Peaking Attenuation of High-Gain Observers Using Adaptive Techniques: State Estimation and Feedback Control

Transcription

1 Peaking Attenuation of High-Gain Observers Using Adaptive Techniques: State Estimation and Feedback Control Mehran Shakarami, Kasra Esfandiari, Amir Abolfazl Suratgar, and Heidar Ali Talebi Abstract A new state estimation scheme using the secondlevel adaptation technique and Multiple High-Gain Observers (MHGO) is presented in this paper, which overcomes the drawback of High-Gain Observer (HGO) associated with the existence of undesired peaks in transient response. The proposed method considers state estimation as a convex combination of provided information by multiple high-gain observers. In this regards, it is shown that there exist some constant parameters in such combination that provide a perfect state estimation. Then, a modified version of the recursive least squares algorithm is employed for estimating these parameters. It is shown that the proposed observation scheme is stable, and the state estimation converges to the actual state of plant. In addition, MHGO is proved to be able to provide a better estimation as compared to a single HGO. Moreover, the output feedback control problem is considered, and it is demonstrated that a separation principle is valid for MHGO. Therefore, nonlinear systems can be stabilized using MHGO state estimation fed into a bounded controller. Finally, simulation results illustrate that MHGO provides an accurate state estimation, and MHGO-based controller is able to recover the performance of state feedback controller. Index Terms High-gain observers, second-level adaptation, output feedback. I. INTRODUCTION HIGH-Gain Observers (HGOs) are able to reconstruct system states from the output measurements [], [2]. Many researches have employed such observers in solving different problems, such as control and state estimation of nonlinear systems [3], [4], [5] and fault detection and isolation [6]. One factor that resulted in the wide utilization of HGOs is the satisfaction of the separation principle in HGO-based control problems. This feature is introduced and proved in [7], [8], and the improvement of its conditions is made in [9]. Although HGOs recover plant states, they exhibit an inherent drawback which is the existence of undesired peaks in the transient response, known as the peaking phenomenon. The interaction of this behaviour with system nonlinearities could result undesired performance of the observer and even instability of the closed-loop system []. One method that is widely employed in system theory for recovering state variables is multi observers [], [2]. In this M. Shakarami is with Engineering and Technology Institute Groningen, University of Groningen, 9747 AG Groningen, The Netherlands ( m.shakarami@rug.nl). K. Esfandiari is with the Center for Systems Science, Yale University, New Haven, CT, USA (kasra.esfandiari@yale.edu). A.A. Suratgar and H.A. Talebi are with the Center of Excellence on Control and Robotics, Department of Electrical Engineering, Tehran Polytechnic, Tehran, Iran ( {a.suratgar;alit}@aut.ac.ir). approach, multiple observers are employed to obtain multiple estimations of system states. Then, a supervisor selects one of the observers at each time instant based on an appropriate criteria. This method is able to provide preferable estimations; however, it has two drawbacks. Although the structure has multiple observers, only the obtained information from one observer is used at each time instant. Furthermore, the method needs to employ c n observers, where n is the number of states and c >, to be able to reconstruct system s state [2]. Consequently, the required number of observers grows exponentially by an increase of state variables. In adaptive systems based on single adaptive model, the transient response is oscillatory when the system uncertainty is large [3]. A proposed solution to this problem is the utilization of multiple models for identification of the plant and a supervisor for selection of the closest model to plant at any time instant [4]. On the other hand, this method has similar drawbacks to multi observers: a large number of models is required and the available information of all models is not employed efficiently. In [5] a novel scheme, called second-level adaptation technique, is presented for Linear Time Invariant (LTI) systems in companion form. In this approach, switching between models is eliminated and a convex combination of all information is employed to obtain an estimation of unknown system parameters. Afterwards, the obtained estimation is used in the design procedure of a controller for the system. The robustness of the method is investigated in [6] and an extension to single-input singleoutput systems is presented in [7]. Moreover, in different research, this concept is applied to systems with unknown parameters for solving problems raised in control theory, e.g., LTI systems [8], linear systems with periodic parameters [2], [9], nonlinear systems [2], [22], and twin-rotor system [23]. Besides, the state estimation problem for nonlinear systems is addressed in [24]; however, the more challenging observerbased control problem is remained intact there. In this paper, the second-level adaptation technique is employed together with multiple HGOs to recover the state of a special class of nonlinear systems, accurately, and the obtained estimation is employed in an observer-based controller. In this regard, multiple HGOs (MHGO) with a new structure are presented, and the state estimation is considered as a convex combination of individual HGOs state estimations. Then, the required unknown parameters of this combination are estimated using a modified version of the Recursive Least Squares (RLS) algorithm. Moreover, the obtained MHGO estimation is

2 2 employed for solving the control problem. The main contributions of this paper are: I) A new methodology using multiple HGOs and the second-level adaptation technique is proposed for state estimation of nonlinear systems. II) By employing the properties of the proposed structure, state estimation problem is converted to a parameter and state estimation problem. III) A modification of the RLS algorithm is presented for estimation of the parameters. IV) It is shown that state estimation obtained from the proposed scheme converges to system state, and MHGO is able to provide an estimation with a better transient response in comparison to a single HGO. V) The observerbased control problem is considered and it is proved that the separation principal is valid when MHGO state estimation is employed. The rest of the paper is structured as follows. The problem statement and preliminaries are presented in Section 2. In Section 3, the proposed observer scheme is introduced; then, its stability and performance are investigated. Afterwards, it is shown that MHGO is able to provide better estimation as compared to a single HGO. Furthermore, MHGO-based control problem is addressed. Simulation results are included in Section 4, and finally, Section 5 provides the conclusions. II. PROBLEM STATEMENT AND PRELIMINARIES This sections provides problem formulation, high-gain observers, and preliminaries on convex analysis. First, the considered special class of nonlinear systems is introduced. Then, the structure of conventional HGOs and their ability in reconstructing plant states are presented. In addition, the required properties of convex sets are discussed. A. Problem Formulation Consider the following nonaffine nonlinear system ẋ = Ax + Bf(x, u) y = Cx where x R n is the state vector, u R and y R denote the input and output of the system, respectively, and the matrices A R n n, B R n, and C R n are defined as follows:. A = C = [ ], B = In addition, f(x, u) is a nonlinear function which is locally Lipschitz in (x, u) for all x D R n and u B R, and f(, ) =. This ensures that the origin is an equilibrium point of the system [9]. B. High-Gain Observers In order to present the design procedure of the proposed MHGO, let us have a quick review on the structure of conventional HGOs. Consider the structure of a single HGO as follows [2], [25]: () ˆx = Aˆx + Bf (ˆx, u) + H(y C ˆx) (2) where ˆx is the state estimation vector, f ( ) is a saturated version of f( ) and agrees with that on D B, and κ / κ 2 / 2 H =, (, ],. κ n / n with κ i s are chosen such that all the eigenvalues of the matrix A HC have negative real parts, i.e., A HC is Hurwitz. To obtain the required conditions for reconstructing system states, the estimation error dynamics is considered as follows by subtracting (2) from (): x = A x + B[f(x, u) f (ˆx, u)] (3) where x = x ˆx and A = A HC. As it is proved in [2], [25], there exists > such that for every < and any admissible x D and u B, the effect of f(x, u) f (ˆx, u) on x is rejected. Also since A is a Hurwitz matrix, the state estimation converges to the state of the plant, i.e., lim t x(t) = C. Convex Sets and Convex Hull Throughout the paper, properties of convex sets are required to be utilized. Therefore, to accomplish the goals of paper, the following definition and lemma are required to be considered. Definition ([26]): A set K in a linear space L is called convex if the line segment ab is contained in K for any elements a, b K, i.e., ( λ)a + λb K for any pair (a, b) K and any λ [, ]. Lemma ([26]): Let a, a 2,, a m L where L is a linear space. The intersection of all convex sets in L containing a i s is called the convex hull K of {a i (i =, 2,, m)} and any element of which, a, can be expressed as follows: a = m β i a i i= where β i [, ] is a constant satisfying m i= β i =. III. THE MAIN RESULTS In this section, the structure of the proposed observer, MHGO, is introduced which uses information provided by several HGOs. To elaborate more on this issue, MHGO estimation is considered as a convex combination of individual HGOs estimations. First, the structure of the proposed observer is presented, and the existence of some unknown constant parameters, enabling us to reconstruct state variables precisely, is guaranteed. Thereafter, the deriving procedure of the required parameters is presented. The stability and performance of the suggested observation scheme are also investigated in details. Furthermore, the output feedback control problem using MHGO state estimation is addressed.

