξ f et Pb c b T c Pe + ξ f and b c α f α where α is a given upper bound) in 6) we obtain V φ x)k et Qe + φ x)k et Pe + α e T Pe+ et Pb c b T c Pe+ ξ f 7) which using C and the fact that φ x) can be upper bounded by V κ e T Qe + ϕ κ e T Pe V κ e T Q + α e T Pe+ et Pb c b T c Pe+ ξ f 8) Finally 8) can be rearranged as ϕ + α κ ) P κ Pb c b Tc P ) e + ξ f 9) If the matrices Q and P are chosen such that Q ϕ + α ) P Pb κ κ c b T c P Q 4) for some positive definite matrix Q then 9) becomes V κ e T Q e + ξ f 4) which can be upper bounded by V κ λ Q e + ξ f 4) where λ Q is the smallest eigenvalue of Q Then V outside the bounded region defined by e κ λ Q ξ f 4)

7 Neural network-based MRAC control of dynamic nonlinear systems 5 are bounded ė L and therefore e is uniformly continuous Integrating both sides of 4) yields e dt κ λ Q V ) + V T ) + ξ f dt 5) κ λ Q Fig Full state-based adaptive neural control structure The properties of the proposed adaptive neural state feedback approach are summarized by the following theorem: Theorem The control system of Fig composed by the nonlinear system 7) 8) satisfying P P the reference model 9) ) the neural controller ) satisfying C C the update laws 6) 8) and the matrices P Q satisfying 4) 5) and 4) guarantee the following: i) ii) iii) x x m + c ξ f 44) u κ x m + κ r + c ξ f 45) e dt c + c 4 ξ f dt 46) Proof i) Using 4) and the fact that x x m + e we have x x m + ξ f 47) κ λ Q Then 44) holds by setting c =/ κ λ Q ii) From ) and the parameter bounds the control input is bounded by u κ x + κ + κ r 48) where the fact that φ i x) i= is used Then introducing 47) in 48) we get u κ x m + κ + κ r + κ c ξ f 49) Therefore 45) is proved with c = κ c iii) From 4) we have that e L and since K k k L we obtain V L Since all terms in ) Hence setting c = κ λ Q V ) + V T ) and c 4 = κ λ Q yields 46) Remarks: The control law ) is a nonlinear form of the linear MRAC control structure Various terms may be included such as a term proportional to the tracking error or a switching control term Note that different control laws may be used in different control theory frameworks such as feedback linearization or adaptive sliding mode control In the proposed approach RBF or MLP networks may be used depending on the application context RBF networks are simple to design and rapid in the training process but they are not suitable for high dimensional systems since RBF suffers from the course of dimensionality MLP networks are more suitable for highly nonlinear systems due to their parallelism and approximation capabilities The bounds κ κ and κ are selected to force the realization of 5) 7) ie they should be such that κ > max a m + ax)) /gx) κ > max a x)/gx) and κ > max b m /gx) Since the system s nonlinearities are poorly known these bounds should be large enough to ensure that the above limits are included 4 It would be simpler to choose the same set of activation functions in ) ie φ x) = φ x) = φ x) =φx) But in practice fx) and gx) may depend on different state variables and therefore the proposed approach provides a mean to design the neural control input with a reduced number of activation functions and adjustable parameters 5 Compared with the direct approach presented in Spooner and Passino 996) this approach is simpler since only neural state feedback is used and no supervisory control term is required to ensure the boundedness of variables 6 Compared with Ge et al 999; Spooner and Passino 996) where an upper bound on ġx) is required to achieve the control objective this scheme needs only an upper bound on φ x) which is a user designed parameter and therefore is available for evaluation

