1 Introduction Almost every physical system is subject to certain degrees of model uncertainties, which makes the design of high performance control a

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1 Observer Based Adaptive Robust Control of a Class of Nonlinear Systems with Dynamic Uncertainties Λ Bin Yao + Li Xu School of Mechanical Engineering Purdue University, West Lafayette, IN 47907, USA + Tel: (765) Fax:(765) byao@ecn.purdue.edu Abstract In this paper, a discontinuous projection based adaptive robust control (ARC) scheme is constructed for a class of nonlinear systems in an extended semi-strict feedback form by incorporating a nonlinear observer and a dynamic normalization signal. The form allows for parametric uncertainties, uncertain nonlinearities, and dynamic uncertainties. The unmeasured states associated with the dynamic uncertainties are assumed to enter the system equations in an affine fashion. A novel nonlinear observer is first constructed to estimate the unmeasured states for a less conservative design. Estimation errors of dynamic uncertainties, as well as other model uncertainties, are dealt with effectively via certain robust feedback control terms for a guaranteed robust performance. In contrast with existing conservative robust adaptive control schemes, the proposed ARC method makes full use of the available structural information on the unmeasured state dynamics and the prior knowledge on the bounds of parameter variations for high performance. The resulting ARC controller achieves a prescribed output tracking transient performance and final tracking accuracy in the sense that the upper bound on the absolute value of the output tracking error over entire time-history is given and related to certain controller design parameters in a known form. Furthermore, in the absence of uncertain nonlinearities, asymptotic output tracking is also achieved. Λ Part of the paper has been presented in the 1999 IEEE Conference on Decision and Control. The work is supported by the National Science Foundation under the CAREER grant CMS

2 1 Introduction Almost every physical system is subject to certain degrees of model uncertainties, which makes the design of high performance control algorithms a very challenging job. Normally, the causes of model uncertainties can be classified into two categories: (i) repeatable or constant unknown quantities such as the unknown physical parameters (e.g., the inertia load of any industrial drive systems), (ii) non-repeatable unknown quantities such as external disturbances and imprecise modeling of certain physical terms. To account for these uncertainties, two nonlinear control methods, the deterministic robust control (DRC) [1]-[4] and the adaptive control (AC) [5]-[7] (or its robustified version, the robust adaptive control [8]-[11]), may apply. In general, the deterministic robust controllers can achieve a guaranteed transient and final tracking accuracy in the presence of both parametric uncertainties and uncertain nonlinearities. However, since no attempt is made to learn from past behavior to reduce the effect of constant unknown quantities, the design is conservative and involves either switching [1] or infinite-gain [4] feedback for asymptotic tracking; both means are impractical and unattainable. In contrast, the adaptive controllers [5, 6, 7] are able to achieve asymptotic tracking in the presence of parametric uncertainties without resorting to infinite gain feedback. Recently, as in the robust adaptive control (RAC) of linear systems, much of the effort in nonlinear adaptive control area has been devoted to robustifying the adaptive backstepping designs [5] with respect to disturbances and significant progress has been made [8]-[11]. In [12]-[15], an adaptive robust control (ARC) approach has been proposed for the design of a new class of high-performance robust controllers. By exploiting practically reasonable prior information on a physical system such as the bounds of parameter variations as much as possible, the approach effectively combines the design methods of DRC and AC. The resulting ARC controllers achieve the results of both DRC and AC while naturally overcoming their practical limitations. The approach has been applied to various applications and comparative experimental results [16]-[18] have verified the effectiveness and the high performance nature of the proposed ARC strategy; for different applications [16, 17], tracking errors have been consistently reduced almost down to measurement resolution level during most of the execution period. In addition to parametric uncertainties and uncertain nonlinearities, some systems are further subjected to dynamic uncertainties that depend on the unmeasured states of exogenous dynamic systems [19, 20]. Practical examples include the dynamic friction model in [21] and the control of eccentric rotor in [22, 19]. This class of systems has received a lot of attention in recent years since there are few results available on the general problem of robust control of nonlinear systems with partial state feedback. In [19], Freeman and Kokotović constructed an adaptive controller for a class of extended strict feedback nonlinear systems in which the unmeasured states enter the systems in a linear affine fashion. As pointed out in the paper, it is unclear how the approach can be made robust to modeling errors such as uncertain nonlinearities. In [23], Jiang and Praly generalized the idea of using a dynamic signal to dominate dynamic disturbances to the robust adaptive control of nonlinear systems subjected to dynamic uncertainties. Using a small-gain type of argument, they also presented in [20] a modified robust adaptive control (RAC) procedure [9] for a class of uncertain nonlinear systems subject to dynamic uncertainties satisfying an input-to-state stability property. In this paper, by incorporating a nonlinear observer design and the dynamic signal introduced in [20], the discontinuous projection based ARC approach [12] will be extended to a class of nonlinear systems 2