3 3 A. The Proposed MHGO The proposed method reconstructs the state of the plant at any given time instant using full knowledge provided by multiple observers. Hence N HGOs with the structure of (4) are considered to overcome the aforementioned demerits of multi observers and conventional HGOs. ˆx i (Λ, t) = Aˆx i (Λ, t) + H(y(t) C ˆx i (Λ, t)) + Bf ( α iˆx i (Λ, t), u(t)), ˆx i (Λ, ) = ˆx i () i= where i =,, N and ˆx i denotes state estimation from the ith observer, α i [, ] is a constant term satisfying N i= α i =, and Λ = [ ] T α α 2 α N. The notation of ˆx i (Λ, t) is employed to show that ˆx i s are functions of α i s. The final estimation is considered as a combination of N estimations as follow: ˆx o (t) = α iˆx i (Λ, t) (5) i= From (4), it can be seen that at any given time, each observer utilizes state estimations provided by the other observers. In another word, the estimations of other observers are fed into the ith s observer. Having access to such information at any given instant helps them to provide a more reliable estimation. In addition, a combination of individual observers are considered as the final estimation (refer to (5)) which assists in having a more accurate estimation. Although the proposed observer structure (4) seems to provide promising results, a question which is the matter of debate is whether one can mathematically prove its capability for estimating plant states, accurately. Towards this end, it is essential to show that there exist constant αi s such that using such parameters results in an estimation which satisfies x(t) = ˆx o (t) for all t. In this regard, the following lemma is presented. Lemma 2: Consider nonlinear system (), N high-gain observers (4), equation (5), and let x D R n and u B R. If initial conditions of HGOs (4) are selected such that x() is in the convex hull K of ˆx i (), then there exist some αi s such that for α i = αi, the equality of x(t) = ˆx o(t) holds for all t. Proof. The proof is divided into two steps. In the first step, it will be shown that there exist αi s such that x(t = ) = ˆx o (t = ) if α i = αi s. Next, it will be also proved that the equality, x(t) = ˆx o (t), holds for all t >. This implies that there exist constant terms αi s providing perfect estimation. To accomplish the first step, by employing Lemma, it can be considered that since ˆx i ()s are chosen such that x() is in the convex hull K of {ˆx i ()(i =, 2,, N)}, then some α i s exist such that x() = αi ˆx i () i= The preceding equation implies that by defining estimation error x o = x ˆx o and choosing α i = α i s, we have x o() =, and this concludes the first step. (4) Since the second step is aimed at providing the analysis for t >, it is required to obtain the observation error dynamics. Subtracting (4) from (), results in x i (Λ, t) = A x i (Λ, t) + B[f(x(t), u(t)) f ( α iˆx i (Λ, t), u(t))] i= where A = A HC and x i = x ˆx i denotes observation error corresponding to the ith observer. In order to obtain the error dynamics of MHGO, one can use (5) and the fact that N i= α i = to get x o (t) = α i x i (Λ, t) i= By considering (6) and the fact that α i s are constant, the time derivative of the preceding equation can be obtained as follows (6) x o = A x o + B[f(x, u) f (ˆx o, u)] (7) Since f agrees with f on D B, it can be concluded that x o = is an equilibrium point of the preceding equation. On the other hand, as shown in the first step of proof, choosing α i = αi s results in x o() = ; and therefore, x o (t) = for all t. The following assumption can be considered, which is obtained from Lemma 2. Assumption : The initial conditions of HGOs (4), ˆx i ()s, are chosen such that the initial condition of plant (), x(), is in their convex hull K. Comment : For satisfaction of Assumption, at least N = n + observers are required, where n is the number of states. As a result, in comparison to multi observers that need c n observers with c >, MHGO requires fewer observers, especially when n is a large number. From Lemma 2 it can be concluded that there exist some unknown constant αi s that result in a perfect state estimation. Therefore, the following equation can be obtained x(t) = αi ˆx i (Λ, t), t (8) i= where Λ = [ ] α α2 αn T. Although choosing αi s identical to αi s results in the perfect state estimation, such a selection is impossible due to the fact that αi s are unknown terms. Thus to access an accurate state estimation, deriving an appropriate estimation of αi s is required. In another word, the understudy state estimation problem can be transformed into a combination of state and parameter estimation problem. In order to estimate αi s, let us use the fact that N i= α i = and rearrange (8) and obtain the following equation i= α i x i (Λ, t) = (9) By considering αn = N i= α i and defining θ = [ ] α α2 αn T, one can rewrite (9) as follows: N i= α i ( x i (θ, t) x N (θ, t)) = x N (θ, t) ()

4 4 On the left hand side of (), x i (θ, t) x N (θ, t) is equal to ˆx N (θ, t) ˆx i (θ, t). Moreover, by using (4) one can get ˆx N (θ, t) ˆx i (θ, t) = A (ˆx N (θ, t) ˆx i (θ, t)) It can be seen that ˆx N (θ, t) ˆx i (θ, t) is the state of a LTI system, and in turn, it is not dependent on θ ; thus if a matrix M is defined such that its ith column is ˆx N (θ, t) ˆx i (θ, t), it is independent of θ, and one can write the regression form of () as follows: M(t)θ = x N (θ, t) () Although the matrix M(t) is completely known, the right hand side of () is composed of x and ˆx N (θ ) which are unknown. Therefore, one cannot obtain an estimation of θ using conventional adaptive approaches merely based on (). To overcome the aforementioned difficulty, let us premultiply () by C, and get CM(t)θ = ỹ N (θ, t) where ỹ N (θ, t) = y(t) C ˆx N (θ, t). The right hand side of the preceding equation is still unknown since θ is not available, i.e., ˆx N (θ, t) is unavailable. Therefore, the RLS algorithm which