An Estimated Replacement Approach for Stable Control of a Class of Nonlinear Systems with Unknown Functions of States

In this paper, we propose an approach for stable control of a class of nonlinear systems, which can be expressed in a state-feedback linearizable form with unknown nonlinear functions of states. The idea is to replace the unknown functions with estimated (not need to be accurate) functions and to use a universal approximator to compensate for the error caused by the replacement. For achieving a stable controller with a continuous control signal, a bisigmoid function based compensator is used and studied. In addition, the paper also deals with the control problem of input constraints and the way to examine this subject

Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA An Estimated Replacement Approach for Stable Control of a Class of Nonlinear Systems with Unknown Functions of States Nguyen Duy Hung and Nguyen Thi Huong Lan, VIELINA Abstract—In this paper, we propose an approach for stable control of a class of nonlinear systems, which can be expressed in a state-feedback linearizable form with unknown nonlinear functions of states The idea is to replace the unknown functions with estimated (not need to be accurate) functions and to use a universal approximator to compensate for the error caused by the replacement For achieving a stable controller with a continuous control signal, a bisigmoid function based compensator is used and studied In addition, the paper also deals with the control problem of input constraints and the way to examine this subject In general, the unknown function(s) g ( x) or g (x), f (x) is/are approximated by adjustable function approximator(s) g (x, θ g ) or g (x, θ g ), f (x, θ f ) respectively, where θ g , θ f are weights or parameter vectors As the aim is to design a stable adaptive controller with suitable adaptation law to reduce uncertainties in each case, g must be other than zero on the Index Terms—nonlinear control, unknown functions, estimated replacement, universal approximators domain Ω x to avoid singularities at g = during adaptation To deal with such a problem a parameter projection method is employed ([10], [11]), but this situation can also be avoided when using techniques presented in some schemes, such as a modified Lyapunov function ([7]) or a modified term ([8]) I PROBLEM FORMULATION II AN ESTIMATED REPLACEMENT APPROACH Consider a SISO nonlinear system in its full state-feedback linearizable form [3] x1 = x2 Suppose that, from a knowledge of the system we can find out continuous and bounded functions f (x) and g (x) > xn −1 = xn (1) xn = f ( x ) + g ( x )u Δ dxn (x, u ) ≤ W holds for all x ∈ Ω x , u ∈ Ωu where Δ dxn ( x, u ) = f ( x ) − f ( x ) + ( g ( x ) − g ( x ) ) u y = x1 where x = [ x1 , x2 ,…, xn ] ∈ Ω x ⊂ ℜn is a state vector, T u(t ) ∈ Ωu ⊂ ℜ is an input ( Ω x , Ωu are compact sets), y (t ) ∈ℜ is an output, and f ( x ) ∈ℜ , g ( x ) ∈ℜ are unknown, but continuous and bounded functions The control objective is to design a locally stable controller for tracking a reference trajectory r (t ) ∈ ℜ with bounded error Because g (x) can not be zero, without loss of generality, we can assume that g (x) > for all x ∈ Ω x Additionally, it also assumes that x are measurable whereas r (t ) and its derivatives up to the n-th one are bounded and known For the given control problem, many adaptive designs have been developed as shown in [7]-[12] and the references