Adaptive Robust Control of MIMO Nonlinear Systems in Semi-Strict Feedback Forms Λ Bin Yao + and Masayoshi Tomizuka ++ + School of Mechanical Engineeri

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1 Adaptive Robust Control of MIMO Nonlinear Systems in Semi-Strict Feedback Forms Λ Bin Yao + and Masayoshi Tomizuka ++ + School of Mechanical Engineering Purdue University West Lafayette, IN 47907, USA ++ Mechanical Engineering Department University of California at Berkeley Berkeley, CA 9470, USA Abstract The paper considers the construction of adaptive robust controllers for a class of multi-input multioutput (MIMO) nonlinear systems transformable to two semi-strict feedback forms The forms can have both parametric uncertainties and uncertain nonlinearities such as modeling errors and external disturbances In addition, the forms allow coupling and appearance of parametric uncertainties in the input matrix of each layer Furthermore, the usual assumption on the linear parametrization of the state equations is relaxed to the extent that the forms are applicable to the control of some mechanical systems To deal with the complexity and difficulties caused by the strong coupling and the appearance of parametric uncertainties in the input matrices, the concept of adaptive robust control (ARC) Lyapunov functions is first introduced to formalize the recently proposed ARC approach Such a formulation systematizes and simplifies the ARC controller design, and allows some systematic design procedures such as the backstepping design to be employed to enlarge the class of applicable nonlinear systems Two backstepping designs via ARC Lyapunov functions are presented The results are then used to construct specific ARC control laws for MIMO nonlinear systems in the semi-strict-feedback forms By using trajectory initialization, the resulting ARC law achieves a guaranteed output tracking transient performance and final tracking accuracy in general, while keeping all physical states and control inputs bounded In addition, the control law achieves asymptotic output tracking in the presence of parametric uncertainties without using a discontinuous or infinite-gain feedback term Applications include the robust control of robot manipulators in various applications such as the constrained motion and force tracking control, coordinated motion and force tracking control of multiple robots grasping a common object, and motion and force tracking control of robot manipulators in contact with stiff surfaces with unknown stiffness approach Simulation results are presented to illustrate the Keyword Adaptive Control; Robust Control; Sliding Mode Control, Nonlinear System, Uncertain Dynamic System Λ Parts of the paper were presented at the 1995 IEEE Conference on Decision and Control and at the 1996 IFAC World Congress The work is supported in part by the National Science Foundation under the CAREER grant CMS

2 I Introduction The following two types of uncertainties are of major concern in the control of uncertain nonlinear systems: parametric uncertainties (eg, gravitational load for robots) and general uncertainties coming from modeling errors (eg, ignored nonlinear friction) and external disturbances, which are referred to as uncertain nonlinearities or unknown nonlinear functions in this paper To account for these uncertainties, two nonlinear control methods have been popular: adaptive control [1,, 3, 4, 5] and deterministic robust control [6, 7, 8, 9, 10, 11, 1] The adaptive control achieves asymptotic tracking for reasonably large classes of nonlinear systems without using discontinuous or infinite-gain feedback [1] terms However, adaptive controllers only deal with the ideal case of constant parametric uncertainties and the adaptation law may lose stability even when a small disturbance appears [13] Every physical system is subject to some form of disturbance Additional effort has to be made to safely implement such adaptive nonlinear controllers One may apply remedies similar to those used in robust adaptive control of linear systems [14, 15] However, although asymptotic tracking is still preserved in the absence of disturbances, such modifications [13, 16] do not guarantee tracking accuracy in the presence of