requires ỹ N (θ, t) for estimating θ cannot be utilized. Nonetheless, to find an estimation of θ, a modified version of the RLS algorithm can be employed as follows ˆθ(t) = P (t)m(t) T C T (ỹ N (ˆθ, t) + CM(t)ˆθ(t)), P (t) = P (t)m(t) T C T CM(t)P (t), ˆθ() = ˆθ P () = γi (2) where ˆθ represents an approximation of θ, ỹ N (ˆθ, t) = y(t) C ˆx N (ˆθ, t), γ is a positive constant, and I is the identity matrix. It can be seen that the modified RLS algorithm uses ỹ N (ˆθ, t) instead of ỹ N (θ, t), and it will be shown in the next sections that this structure is appropriate for obtaining an accurate state estimation. It is worth noting that since M(t) is independent of θ, its ith column is considered as ˆx N (ˆθ, t) ˆx i (ˆθ, t) that can be easily obtained. After obtaining ˆθ, it is employed for state estimation as follows ˆx i (ˆθ) = Aˆx i (ˆθ) + H(y C ˆx i (ˆθ)) + Bf (ˆx o, u) ˆx o = N i= N ˆα iˆx i (ˆθ) + ( i= ˆα i )ˆx N (ˆθ) (3) It can be seen that the estimation is obtained from two interconnected systems (2) and (3). Hence it is needed to investigate estimation error as well as the stability of these systems, which is the goal of next section. B. Performance Investigation In the previous section, it was shown that if α i = αi s, a perfect state estimation can be obtained. Then, a method for estimating αi was proposed. This section is aimed at answering the challenging question that whether the interconnected systems (2) and (3) are stable and the obtained state estimation converges to system s state. Afterwards, it is investigated that whether the proposed observer is able to provide a better estimation in comparison to a single HGO. The following theorem is presented for investigation of the stability of the proposed observer and the convergence of ˆx o (t) to x(t). Theorem : Consider system (), N high-gain observers (3), and the modified RLS algorithm (2), and let x D R n and u B R. Then there exists > such that for <, ˆθ and P are bounded, ˆx i s are uniformly ultimately bounded, and ˆx o converges to x. Proof. Consider the following scaled estimation error η (i) = x (i) ˆx o(i), i =,, n n i where x (i) and ˆx o(i) represent the ith elements of x and ˆx o, respectively. Using the preceding equation, one can obtain D()η = x ˆx o D() = diag( n, n 2,,, ) (4) where η = [ ] T η () η (n). By using N i= ˆα i =, the scaled error (4), and the fact that CD() = n C, the dynamics of MHGO, including the modified RLS algorithm (2) and N high-gain observers (3), can be rewritten as follows: η = N i= N ˆα i η i (ˆθ) + ( i= ˆα i )η N (ˆθ) (5) ˆθ = 2(n ) P M T C T C(η N (ˆθ) + M ˆθ) (6) P = 2(n ) P M T C T CM P (7) η i (ˆθ) = A η i (ˆθ) + B[f(x, u) f (x D()η, u)] (8) where Dη i = x ˆx i is scaled state estimation error for the ith observer, the ith column of M is η i (ˆθ) η N (ˆθ), and κ. A = κ n κ n Note that since κ i s are chosen such that A is a Hurwitz matrix, and A = D() A D(), it can be concluded that A is also a Hurwitz matrix. Since P (t) and P (t), it can be seen that P (t) P () = γi and P (t) is bounded. Moreover, using the fact that P (t) is symmetric, we have P (t) 2 = λ max (P (t)) 2 ; thus, it can be obtained that P (t) γ. By employing the definition of M, one can get the regression form of η from (5) as follows η = M ˆθ + ηn (ˆθ) (9) By substituting the preceding equation in (6), it can be obtained ˆθ = 2(n ) P M T C T Cη (2) On the other hand, by using (8) and the fact that the ith column of M is η i (ˆθ) η N (ˆθ), it can be obtained Ṁ = A M (2)

5 5 Now by employing (8), (2), and (2), the derivative of (9) is η = A η 2(n ) M P M T C T Cη + B[f(x, u) f (x D()η, u)] (22) In order to investigate the convergence of ˆx o (t) to x(t), a Lyapunov function candidate V (η) = η T P η is considered where P is the positive definite matrix satisfying A T P + P A = I. Therefore, the derivate of V (η) can be obtained using (22) as follows V (η) = ηt η 2 2(n ) η T P M P M T C T Cη + 2η T P B[f(x, u) f (x D()η, u)] (23) Employing D() =, the fact that f is locally Lipschitz in D B and f is a saturated version of which and agrees with that in this domain, it can be concluded that f(x, u) f (x D()η, u) L η, (x, u) D B (24) where L > is a Lipschitz constant. One can use (23), (24), and B = to express V (η) as V (η) η (n ) P M 2 P η 2 + 2L P η 2 It can be seen that by defining choosing <, we have = /(4L P ) and V (η) 2 η (n ) P M 2 P η 2 (25) As it was shown before, P (t) γ. Moreover, from (2), we know that M (t) = e At M (). Since A is a Hurwitz matrix, a Lyapunov function candidate W = Tr[e AT t P e At ] can be considered and obtain Furthermore, we have Ẇ = Tr[e AT t e At ] λ min (P ) Tr[e AT t e At ] W λ max (P ) Tr[e AT t e At ] (26) where λ min (P ) and λ max (P ) are the smallest and the largest eigenvalues of P, respectively. By employing the previous equation, it can be obtained Therefore, one has Ẇ λ max (P ) W W (t) e λmax(p ) t W () (27) Now, one can consider the following equation e At 2 Tr[e AT t e At ] n e At 2 Consequently, (26), (27), and the preceding equation can be used to get e At ke λ t (28) with k = nλ max (P )/λ min (P ) and λ = /(2λ max (P )). By using (28), it can be obtained M (t) k M () e λ t. Then, the obtained upper bounds of P (t) and M (t) and (25) can be utilized to obtain the following equation V (η) 2 η 2 + ρe 2 λ t η 2 with ρ = 2k 2 γ 2(n ) P M () 2. Note that the following inequality always holds. λ min (P ) η 2 V (η) λ max (P ) η 2 (29) Thus, (29) can be used to obtain V (η) ( 2λ max (P ) + λ min (P ) ρe 2 λ t )V (η) By solving the preceding equation, we have V (t) e 2λmax(P ) t e 2λλ min (P ) ρ( e 2 λ t) V () Using the fact that e 2 λ t, the preceding equation is rewritten as follows V (t) k e 2λmax(P ) t V () (3) 2λλ with k = e min (P ) ρ. As a result, lim η(t) =, and ˆx o (t) t converges to x(t). In order to show that ˆθ is bounded, (2) can be considered to get ˆθ(t) = ˆθ 2(n ) P (τ)m T (τ)c T Cη(τ)dτ (3) Furthermore, one can get the following inequality by employing (29) and (3). η(t) k 2 e 4λmax(P ) t η() (32) where k 2 = k λ max (P )/λ min (P ). By utilizing the preceding equation, the upper bounds of P (t) and M (t), and (3), one can get ˆθ(t) ˆθ + k 3 e ( λ + 4λmax(P ) )τ dτ with k 3 = kk 2 γ 2(n ) M () η(). Now, it can be seen that ˆθ(t) ˆθ + k 3 λ + 4λ max(p ) Therefore, ˆθ is bounded. To show that ˆx i s are bounded, using the boundedness of x, it is required to show that η i s are bounded. Towards this end, a Lyapunov function candidate V i (η i ) = η T i P η i and (8) are employed to obtain V i (η i ) η i 2 + 2L P η i η By using (32), it can be concluded that V i (η i ) < for η i > 2L k 2 P η(). As a result, η i s are uniformly ultimately bounded. From Theorem, it can be seen that the proposed observer is stable and also its estimation converges to system s state. On the other hand, the main purpose of the proposed observer is

6 6 obtaining a better estimation in comparison to a single HGO. In order to investigate that, the analysis of estimation error of MHGO and a single HGO needs to be performed. Towards this end, suppose that Assumption holds. Therefore, from () it can be seen that M (t)θ +η N (θ, t) =, and one can rewrite (9) to get η = M θ + σ(ˆθ) (33) where θ = ˆθ θ and σ(ˆθ) = η N (ˆθ) η N (θ ). The goal is to find an upper bound for the norm of η and show that it can become smaller than a single HGO. Therefore, it is required to find θ and in this regard, (2) and (33) can be utilized to obtain θ = 2(n ) P M T C T C(M θ + σ(ˆθ)) (34) Now by considering d(p θ) dt and (34), one can get = P P P θ + P θ, (7), d(p θ) = 2(n ) M T C T Cσ(ˆθ) dt By taking the integral of preceding equation and premultiplying by P (t), θ(t) can be obtained as follows θ(t) = P (t)p () θ 2(n ) P (t) M (τ) T C T Cσ(ˆθ(τ), τ)dτ (35) In the preceding equation, it is required to obtain P (t). dp Towards this end, dt = P P P and (7) can be utilized to get dp dt = 2(n ) M T C T CM Taking the integral of previous equation and employing M (t) = e At M () results in [ ] P (t) = P () + 2(n ) M () T Γ o (t)m () where Γ o (t) = e AT τ C T Ce Aτ dτ is the observability Gramian corresponding to the observable pair (A, C). Hence Γ o (t) is always positive definite. Now, by considering (35), it can be seen that one needs to calculate P (t)p () and 2(n ) P (t). Towards this end, the preceding equation and P () = γi is employed to obtain [ ] P (t)p () = I + γ 2(n ) M () T Γ o (t)m () By employing the matrix inversion lemma [27], we have P (t)p () = I M () T ( γ 2(n ) Γ o(t) + M ()M () T ) M () (36) In order to proceed with the analysis, the following assumption requires to be satisfied. Assumption 2: The initial conditions of HGOs (4) are chosen such that the matrix M()M() T has full rank. Comment 2: Since M() is n (N ) and N n, Assumption 2 is not a restricting assumption. Given Assumption 2, M ()M () T matrix. Thus, we can get is also a full rank ( γ Γ o(t) +M 2(n ) ()M () T ) ( = M ()M () T ) ( I + γ Γ o(t) (M 2(n ) ()M () T ) ) (37) The goal is to obtain the effect of γ 2(n ) on the preceding equation. Therefore, it is required to rewrite (37) using the Neumann series [28]. Towards this end, one has to show that there exists γ > such that for every γ > γ, the following equation is satisfied γ 2(n ) Γ o(t) (M ()M () T ) < (38) In this regard, let γ = ξ/ 2(n ) with the positive constant ξ and (M ()M () T ) = k 4. Moreover, since Γ o (t) is symmetric and positive definite, there exists a positive constant k 5 such that < k 5 λ min (Γ o (t)); hence k 5 I Γ o (t) which results in Γ o (t) k 5 I. In addition, since Γ o (t) is a symmetric matrix, similar to P (t), it can be shown that Γ o (t) k 5. By employing this upper bound, we can get γ Γ o(t) (M 2(n ) ()M () T ) k 4 ξ k 5 As a result, by defining ξ = k4 k 5 and choosing ξ > ξ, the condition (38) will be satisfied. Now the Neumann series can be employed to obtain ( I + γ Γ o(t) (M 2(n ) ()M () T ) ) = ( ) k k= ( ξ )k ( Γ o (t) (M ()M () T ) ) k By employing the preceding equation and (37), one can rewrite (36) as follows P (t)p () = I M () T (M ()M () T ) M () where G (t) = + M () T (M ()M () T ) G (t)m () (39) ( ) k+ ( ( ξ )k Γ o (t) (M ()M () T ) ) k k= As it can be seen from (33) and (35), it is required to calculate M (t)p (t)p () and 2(n ) M (t)p (t). In this regard, one can employ (39) and M (t) = e At M () to obtain In addition, we have M (t)p (t)p () = e At G (t)m () (4) 2(n ) P (t) = γ 2(n ) P (t)p () Therefore, by using γ 2(n ) = ξ and (39), one can get 2(n ) M (t)p (t) = e At Γ o (t) G 2 (t) (M ()M () T ) M () (4)

7 7 where G 2 (t) = ( ) k ( ( ξ )k (M ()M () T ) Γ o (t) ) k k= It is worth noting that G (t) G 2 (t) k= k= ( ξ ( ξ ξ )k = ξ )k = ξ ξ ξ ξ ξ ξ (42) Now (35), (4), and (4) can be employed to obtain the following equation M (t) θ(t) = e At G (t)m () θ e At Γ o (t) G 2 (t) Then, one can use (28) and (42) to get ξ M (t) θ(t) k ξ ξ M () θ e λ t + k2 k 5 ξ ξ ξ sup τ t Therefore, by using the fact that e λ t e 2 λ t /4 we can say ξ M (t) θ(t) k ξ ξ M () θ + k2 k 5 ξ 4λ ξ ξ e AT τ C T Cσ(ˆθ(τ), τ)dτ σ(ˆθ(τ), τ) e λ t e λ τ dτ and e λ t sup σ(ˆθ(τ), τ) τ t (43) In order to analyze the preceding equation, it is required to obtain the supremum of σ(ˆθ). Towards this end, Lemma 2 can be employed to conclude that η N (θ ) = A η N (θ ). As a result, one can use the definition of σ(ˆθ) = η N (ˆθ) η N (θ ) and get σ(ˆθ) = A σ(ˆθ) + B [f(x, u) f (x Dη, u)] Thus we have σ(ˆθ(t), t) =e At σ(ˆθ(), ) + τ= e A(t τ) B [f(x(τ), u(τ)) f (x(τ) Dη(τ), u(τ))] dτ Moreover, from the definition of σ it can be seen that σ( θ(), ˆ ) =. Therefore, (24) and (28) are utilized to obtain σ(ˆθ(t), t) kl sup τ t η(τ) kl λ sup η(τ) τ t τ= e λ (t τ) dτ Since the right hand side of the preceding equation is nondecreasing, one can employ the fact that the supremum of a function is its least