EXTENSIONS ON ADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR DYNAMICAL SYSTEMS USING NEURAL NETWORKS

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1 EXTENSIONS ON ADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR DYNAMICAL SYSTEMS USING NEURAL NETWORKS Fabrício Galene Marques e Carvalho, Eler Moreira Hemerly Instituto Tecnológico e Aeronáutica - IEE-IEES, Praça Mal Euaro Gomes, 50, Vila as Acácias, São José os Campos-SP s: fgmc@itabr, hemerly@itabr Abstract This paper reviews a recently evelope aaptive output feeback control methoology using neural networks It is shown how to remove the contraction mapping assumption use in the original paper it is clarifie the impact of this removal in the approximate moel selection Moreover, it is illustrate how to explore the previous information, available in the form of an approximate moel, to improve the controlle system performance Keywors Output feeback, neural networks, aaptive control, input-output linearization Resumo Neste trabalho faz-se uma revisão e uma técnica recentemente esenvolvia e controle aaptativo utilizo realimentação e saía e rees neurais É mostrao como se poe remover a suposição e mapeamento tipo contração utilizaa no trabalho original, além e se esclarecer o impacto este resultao na seleção e um moelo aproximao a planta Ilustra-se também como se poe explorar a informação prévia isponível, sob a forma e um moelo aproximao, na melhoria e esempenho o sistema controlao Palavras-chave Realimentação e saía, rees neurais, controle aaptativo, linearização entraa-saía Introuction Research on aaptive control using neural networks NN is motivate by novel actuator evices the existence of a large class of nonlinear systems for which a systematic constructive control proceure has not been evelope yet The first works in aaptive control using neural networks ate back 980s early 990s A pioneer work in this fiel that use NN in ientification control of nonlinear ynamical systems is that reporte by Narenra Parthasarathy 990 Following Narenra Parthasarathy s work, Sanner Slotine 99 use raial basis function RBF NN to control affine nonlinear systems by state feeback Furthermore, Sanner Slotine combine NN with proportional-erivative PD nonlinear sliing moe controllers prove system stability through a Lyapunov base analysis A similar approach was use by Lewis et al 995 to control a planar robotic manipulator As in the work of Sanner Slotine, in contrast to Narenra Parthasarathy s work, this approach oes not nee an off-line training phase the control architecture was compose of linear in the parameters neural networks All the works that were mentione so far are characterize by NN that operates in conjunction with other linear or nonlinear controllers are thus a kin of assiste control techniques A ifferent approach that uses NN as the main centralize controller can be foun in the works of Rovithakis 999, Ge et al 999 Ge Zhang 003 The main rawback share by these approaches lies on the centralize control architecture that can lea to unesirable or completely intolerable transient behavior, specially in critical applications like aircraft control Stevens Lewis, 003 An approach that can use prior knowlege about the plant combines input-output linearization Isiori, 995, NN linear control is reporte in Calise et al, 00 In this paper NN compensates the inversion error that arises when an approximate moel is use in ynamic inversion A contraction mapping is assume between the aaptive control the inversion error, which poses some restrictions in the moel selection Base in the work of Hovakimyan et al 004, which is similar but uses observers to calculate the error vector, we remove the contraction mapping assumption show the impact of this result in moel selection It is shown that the same upate law, propose by Calise et al 00, can be use without the contraction mapping assumption provie that some aitional knowlege about plant ynamics is available This article also aress the problem regaring the moel choice in a way to satisfy the conitions use in the stability proof Problem Statement Consier a SISO, continuous-time observable nonlinear ynamical system given by ẋ = fx, u, y = hx, x Ω R n is the state, u R is the input y R is the output In - f, : R n R R h : R n R are both continuous ifferentiables n nee not be known