disturbances since the steady state tracking error can only be shown to stay within an unknown region, whose size depends on the disturbances Furthermore, transient performance is unknown In contrast, the deterministic robust control eg, sliding mode control [6]-can be used to achieve a guaranteed transient performance and a guaranteed final tracking accuracy in the presence of both parametric uncertainties and uncertain nonlinearities However, it usually involves switching [6] or infinite-gain feedback [9] terms, which introduces control chattering Chattering may be avoided at the expense of degraded tracking performance by using some smoothing techniques [10, 17] In [17], we presented a systematic way to combine the adaptive control and the sliding mode control (SMC) for the trajectory tracking control of robot manipulators to preserve the advantages of the two methods while overcoming their drawbacks 1 Comparative experimental results for the motion control of robot manipulators [18] and the high-speed/high-accuracy trajectory tracking control of machine tools [19] have demonstrated the substantially improved performance of the suggested adaptive robust control (ARC) approach In the motion control of rigid robots [18], the design was for multivariable nonlinear differential equations with relative degree of one In [0], the methodology was extended to a class of single-input single-output (SISO) nonlinear systems with arbitrary known "relative degrees" in a semi-strict feedback form [16] by combining the backstepping adaptive control [] with the general deterministic robust control Recently, other researchers have also approached the control of SISO nonlinear systems in the semistrict feedback form from various perspectives and excellent results have been obtained Specifically, in [1], Santosuosso considered the regulation problem of SISO nonlinear systems in a semi-strict feedback form; aside from zero-dynamics, the form in [1] has the same structure as that studied in [0] but the allowable disturbance is restricted to the class of L -norm bounded disturbances In [], Pan and Basar cast the problem of robust adaptive control of SISO nonlinear systems in the framework of nonlinear H 1 optimal control, where 1 In this paper, the terminology of adaptive robust control (ARC) is used to represent such a combined design approach; the use of this terminology is to differentiate the approach from conventional robust adaptive control approach for reasons which will become clear later 1

3 specific measures of asymptotic tracking, transient behavior, and disturbance attenuation were all incorporated into a single cost functional, and an explicit solution was presented The presented algorithm suffers from the over-parametrization problem and gives the H 1 -gain that relates the L norm of the tracking error tothel norm of the disturbance only; in other words, transient performance in terms of L 1 norms of the tracking error and the disturbance is not clear In [3], Freeman, Krstic, and Kokotovic robustify the well-known backstepping nonlinear adaptive control algorithms developed in [1] for bounded uncertainties/disturbances The L 1 =L estimates were also given on the effects of bounded uncertainties/disturbances on the tracking error a much stronger performance result than those achieved in the conventional robust adaptive control schemes [16, 13] It is noted that the same strong performance results are also obtained in a recent paper by Marino and Tomei [4] It should be realized that there are some subtle but fundamental differences between the proposed adaptive robust control (ARC) approach [17, 0] and the tuning function based robust adaptive control (RAC) approach [3] Firstly, in terms of fundamental view point, the proposed ARC [17, 0] puts more emphasis on the underline robust control law design in achieving a guaranteed output tracking robust performance In fact, the parameter adaptation law in ARC [17, 0] can be switched off at any time without affecting system stability and sacrificing the guaranteed output tracking transient performance since the resulting controller becomes a deterministic robust controller Secondly, in terms of the achievable performance, in the proposed ARC [17, 0], the upper