3 subjected to parametric uncertainties, uncertain nonlinearities, and dynamic uncertainties. The motivation for this research is that, in most situations, the structural information on the unmeasured state dynamics and on the way itinteracts with the rest of the system dynamics is known. If these prior structural information can be utilized effectively, a less conservative controller can be synthesized and a better performance can be achieved. With this fact in mind, departing from [20], in the absence of uncertain nonlinearities, the nominal structure of the unmeasured state dynamics are assumed to be known and the unmeasured states are assumed to enter the system equations in a linear affine fashion as in [19]. With a mild assumption which is equivalent to the detectability assumption in linear systems, a nonlinear observer is first constructed to recover the unmeasured states of dynamic uncertainties. By doing so, the estimates of the unmeasured states can be used in the controller design to eliminate the effect of dynamic uncertainties for an improved achievable nominal performance asymptotic tracking is obtained in the presence of parametric uncertainties and dynamic uncertainties. In addition, in the presence of uncertain nonlinearities, the observer error dynamics is made to be input-to-state stable. Estimation errors, as well as model uncertainties, are dealt with effectively via certain robust feedback as in the ARC design in [12] to achieve a guaranteed robust performance. Consequently, the contributions of the paper are as follows. Firstly, compared to the adaptive control design [19] where uncertain nonlinearities are not considered, the proposed scheme is observer based and is robust to uncertain nonlinearities; in fact, a guaranteed robust performance is obtained even in the presence of uncertain nonlinearities. Secondly, compared to the RAC approach [20], the proposed scheme explicitly utilizes the structural information of the system and a better nominal performance, asymptotic tracking, is obtained in the absence of uncertain nonlinearities. In addition, the approach puts more emphasis on the robust control law design; in fact, the parameter adaptation law in the proposed ARC design can be switched off at any time without affecting global stability or sacrificing the prescribed transient performance result, since the resulting nonadaptive" controller becomes a deterministic robust controller. Because of this design philosophy, the proposed controller achieves a prescribed transient performance and final tracking accuracy, i.e., the upper bound on the absolute value of the tracking error over the entire time-history is given and related to certain controller design parameters in a known form, which is much more transparent than that in RAC design [20]. Lastly, the nonlinear observer design in this paper is motivated by the recent excellent research done in [24]. However, in [24], the effect of parametric uncertainties and uncertain nonlinearities are not considered. Thus, the paper also extends the nonlinear observer design. The paper is organized as follows: Problem statement is presented in Section 2. A nonlinear state observer design is presented in Section 3. The proposed observer based ARC controller is shown in Section 4. A design example and comparative simulation results are presented in Section 5, and conclusions are drawn in Section 6. 2 Problem Statement The following nonlinear system will be considered in this paper _ = F (μx l ) + G (μx l ) + μ (x; ; u; t); _x i = x i+1 + T ' i (μx i )+ i(x; ; u; t); 1» i» l 1; _x i = x i+1 + T ' i (μx i )+' T i (μx i) + i(x; ; u; t); l» i» n 1; _x n = u + T ' n (x)+' T n(x) + n(x; ; u; t); y = x 1 ; (1) 3

4 where μx l =[x 1 ; ;x l ] T 2 IR l,μx i =[x 1 ; ;x i ] T 2 IR i, and x =[x 1 ; ;x n ] T 2 IR n. y 2 IR and u 2 IR are the control input and the output, respectively. 2 IR m represents the unmeasured states, and 2 IR p is a vector of unknown constant parameters. F 2 IR m p, G 2 IR m m, ' i 2 IR p and ' i 2 IR m are matrices or vectors of known smooth functions, which are used to describe the nominal model of the system. μ and i represent the lumped unknown nonlinear functions such as disturbances and modeling errors. Throughout the paper, the following notations will be used. In general, ffl i represents the i-th component of the vector ffl and the operation < for two vectors is performed in terms of the corresponding elements of the vectors. The following practical assumptions are made: Assumption 1 The extent of parametric uncertainties and uncertain nonlinearities are known. In other words, parametric uncertainties and uncertain nonlinearities μ and i satisfy 2 Ω = f : min < < max g; μ 2 μω = f μ : j μ (x; ; u; t)j» ffi(μx μ l )g; = f i : j i(x; ; u; t)j»ffi i (μx i )g; i 2 Ω where min, max, μ ffi(μx l ) and ffi i (μx i ) are known. 1 (j j denotes the usual Euclidean norm.) 3 (2) Assumption 2 The -subsystem, _ = F (μx l ) + G (μx l ) + μ (x; ; u; t), with as the state and μx l (t) as the input, is bounded-input-bounded-state stable in the sense that for every 0 2 IR m and every μx l (t) 2 L l 1[0; 1), the solution (t) starting from the initial condition 0 is bounded, i.e., (t) 2 L m 1[0; 1). 3 Let y d (t) be the desired output trajectory, which is assumed to be known, bounded with bounded derivatives up to n-th order. The objective is to synthesize a control input u such that the output y tracks y d (t) as closely as possible in spite of various model uncertainties. Before leaving this section, we would like to make the following remark concerning the connection and the difference between the system (1) and the systems extensively studied in the literature. Remark 1 (i) In the absence of dynamic uncertainties, i.e., if there are no unmeasured state in (1), the system (1) with Assumption 1 reduces to the semi-strict feedback form studied in [12, 15], where a discontinuous projection based ARC scheme has been presented for a guaranteed output tracking transient performance and final tracking accuracy [12]. (ii) If there are no uncertain nonlinearities, i.e., μ =0, and i =0; 8i, the system (1) with Assumption 2 reduces to the extended strict feedback form studied by Freeman and Kokotović [19], where an adaptive controller was constructed. (iii) In [20], Jiang and Praly considered a similar system which allows uncertain nonlinearities to depend on nonlinearly appearing parametric uncertainties also. However, the -dynamics is assumed to be input-to-state practically stable, which is more stringent than Assumption 2. Also, the nominal structure of the -dynamics is ignored. } 1 The design can be extended to the case where μ ffi and ffii depend on t explicitly. 4