therein Manuscript received July 6, 2007 VIELINA is the Vietnam Institute of Electronics, Informatics, and Automation Address: 156A Quan Thanh St., Hanoi, Vietnam The authors are with the Center of Automation and Control, VIELINA (e-mail: ndhung@vielina.com) ISBN:978-988-98671-6-4 such that if we replace f (x) , g (x) in (1) with f (x) , g ( x ) respectively, we can approximate xn with bounded error, i.e., = Δ f ( x ) + Δ g ( x )u and W > is a bounded constant Based on a method mainly derived from [3], let us define an error system E (t , x ) = k T e (2) where e = x − r , rT = ⎡ r , r , … , r ( n −1) ⎤ , k T = ⎡⎣k1 , … , kn −1 ,1⎤⎦ ⎣ ⎦ with s n −1 + kn −1s n −2 + … + k1 is a Hurwitz polynomial In the sense of performance analysis, the error system provides a quantitative measure of the closed-loop system performance Hence, once the system dynamics are used with the definition of the error system to define the error dynamics, a Lyapunov candidate V ( E ) is then used to provide a scalar measurement of the error system In addition, in terms of boundedness, the error system and the Lyapunov candidate are also chosen such that bounding V will place bounds on the error system E and the system states x too To focus on the main idea of this paper, we accept without WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA proof that (2) satisfies the error system assumption (see Appendix A) Additionally, for the time being, we ignore the local stabilization case and not take the state and input bounding conditions into consideration Thus if we denote k TE = [ k1 ,…, kn −1 ] and dTE = ⎡ x1 − r, x2 − r ,…, xn −1 − r ( n −2) ⎤ , ⎣ ⎦ the error system (2) can be rewritten as ) ( E (t , x ) = k TE d E + xn − r ( n −1) and its time derivative (i.e., the error dynamic) becomes E = k TE d E + xn − r ( n) (3) = k TE d E − Δ dxn − r ( n) + f + gu u=u= g where η > V (E) = ( −k TE d E and +r (n) − f −η E consider the ) Lyapunov (4) candidate E , then the time derivative of the Lyapunov function along the solution of the error dynamic (3) is bounded by ≤ −ηV − 12 η E + W E = −ηV + W2 − (W − η E ) 2η 2η ≤ −ηV + W2 2η V≤ W 2η ⇒ E ≤ and lim VH = t →∞ η2 W2 2η enough, the closed-loop system performance depends only on the error bound W in approximating xn without considering about how large the individual approximation errors Δ f (x) and Δ g (x) in replacing the unknown functions are This means that we can replace the unknown functions with preferred estimated functions at our convenience provided that the approximation error Δ dxn is bounded by W Theorem 1: The state-feedback control law (5) ensures that the solution of the error dynamic (3) is uniformly ultimately bounded by (6) if there exist continuous and bounded functions f (x) and g (x) > such that Δ dxn (x, u ) ≤ W holds for all x ∈ Ω x , u ∈ Ωu where W > is a known bounded constant such that (7) ∂V E ≤ −γ ( E ) ∂E for ∀ E ≥ R and t ≥ with knowing that V ( E ) is V (E) = − t = VH (1 − e η ) + E e η − t continuously differentiable on E ≥ R Choosing γ ( E ) = γ ( E ) = V ( E ) = (1 − e η ) + V e η W2 Remark 3: From (6), we see that if we choose E0 small γ1 ( E ) ≤ V ( E ) ≤ γ ( E ) W2 , we obtain 2η − t for all t ≥ [0, ∞ ) Let V0 and E0 denote the V and E at t = , thus according to the lemma of ultimate bound (Appendix B) with 2 − t = V∞ , lim EH = t →∞ W η V ( E ) ≤ −ηV + = EH = −εηγ ( E ) − (1 − ε )ηV + V ( E ) ≤ −γ ( E ) + η E = η e and the control law (4) can be then formulated as T ( ) (5) Remark 