bound on the absolute value of the output tracking error over entire time-history is given and is related to certain controller design parameters in a known form, which is more transparent than that in RAC [3, 16] Finally, in terms of specific approaches used for the controller design and the proof of achievable performance, the proposed ARC uses two Lyapunov functions; one the same as that in the deterministic robust control [6, 7, 9, 10, 11, 1] and the other the same as that in adaptive control [1], while the robust adaptive control [3] uses the same Lyapunov function as in adaptive control [1] only Because of these subtle differences, the terminology of "adaptive robust control (ARC)" is used for the proposed combined design method to differentiate the approach from the conventional robust adaptive control approach and to reflect the strong emphasis on the robust control law design for robust performance Although a large amount of work has been carried out on the construction of backstepping adaptive or robust controllers for SISO nonlinear systems in semi-strict feedback form [1, 16, 0, 1,, 3], very few results are available for MIMO nonlinear systems In [1], Krstic, et al, considered the backstepping adaptive control of MIMO nonlinear systems in a parametric strict feedback form However, the form assumes no parametric uncertainties in the input matrix This paper concerns the systematic construction of adaptive robust controllers (ARC) for a class of MIMO nonlinear systems transformable to two semi-strict feedback forms Compared with the MIMO parametric strict feedback form studied in [1], the presented forms have the following additional complexities and design difficulties Firstly, the proposed semi-strict feedback forms allow strong coupling and appearance of parametric uncertainties in the input matrices of all intermediate layers As a result, the following new problems have to be solved: (i) At each layer, new ARC design techniques have to be developed to determine the ARC control functions for all virtual inputs simultaneously in order to attack the problem of coupling among input

4 channels, which is qualitatively different from that in [1] In [1], since input matrix assumes no parametric uncertainties, realizable nonlinear state transformations and input transformations can be constructed to decouple the coupling among input channels The resulting MIMO system essentially consists of a bunch of "uncoupled" input-output pairs, and the corresponding SISO backstepping adaptive design can be applied to each pair of input-output independently without caring about the control functions needed for other inputoutput pairs; (ii) New backstepping ARC design techniques have to be developed to deal with the appearance of parametric uncertainties in the input matrices Our early results on the adaptive robust control of SISO nonlinear systems [0] cannot handle parametric uncertainties in the input channels It is well-known that the appearance of parametric uncertainties in the input matrix normally complicates the controller design significantly Even in the absence of uncertain nonlinearities, backstepping adaptive control of the proposed MIMO semi-strict feedback forms has not been solved; (iii) Due to the complexity associated with strongly coupled uncertain MIMO nonlinear systems, even the presentation of MIMO semi-strict feedback forms in a meaningful manner becomes difficult The paper gives two typical representations of MIMO semi-strict feedback forms Both forms are concise and useful since they admit certain practically significant applications The second unique feature of the proposed two forms is that the usual requirement on the linear parametrization of the state equations is relaxed to the extent that the proposed forms are applicable to the control of some multiple degrees-of-freedom (DOF) mechanical systems For most mechanical systems, such as robot manipulators, their state equations cannot be linearly parametrized in terms of a set of unknown parameters, which prohibits the direct application of the MIMO backstepping adaptive control results in [1] Lastly, the proposed forms allow the presence of certain uncertain nonlinearities This significantly extends the applicability of