5 3 State Estimation Since states are not measurable, a nonlinear observer need to be constructed to provide their estimates. Motivated by [24], we first introduce a transformation of coordinate. Define a vector ο =!(μx l ); (3) where!(μx l )=[! 1 (μx l ); ;! m (μx l )] T is the vector of design functions yet to be determined. Its derivative is computed as For simplicity, let _ο = _ _!(μx l ) = F (μx l ) + G (μx l ) + μ P i=1 i i+1 + T ' i + ' l l = F (μx l ) +(G (μx l ' T l ) P i (x i+1 + T ' i )+ μ P i i Substituting (3) and (5) into (4), we have _ο = A(μx l )(ο +!(μx l )) + F (μx l ) A(μx l )=G (μx l ' T l : (5) lx i=1 If were known, we would design a nonlinear observer _^ο = A(μx l )(^ο +!(μx l )) + F (μx i (x i+1 + T ' i )+ μ lx lx i i: i (x i+1 + T ' i ): (7) Then, the state estimation error " = ^ο ο would be governed by the following dynamic system _" = A(μx l )" + "; " = μ + i i: (8) Since is not known, the observer (7) is not implementable but it provides motivation for the design of following nonlinear filters _ 0 = A(μx l ) 0 + A(μx l )!(μx l ) P i=1 i i+1 ; _ j = A(μx l ) j + F j (μx l ) P i=1 i i;j ; 1» j» p; where j 2 IR m, F j represents the jth column of F, and ' i;j is the jth element of the vector ' i. The state estimate can thus be represented by ^ο = 0 + px i=1 (9) j j = 0 + ; (10) where =[ 1 ; ; p ] 2 IR m p. >From (9) and (10), it can be verified that the observer error dynamics is still described by (8). Therefore, the equivalent expression for the unmeasurable state is = 0 + +!(μx l ) ": (11) In viewing the observer error dynamics (8), the following additional assumption is made. 5

6 Assumption 3 : A There exists an exp-isps Lyapunov function 2 V " for the observer error dynamics (8), i.e., there exists a Lyapunov function V " such that fl 1 (j"j)» V " (")» fl 2 (j"j); 8" 2 IR m ; _V "» c " V " (")+fl " (jμx l j)+d " ; (12) where fl 1, fl 2 and fl " are class K 1 functions, c " > 0 and d " 0 are two constants. B The unperturbed system of (8) is assumed to be exponentially stable, i.e., when " =0, the observation error " converges to zero exponentially. 3 Remark 2 In [25], it is shown that A of Assumption 3 is equivalent to the assumption that the observer error dynamics (8) with " as the state and μx l as the input is input-to-state practically stable (ISpS). A control system _x = f(x; u) is ISpS if there exist a class KL function fi, a class K function fl, and a nonnegative constant d such that, for any initial condition x(0) and each input u 2 L 1 [0;t), the corresponding solution x(t) satisfies, for all t 0, jx(t)j»fi(jx(0)j;t)+fl(ku t k)+d; (13) where u t is the truncated function of u at t and k k stands for the L 1 supremum norm. In [20], a similar assumption as A of Assumption 3 is made with respect to the dynamic uncertainties. } Remark 3 It should be noted that Assumption 3 is a quite mild assumption and is in fact a nonlinear generalization of the detectability of the unmeasured -dynamics. To see this, let us assume that ' T l is a constant output vector and G is a constant matrix. It is proved in the following that the detectability of the pair (' T l, G ) implies the Assumption 3. Since the pair (' T l, G )isassumed to be detectable, there exists a constant vector L 2 R m such that A = G L' T l (14) is Hurwitz. By choosing!(μx l ) = Lx l, from (5), it follows that A(μx l ) = A. Hence there exist two positive-definite matrices P > 0 and Q>0 such that Let Then we have PA + A T P = Q: (15) V " (") =" T P": (16) fl 1 (j"j)» V " (")» fl 2 (j"j); (17) (18) where fl 1 (j"j) = min (P )j"j 2 and fl 2 (j"j) = max (P )j"j 2. Noting Assumption 1, from (8), the derivative of V " (") satisfies _V "» min (Q)j"j 2 +2j"j max (P )(j μ j + jljj lj)» min (Q)j"j 2 +2j"j max (P )( ffi μ + jljffi l )» c " max (P )j"j 2 + max (P ) min (Q) c" max(p ffi + jljffi ) l ) 2 2 The ISpS notion was first introduced in [25] 6

7 where c " > 0 is any positive scalar satisfying c " < max(p ). Noting that ffi μ min + jljjffi (Q) l is a function of μx l and t only and is bounded with respect (w.r.t) to t, there exist a class K 1 function fl " (jμx l j) and a positive constant d " such that fl " (jμx l j)+d " 2 max (P ) (μ min (Q) c " max(p ffi + jljffi ) l ) 2. Thus, (18) becomes _V "» c " V " (")+fl " (jμx l j)+d " ; (19) which implies that A of Assumption 3 is satisfied. Since A satisfied. is Hurwitz, B of Assumption (3) is also } Remark 4 If the unmeasured state is of dimension 1, i.e., m =1, then, as long as ' l (μx l ) is non-zero, a nonlinear!(μx l ) can be explicitly constructed as follows. Let k beapositive design constant and choose!(μx l ) to satisfy k = g (μx l ' l (μx l ): l The differential equation (20) has the following explicit solution:!(μx l )= Z xl 0 g(μx l )+k dx l : (21) ' l (μx l ) In such a case, A(μx l ) = k. It is thus obvious that Assumption 3 is satisfied. Furthermore, the exponential convergence rate for the unperturbed observer error dynamics (8) can be arbitrarily placed. } The following lemma, which is proved in [20], will be used in the subsequent ARC controller design. Lemma 1 3 If (12) holds, then, for any constants μc 2 (0;c " ), any initial condition " 0 = "(0) and r 0 > 0, for any function μfl such that μfl(μx l ) fl " (jμx l j), there exists a finite T 0 = T 0 (μc; r 0 ;ffl 0 ) 0, a nonnegative function D(t) defined for all t 0 and a dynamic signal described by such that D(t) =0for all t T 0 and _r = μcr +μfl(μx l )+d " ; r(0) = r 0 ; (22) V " (")» r(t)+d(t); (23) for all t 0 where the solutions are defined. 4 >From (12) and (23), it follows that [20] j"(t)j»fl 1 1 (r(t)+d(t))» fl 1(2r(t)) + fl 1(2D(t)): (24) Discontinuous Projection Based ARC Backstepping Design 4.1 Parameter Projection Let ^ denote the estimate of and ~ the estimation error (i.e., ~ = ^ ). Under Assumption 1, a discontinuous projection based ARC design [12] will be constructed to solve the robust tracking control 3 see [20], Lemma 3.1 7