2: If V0 ≤ V ( E∞ ) then ≤ V ≤ V ( E∞ ) for all t ≥ since V is positive definite so that it can not grow greater than V ( E∞ ) Furthermore, in the case of V0 > V ( E∞ ) we have V ≤ until V ≤ V ( E∞ ) , thus we find ISBN:978-988-98671-6-4 E we have W2 2η for ε satisfying < ε < Let γ ( E ) = εηγ ( E ) we see that = E∞ Remark 1: If we denote η = ⎡⎣η k1 , k1 + η k2 , … , kn −1 + η ⎤⎦ u = u = g −1 r ( n ) − ηT e − f W2 2η if and T k TE d E (6) Proof: According to a theorem of condition for uniform ultimate boundedness ([2]), in proving Theorem we wish to find some γ ( E ), γ ( E ) ∈ K ∞ and γ ( E ) ∈ K defined on V = EE = −η E − Δ dxn E m1 = η and m2 = ) Above results lead to the state of the following theorem In terms of feedback linearization, use the control law −1 ( ≤ V ≤ max (V0 , V ( E∞ ) ) ⇒ E ≤ max E0 , E∞ equivalently, E ≥ only if V ≥ W2 2(1 − ε )η , or W = R As the chosen functions 1− ε η fulfill requirement (7), Theorem is thus proved Theorem shows that it is possible to define (static) stabilizing controllers by applying the method of estimated replacement if we could find substitution functions satisfying the bounding condition over a valid region But a problem arises when W is large, since though the error system bound may be decreased by choosing η large, the control signal may WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA increase in amplitude and may start to oscillate To dealing with such a problem, the usual approach is to compensate for error effects caused by the replacement For this purpose, a number of techniques, such as nonlinear damping and dynamic normalization ([3]) may be used In this sense, here we propose a method which comes from the notion that if we can approximate Δ dxn (x, u ) with sufficiently small error, it is possible to include an additional stabilizing component to increase the robustness of the closed-loop system Because Δ dxn (x, u ) is a continuous and bounded function defined on compact sets, it can be approximated by a universal approximator (such as a fuzzy system or a neural network) with arbitrary accuracy Therefore by assumption that there are data available for tuning of an approximator to match certain condition, we can use it as a compensation component to form a robust state-feedback control law This subject will be studied in more detail later in this paper Now, before turning to developing a stable controller for making the closed-loop system more robust to system uncertainties, we will investigate some mathematical base or equivalently − x + ( ρ + 1) = −( ρ + 1)e − x This equation is in form of x + b = ae x where a ≠ , thus according to [13] it has the single root, equal to −b − w( −ae −b ) where w( x ) is the Lambert w-function (Note that the Lambert w-function is the inverse function of x = w( x )e w( x ) ) The substitution for a = −( ρ + 1) and b = ρ + leads to the solution of (10), afterward denote as x0 = ρ + + w(p) where p( ρ ) = ( ρ + 1)e −( ρ +1) Because of dp = − ρ (2 + ρ )e −( ρ +1) < , p is decreasing dρ ) for ρ ∈ ( 0,1] , therefore p(1) ≤ p( ρ ) < p(0) or p ∈ ⎡ e2 ,1 e ⎣ Consequently μ x ( ρ , x ) has the unique extremum at x0 and if we denote μ = μ x ( ρ , x0 ) then μ = ( ρ + 1)e−2 x0 + 2( ρ − x0 )e− x0 + ρ − ⎛ ⎞ = ( ρ + 1)e−2( ρ +1) ⎜ ⎟ ⎝ e w(p) ⎠ III MATHEMATICAL BASE Define a real-valued scalar function μ E ( ρ , κ , E ) = E ( ρ − sgn( E ) bsig(κ , E ) ) (10) +2 [ ρ − ( ρ + 1) − w(p)] e−( ρ +1) (8) where < ρ ≤ , κ > are parameters, E ∈ℜ is a