2 Assumption The ynamical system given by Eqs - is feeback linearizable with relative egree r That is y = hx = L 0 f hx = h 0 x T h ẏ = fx, u = L f hx = h x x T Lf hx ÿ = fx, u = L f hx = h x x y r = L r f hx, u = h r x, u, 3 with h k u = 0 for 0 k < r As in Calise et al, 00, it is esire that the output y follows y generate by a stable reference moel is also assume that only the output measurements are available Moreover, any internal ynamics is assume stable The control strategy is base in input-output linearization is shown in section 3 3 Controller Architecture Input-output linearization is one using a pseuo control variable efine as follows y r = v, 4 v = h r x, u 5 u = h r x, v 6 An approximate inversion can be mae with v = ĥry, u, 7 u = ĥ r y, v 8 In Calise et al, 00, the output of the NN is esigne to cancel efine by = x, u = h r x, u ĥry, u 9 v is given by v = y r + v c v a, 0 v c is the output of a single-input multioutput SIMO linear compensator, v a is the output of a NN In this way, v a is esigne to cancel, which epens of v a through v To guarantee existence uniqueness of a solution for v a a contraction mapping between it is assume the following two conitions restrict the choice of an approximate moel: sgn h r / = sgn ĥr/, ĥr/ > h r/ > 0 To eliminate the nee of conition, base in the work Hovakimyan et al, 004, we efine v a = v c + y r 3 v b = ĥry, h r x, v a 4 Because of the assume invertibility of ĥr, h r, with respect to the secon argument h r x, v a = ĥ r y, v b v a = h r x, ĥ r y, v b 5 Now, base on Eqs 9,0, 4 5 we can write v a = v a h r x, u + ĥry, u = v a h r x, u + v a v a = h r x, u + v a = h r x, ĥ r y, v + v a = h r x, ĥ r y, v b h r x, ĥ r y, v 6 If h r is continuous ifferentiable over the interval v, v b, then, by applying the mean value theorem, there exists a v such that h r v v= v = h rx, ĥ r y, v b h r x, ĥ r y, v v b v 7 v = v + θv b v, 0 θ 8 Equation 6 can now be rewritten as v a = hr v v= vv b v = v b v = ĥry, h r x, v a v a + v a = v a x, y, v a, 9 x, y, v a = ĥry, h r x, v a v a, 0 = h r v v= v Accoring to Assumption, h r 0 v = v, therefore, in the light of 7 we have h r v = h r If conition is satisfie, then h r v = h r ĥ r ĥr > 0 3 the error ynamics can be expresse as ỹ r = v c + v a 4

3 Figure : Control System block iagram Now, the aaptive signal v a is esigne to cancel, which oes not epen explicitly on u Figure shows a schematic iagram of the controller architecture It shoul be note that Fig is similar to the controller structure in Calise et al, 00, but now the input to the neural network is not v but v a For systems with stable zero ynamics an aitional elaye input vt may be require to account for the unobservable states The elaye v signal avois the fixe point solution problem at the expense of increase NN approximate error boun Kim, 003 The SIMO linear controller is given by vc s ỹ a s = Nc s D c s N a s ỹs 5 ỹ = y y 6 Assumption The roots of D c s are locate at the open left half complex plane The transfer function between ỹ v a is given by ỹs = D cs v a s s r D c s + N c s 7 Therefore, from Eqs 5 7 the transfer function between ỹ a v a is given by ỹ a s = N asv a s s r D c s+n c s = Gs v a s 8 A stable low pass filter T s is introuce to make the transfer function between ỹ a v a strictly positive real SPR: ỹ a s = ḠsT s v a s, 9 Ḡs = GsT s A linear in the parameters NN can be use to approximate the inversion error The output of such a network is given by y nn = W T φx, 30 W are the NN weights φ is a vector basis function over the omain of approximation For a general continuous k times ifferentiable function gx with x D R n gx = W T φx + ɛx 3 In Eq 3, ɛx is the reconstruction error Definition: The functional range of a NN that have its output given by 30 is ense over a compact omain x D if for any continuous k times ifferentiable g ɛ there exists a finite set of boune weights W such that Eq 3 hols with ɛx < ɛ Theorem Calise et al, 00 Given ɛ > 0, there exists a set of boune weights W such that x, y, v a can be approximate over a compact omain D Ω R by a linearly parameterize neural network = W T φη + ɛη, ɛ < ɛ 3 using the input vector ηt = v T ȳ T T, 33 v T = v at v a t v a t n r T, ȳ T = yt yt yt n T 34 with n n > 0, provie there exists a suitable basis of activation functions φ on the compact omain D Proof: See Calise et al, 00 Theorem allow Eq 9 to be rewritten as ỹ a s = ḠsT s Ŵ T φ W T φ ɛs = ḠsT s W T φ ɛs = Ḡs{ W T φ f s + δs ɛ f s} 35 T s W T φs = T s W T φs+ W T φ f s W T φ f s T s W T φs = W T φ f s + δs, 36 δs = T s W T φs W T φ f s, 37 φ f s = T sφs, 38 ɛ f s = T s ɛs 39 Provie that is continuous over the interval v a, v, it reaches its maximum value h vmax on this interval if φ is a squashing function, an upper boun for δ can be written as δ c W F h vmax 40 The following upate law Calise et al, 00 can be use to ajust the NN free parameters Ŵ = F ỹ a φ f + λŵ, 4 F is a positive efinite matrix λ is the aaptation gain Next section shows how the relaxation of the contraction mapping assumption alters the stability proof