8 problem for (1). Specifically, the parameter estimate ^ is updated through a parameter adaptation law having the form given by _^ = Proj^ ( fi); (25) where is a symmetric positive definite (s.p.d.) diagonal adaptation rate matrix, fi is an adaptation function to be synthesized later, and the projection mapping Proj^ (ffl) = (ffl [Proj^ 1 1 (ffl );:::;Proj^ p p )] T is defined by [26, 27] (for simplicity, assume that is a diagonal matrix in the following) Proj^ i (ffl i )= 8 >< >: 0 if ^ i = i max and ffl i > 0 0 if ^ i = i min and ffl i < 0 ffl i otherwise It can be shown [13] that for any adaptation function fi, the projection mapping (26) guarantees (26) P1 ^ 2 μω = f^ : min» ^» max g P2 ~ T ( 1 Proj^ ( fi) fi)» 0; 8fi (27) 4.2 ARC Controller Design In this subsection, the ARC backstepping design [12] will be employed and extended to the present case with dynamic uncertainties. In the first l 1 steps, the design procedure is the same as that in [12], since the first l 1 steps do not involve the unmeasured dynamic uncertainties. From step l, we will deal with the effect of the unmeasured state by replacing it with its estimate for high performance. The effect of state estimation error will be dealt with via certain robust feedback terms through the incorporation of a dynamic normalization signal for a guaranteed robust performance Step 1» i» l 1 At step i, 81» i» l 1, we construct a control function ff i for the virtual input x i+1 such that x i tracks its desired ARC control law ff i 1 synthesized at step i 1 (for simplicity, denotes ff 0 (t) = y d (t)). Let z i = x i ff i 1, ~ 1(x; t) = 1(x; t), ffi 1 (x 1 )=' 1 (x 1 ), and recursively define the following functions ffi i (μx i ; ^ ) = P i i 1 ~ i = P i 1 ' j + ' i (μx i i 1 j + i(x; t): (28) Then we have the following lemma, which is proved in [12]. Lemma 2 At step i, 81» i» l 1, choose the desired control function ff i as ff i (μx i ; ^ ; t) =ff ia + ff is ; ff ia = z i 1 ^ T ffi i + P i i 1 x j+1 i 1 ff is = ff is1 + ff is2 ; ff is1 = k is z i ; k is g i + i 1 C i j 2 + jc ffii ffi i j 2 ; (29) 8

9 where g i > 0 is a constant, C i and C ffii are positive definite constant diagonal matrices, and ff is2 is a robust control term satisfying the following two conditions i. z i (ff is2 ~ T ffi i + ~ i)» ffl i ii. z i ff is2» 0 (30) in which ffl i is a positive design parameter. Then the i-th error subsystem is _z i = z i+1 z i 1 k is z i +(ff is2 ~ T ffi i + ~ i 1 _^ ; (31) and the derivative of the augmented p.s.d. function V i = V i z2 i ; V 0 =0 (32) satisfies _V i = z i z i+1 + ix f k js z 2 j + z j (ff js2 ~ T ffi j + ~ j 1 _^ z j g: (33) (The fact 0 =0is used in the above descriptions.) Step l Noting (1) and (11), the derivative ofz l = x l ff l 1 is _z l = x l+1 + T ' l + ' T l( 0 + +!(μx l ) ")+ l _ff l 1 ; (34) where _ff l 1 = P l l 1 (x j+1 + T ' j + j) l 1 _^ l 1. If we treat l+1 as the input, we can synthesize a virtual control law ff l for x l+1 such that z l is as small as possible. Since (34) contains unknown parameters, uncertain nonlinearity l, and the estimation error ", the ARC approach proposed in [12] will be generalized to accomplish the objective. The control function ff l consists of two parts given by ff l (μx l ;r; 0 ; ;^ ; t) =ffla + ff ls ; ff la = z l 1 ^ T ' l ' T l ( 0 + ^ +!(μx l )) + P l l 1 (x j+1 + ^ T ' j l 1 ff ls = ff ls1 + ff ls2 ; ff ls1 = k ls z l ; k ls g l + l 1 C l j 2 + jc ffil ffi l j 2 + c jψ l j 2 ; (35) where g l and c are positive constants, ψ = l ' l, C l and C ffil are positive definite constant diagonal matrices to be specified later. Let z l+1 = x l+1 ff l denote the input discrepancy. Substituting (35) into (34) leads to _z l + k ls z l = z l+1 z l 1 + ff ls2 ~ T ffi l ψl T " + ~ l 1 _^ ; l 1 ' j + ' l + T ' l and ~ l where ffi = P l 1 l From (33) and (36), its time derivative is _V l = z l z l+1 + lx = P l l 1 j + l. Choose V l = V l z2 l. f k js z 2 j + z j (ff js2 ~ T ffi j ψj T " + ~ j 1 _^ z j g; (37) 9