upper bound and obtain sup σ(ˆθ(τ), τ) kl τ t λ sup η(τ) (44) τ t Therefore, (33), (43), (44), and η() = M () θ can be used to get ξ η(t) k ξ ξ η() + ( k2 k 5 ξ 4λ ξ ξ + )kl λ sup τ t η(τ) It can be seen that the right hand of the preceding equation is nondecreasing, therefore, it is also greater or equal to sup η(τ) for τ t. Moreover, there exist some ξ > and > such that for ξ > ξ and < <, we have ( k2 ξ k 5 4λ ξ ξ + )kl λ > As a result, one can write sup η(τ) τ t ( k2 k 5 k ξ ξ ξ η() ξ 4λ ξ ξ On the other hand, from (4) it is obtained 2(n ) η 2 x o 2 η 2 + )kl λ (45) Finally, the preceding equation and (45) can be employed to get sup x o (τ) τ t k n ( k2 k 5 ξ ξ ξ x o () ξ 4λ ξ ξ + )kl λ (46) Now for comparison, it is required to employ a similar approach for obtaining an upper bound of the estimation error of a single HGO. Towards this end, by considering (3) and (7), it can be easily seen that state estimation obtained from (4) and (5) with fixed α i s is equal to the estimation of a single HGO. Therefore, one can conclude that a single HGO with the initial condition of ˆx() = N i= ˆα i()ˆx i () performs like a MHGO with the same initial condition and fixed ˆα i s, i.e., ˆx(t) = N i= ˆα i()ˆx i (Λ, t). With regard to this fact, the scaled state estimation error of a single HGO, η s = D() x, can be obtained as follows η s (t) = M (t) θ + σ(ˆθ, t) where σ(ˆθ, t) = η N (ˆθ, t) η N (θ, t). Note that in the preceding equation ˆθ does not change, i.e., θ(t) = θ for all t. By using a similar approach to MHGO and η s () = M () θ, we can get M (t) θ() k η s () sup σ(ˆθ, τ) kl τ t λ sup η s (τ) τ t Therefore, by choosing similar to MHGO, one can obtain the following equation for the estimation error of a single HGO sup τ t x(τ) k n x() kl λ (47) By comparing (46) and (47) and using x o () = x(), it can be seen that by choosing ξ big enough which is equivalent to taking γ big, when tends to zero, the transient estimation obtained from MHGO peaks to O(/(ξ n )) which is much less than the peak of a single HGO estimation, i.e., O(/ n ).

8 8 As a result, the proposed MHGO can provide state estimations with more preferable transient response in comparison to a single HGO. Comment 3: One can see from Theorem and the presented performance investigation that the satisfaction of Assumptions and 2 is not required for the convergence of state estimation. However, if the initial conditions of HGOs are selected such that they are held, then it can be guaranteed that by choosing γ big enough, the proposed observation scheme is able to provide better state estimation than a single HGO. C. Control This section considers output feedback control problem using the obtained state estimation from MHGO. The proposed output feedback control strategy is inspired by a theorem which considers output feedback control problem of nonlinear systems using a single HGO. To elucidate more, the details of this theorem are presented here. Theorem 2 ([2]): Consider system () and the following state feedback controller u = ω(x) (48) where ω(x) is a locally Lipschitz function in x over the domain of interest and globally bounded function of x, also ω() =. Suppose that the origin of system () under the state feedback (48) is asymptotically stable and Ω is its region of attraction. Now, consider the following output feedback controller u = ω(ˆx) where ˆx is generated by the single HGO (2). Let S be any compact set in the interior of Ω and Q be any compact subset of R ρ. Then, (i) there exists > such that, for every <, the solution (x(t), ˆx(t)) of the closed-loop system, starting in S Q, is bounded for all t. (ii) given any µ >, there exist 2 > and T 2 >, both dependent on µ, such that, for every < 2, the solutions of the closed-loop system, starting in S Q, satisfy x(t) µ and ˆx(t) µ, t T 2 (iii) given any µ >, there exists 3 >, dependent on µ, such that, for every < 3, the solutions of the closed-loop system, starting in S Q, satisfy x(t) x r (t) µ, t where x r (t) is the solution of system () under the state feedback (48), starting at x(). (iv) if the origin of system () under the state feedback (48) is exponentially stable and f(x, u) is continuously differentiable in some neighborhood of x =, then there exists 4 > such that, for every < 4, the origin of the closed-loop system is exponentially stable and S Q is a subset of its region of attraction. The aforementioned theorem posits that by choosing sufficiently small, the output feedback controller that employs ˆx from (2), is capable to recover the performance of the state feedback controller. In the sequel, a theorem corresponding to the performance of the observer-based control scheme that utilizes state estimation obtained from the proposed MHGO, is provided. Theorem 3: Let conditions of Theorem 2 be satisfied, and consider the proceeding MHGO ˆx i (ˆθ) = Aˆx i (ˆθ) + H(y C ˆx i (ˆθ)) + Bf (ˆx o, ω(ˆx o )) ˆx o = N i= N ˆα iˆx i (ˆθ) + ( i= ˆα i )ˆx N (ˆθ) where i =,, N and ˆx o denotes the reconstructed system s state; moreover, ˆθ = [ˆα ˆα N ] T is obtained from the modified RLS algorithm (2). Then, the output feedback controller u = ω(ˆx o ) recovers the performance of the state feedback controller, in the sense of Theorem 2. Moreover, ˆθ and P are bounded and ˆx i s are uniformly ultimately bounded. Proof. Let us rewrite the dynamics of the overall closedloop system as follows: ẋ = Ax + Bf(x, ω(ˆx o )) ˆx o = N i= N ˆα iˆx i (ˆθ) + ( i= ˆα i )ˆx N (ˆθ) ˆθ = P M T C T C( x N (ˆθ) + M ˆθ) P = P M T C T CMP ˆx i (ˆθ) = Aˆx i (ˆθ) + H(y C ˆx i (ˆθ)) + Bf (ˆx o, ω(ˆx o )) By using the scaled error as defined in (4) and the fact that CD() = n C, one has ẋ = Ax + Bf(x, ω(x D()η)) (49) η = M ˆθ + ηn (ˆθ) (5) ˆθ = 2(n ) P M T C T Cη (5) P = 2(n ) P M T C T CM P (52) η i = A η i + B (x, D()η) (53) where A = D() (A HC)D(), the ith column of M is η i (ˆθ) η N (ˆθ), and (x, D()η) = f(x, ω(x D()η)) f (x D()η, ω(x D()η)) Similar to the proof of Theorem, since P and P, it can be concluded that P is bounded and P γ. In addition, one can consider (5) and use (2), (5), and (53) to obtain η = A η 2n M P M T C T Cη + B (x, D()η) (54) It can be seen that (49) and (54) construct a singular perturbation model. Therefore, according to the perturbation theory, the analysis is divided into two stages: finding the reduced model and finding the boundary-layer model. To obtain the reduced system, it is required to set = in (49) and (54). On the other hand, since A is a Hurwitz matrix, it has full rank. Thus by