variable, e w(p) + ρ −1 Lemma The function (8) reaches its positive maximum value of μ E _ max ( ρ , κ ) = μ E ( ρ , κ , ± Em ) at ± Em where ⎡ ⎤ w(p) = ( ρ + 1)e −2( ρ +1) ⎢ ⎥ ⎢⎣ ( ρ + 1)e−( ρ +1) ⎥⎦ w(p) −2 (1 + w(p) ) e −( ρ +1) + ρ −1 ( ρ + 1)e−( ρ +1) x = κ Em is the unique solution of the equation = sgn( E ) is the sign function, and bsig(κ , E ) = /(1 + e −κ E ) − is bisigmoidal μ x ( ρ , x ) = ( ρ + 1)e −2 x + 2( ρ − x )e − x + ρ − = (9) Proof: Because (8) is an even function, thus we can take only the case E ≥ , i.e., μ E + ( ρ , κ , E ) = E ( ρ − bsig(κ , E ) ) into account It follows that the derivative of μ E + with respect to E can be calculated as ( ρ + 1)e−2 x + 2( ρ − x)e− x + ρ − μ x ( ρ , x) d = μE+ = 2 dE −x 1+ e + e− x ( ) ( ) where x = κ E ≥ Obviously, μ E+ has its extremum at xm = κ Em if satisfies μ x ( ρ , κ Em ) = Next we will show that, x = xm is the unique solution of (9) and μ E _ max ( ρ , κ ) is a positive maximum Take the derivative of μ x ( ρ , x ) with respect to x , we obtain d μ x ( ρ , x ) = 2e − x ⎡ x − ( ρ + 1) − ( ρ + 1)e − x ⎤ ⎦ ⎣ dx d μ x ( ρ , x) = For studying μ x ( ρ , x ) , solve the equation dx ISBN:978-988-98671-6-4 = w (p) w(p) − (1 + w(p) ) + ρ −1 ρ +1 ρ +1 ρ − ( w(p) + 1) ρ +1 Since the Lambert w-function is strictly increasing on [− e , ∞ ) we get w(2 e ) ≤ w(p) < w(1 e) , thus x0 > and μ < for all ρ ∈ ( 0,1] In addition μ x ( ρ , 0) = ρ > and μ x ( ρ , ∞) = ρ − ≤ so that the graph of μ x ( ρ , x ) cuts the x-axis only at xm ∈ ( 0, x0 ) as well as the extremum μ is the minimum of μ x ( ρ , x ) Note that μ ( ρ , x) d μE+ = x , we can infer that μ E+ dE + e− x ( ) reaches its maximum value of μ E + ( ρ , κ , Em ) = μ E _ max ( ρ , κ ) at Em = xm κ > and as μ E + ( ρ , κ , 0) = , the unique maximum is positive This proves Lemma For a better understanding of Lemma 1, Fig shows graphs WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA of (8) in cases of κ = and κ = 10 with ρ = 0.5, 0.9,1 in each example whereas Fig illustrates the graph of Em ( ρ , κ ) and μ E _ max ( ρ , κ ) with respect to ρ and κ in the case of κ = and ρ = respectively κ =5 κ = 10 0.06 0.1114 0.0557 0.0879 0.0440 Δ dxn (x, u ) = f (x) − f (x) + ( g (x) − g (x) ) u ≤ W for all x ∈ Ω x , u ∈ Ωu where W > W We will search for a solution to cope with this problem As mentioned above, the error function Δ dxn can be approximated by the universal approximator within a compact Fig Graphs of μ E ( ρ , κ , E ) 0.12 enough) However the estimated functions available for use only guarantee that set, which hereafter we denote as FΔ (x, u , θ) where θ ∈ ℜ p is an adjustable parameter vector and FΔ (x, u, θ) ∈ ℜ Right now 0.08 μE μE ←ρ = 0.04 0.0256 tunable parameters θ Assume that WΔ > be the known approximation error bounding constant, which satisfies FΔ (x, u, θ) − Δ dxn (x, u ) ≤ WΔ 0.02 ← ρ = 0.9 ← ρ = 0.5 -0.5 E 0.5 ← ρ = 0.5 -0.02 -0.5 -0.25 E 0.25 0.5 Fig Graphs of Em ( ρ , κ ) and μ E _ max ( ρ , κ ) κ =1 ρ =1 1.4 0.4 1.2785 1.2 1.0769 0.3 0.2557 0.8 ← Em ← Em 0.2 0.6 0.5569 0.5229 0.1278 0.1114 0.1 0.4396 0.4 ← μE_max 0.2 ← μE_max 0.0557 0.1278 0.2 0.4 0.5 0.6 0.8 0.9 0 for all x ∈ Ω x , u ∈ Ωu and θ ∈ℜ p is the best known parameter vector available from adjusting the parameters of the approximator Therefore the problem for approximating xn with error bound W can be considered as the problem for approximating Δ dxn with error bound WΔ Thus, we can avoid the difficulty of dealing with choosing estimated functions