10 where ψ j = 0; 8j < l. Noting that j ~ lj» ffi ~ l (μx l ;t) = P l 1 l 1 jffi j + ffi l, the ARC design [14] can be applied to synthesize a robust control function ff ls2 satisfying the following two conditions i. z l (ff ls2 ~ T ffi l ψl T " + ~ l)» ffl l (1 + ρ 2 ) ii. z l ff ls2» 0 (38) where ffl l is a positive design parameter which represents the level of attenuation, and ρ(t) = fl 1 1 (2D(t)). Remark 5 One smooth example of ff ls2 satisfying (38) can be found in the following way. Let h l be any smooth function satisfying h l j M jjffi l j + jψ l jfl 1 1 (2r)+~ ffi l ; (39) where M = max min. Then, ff ls2 can be chosen as ff ls2 = 1 4ffl l (h 2 l + jψ l j 2 )z l : (40) It is obvious that Condition ii of (38) is verified. From (24), it follows that z l (ff ls2 ~ T ffi l ψl T " + ~ l)» z l ff ls2 + jz l jj M jjffi l j + jz l jjψ l jfl 1 1 (2r)+jz ljjψ l jρ + jz l j ffi ~ l = ( 1 2 p h ffl l l jz l j p ffl l ) 2 ( 1 2 p jψ ffl l l jjz l j p ffl l ρ) 2 + ffl l + ffl l ρ 2» ffl l (1 + ρ 2 ) (41) Condition i of (38) is thus satisfied. Other smooth or continuous examples can be worked out in the same way as in [13, 15, 12]. } Step i (l +1» i<n) >From (35) and (1), where _ff i 1 = P i i 1 (x j+1 + T ' j + ' T j + i 1 i P i 1 j j i 1 _^ i 1 = _ff (i 1)c + _ff (i 1)u _ff (i 1)c = P i i 1 i 0 _ff (i 1)u = P i 1 fx j+1 + ^ T ' j + ' T j ( 0 + ^ +!(μx l ))g i 1 0 _ + P i 1 j j i 1 f ~ T ' j ' T j ( ~ + ")+ jg i 1 _^ In (42) and (43), noting (9) and (22), _ff (i 1)c is calculable and can be used in the design of control functions, but _ff (i 1)u cannot due to various uncertainties. Therefore, it has to be dealt with via robust feedback in this step design. The details are given below. To make the development general, mathematical induction will be used to prove the general results for all intermediate step designs. At step i, the ARC design similar to that used in step l will be employed to (43) 10

11 construct a control function ff i for x i+1. Let z i = x i ff i 1 and recursively define the following functions ffi i = P P i i 1 i 1 ' j ~ i = P i i 1 j + i; ψ i = P i i 1 ' j + ' i i 1 T ' j + T ' i + ' i ; (44) Lemma 3 At step i, 8 l +1» i<n, choose the desired control function ff i as ff i (μx i ;r; 0 ; ;^ ; t) =ff ia + ff is ; ff ia = z i 1 ^ T ' i ' T i f 0 + ^ +!(μx l )g + _ff (i 1)c ; ff is = ff is1 + ff is2 ; ff is1 = k is z i ; k is g i + i 1 C i j 2 + jc ffii ffi i j 2 + c jψ i j 2 ; where g i > 0 are constants, C i and C ffii are positive definite constant diagonal matrices, and ff is2 satisfies (45) i. z i (ff is2 ~ T ffi i ψi T " + ~ i)» ffl i (1 + ρ 2 ) ii. z i ff is2» 0 (46) Then the i-th error subsystem is _z i = z i+1 z i 1 k is z i +(ff is2 ~ T ffi i ψi T " + ~ i 1 _^ ; (47) and the derivative of the augmented p.s.d. function V i = V i z2 i (48) satisfies _V i = z i z i+1 + ix f k js zj 2 + z j (ff js2 ~ T ffi j ψj T " + ~ j 1 _^ z j g: (49) Proof: It is easy to check that the steps l +1and l +2 satisfy the Lemma. So let us assume that the lemma is valid for step j, 8j» i 1, and show that it is also true for step i to complete the induction process. From (44), we have j ~ i j» ~ ffi i = Xi 1 i 1 jffi j + ffi i : (50) Since ~ ffi i is known, there exist ff is2 (μx i ;r; 0 ; ;^ ; t) satisfying (46) as in the step l design. The control law (45) can then be formed. We express the derivative ofz i as _z i = _x i _ff i 1 = x i+1 P + T ' i + ' T i ( 0 + +!(μx l ) ")+ i _ff (i 1)c i i 1 f ~ T ' j ' T j ( ~ + ")+ i 1 _^ (51) Substituting x i+1 = z i+1 + ff i, (45) and (44) into (51), it is straightforward to verify that (47) and (49) are satisfied for i. This completes the induction process. 2 11