9 9 setting = and performing some basic manipulations, one can get ẋ = Ax + Bf(x, ω(x)) From the preceding equation, it is obvious that the reduced system is the closed-loop system under the state feedback. Thus this system is asymptotically stable with the region of attraction Ω, and according to the converse Lyapanouv s theorem [2], there exists a Lyapanouv function V (x) and a positive definite function U(x), defined for x Ω, such that V (x) as x Ω V x [Ax + Bf(x, ω(x))] U(x), x Ω (55) and for any c >, {V (x) c} is a compact subset of Ω. Define S as a compact set in the interior of Ω, then S Ω c = {V (x) c} Ω In the next step, to achieve the boundary-layer model, the change of time variable τ = t/ is employed. Then, by setting =, the model is derived as follows dη dτ = A η Since all eigenvalues of A have negative real parts, there exists a Lyapunov function W (η) = η T P η such that W η A η = η 2 ; where the matrix P satisfies A T P + P A = I. Let Σ = {W (η) ρ 2 } and Π = Ω c Σ. The proof of (i) is divided in two steps. In the first step, it is shown that there exist > such that by selecting <, the set Π is a positively invariant set. On the other hand, it is not possible to guarantee that η() Σ. To address this challenge, in the second step, it is shown that there exists 2 > such that for < 2 and (x(), ˆx o ()) S Q, the trajectory (x(t), η(t)) enters Π in finite time. For the first step, we can use (49) and consider the derivative of V as follows V = V [Ax + Bf(x, ω(x Dη))] (56) x Since f and ω are locally Lipschitz functions, we can consider the following inequality B[f(x, ω(x Dη)) f(x, ω(x))] k η, (x, η) Π (57) where k is a Lipschitz constant. Moreover, over Ω c, the following equation is valid V x k 2 (58) Now by adding and subtracting V x Bf(x, ω(x)) to the right hand side of (56) and using (55), (57), and (58), one can get V U(x) + k k 2 η (59) On the other hand, in the set Π we have W (η) ρ 2, and consequently ρ η λmin (P ) By using the preceding equation, one can rewrite (59) as follows ρ V U(x) + k k 2 λmin (P ) By selecting ρ = β/(k k 2 ) where β = λmin(p ) min x Ωc U(x), it can be shown that V. For W (η), by using (54) we have Ẇ = ηt η 2 2(n ) η T P M P M T C T Cη + 2η T P B (x, D()η) Now P (t) γ, M (t) k M (), and the fact that (x, D()η) L η for (x, η) Π can be employed to get Ẇ ( + 2(n ) k 3 + k 4 ) η 2 where k 3 = 2k 2 γ P M () 2 and k 4 = 2L P. Consequently, there exists > small enough such that for any < one has 2(n ) k 3 + k 4 2 By using the preceding equation, we have Ẇ 2 η 2 < (6) Therefore, for all (x, η) Π, we have V and Ẇ <, i.e., Π is a positively invariant set. Now, in the second step, it is considered that (x(), ˆx o ()) S Q. Since f(x, ω(x Dη)) is locally Lipschitz and ω(x Dη) is a globally bounded function of x Dη, for all x Ω c we have Ax + Bf(x, ω((x Dη))) k 5 (6) where k 5 is non-negative and independent of. Therefore, by using (49), (6), and the fact that x() is in the interior of Ω c, we have ẋ(τ)dτ ẋ(τ) dτ k 5 t From the preceding equation it can be seen that for x(t) Ω c, it is valid to say x(t) x() k 5 t (62) This means that there exists T >, independent of, such that x(t) is in the interior of Ω c for all t [, T ]. During this time interval, by choosing <, the equation (6) is satisfied. Thus, it can be shown that Consequently, we have Now by defining T () as Ẇ 2λ max (P ) W W (t) e 2λmax(P ) t W () (63) T () = 2λ max (P ) ln( W () ρ 2 )

10 it can be seen that there exists 2 > such that the inequality T () 2 T is satisfied for any < 2. It should be noted that such a selection exists since if tends to zero, T () tends to zero too. Therefore, by selecting < 2 it can be guaranteed that W (T ()) ρ 2. In other words, η(t) enters Σ before x(t) leaves Ω c. Now by considering = min(, 2 ) and selecting <, it is valid to say that in the time interval [, T ()], (x(t), η(t)) enters Π and stays in the interior of Π for all t T (). Thus, the trajectory (x(t), η(t)) is bounded for all t T (). In addition, it can be seen from (62) and (63) that (x(t), η(t)) is also bounded for t [, T ()], and this concludes the proof of (i). As it was shown, by selecting <, (63) is valid. Therefore, similar to the proof of Theorem, one can get ˆθ(t) ˆθ + k 6 λ+ 4λmax(P ) with k 6 = kk 7 γ 2(n ) M () η() and k 7 = λmax (P )/λ min (P ). In addition, by considering a Lyapunov function candidate W i (η i ) = ηi T P η i, it can be obtained that W i (η i ) < for η i > 2L k 7 P η(). Hence, ˆθ is bounded and η i s are uniformly ultimately bounded. On the other hand, since the boundedness of x(t) is guaranteed, ˆx i s are also uniformly ultimately bounded. The proof of (ii)-(iv) is similar to Theorem 2 and can be found in [2]. Comment 4: The theorem states that the separation principle proved for nonlinear systems and a single HGO is still valid for the proposed MHGO. As a result, a globally bounded state feedback controller and the proposed MHGO can be employed to stabilize nonlinear systems. From the preceding theorem it can be concluded that the obtained state estimation from the proposed MHGO can be utilized for control of nonlinear systems. On the other hand, it has been shown that MHGO is able to provide a better state estimation in comparison to a single HGO. This means that the utilization of MHGO state estimation for