correctly by working with a proper approximator to compensate for the effect of the replacement error But one must determine how small WΔ must be to achieve the desired closed-loop system performance In order to solve this problem, now we introduce the compensation component defined as F ( x, u, θ) uc = − Δ bsig κ , EFΔ ( x, u, θ) (12) ρ g (x) where ρ , κ are constants satisfying < ρ ≤ , κ > and u is specified by (5) Then adding the component (12) together with the state-feedback control law (5) forms the new control law (13) u = u + uc and consequently the following theorem is the extension of Theorem to this case ( 0 (11) 0.0128 ← ρ = 0.9 -0.04 -1 let FΔ (x, u , θ) represent a neural network or fuzzy system with 0.04 ←ρ = ρ 10 15 20 κ ) Theorem 2: If there exist an approximator FΔ (x, u , θ) and a parameter vector θ such that FΔ (x, u , θ) can approximate IV CONTROLLER DESIGN Recall from previous studies that we are going to develop a stable controller in the proposed approach called estimated replacement The main concept in this approach is to seek estimated functions fitting the bounding requirement and to use a compensation technique to make the controller robust to uncertainties The later problem can be considered in this section as follows Suppose that we have to design a controller for the tracking problem with the aim to keep the error system bounded by W E∞ = (it is assumed that E0 can be selected small η ISBN:978-988-98671-6-4 Δ dxn (x, u ) with error bounded by WΔ satisfying < WΔ ≤ W − 2η ρ μ E _ max ( ρ , κ ) (14) for all x ∈ Ω x , u ∈ Ωu where < ρ ≤ , κ > and η > then the state-feedback control law (13) ensures that the solution of the error dynamic (3) is uniformly ultimately bounded by (6) Proof: For simplicity, denote FΔ = FΔ (x, u, θ) , FΔ = FΔ ( x, u, θ) WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA then from E = −η E + g (x )uc − Δ dxn ( x, u ) we have the system error is uniformly ultimately bounded by (6) while satisfying input constraints u ∈ Ωu where Ωu is defined as V = −η E + Eg ( x )uc − E Δ dxn ( x, u ) ≤ −η E − ≤ −η E + ρ (15) if uM > EFΔ bsig(κ , EFΔ ) + E Δ dxn ρ ( ) E ρ Δ dxn − FΔ sign( E ) bsig(κ , EFΔ ) Since sgn( E ) sgn( FΔ ) = sgn( EFΔ ) and Δ dxn ≤ FΔ + WΔ Proof: By assumption, the estimated functions f (x) , g (x) are locally Lipschitz continuous, therefore we can find constants K f , K g such that so we obtain V ≤ −η E + E WΔ + = −η E + E WΔ + = −ηV W2 + Δ 2η − E ρ ρ ( ρ − sgn( EFΔ ) bsig(κ , EFΔ ) ) r ( n) − ηT e − f (x) − u = (WΔ − η E ) + ρ1 μ E (κ , EFΔ ) 2η μ E (ρ ,κ , E ) is defined as in (8) E ∈ℜ as stated in Lemma Clearly, to have the error system bounded by (6), we need W , hence it follows that the requirement (14) 2η holds Notice that because WΔ > , we must choose ρ , κ and η such that ρ ( FΔ bsig κ , EFΔ ) g ( x) = r ( n) − ηT e − f (e + r ) + f (r ) − f (r ) FΔ bsig(κ , EFΔ ) − ρ g (r ) g (r ) × g (r ) g (e + r ) and μ E ( ρ , κ , E ) ≤ μ E _ max ( ρ , κ ) for all < ρ ≤ , κ > and V ≤ −ηV + g ( x) − g ( x) ≤ K g x − x for ∀x, x ∈ Ω x From (13) and note that x = e + r we have W2 ≤ −ηV + Δ + μ E _ max ( ρ , κ ) 2η ρ where f ( x) − f ( x) ≤ K f x − x ( ρ FΔ − FΔ sgn( E ) bsig(κ , EFΔ ) ) EFΔ W + WΔ and the condition (17) holds ρ gL 2η μ (ρ , κ ) < W ρ E _ max Then similar to the proof of Theorem 1, we come to that the new control law (13) makes the solution of the error system (3) uniformly ultimately bounded by (6) This proves Theorem V INPUT CONSTRAINTS ANALYSIS