8 6 G Debbache et al AT m P PA m ϕ + α κ ) P κ Pb c b T c P Pb c ct b T c P c P> 5) > 5) 7 Using Barbalat s lemma Sastry and Bodson 989) it can be shown that if ξ f L then the tracking error asymptotically converges to zero ie lim t et) = 8 The stability conditions encountered in 4) 5) and 4) can be expressed in the LMI form 5) and 5) where the LMI 5) is introduced to force the positivity of P Then an LMI-based stability analysis is carried out to check whether the stability condition is satisfied In the case where 5) and 5) are not satisfied the reference model dynamics and/or the filter H s) should be modified so that the above LMIs are feasible 4 Observer-Based Design The above design was based on full state information Since this is seldom the case in many practical situations the problem will be solved using only the system s output as the available information To estimate the system states define the following observer: ê = Aê + l e y ê y ) 5) ê y = c T ê 54) where ê = x m x is the estimated tracking error ê y = y m ŷ is the output estimated tracking error x ŷ are the estimated states and output respectively and l T = l l l n R n is selected such that A lc T is a Hurwitz matrix The observer 5) 54) can be further arranged as ê = A lc T ê + lc T e 55) ê y = c T ê 56) The control input is now defined using the estimated states as u = φ x) K x + φ x) k + φ x) k r 57) Introducing the control input 57) in ) and using 5) 7) yields e = A m e + b m φ x)k b φ x) K x + φ x) k + φ x) k r with + φ x)k b α 4 e + α 5 ê + ζ 58) ζ = α 6 x m + α 7 + α 8 r α 4 = φ x)k gx)φ x)k ε x) φ x) K α 5 = φ x)k α 6 = α 4 + α 5 ) ε x) gx) φ x)k + φ x) K )) α 7 = ε x) φ x) k + φ x)k gx) φ x)k +φ x) k ε x) α 8 = ε x) φ x) k + φ x)k gx) φ x)k +φ x) k ε x) where φ i x) =φ i x) φ i x) i = Then dividing and multiplying 58) by Hs) yields e = A m e + b m φ x)k b c φ f x) K x+φ f x) k +φ f x) k r + φ x)k b c α4f ê + α 5 e + ζ f 59) where ζ f = H s) ζ α 4f = H s) α 4 and α 5f = H s) α 5

9 Neural network-based MRAC control of dynamic nonlinear systems Using ) 59) and 55) 56) the augmented dynamics of the estimated error and tracking error is given by v = A a v + b m φ x)k b a φ f x) K x + φ f x) K r + φ x)k b a αf v + ζ f 6) e y = c T a v 6) + v T P a b a αf v + ζ f + tr K T γ K 7 + kt k γ + kt k γ 65) Then exploiting 6) 65) can be arranged as V = φ x) k vt Q a v + φ x) k vt P a v + v T P a b a αf v + ζ f + tr KT γ γ b m e y φ Tf x) xt ) K with v T = ê T e T b T a = n b T c c a = n c T α f = α 4f α 5f and A a = A lc T lc T n n A m Note that for the state space realization 6) 6) there exist two symmetric positive definite matrices P a and Q a such that A T a P a + P a A a = Q a 6) P a b a = c a 6) This can be proved by observing that c T a si n A a b a = G s) and on the other hand that G s) can be always made SPR by a proper choice of H s) which means that 6) and 6) are satisfied The parameters of the neural controller 57) are adjusted using the same update laws 6) 8) except that x and φ if x) are replaced by x and φ if x) respectively It can be verified using the same analysis that the update laws 6) 8) with the indicated changes) guarantee that K κ k κ and k κ t Consider the Lyapunov function V = φ x) k vt P a v + tr KT γ K + kt γ k + kt γ k 64) where P a is the solution of 6) and 6) for a given positive definite matrix Q a The differentiation of 64) along the trajectory of 6) yields V = φ x) k vt Q a v + φ x) k vt P a v + b m v T P a b a φ f x) K x + φ f x) k + φ f x) k r + kt γ b m e y φ Tf γ x) ) k + kt γ b m e y φ Tf γ x) r ) k 66) Further using the update laws 6) 8) and the same analysis as that performed in the previous section it can be shown that the last three terms in 66) are always nonpositive Hence 66) becomes V φ x) k vt Q a v + φ x) k vt P a v + v T P a b a α f v + v T P a b a ζ f 67) which is upper bounded by V κ v T Q a v + ϕ κ v T P a v + ρ v T P a v + vt P a b a b T a P av + ζ f 68) where the upper bounds b a α f ρ and φ x) are used Further 68) can be arranged as V κ v Q T a ϕ + ρ ) P a ) P a b κ κ a b T a P a v + ζ f 69) Hence if the matrices Q a and P a are chosen such that Q a ϕ + ρ ) P a P a b κ κ a b T a P a Q 7) for some positive definite matrix Q then V κ v T Q v + ζ f 7) Hence 7) can be upper bounded by V κ λ Q v + ζ f 7) where λ Q is the smallest eigenvalue of Q