the proposed approach since most physical systems have uncertain nonlinearities in one form or another To overcome the control design difficulties mentioned above, in this paper, the concept of adaptive robust control (ARC) Lyapunov functions is first introduced to formalize and systematize the proposed ARC approach Specifically, the adaptive robust control of a system is reduced to the problem of finding an ARC Lyapunov function for the system Such a formulation simplifies the controller design, especially when some systematic design procedures such as the backstepping design [1] are used to enlarge the applicable class of nonlinear systems As a result, ARC controllers can be synthesized for much more complicated nonlinear systems The formulation also makes it possible to relax the assumption on the linear parametrization of the state equations Two general backstepping designs via ARC Lyapunov functions are presented to construct specific ARC controllers The results are then used to systematically construct ARC controllers for MIMO nonlinear systems in the proposed two semi-strict feedback forms by employing an additional tool trajectory initialization Various applications will be mentioned and simulation results will be presented to illustrate the advantages of the suggested approach 3

5 II General Formulation of Adaptive Robust Control In this section, adaptive robust control (ARC) Lyapunov functions are introduced to concisely describe the objective and the achievable results of the proposed adaptive robust control method Consider the following MIMO nonlinear system _x = f(x; ; t)+b(x; ; t)u + D(x; t) (x; ; u; t) y = h(x; t) (1) where y R m and u R m are the output and input vectors respectively, x R n is the state vector, R p is the vector of unknown parameters, h(x; t);f(x; ; t);b(x; ; t), and D(x; t) R n l d are known, and (x; ; u; t) R l d represents the vector of unknown nonlinear functions such as disturbances and modeling errors The following reasonable and practical assumptions are made, which are satisfied by most applications [17, 18, 19]: AII1 It is assumed that if a function in the system dynamics depends on t explicitly (eg, f(x; ; t) in (1)), then, the function and all its partial derivatives are bounded with respect to time t; 3 a function f(x; ; t) is called bounded with respect to time t iff there exists a positive function f u (x; ) such that kf(x; ; t)k»f u (x; ); 8t } AII The extent of parametric uncertainties and uncertain nonlinearities is known, ie, Ω = f : min < < max g Ω = f : k (x; ; u; t)k»ffi(x; t) g () where min ; max and ffi(x; t) are known } 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 Let ^ffl denote the estimate of ffl (eg, ^ for ) For any unknown parameter vector ffl lying in a known bounded region Ω ffl =fffl : ffl min < ffl < ffl max ; g (eg, Ω ), a smooth projection map ß ffl can be defined for ^ffl and has the following properties: P1 8^ffl Ω ffl ; ß ffl (^ffl) =^ffl P 8^ffl, ß ffl (^ffl) Ω^ffl =fμ : ffl min " ffl» μ» ffl max + " ffl g where " ffl is a known vector of positive numbers which can be arbitrarily small P3 ß ffli (^ffl i ) is a nondecreasing function of ^ffl i P4 The derivatives of the projection are bounded up to a sufficiently high-order, ie, 8j» n, ß ffl (j) (^ffl) is bounded See [0] for further details of the smooth projection For convenience, define ^ffl ß as ^ffl ß = ß ffl (^ffl) and the projected estimation error as~ffl ß = ^ffl ß ffl For any index j, let μffl (j) ß denote μffl (j) ß =[ß(^ffl) T ;:::;ß (j) (^ffl) T ] T Let y d (t) R m be the desired output trajectories, and the output tracking errors as e y = y y d (t) The objective of the proposed adaptive robust control is to construct a bounded control law such that, under the Avector or matrix is known if all its elements are known functions with respect to their variables 3 Noted that this Assumption is different from the assumption that all system functions are bounded 4