12 4.2.4 Step n This is the final design step. By letting u = x n+1, the last equation of (1) has the same form as the intermediate equations l +1» i» n 1. Therefore, the general form (44)ο(49) applies to Step n. Since u is the actual control input, we can choose it as where ff n is given by (45). u = ff n (x; r; 0 ; ;^ ; t); (52) Theorem 1 Let the parameter estimates be updated by the adaptation law (25) in which fi is chosen as fi = ffi j z j : (53) Let c ji and c ffiki be the ith diagonal elements of the diagonal matrices C j and C ffik respectively. If the P controller parameters C j and C ffik are chosen such that c 2 ffiki n nj=2 1 4 c 2 ; 8k; i, then, the control law ji (45) guarantees that A. In general, the control input and all internal signals are bounded. Furthermore, V n is bounded above by Z V n (t)» exp( n t)v n (0) + ffl [1 exp( n t)] + ffl t exp[ n (t ν)]ρ 2 (ν) dν; (54) n n where n = 2 minfg 1 ; ;g n g and ffl = P n ffl j. Noting that ρ(t) = 0 for all t T 0, V n (t) is ultimately bounded by V n (1)» ffl n : (55) B. If after a finite time t f, μ = 0 and i = 0, i.e., in the presence of parametric uncertainties and dynamic uncertainties only, then, in addition to results in A, asymptotic output tracking (or zero final tracking error) is also achieved. 4 0 Proof of the Theorem is given in Appendix. Remark 6 Results in A of Theorem 1 indicate that the proposed controller has an exponentially converging transient performance with the exponentially converging rate n and the final tracking error being able to be adjusted via certain controller parameters (g j and ffl j ) freely in a known form. Theoretically, this result is what a well-designed robust controller can achieve. In fact, when the parameter adaptation law (25) is switched off, the proposed ARC law becomes a deterministic robust control law and Results A of the Theorem remain valid as in [13, 15]. Result B of Theorem 1 implies that the proposed controller is able to make full use of the nominal structure of the system and eliminates the effect of parametric uncertainties and dynamic uncertainties through certain parameter adaptation laws. As a result, asymptotic output tracking is obtained without using infinite-gain feedback (i.e., none of the nonlinear gains used in the control law approaches infinity as time goes infinity). Theoretically, Result B is what a well-designed adaptive controller can achieve. } 12

13 Remark 7 It is seen from (54) that the transient tracking error is affected by the initial value V n (0) also. To further reduce transient tracking error, the idea of filter initialization [5, 15] can be used to render V n (0) = 0. } 5 Design Example and Comparative Simulation Results In order to illustrate the above ARC algorithm and compare it with the previous RAC design algorithms, simulation results are obtained for the following simple example which is used in [20]: _ = + x μ ; (56) _x 1 = x 2 + x ; (57) _x 2 = u; (58) y = x 1 : (59) where is an unknown constant parameter, μ and 1(t) are two unknown bounded disturbances and is the unmeasured state. For comparison purpose, we take exactly the same simulation values as in [20] for, μ and 1(t), i.e., =0:1, μ =0:5 and 1(t) =0:6 sin(2t). The bounds describing the uncertain ranges in (2) are Ω =( 1; 3), μ ffi =2and ffi 1 =2. The control objective is to track a desired trajectory x d (t). Let ο =!(x 1 ), with! = 1+A 2 x 1 and A is a negative design parameter. Then, the observer design in Section 3 can be applied. Specifically, the filters are implemented as _ 0 = A 0 + A!(x 1 )+x 2 The equivalent expression for the unmeasured state z 1 x 2 ; (60) _ 1 = A x 2 ; (61) 1 = !(x 1 ) "; (62) where " is the state estimation error which is governed by the following dynamic system _" = A" μ 1+A 1(t): (63) 2 Taking V " = 1 2 "2, we see that (12) is satisfied with fl " 0. Thus Assumption 3 is satisfied, and the general design procedure in Section 4 can be applied to obtain an ARC controller. Furthermore, since " is bounded above by an unknown constant as seen from (63), the dynamic normalization signal r is not needed and the resulting simplified ARC controller is given below: Step 1 z 1 = x 1 x d ; (64) ffi 1 = x ; (65) 13

14 ff 1a = ^ x 2 1 2( ^ +!(x1 )) + _x d ; (66) ff 1s1 = [g 1 +(C ffi1 ffi 1 ) 2 ]z 1 ; (67) ff 1s2 = [( M ffi 1 ) 2 + ffi ]z 1 =(4ffl 1 ); (68) ff 1 = ff 1a + ff 1s1 + ff 1s2 ; (69) Step 2 z 2 = x 2 ff 1 ; ffi 2 1 (x ); (71) ff 2a = z (x 2 + ^ x ^ +2!)+ _ 0 1 _ 1 ; (72) ff 2s1 = [g 2 1 (70) C 2) 2 +(C ffi2 ffi 2 ) 2 ]z 2 ; (73) ff 2s2 = [( M ffi 2 ) 2 1 ) 2 (ffi )]z 2 =(4ffl 2 ); (74) u = ff 2a + ff 2s1 + ff 2s2 : (75) The parameter adaptation law is given by The following two cases are considered: Case 1: Set-Point Regulation _^ = Proj^ ( fi); fi = ffi 1 z 1 + ffi 2 z 2 : (76) By setting x d 0, trajectory tracking comes to the set-point regulation problem considered in [20]. For the sake of easy comparison, the ARC scheme is adjusted to employ about the same degree of control effort as that of the RAC scheme [20] for the same initial conditions. The design parameters and initial conditions for ARC are given as follows: (0) = x 1 (0) = x 2 (0) = 1; 0 (0) = 1 (0) = 0; ^ (0) = 0:5; A = 10; g 1 = g 2 =10; =1; C ffi1 = C ffi2 =0:25; (77) C 2 =2; ffl 1 =30; ffl 2 = 300: The plots in Figures 1 and Figure 2 show that the ARC scheme achieves a better performance without using large control input. The parameter estimates of both controllers are given in Figure 3. It is seen that both estimates do not converge to the true value. The reason is that the persistent excitation condition is not satisfied. Case 2: Trajectory Tracking To test the tracking capability of the proposed algorithm, a sinusoidal desired trajectory given by x d = 0:5(1 cos(1:4ßt)) is used. All design parameters and initial conditions remain unchanged except (0) = 0 and x 1 (0) = x 2 (0) = 0. The following two conditions will be considered: (i) No Disturbance 14