10 8 G Debbache et al which is upper bounded by V o λ Q o ê + λ Po e ê 8) where λ Po are the smallest and the largest eigenvalues of Q o and P o respectively Then V o whenever ê is outside the region defined by ê λ P o e 8) Fig Observer-based neural adaptive control structure Then V outside the bounded region defined by v κ λ Q ζ f 7) The properties of the observer-based neural adaptive approach are summarized by the following result: Theorem The control system of Fig composed of the nonlinear system 7) 8) satisfying P P the reference model 9) ) the neural controller 57) satisfying C C the state observer 5) 54) the matrices P a and Q a satisfying 6) 6) and 7) and the update laws 6) 8) with the indicated changes) guarantee the following: i) ii) iii) x x m + c 5 ζ f 74) x x m + c 6 ζ f 75) u κ x m + κ + κ r + c 7 ζ f 76) ê dt c 8 + c 9 ζ f dt 77) e dt c + c ζ f dt 78) Proof i) Define the following Lyapunov function: V o = êt P o ê 79) where for some positive definite matrix Q o P o is a solution of the equation A lc T T Po + P o A lc T = Q o 8) The differentiation of 79) along 55) yields V o = êt Q o ê + ê T P o lc T e 8) On the other hand we have v = e + ê If combined with 8) this yields e +4 ) v 84) Further 7) and 84) yield the following upper bound: e κ λ Q +4 ζ f 85) Then introducing 85) in 8) gives ê λ Po / κ λ Q +4 ζ f 86) From 86) and the fact that x x m + ê we get x x m + which yields 74) with c 5 = λ Po / κ λ Q +4 ζ f 87) λ Po / κ λ Q +4 Now from x e + x m and 85) it follows that x x m + κ λ Q +4 ζ f 88) Then 75) is satisfied for c 6 = κ λ Q +4

11 Neural network-based MRAC control of dynamic nonlinear systems 9 P a > 94) AT a P a P a A a ϕ + ρ ) P a P a b κ κ a b T a P a P a b a c T a b T a P a c a > 95) ii) Using 87) and u κ x + κ + κ r weget u κ x m + κ + κ r + κ c 5 ζ f 89) Hence 76) is proved with c 7 = κ c 5 iii) Introducing 85) in 8) gives V λ Q o ê + λ Q o c 5 ζ f 9) Then integrating both sides of 9) yields ê dt V ) + V T ) + c 5 ζ f dt 9) which gives 77) with c 8 = V ) + V T ) and c 9 = c 5 To prove 78) integrating both sides of 7) yields v dt κ λ Q V ) + V T ) + ζ f dt 9) κ λ Q Further using 84) we get e dt T ) +4 Then combining 9) and 9) yields 78) with and V ) + V T ) c = +4 κ λ Q c = +4 ) ) κ λ Q v dt 9) Remarks: From 77) and 78) we have that if ζ f L then by Barbalat s lemma the tracking and estimated tracking errors converge to zero Other observers such as a high gain observer may be used but in this case the control input should be saturated to avoid the effect of large transient values of the estimated states Parts i) iii) of Theorem and point out the following facts: First the tracking and estimation errors are bounded and their bounds depend on some design parameters that can be tuned by the designer Second the input control is also bounded and the amount of energy can be controlled by the design parameters Third if the uncertainty terms are square integrable then the tracking and estimation errors converge to zero 4 The satisfaction of the stability conditions 6) 6) and 7) can be expressed as an LMI problem of the form 94) and 95) The feasibility of 94) and 95) depends on the choice of the reference model 9) and ) the filter Hs) and the observer dynamics 5) and 54) The reference model and observer eigenvalues influence the magnitudes of the eigenvalues of matrix P which affects the positivity of 95) On the other hand an increase in the norm of Hs) reduces the norm of the uncertain terms α 4f and α 5f which reduces the magnitude of ρ but increases the filtering dynamics 5 Simulation The validity of the proposed neural adaptive control scheme is verified on an inverted pendulum Fig 4) described by with x = x x = fx x )+gx )u + d 96) fx x )= m c + m)g sin x mlx cos x sin x l 4 m ) c + m) m cos x