6 assumption (), the system state x is bounded and the output tracking possesses certain desirable features Specifically, the control law to be sought consists of two parts and is given by u(x; μ (lu) ß ;t)=u a (x; μ (lu) ß ;t)+u s (x; μ (lu) ß ;t) (3) where l u is an index, u a functions as an adaptive control law and u s a robust control law to be designed within an allowable set Ω u The control functions u a and u s and an adaptation law are determined to make a positive semi-definite (psd) function an ARC Lyapunov function defined as follows Definition 1 Let V (x; ; μ (l V ) ß ;t) be a psd function with continuous partial derivatives (l V is any index) V is called an adaptive robust control (ARC) Lyapunov function for (1) if it satisfies the following three requirements for some continuous control functions u a (x; μ (lu) ß fi(x; μ ß (lr) ;u;t) ;t) and u s (x; μ ß (lu) ;t) and an adaptation function R1 Bounded V means bounded state x, and guaranteed transient performance of V (t) is equivalent to the guaranteed transient performance of output tracking error e y (eg, V (t)! 0 means e y! 0 or guaranteed exponential convergence of the upper bound of V (t) means guaranteed exponential convergence of the upper bound of je y j) 4 R There exists a continuous control law u a such that 8u s Ω u [f(x; ; t)+b(x; ; a + u s W (x; ; μ (lr) ß ;t)+ ~ T ß fi(x; μ (lr) ß fi (4) or, equivalently, _ V j =0» W + ~ T ß fi (_^ + fi) (5) where l r = maxfl V +1;l u g, fi(x; μ ß (lr) ;u;t) is a known function, V _ j =0 represents the derivative of V when =0, and W (x; ; μ ß (lr) ;t) is any continuously differentiable psd function and satisfies the condition that asymptotic convergence of W means asymptotic output tracking (ie, W! 0 =) e y! 0) R3 There exists a u s Ω u such that 8 Ω and 8 Ω : or, equivalently, [f + B(u a + u s )+D (x; ; u; @V fi(x; μ (lr) ß ;u;t)» V V + c V (t) (6) _V» V V + c V (t) (_^ + fi) (7) > 0 and c V (t) is a bounded positive scalar, ie, 0» c V (t)» c V max It is further required that both V and c V can be adjusted freely by certain controller parameters in a known form without affecting V (0), the initial value of V } 4 For simplicity of notation, ffl! 0 means that ffl! 0ast! 1 5

7 Remark 1 Introduction of the above ARC Lyapunov function concept summarizes the type of control laws and the type of responses that the proposed ARC approach is looking for Specifically, Requirement R1 converts the stability and output tracking performance of the nonlinear system (1) into the study of the stability and performance of the scalar function V, which is much easier to handle Requirement R guarantees that there exists an adaptive control law to achieve asymptotic output tracking in the presence of parametric uncertainties as shown in the following theorem Requirement R3 states that a robust control law can be synthesized to attenuate the effect of both parametric uncertainties and uncertain nonlinearities to achieve a guaranteed output tracking transient performance as well as final tracking accuracy 4 Theorem 1 If there exists an ARC Lyapunov function V for (1), then, by using the control law (3) and the adaptation law the following results hold: _^ = fi(x; μ ß (lr) ;u(x; μ ß (lu) ;t);t) (8) A In general, the control input and the system state are bounded with V bounded above by V (t)» exp( V t)v (0) + R t exp( 0 V (t ν))c V (ν)dν» exp( V t)v (0) + c V max [1 exp( V V t)] (9) Output tracking is guaranteed to have arbitrary good transient performance and final tracking accuracy in the sense that both V, the exponentially converging rate, and c V max, the bound on V (1), can be V freely adjusted via certain controller parameters in a known form without affecting V (0) B If, after a finite time, there are no uncertain nonlinearities, ie, (x; ; u; t) =0; 8t t 0 ; for some finite t 0, then, in addition to the results in A, asymptotic output tracking is achieved 4 Proof of Theorem 1 See Appendix A Remark In the absence of parameter adaptation (ie, =0), the proposed ARC law reduces to a DRC law and Result A of Theorem 1 still holds Therefore, the adaptation loop can be switched off at any time without affecting the stability and the guaranteed output tracking transient performance However, such a control law does not discriminate the difference between parametric uncertainties and uncertain nonlinearities as in [6, 7, 8, 9, 11, 1] and results in a conservative design since Result B of Theorem 1 is lost As for adaptive control [1,, 3, 4, 5], the proposed ARC uses certain coordination mechanisms (eg, smooth projection) and robust feedback control u s to achieve a guaranteed output tracking transient performance even in the presence of uncertain nonlinearities (A of Theorem 1) while without losing its nominal performance (B of Theorem 1) 4 Corollary 1 If there exists an ARC Lyapunov function V (x; ; t), which is not a function of ^, then, by using the control law (3) and the modified adaptation law _^ = [l (^ ) +fi(x; μ ß (lr) ;u(x; μ ß (lu) ;t);t)] (10) 6