15 To test the nominal performance of the two controllers, simulation is first run for the system without uncertain nonlinearities, i.e., μ = 1 = 0 is used in the simulation. It is seen from Figure 4 that the ARC scheme has a much better transient performance and final tracking accuracy, which agrees with the theoretical results obtained in Theorem 1. Parameter estimation is given in Figure 5, and it shows that the parameter estimate of ARC converges to the true value. However, the parameter estimate of RAC does not converge because of the dynamic uncertainties. It also can be seen from Figure 6 that the state estimate ^ converges to the true value. (ii) With Disturbance To test the performance robustness of the controllers to uncertain nonlinearities, the assumed disturbances are used in the simulation, i.e., μ =0:5 and 1(t) =0:6 sin(2t). The tracking errors of both controllers are shown in Figure 7 with the control inputs shown in Figure 8. The plots in Figures 9 shows that the state estimate does not converge, but it is robust with respect to the disturbances. 6 Conclusions In this paper, an observer based adaptive robust control (ARC) scheme is presented for a class of nonlinear systems in an extended semi-strict feedback form, in which the unmeasured states enter the system equations in an affine fashion. The form allows for parametric uncertainties, uncertain nonlinearities, and dynamic uncertainties due to the unmeasured states. In contrast to other existing robust adaptive control (RAC) schemes, the proposed ARC scheme utilizes the structural information of the unmeasured state dynamics to construct a nonlinear observer to recover the unmeasured states. By doing so, in the absence of uncertain nonlinearities, the effect of dynamic uncertainties associated with the unmeasured states is eliminated and an improved performance is obtained asymptotic output tracking is achieved in the presence of both parametric uncertainties and the unmeasured states. In addition, the state estimation errors and uncertain nonlinearities are handled effectively via certain robust feedback to achieve a guaranteed robust performance. References [1] V. I. Utkin, Sliding modes in control optimization. Springer Verlag, [2] M. J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems," IEEE Trans. on Automatic Control, vol. 26, no. 10, pp , [3] A. S. I. Zinober, Deterministic control of uncertain control system. London, United Kingdom: Peter Peregrinus Ltd., [4] Z. Qu, Robust control of nonlinear uncertain systems under generalized matching conditions," Automatica, vol. 29, no. 4, pp , [5] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and adaptive control design. New York: Wiley, [6] R. Marino and P. Tomei, Global adaptive output-feedback control of nonlinear systems, part i: Linear parameterization; part ii: nonlinear parameterization," IEEE Trans. on Automatic Control, vol. 38, no. 1, pp ,

16 [7] J. B. Pomet and L. Praly, Adaptive nonlinear regulation: estimation from the lyapunov equation," IEEE Trans. on Automatic Control, vol. 37, pp , [8] J. S. Reed and P. A. Ioannou, Instability analysis and robust adaptive control of robotic manipulators," IEEE Trans. on Robotics and Automation, vol. 5, no. 3, [9] M. M. Polycarpou and P. A. Ioannou, A robust adaptive nonlinear control design," in Proc. of American Control Conference, pp , [10] R. A. Freeman, M. Krstic, and P. V. Kokotovic, Robustness of adaptive nonlinear control to bounded uncertainties," in IFAC World Congress, Vol. F, pp , [11] Z. Pan and T. Basar, Adaptive controller design for tracking and disturbance attenuation in parametricstrict-feedback nonlinear systems," in IFAC World Congress, Vol.F,, pp , [12] B. Yao, High performance adaptive robust control of nonlinear systems: a general framework and new schemes," in Proc. of IEEE Conference on Decision and Control, pp , [13] B. Yao and M. Tomizuka, Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance," in Proc. of American Control Conference, pp , The full paper appeared in ASME Journal of Dynamic Systems, Measurement and Control, Vol. 118, No.4, pp , [14] B. Yao and M. Tomizuka, Adaptive robust control of nonlinear systems: effective use of information," in Proc. of 11th IFAC Symposium on System Identification, pp , (invited). [15] B. Yao and M. Tomizuka, Adaptive robust control of siso nonlinear systems in a semi-strict feedback form," Automatica, vol. 33, no. 5, pp , (Part of the paper appeared in Proc. of 1995 American Control Conference, pp ). [16] B. Yao, F. Bu, J. Reedy, and G. Chiu, Adaptive robust control of single-rod hydraulic actuators: theory and experiments," IEEE/ASME Trans. on Mechatronics, vol. 5, no. 1, pp , [17] B. Yao, M. Al-Majed, and M. Tomizuka, High performance robust motion control of machine tools: An adaptive robust control approach and comparative experiments," IEEE/ASME Trans. on Mechatronics, vol. 2, no. 2, pp , (Part of the paper also appeared in Proc. of 1997 American Control Conference ). [18] B. Yao and M. Tomizuka, Comparative experiments of robust and adaptive control with new robust adaptive controllers for robot manipulators," in Proc. of IEEE Conf. on Decision and Control, pp , [19] R. A. Freeman and P. V. Kokotovic, Tracking controllers for systems linear in the unmeasured states," Automatica, vol. 32, no. 5, pp , [20] Z. P. Jiang and L. Praly, Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties," Automatica, vol. 34, no. 7, pp , [21] C. C. de Wit, H. Olsson, K. J. Astrom, and P. Lischinsky, A new model for control of systems with friction," IEEE Trans. on Automatic Control, vol. 40, no. 3, pp , [22] C. J. Wan, D. S. Bernstein, and V. T. Coppola, Global stabilization of the oscillating rotor," in Proc. of IEEE Conf. on Decision and Control, pp , [23] Z. P. Jiang and L. Praly, Preliminary results about robust lagrange stability in adaptive regrlation," Int. J. Adaptive Control Signal Process, vol. 6, pp , [24] Y. Tan, I. Kanellakopoulos, and Z. P. Jiang, Nonlinear observer/controller design for a class of nonlinear systems," in Proc. IEEE Conf. on Decision and Control, pp , [25] Z. P. Jiang, A. Teel, and L. Praly, Small-gain theorem for iss systems and applications," Mathematics of Control Signals Systems, no. 7, pp , [26] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ 07632, USA: Prentice Hall, Inc.,