control of nonlinear systems provides a more accurate estimation for the controller, and this can result in a more preferable performance. IV. SIMULATION RESULTS This section is aimed at providing some insights assisting us in having a more comprehensive knowledge about the proposed strategy. Towards this end, simulations are performed on two practical systems, Van der Pol oscillator and robotics systems. The first example discusses the capability of the proposed MHGO in providing an accurate state estimation. In the second example, the output feedback control problem is addressed. Towards this end, a state feedback controller is considered by presuming that full state variables are measurable. Then, the state of the plant is reconstructed using the proposed MHGO and fed into the controller. The simulation results validate the aforementioned discussions in terms of stability and accuracy. A. Example In this simulation, the problem of estimating the state of Van der Pol oscillator will be addressed. The dynamical equation of this system is as follows [2]: ẍ = α 2 x + β( x 2 )ẋ y = x where it is considered that α = and β =.5. The initial condition of system is x() = 3 and ẋ() = 2. In order to represent the system in the form of (), one can choose x () = x, x (2) = ẋ, and f(x) = α 2 x () + β( x 2 () )x (2). To estimate the system states, a saturated version of f(x) is considered as f (ˆx o ) = 2 tanh(f(ˆx o )/2). Also, since N n + observers are required, N = 3 observers with the initial conditions of ˆx () = [ ] T, ˆx2 () = [ 5 +5 ] T, ˆx3 () = [ +5 5 ] T and design parameters κ = 2, κ 2 =, and =. are utilized. It can be seen that x() is in the convex hull K of ˆx i ()s, and M()M() T has full rank. Moreover, the initial values for estimating θ are chosen as ˆθ() = and P () = γi 2 where I 2 is the 2 2 identity matrix. As it was shown through the paper, by choosing γ big enough, MHGO is able to provide better estimations. To demonstrate that, γ = 2 and γ = are employed for estimation and the results are depicted in Fig.. From this figure, it can be easily seen that by choosing γ big, the proposed scheme provides a more preferable estimation. In order to investigate the performance of the proposed strategy, simulation results are also provided using a single HGO. To provide a reasonable comparison, f and the design parameters of the HGO are selected the same as the MHGO, i.e., κ i s and, and the initial condition of HGO as ˆx() = 3 i= ˆα i()ˆx i () = [ +5 5 ] T. The obtained estimation error using the single HGO is presented in Fig. 2. Form figures, it can be seen that the proposed method results in an estimation with more desirable features in the transient response. To elucidate more on this issue, it is worth pointing out that the proposed methodology yields a faster convergence rate, which is of a great significant in practice. Moreover, in the sense of the peaking phenomenon, MHGO produces estimations with smaller peaks in the transient response. B. Example 2 This section evaluates the performance of output feedback controllers using MHGO. In this regard, a single-link flexible joint manipulator with the following dynamic equation is considered as a case study [29], ẋ () = x (2) ẋ (2) = MgL I i sin x () k I i (x () x (3) ) ẋ (3) = x (4) ẋ (4) = k J (x () x (3) ) + J u y = x () where the system parameters are defined in Table I. (64)

11 x o() x o(2) 2 γ = γ = γ = -5 γ = Fig. : Estimation error of Van der Pol oscillator s state using MHGO. x () x (2) Fig. 2: Estimation error of Van der Pol oscillator s state using a single HGO. TABLE I: Parameters of the robotic system. Parameter Description Value M Link mass 2 kg k Joint elastic constant N/m L Distance between rotation axis and the link center of mass m g Gravitational acceleration 9.8 m/s 2 I i Link inertia moment.5 kg.m 2 J Rotor inertia moment.5 kg.m 2 The system can be transformed into the canonical form, as follows [2]: ż = Az + Bf(z, u) (65) y = Cz where z () = x (), f(z, u) = φ(z) + ku/(i i J), and φ(z) = MgL sin z () I i ( z 2 (2) + MgL ( z (3) + MgL I i sin z () ) cos z () + k I i I ) ( i k + k I i J + MgL ) cos z (). J The initial condition of the system is considered as z() = [..2 ] T. Assuming full state measurement available, the following controller can stabilize the system states and force its output to track the desired trajectory u(z) = tanh( I ij k ( φ(z) z () 9z (2) 3z (3) 5z (4) + sin t)) (66) Since the state variables are not measurable, it is needed to estimate them. Therefore, five HGOs (N = 5) with the following initial conditions and f are considered ẑ () = [ + ] T ẑ 2 () = [ + + ] T ẑ 3 () = [ ] T ẑ 4 () = [ + + ] T ẑ 5 () = [ ] T f (ẑ o ) = 2 tanh( 2 (φ(ẑ o) + k I i J u(ẑ o))) (67) Therefore, z() is in the convex hull K of ẑ i ()s, and M()M() T is a full rank matrix. Furthermore, the design parameters of the HGOs are selected as κ = 4, κ 2 = 6, κ 3 = 4, κ 4 =, and =.. For obtaining an estimation of θ, the initial values of ˆθ() = and P () = γi 4 are employed. Simulation results for γ = 2 and γ = (y(t)), the output of the plant under the state feedback controller (y r (t)), and the error between y r (t) and y(t) (ỹ r (t) = y r (t) y(t)) are illustrated in Fig. 3. It can be seen that when γ is big, the output feedback controller u(ẑ o ) recovers the performance of state feedback controller, and that is because by choosing γ big, MHGO provides more accurate state estimations for the controller. As a comparison, control is also performed using a single HGO with the same f, κ i s,, and initial condition, i.e. ẑ() = [ ] T. The simulation result (ys (t)) along with the error between y r (t) and y s (t) (ỹ rs (t) = y r (t) y s (t)) are depicted in Fig. 4. By comparing Fig. 3 to Fig. 4, it can be clearly seen that the proposed MHGObased control strategy can reconstruct the behaviour of full state-based control scheme more rapidly and accurately.