Up to this point we have not taken a state boundedness and input constraints into account However, for state boundedness, we can examine it using the error system boundedness In this section, we only consider the case of input constraints Notice that the original work on stabilization and tracking of feedback linearizable systems under input constraints in which we have utilized its concepts can be reviewed in [6] The problem of input constraints can be stated here as how to select parameters (if they exist) for the control design so that the control input (13) always remains in a valid region Ωu , which is defined as Ωu = {u ∈ ℜ : u ≤ uM } (15) where uM is positive bounded constant Additionally, it assumes < g L ≤ g (x) and f (x) , g (x) can be chosen so that they are locally Lipschitz in x Since FΔ ≤ W + WΔ and recall that W > W , we get ⎛ r ( n ) − f (r ) ηT e f (r ) − f (e + r ) u ≤⎜ + + ⎜ g (r ) g (r ) g (r ) ⎝ F bsig(κ , EFΔ ) ⎞ g (e + r ) − g (r ) + Δ ⎟× 1− ⎟ g (e + r ) ρ g (r ) ⎠ ⎛ r ( n ) − f (r ) K e FΔ η e f ≤⎜ + + + ⎜ g (r ) g (r ) g (r ) ρ g (r ) ⎝ ⎞⎛ Kg e ⎟ ⎜1 + ⎟⎜ g (e + r ) ⎠⎝ ⎞ ⎟ ⎟ ⎠ ⎛ r ( n ) − f (r ) e W + WΔ ⎞ ⎛ e ⎞ ⎟ ⎜1 + K g ≤⎜ + η +Kf + ⎟ ⎜ ⎜ g (r ) gL ρ gL ⎟ ⎝ g L ⎟⎠ ⎝ ⎠ In order to have the control input remain in Ωu , we need ( r ( n) − f (r ) ≤u = g (r ) ) uM 1+ Kg e ( − η +Kf )g e L − W + WΔ (16) ρ gL gL In addition, as E = k e ≤ max ( E0 , E∞ ) so we can write T ( ) e(t ) ≤ K max E0 , E∞ = eM where eM > Let’s define M = uM e 1+ Kg M gL ( − η +Kf ) gM e L − W + WΔ ρ gL then M ≤ u and we see that if M > then (16) always holds To have M > , it requires Theorem 3: The state-feedback control law (13) ensures that ISBN:978-988-98671-6-4 WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA ⎛ e ⎞⎛ e W + WΔ uM > ⎜ + K g M ⎟ ⎜ η + K f M + ⎜ ρ gL gL ⎠ ⎝ gL ⎝ ( ) y (t ) → r (t ) ⎞ ⎟⎟ ⎠ ⎛e ⎞ ⎛ W + WΔ ⎞ eM ⇔ Kg η + K f ⎜ M ⎟ + ⎜ η + K f + Kg ⎟ ⎜ ρ g L ⎟⎠ g L ⎝ gL ⎠ ⎝ W + WΔ + − uM < ρ gL The above quadratic inequation is in the form of e Az + Bz + C < where z = M > and gL ( ) ( ) A = Kg η + K f > B = η + K f + Kg C= W + WΔ − uM ρ gL W + WΔ so that it has a positive one It need C < , i.e., uM > ρ gL K − B + B − AC max ( E0 , E∞ ) < z2 = gL 2A satisfies APPENDIX B A ULTIMATE BOUND STUDY (LEMMA 2.1 IN [3]) If V (t , E) : ℜ+ × ℜn → ℜ+ is positive definite and V ≤ − m1V + m2 where m1 > and m2 ≥ are bounded [1] [2] [3] [4] [5] [6] and therefore we must choose (if it exists) ) E(t , x ) to e ∈ ℜ+ for each fixed t follows that ( function m2 ⎛ m ⎞ + ⎜ V (0) − ⎟ e − m1t for all t m1 ⎝ m1 ⎠ REFERENCES the solution of the quadratic inequation is z1 < z < z2 Since if C ≥ , the mentioned polinomial has non-positive roots so we max E0 , E∞ the x ≤ ψ x (t , E ) for all t , where ψ x : ℜ+ × ℜ+ → ℜ is bounded for any bounded E and ψ x (t , e) is nondecreasing with respect W + WΔ >0 ρ gL g r ( n) − f (r ) < L z2 and ≤M g (r ) K that constants, then V (t , E) ≤ Let z1 < z2 are roots of the polynomial Az + Bz + C then 0< z= and [7] (17) [8] for solving the problem of input constraints (Q.E.D.) VI CONCLUSION In summary, the proposed approach gives a new concept to design stable controllers for state-feedback linearizable systems with unknown functions of states In this way we can also avoid the problem of singularities mentioned above because the estimated functions for replacement can be chosen at our intention and they are known in advance However the controller we have developed in this paper is static, that is its parameters