4 4 Stability Proof Let {A c, B c, C c }, {A f, B f, C f } be the controller canonical state space realization of Ḡs all the cast filters T s use to filter φ, respectively z z f be the corresponing state variables associate with these realizations Since the filter T s is stable Ḡs is SPR, it follows that A T f P f + P f A f = Q f, 4 A T c P + P A c = Q 43 P B c = C T c, 44 for positive efinite Q f, P f, Q P Equation 44 follows from the Kalman-Yakuvovich lemma Astrom Wittenmark, 995 The following theorem is the main result of this paper assures ultimate bouneness of the close loop signals Theorem Subject to Assumptions - if sgn h r ĥr = sgn, the error signals of the system comprise of the ynamics in Eq -, together with the ynamics associate with the realization of the controller the NN aaptation rule, are uniformly ultimately boune, provie that the following conitions hol: Q m = Q m > C c 45 λ > c /4, 46 Q m h vmax H v P MAX 47 c = c h vmax 48 In Eqs Q m is the minimum eigenvalue of Q, P MAX is the maximum eigenvalue of P, is the minimum value of h v H v is the maximum value of t Proof: Consier the following positive efinite ecrescent function V = z T P z + zt f P z f + W T F W 49 Differentiating 49 with respect to time gives V = t z T P z + P z + P T z T ż T + P f z f + Pf T z f zf T + F T W + F W W V = t z T P z + P z + P T z T A c z+ +B c W T φ f + δ ɛ f T + P f z f + Pf T z f Af z f + B f φ T F T W + F W W + 50 Because of the symmetry of the matrices F, P P f, using the fact that the transposition of a scalar is equal to the same scalar, substituting the upate law in 50 we get V = t + zt C T c z T P z zt Qz W T φ f + δ ɛ f +z T P f B f φ + W T F Ŵ V = + ỹa t zt f Q f z f z T P z zt Qz + ỹ a W T φ f δ ɛ f zt f Q f z f + z T P f B f φ ỹ a W T φ f W T λŵ V = zt Qz + ỹa δ ɛ f zt f Q f z f +zf T P f B f φ λ W T W + W + t z T P z 5 Assuming that the filter T s is scale so that its maximum gain is unity, the right sie of Eq 5 can be upper boune as V Qm z h vmax Q fm z f + HvP MAX z + z f P f B f φ + c ỹ a W F h vmax + ỹ a ɛ h vmax λ W W F W V Qm ỹ a Cc h vmax + HvP MAX ỹ a Cc + c ỹ a W F + ỹ a ɛ Qfm z z f f P f B f φ λ W W F W V Q m ỹ a Cc + c ỹ a W F + ỹ a ɛ Qfm z z f f P f B f φ λ W W F W, 5 ɛ = ɛ h vmax 0 53 Inequality 5 is the same as inequality 40 of Calise et al, 00 with Q m, ɛ c replace by Q m, ɛ c respectively Therefore, the remaining of the proof is straightforwar Remark As can be seen in 5, in Calise et al, 00 there is an error which was carrie out throughout the stability proof Therefore, the correcte conition that restricts the choice of Q is Q m > C c Remark For systems with unknown but constant control effectiveness, H v = 0 conition 45 becomes Q m > h vmax C c 54 The same conition can be use as an approximation to systems { } with unknown slowing varying 0 t

5 Section 5 shows a esign example to clarify how this result affects the inversion moel choice the aaptive control esign 5 Design Example Most of papers that employ techniques similar to that presente in works like Calise et al, 00; Hovakimyan et al, 00; Hovakimyan et al, 004, assume that contraction mapping hols use an approximate inverting moel given by v = u 55 Sometimes, an approximate linear moel aroun an operating point is use This occurs specially in the state feeback case Rysyk Calise, 005 In both cases, the question that arises about the existence of a contraction mapping between v a is not touche upon The following example shows that the contraction mapping assumption can be easily violate in both cases Example: Consier the nonlinear system given by x = x x = x 3 + cosx + K + K sinµx u, 56 with µ = 00, K = K = Linearization aroun the operating point x op = 5 0 T furnishes an approximate control effectiveness ĥr 0506, which clearly violates the contraction mapping assumption if x = π/µ The same thing occurs if Eq 55 is use in the inversion Suppose that an inverting control law of the form v = K u is use, then ĥr = K On the other h { H v = max t = K + K sinµx K v= v, 57 { H v = µ K K max h vmax = K + K K 58 K K + K sinµx cosµx x K + K sinµx }, 59 } 60 This suggests that an increasing in K ecreases increases H v, therefore this might not be the best way to assure a solution for Now suppose that v = K + K sinµx u Carrying out the same calculations for h vmax H v we get { } K + K sinµx h vmax = max K + K, 6 sinµx { } µk K K K cosµx x H v = max K + K sinµx 6 If we have, K =, K =, K = maxx =, for the case of linear inversion v = K u, h vmax 83 H vmax 006 On the other h, for the nonlinear inverting control law with K = 08 K = 06, h vmax 57 H v 0004 Therefore, an approximate nonlinear inverting control law, that uses only output measurements, can reuce simutaneously h vmax H v more than a linear inverting control law