12 Fig 4 Inverted pendulum cos x gx )= l 4 m ) c + m) m cos x where x x and u are the pendulum angular position velocity and input force respectively Moreover d is an uncertainty term Remark that fx) can be written as fx) =ax)x with x T = x x and m c + m)g a x) = l 4 m ) sin x c + m) m cos x x mlx cos x sin x a x) = l 4 m ) c + m) m cos x For the simulation the system parameters are chosen as follows: m c = Kg m = Kg l = 5 m and g = 98 m/s The reference model is given by a m = and b m = The control system is discretized using the sampling step T =ms We note that since T is small the stability conclusions drawn on the continuous-time basis remain valid If this is not the case the proposed approach should be redesigned in a discretetime framework The steps of the observer-based adaptive neural control design are as follows: For the observer design select l T = 8 6 and to ensure the SPR condition select Hs) =s +) Regarding the reference model dynamics the reference variables are bounded within the interval along x and x respectively) As for the form of f x) and g x ) the control input is given by u = φ x x ) K x + φ x ) k r 97) where the first term in 97) is constructed by a twooutput RBF network defined by nine RBF functions ie φ x x ) R 9 having standard deviations of / and and using mesh spacings of and with respect to x and x respectively The second term in 97) is constructed by a single-output G Debbache et al RBF network with three RBFs ie φ x ) R having standard deviation of / and using mesh spacings of with respect to x K R 9 and k R are the two adjustable parameters of neural networks Note that the pendulum velocity x is replaced by the estimated one x The required bounds are fixed as κ = 6 and κ = which implies a x) < 6 a x) < 5 and 98 <gx ) < 47 respectively The evaluation of φ x ) over the interval defined for x and x yields φ x ) ϕ =8 The upper bounds on the approximation errors are fixed as ɛ =5 and ɛ = 4 Using the above bounds the solution of the LMIs 94) and 95) yields P a = > which gives with b T a = and c a = the following result: b T a P a c a = which is practically zero 5 The free control parameters are adjusted using 6) and 8) with γ = 5 and γ = The tracking performance for the nominal plant ie no uncertainty is introduced) is shown in Fig 5 It is clear that the tracking errors are rapidly reduced and the control input is moderate and smooth Further the pendulum position and velocity are rapidly estimated by the observer used Figure 6 shows the tracking performance under the disturbance term d =signx )+x where after a short transient period the tracking errors converge to small bounded values Figure 7 illustrates the effect of parametric uncertainties where the masses and length are reduced by 5% Case a)) and increased by 5% Case b)) Finally Fig 8 shows the combined effect of the disturbance term with two parametric uncertainty cases a) and b) Those results illustrate the neural controller robustness against disturbances and uncertainties and its effectiveness in both tracking the reference model and estimating the pendulum state vector As predicted by theory the evolution of the norms of the neural adaptive controller parameters converges to bounded values

13 Neural network-based MRAC control of dynamic nonlinear systems Fig 5 Nominal plant tracking performance a) b) 6 Conclusion Fig 6 Effect of disturbance This paper has presented an extension of the model reference control approach to nonlinear plants using adaptive neural networks Both full state and observer-based approaches were analyzed and their stability was proved The advantages of this approach over the previously designed ones are as follows: no supervisory control is needed to ensure closed loop stability and no knowledge of a high-frequency gain derivative bound is required The simulation results conformed the effectiveness and simplicity of the proposed algorithm Future work is directed to extend this control design to multivariable nonlinear systems References Billings SA Jamaluddin HB and Chen S 99): Properties of neural networks with applications to modeling nonlinear dynamical systems Int J Contr Vol 55 No pp 9 4 Fig 7 Effect of parametric uncertainties Cybenko G 989): Approximations by superpositions of a sigmoidal function Math Signals Syst Vol pp 4 Cotter ME 99): The Stone-Weierstrass theorem and its applications to neural nets IEEE Trans Neural Netw Vol No 4 pp 9 95 Chang Y-C and Yen H-M 5): Adaptive output feedback tracking control for a class of uncertain nonlinear systems using neural networks IEEE Trans Syst Man Cybern Part B Vol 5 No 6 pp 6 Ferrari S and Stengel RF 5): Smooth function approximation using neural networks IEEE Trans Neural Netw Vol 6 No pp 4 8 Ge SS Hang CC and Zhang T 999): A direct method for robust adaptive nonlinear control with guaranteed transient performance Syst Contr Lett Vol 7 pp Huang SN Tan KK and Lee TH 6): Nonlinear adaptive control of interconnected systems using neural networks IEEE Trans Neural Netw Vol 7 No pp 4 46 Landau YD 979): Adaptive Control: The Model Reference Approach New York: Marcel Dekker

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