are not adjustable during operation and therefore it is “less robust” to uncertainties than an adaptive equivalent Due to the scope of this topic, we will study adaptive schemes in another paper Additionally, achieved results are intended to be used in real time control systems for industrial applications in the fields of control of chemical processes, water treatment control and robot control [9] [10] [11] [12] [13] Hassan K Khalil, “Nonlinear Systems”, 3rd ed., Prentice Hall, 2001 Horacio J Marquez; “Nonlinear Control Systems: Analysis and Design”, Wiley Interscience, 2003 Jeffrey T Spooner, Mangredi Maggiore, Rẳl Ordónez, and Kelvin M Passino, “Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques”, Wiley Interscience, 2002 Jyh-Shing Roger Jang, Chuen-Tsai Sun, and Eiji Mizutani, “Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence”, Prentice Hall, 1996 Nguyen Duy Hung, “Some Neural Network-based Learning Methods and Problems on Applying in Industrial Control Systems”, Proceedings of 5th Vietnam Conference on Automation (VICA5), 2002, pp.163–168 George J Pappasy, John Lygeros, and Datta N Godbole, “Stabilization and Tracking of Feedback Linearizable Systems under Input Constraints”, Report, Intelligent Machines and Robotics Laboratory, University of California at Berkeley, 34th CDC, 1995 T Zhang, S S Ge, and C C Hang, “Stable Adaptive Control for a Class of Nonlinear Systems using a Modified Lyapunov Function", IEEE Transactions on Automatic Control, vol 45, no 1, Jan 2000 Jang-Hyun Park, Seong-Hwan Kim, and Chae-Joo Moon, “Adaptive Fuzzy Controller for the Nonlinear System with Unknown Sign of the Input Gain", International Journal of Control, Automation, and Systems, vol 4, no 2, Apr 2006, pp 178–186 Hugang Han, Chun-Yi Su, and Yury Stepanenko, “Adaptive Control of a Class of Nonlinear Systems with Nonlinearly Parameterized Fuzzy Approximators", IEEE Transactions on Fuzzy Systems, vol 9, no 2, Apr 2001, pp 315–323 Jun Nakanishi, Jay A Farrell, and Stefan Schaal, “Composite adaptive control with locally weighted statistical learning”, Elsevier Neural Networks 18, 2005, pp 71–90 Jun Nakanishi, Jay A Farrell, and Stefan Schaal, “Learning Composite Adaptive Control for a Class of Nonlinear Systems”, Proceedings of the 2004 IEEE International Conference on Robotics & Automation, New Orleans, LA, pp 2647–2652 Shouling He, Konrad Reif, Rolf Unbehauen, “A Neural Approach for Control of Nonlinear Systems with Feedback Linearization”, IEEE Transactions on Neural Networks, vol 9, no 6, Nov 1998, pp 1409–1421 R.M Corless, G.H Gonnet, D.E.G Hare, and D.J Jeffrey, “On the Lambert's W Function”, Technical Report, Advances in Computational Mathematics, vol 5, 1996, pp 329–359 APPENDIX A AN ERROR SYSTEM ASSUMPTION (ASSUMPTION 6.1 IN [3]) Assume the error system E(t , x ) is such that E = implies ISBN:978-988-98671-6-4 WCECS 2007 ... Control Systems: Analysis and Design”, Wiley Interscience, 2003 Jeffrey T Spooner, Mangredi Maggiore, Rẳl Ordónez, and Kelvin M Passino, “Stable Adaptive Control and Estimation for Nonlinear Systems:... INPUT CONSTRAINTS ANALYSIS Up to this point we have not taken a state boundedness and input constraints into account However, for state boundedness, we can examine it using the error system boundedness... Recall from previous studies that we are going to develop a stable controller in the proposed approach called estimated replacement The main concept in this approach is to seek estimated functions