12 2 5 y r (t) y(t) for γ = y(t) for γ = ỹ r (t) for γ = ỹ r (t) for γ = Fig. 3: Output and output error of the manipulator under MHGO-based controller. ỹrs(t) 5-5 y r (t) y s (t) Fig. 4: Output and output error of the manipulator under single HGO-based controller. V. CONCLUSION This paper deals with state estimation and control of a special class of nonaffine nonlinear systems. HGOs are able to reconstruct system states, and a separation principle is valid for them. However, they have undesired peaks in their transient response which results in a lower performance of state estimation and control. To overcome this demerit, MHGO was presented that considers state estimation as a convex combination of multiple HGOs. It was shown that MHGO is stable and provides an accurate state estimation. Moreover, in contrast to conventional multi observers, MHGO needs less observers and employs all provided information at each time instant. The output feedback control problem was also considered and it was shown that the separation principle is valid for MHGO. As a result, one can employ MHGO along with state feedback controller to stabilize the nonlinear system. It is worth noting that in this paper, it is assumed that a saturated version of f is available. However, it is possible that this condition is not satisfied for some systems. Therefore, state estimation and control of nonlinear systems using MHGO, when f is completely unknown, is worth to be investigated. REFERENCES [] H. K. Khalil and L. Praly, High-gain observers in nonlinear feedback control, International Journal of Robust and Nonlinear Control, vol. 24, no. 6, pp , 24. [2] H. K. Khalil and J. Grizzle, Nonlinear systems. Prentice hall New Jersey, 996, vol. 3. [3] A. A. Prasov and H. K. Khalil, A nonlinear high-gain observer for systems with measurement noise in a feedback control framework, IEEE Transactions on Automatic Control, vol. 58, no. 3, pp , 23. [4] J. Lei and H. K. Khalil, Feedback linearization for nonlinear systems with time-varying input and output delays by using high-gain predictors, IEEE Transactions on Automatic Control, vol. 6, no. 8, pp , 26. [5] K. Esfandiari, F. Abdollahi, and H. A. Talebi, Adaptive control of uncertain nonaffine nonlinear systems with input saturation using neural networks, IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no., pp , 25. [6] L. Torres, C. Verde, G. Besançon, and O. González, High-gain observers for leak location in subterranean pipelines of liquefied petroleum gas, International Journal of Robust and Nonlinear Control, vol. 24, no. 6, pp. 27 4, 24. [7] A. Teel and L. Praly, Global stabilizability and observability imply semi-global stabilizability by output feedback, Systems & Control Letters, vol. 22, no. 5, pp , 994. [8], Tools for semiglobal stabilization by partial state and output feedback, SIAM Journal on Control and Optimization, vol. 33, no. 5, pp , 995. [9] A. N. Atassi and H. K. Khalil, A separation principle for the stabilization of a class of nonlinear systems, IEEE Transactions on Automatic Control, vol. 44, no. 9, pp , 999. [] F. Esfandiari and H. K. Khalil, Output feedback stabilization of fully linearizable systems, International Journal of control, vol. 56, no. 5, pp. 7 37, 992. [] M. S. Chong, D. Nešić, R. Postoyan, and L. Kuhlmann, Parameter and state estimation of nonlinear systems using a multi-observer under the supervisory framework, IEEE Transactions on Automatic Control, vol. 6, no. 9, pp , 25. [2] R. Postoyan, M. H. Hamid, and J. Daafouz, A multi-observer approach for the state estimation of nonlinear systems, in 25 54th IEEE Conference on Decision and Control (CDC). IEEE, 25, pp [3] B. D. Anderson and A. Dehghani, Challenges of adaptive control past, permanent and future, Annual reviews in control, vol. 32, no. 2, pp , 28. [4] K. S. Narendra and J. Balakrishnan, Improving transient response of adaptive control systems using multiple models and switching, IEEE Transactions on Automatic Control, vol. 39, no. 9, pp , 994. [5] Z. Han and K. S. Narendra, New concepts in adaptive control using multiple models, IEEE Transactions on Automatic Control, vol. 57, no., pp , 22. [6] K. S. Narendra, Y. Wang, and W. Chen, Stability, robustness, and performance issues in second level adaptation, in 24 American Control Conference. IEEE, 24, pp [7], Extension of second level adaptation using multiple models to siso systems, in 25 American Control Conference (ACC). IEEE, 25, pp [8] W. Chen, J. Chen, and J. Sun, A combination of second level adaptation and switching scheme for uncertain linear system, in 24 33rd Chinese Control Conference (CCC). IEEE, 24, pp [9] K. S. Narendra and K. Esfandiari, Adaptive control of linear periodic systems using multiple models, 28 IEEE Conference on Decision and Control (CDC), pp , 28. [2], Adaptation in periodically varying environments using second level adaptation, Technical Report, Yale University, arxiv preprint arxiv:, no. 82, 28.

13 3 [2] V. K. Pandey, I. Kar, and C. Mahanta, Adaptive control of nonlinear systems using multiple models with second-level adaptation, in Systems Thinking Approach for Social Problems. Springer, 25, pp [22], Multiple models and second level adaptation for a class of nonlinear systems with nonlinear parameterization, in 24 9th International Conference on Industrial and Information Systems (ICIIS). IEEE, 24, pp. 6. [23], Control of twin-rotor mimo system using multiple models with second level adaptation, IFAC-PapersOnLine, vol. 49, no., pp , 26. [24] M. Shakarami, K. Esfandiari, A. A. Suratgar, and H. A. Talebi, On the peaking attenuation and transient response improvement of high-gain observers, 28 IEEE Conference on Decision and Control (CDC), pp , 28. [25] D. Astolfi and L. Marconi, A high-gain nonlinear observer with limited gain power, IEEE Transactions on Automatic Control, vol. 6, no., pp , 25. [26] I. J. Bakelman, Convex analysis and nonlinear geometric elliptic equations. Springer Science & Business Media, 22. [27] H. V. Henderson and S. R. Searle, On deriving the inverse of a sum of matrices, Siam Review, vol. 23, no., pp. 53 6, 98. [28] W. Cheney, Analysis for applied mathematics. Springer Science & Business Media, 23, vol. 28. [29] M. Vidyasagar, Nonlinear systems analysis. SIAM, 22. Amir Abolfazl Suratgar (M 9 SM 4) was born in Arak, Iran, in 974. He received his B.S. degree in electrical engineering from Isfahan University of technology, Isfahan, Iran, in 996. He was honored M.S. and Ph.D. degrees in control engineering from Amirkabir University of Technology (Tehran polytechnic), Tehran, Iran, in 999 and 22, respectively. He was an Assistant Professor of Electrical Engineering at the University of Arak, Arak, Iran. He received the outstanding research award from Engineering Faculty of University of Arak in 26 and 25. He received the outstanding education award from Engineering Faculty of University of Arak in 29. Currently, he is an Associate Professor in Amirkabir University of Technology (Tehran polytechnic), Tehran, Iran. The areas of his research interests are optimal control and optimization, computational intelligence, fuzzy modeling, fuzzy system stability analysis, neuroscience, and micro robots dynamics analysis and control. He has published more than 3 papers in scientific journals and conferences. Mehran Shakarami received the M.Sc. degree in electrical engineering from Amirkabir University of Technology (Tehran polytechnic), Tehran, Iran, in 24, and B.Sc. and A.Sc. degrees from Shamsipour College, Tehran, Iran, in 2 and 29, respectively. He is currently a Ph.D. student at the University of Groningen, Groningen, The Netherlands, and his current research interests include dynam ical networks, game theory, nonlinear control, adaptive control, robust control, and system identi fication. Heidar A. Talebi (M 2 SM 8) received the B.Sc. degree in electrical engineering from Ferdowsi University, Mashhad, Iran, in 988, the M.Sc. degree in electrical engineering from Tarbiat Modarres University, Tehran, Iran, in 99, and the Ph.D. degree in electrical engineering from Concordia University, Montreal, QC, Canada, in 997. He was a Postdoctoral Fellow and a Research Fellow at Concordia University and The University of Western Ontario. In 999, he joined Amirkabir University of Technology, Tehran, Iran. He is currently a professor in the Department of Electrical Engineering, and his research interests include control, robotics, medical robotics, fault diagnosis and recovery, intelligent systems, adaptive control, nonlinear control, and realtime systems. Kasra Esfandiari (S 3) received the B.Sc. degree in electrical engineering/control systems from Shiraz University, Shiraz, Iran, in 22, and the M.Sc. degree (Hons.) in electrical engineering/ control systems from the Amirkabir University of Technology, Tehran, Iran, in 24. In 27, he received his second M.Sc. degree (Hons.) in electrical engineering from Yale University, New Haven, USA, where he is currently a Ph.D. candidate. His current research interests include adaptive control, intelligent control, nonlinear control, robust control, and robotics.