that has a perfect estimate of K Another possibility to ecrease the eigenvalues of P to reuce C c is to change the magnitue of the eigenvalues of A c but in this case the system performance may experience some egraation Now suppose that v = K u, with K = 05, that the system has to follow a secon orer reference moel given by ÿ = y 8ẏ + u, 63 comme by a square reference signal of amplitue equal to perio 40s To place the close loop of the input-output linearize error ynamics at p 0 = p = 4, p = + j p 3 = j, the following linear lea-lag compensator with an approximate integral control action was use vc s ỹ a s = 339s +48s+348 s +0s s +034s+05 s +0s the stable low pass filter was set to T s = ỹs s + 65 Remark 3 A lea-lag compensator without integral control action leas to a system response with steay state error On the other h, a pure proportional-integral compensator can not be use because it violates Assumption, therefore, a compensator of the form G P I s = K p + K I s+θ, with θ = 00, was ae to the lea-lag controller Remark 4 The reaer can check that this esign easily satisfies 45 47, when H v = 0003, h vmax = 44 Q = I 6 These bouns were calculate supposing that y ẏ follow the reference moel After the esign was carrie out, the estimate signal ranges were verifie through simulations to assure that the esign conitions were fully satisfie It is important to point out that in a practical situation the system may be unknown or partially known Therefore H v h vmax shoul be estimate uring some system ientification proceure

6 y The Gaussian NN comprises six units, with centers romly istribute over the interval 05 5, σ = 6 the network input is ηt = v l yt yt 00 yt 004 yt 006 T The aaptation gains are F = 500I 6 λ = 0005, I 6 is the ientity matrix of orer 6 Simulation results are shown in Fig, it can be seen that the neural network can compensate the inversion error the system has a significant performance improvement without loss of stability, even when the contraction mapping assumption is violate over the signal range weights time s a b without NN with NN RM output time s Figure : Simulation results: a System response; b NN weight evolution 6 Conclusions This paper has aresse the problem of contraction mapping assumption require in a recently evelope approach for nonlinear aaptive output feeback control using neural networks It was shown that the same upate law that was obtaine in the original paper can still be applie without the contraction mapping assumption provie that some aitional knowlege about the control effectiveness is known by the esigner The result was illustrate on a simple example the contraction mapping assumption can be violate if most of the current approaches were use to approximate the control effectiveness term References Astrom, K J Wittenmark, B 995 Aaptive Control, n en, Aison-Wesley, Reaing, MA Calise, A J, Hovakimyan, N Ian, M 00 Aaptive output feeback control of nonlinear systems using neural networks, Automatica 37: 0 Ge, S S, Hang, C C Zhang, T 999 Aaptive neural network control of nonlinear systems by state output feeback, IEEE Trans on Systems, Man Cybernetics - Part B: Cybernetics 96: Ge, S S Zhang, J 003 Neural network control of nonaffine nonlinear system with zero ynamics by state output feeback, IEEE Trans on Neural Networks 44: Hovakimyan, N, Calise, A J Kim, N 004 Aaptive output feeback control of a class of multi-input multi-output systems using neural networks, International Journal of Control 775: Hovakimyan, N, Nari, F, Calise, A Kim, N 00 Aaptive output feeback control of uncertain nonlinear systems using singlehien-layer neural networks, IEEE Trans on Neural Networks 36: Isiori, A 995 Nonlinear Control Systems, 3r en, Springer-Verlag, Lonon Kim, N 003 Improve Methos in Neural Network-Base Aaptive Output Feeback Control with Applications to Flight Control, Georgia Institute of Technology - PhD Thesis Lewis, F L, Liu, K Yesilirek, A 995 Neural net robot controller with guarante tracking performance, IEEE Trans on Neural Networks 63: Narenra, K S Parthasarathy, K 990 Ientification control of ynamical systems using neural networks, IEEE Trans on Neural Networks : 4 7 Rovithakis, G A 999 Tracking control of multi-input affine nonlinear ynamical systems with unknown nonlinearities using ynamical neural networks, IEEE Trans on Systems, Man Cybernetics - Part B: Cybernetics 9: Rysyk, R Calise, A J 005 Robust nonlinear aaptive flight control for consistent hling qualities, IEEE Trans on Control Systems Technology 36: Sanner, R M Slotine, J-J E 99 Gaussian networks for irect aaptive control, IEEE Trans on Neural Networks 36: Stevens, B L Lewis, F L 003 Aircraft Control Simulation, n en, John Wiley Sons, New Jersey