17 [27] G. C. Goodwin and D. Q. Mayne, A parameter estimation perspective of continuous time model reference adaptive control," Automatica, vol. 23, no. 1, pp , Appendix Proof of Theorem 1: Noting z n+1 = 0, from (49), (30), (38) and (46) _V n» By completion of square f (g j + j 1 C j j 2 + jc ffij ffi j j 2 + c jψ j j 2 )z 2 j + z j (ff js2 ~ T ffi j ψ T j " j 1 ~ j) z j _^ g (78) j=2 z j 1 _^» j=2 jz j j 1 C j C 1 j _^ j» j 1 j=2 Noting that C 1 j and are diagonal matrices, from (25) and (26), we have j=2 jc 1 j _^ j 2 = j=2 jc 1 j Proj^ ( fi)j 2» j=2 jc 1 j fij2» ( j=2 k=1 jc 1 j ffi kz k j) 2» n Thus, if C j and C ffik satisfy the conditions in the theorem, from (79) and (80), j=2 z j 1 _^» j=2 j 1 >From (81) and i of (46), (78) becomes, _V n» C j j 2 z j 2 + n 4 k=1 jc 1 j ffi kj 2 z 2 k)» j=2 j 1 C j j 2 z 2 j jc 1 j _^ j 2 ) (79) ( j=2 k=1 C j j 2 z j 2 + jc 1 j ffi kj 2 z 2 k) (80) k=1 jc ffik ffi k j 2 z 2 k (81) f g j z 2 j + ffl j + ffl j ρ 2 (t)g» n V n + ffl + fflρ 2 (t) (82) which leads to (54) and (55). The boundedness of z 1,, z n is thus proved. Using the standard arguments in the backstepping designs [5, 12], it can be proved that all internal signals in the first l 1 steps are globally uniformly bounded. Furthermore, from x l = z l + ff l 1, it follows that x l is bounded. The boundedness of signals x 1 ; ;x l, together with the bounded-input-bounded-state Assumption 2 for the dynamics and Assumption 3 for the observer error dynamics, implies that, ", and r are bounded. Thus, all filter states 0, will be bounded. Thus, recursively using the fact that x i = z i + ff i 1, it is easy to verify that all intermediate control functions ff i and states x i are bounded. From (52) and (45), the boundedness of u is apparent. A of the Theorem 1 is thus proved. In the absence of uncertain nonlinearities, i.e., μ = 0 and i = 0, noting condition ii of (30), (38) and (46), from (78) and (81) Define a new p.s.d. function V as _V n» Noting P2 of (27), from (83), the derivative ofv satisfies ( g j z 2 j c jψ j j 2 z 2 j ~ T ffi j z j z j ψ T j ") (83) V = V n ~ T 1 ~ (84) P n _V» P f g jzj 2 c jψ j j 2 zj 2 z j ψj T "g ~ T fi + ~ T 1 _^ n» P f g jzj 2 c jψ j j 2 zj 2 + jz jjjψ j jj"jg (85) n» g jzj 2 + n 4c j"j 2 By B of Assumption 3, "(t) exponentially converges to zero, and thus "(t) 2L 2 [0; 1). From (85), it is easy to prove that z j (t) 2L 2 [0; 1). It is also easy to check that _" and _z j are bounded. Hence, by the Barbalat's lemma, z! 0ast! 1, which leads to B of Theorem

18 Solid: ARC Dashed: RAC Output Time (sec) Figure 1: Set-point regulation in the presence of various model uncertainties Solid: ARC Dashed: RAC Control Input Time (sec) Figure 2: Control inputs in the presence of various model uncertainties Parameter Estimate Solid: ARC Dashed: RAC Time (sec) Figure 3: Parameter estimation of ARC and RAC 18

19 Solid: ARC Dashed: RAC Tracking Error Time (sec) Figure 4: Tracking errors in the absence of uncertain nonlinearities 0.9 Parameter Estimate Solid: ARC Dashed: RAC Time (sec) Figure 5: Parameter estimation in the absence of uncertain nonlinearities Solid: η Dashed: estimate of η State Estimation Time (sec) Figure 6: State estimation in the absence of uncertain nonlinearities 19

20 Solid: ARC Dashed: RAC Tracking Error Time (sec) Figure 7: Tracking errors in the presence of various model uncertainties Control Input Solid: ARC Dashed: RAC Time (sec) Figure 8: Control inputs in the presence of various model uncertainties Solid: η Dashed: estimate of η 1.1 State Estimation Time (sec) Figure 9: State estimation in the presence of various model uncertainties 20