8 where l (^ ) is any vector of functions satisfying the following conditions i l (^ ) =0 if ^ Ω ii ~ T ß l (^ ) 0 if ^ 6 Ω (11) we have the results in Theorem 1 4 Proof of Corollary 1 See Appendix B Remark 3 The reason for using (10) is that by suitably choosing l (^ ), the parameter estimation process can be made more robust and the boundedness of ^ can be guaranteed since l (^ ) acts like a nonlinear damping term In fact, when a discontinuous modification law l (^ ) is allowed as in the applications studied in [17, 19], the widely used projection method in adaptive systems [7, 8] can be employed since it is shown in [17, 9] that it satisfies (11) For details, see [9] Some continuous modifications are also given in [17] by generalizing ff-modification [13] } III MIMO Semi-Strict Feedback Forms In the previous section, the proposed ARC approach is presented via ARC Lyapunov functions, just like the stability analysis of nonlinear systems via Lyapunov functions Thus, the remaining problem is to construct specific ARC Lyapunov functions for aparticular application to come out specific control laws and adaptation laws, which is the focus of the rest of the paper In particular, specific ARC Lyapunov functions will be constructed for MIMO nonlinear systems transformable to the two semi-strict feedback forms introduced below III1 MIMO Semi-Strict Feedback Form I The proposed MIMO semi-strict feedback form is an inter-connection of r subsystems, which is shown in Fig1 and described by _x i = f i0 (μχ i ;t)+f i (μχ i ;t) + B i (μχ i ; ;t)μx i+1;m i + D i(μχ i ;t) i(χ; ; t) _ i = Φ i0 (μχ i ;t)+φ i (μχ i ;t) ; 1» i» r 1 _x r = M 1 (μχ r 1 ;fi;t)[f r0 + F r + F fi (χ; t)fi + B r (χ; ; fi; t)u + D r (χ; t) r] _ r = Φ r (χ; ; fi; t) y = [y1b T ;:::;yt rb ]T R m (1) where x i R m i ; i R n i, χ i =[x T i ; T i ]T, μχ i =[χ T 1 ; :::;χt i ]T, and χ = μχ r In (1), fi R l fi is a vector of some unknown parameters, which satisfies the same assumption as in (), ie, fi Ω fi = ffi : fimin < fi < fi max g where Ω fi is a known set The form (1) is obtained as follows For all i, the i-th subsystem is a MIMO nonlinear system with the state vector χ i, the input vector v i R m i, and the output vector x i, where it is assumed that 0 = m 0» m 1» m» :::» m r = m The (i +1)-th subsystem is connected to the i-th subsystem in the following way: the first m i outputs of the i +1-th subsystem are connected to 7

9 the inputs of the i-th subsystem (ie, v i = μx i+1;m i = U i+1x i+1 ), and the remaining outputs of the i +1-th subsystem, y i+1b = Ni+1 T x i+1, become the i +1-th block of the system outputs, where U i+1 =[I m i 0] and N i+1 =[0 I m i+1 mi ]T ; 8i The first (r 1) subsystems, which are described by the first two equations of (1), have the same structure The r-th subsystem, which is described by the third and fourth equations of (1), has the additional complexity that it involves the inversion of an unknown positive definite matrix M The vectors or matrices, f i0 ; F i ; B i ; D i ; Φ i0 ; and Φ i are known functions of their variables, which include μχ i 1, the states of all its previous subsystems i is the vector of unknown nonlinear functions Inputs u r r-th SUBSYSTEM Ur r x T Nr χ -th SUBSYSTEM x U N T χ th SUBSYSTEM Outputs x 1 y y y 1b b rb Figure 1: MIMO Semi-strict Feedback Form I The following assumptions are made for the system (1): AIII1 8i» r 1; B i (μχ i ; ;t) is nonsingular for any Ω, and B r is nonsingular for any Ω and fi Ω fi AIII M is an spd matrix and there exist positive scalars k m and k M such that k m I m» M» k M I m AIII3 8i» r 1; B i (μχ i ; ;t) can be linearly parametrized by, ie, B i (μχ i ; ;t)=b i0 (μχ i ;t)+b i (μχ i ; ;t) where B i (μχ i ; ;t) is linear wrt 5 Similarly, B r (χ; ; fi; t) =B r0 (χ; t)+b r (χ; ; t)+b rfi (χ; fi; t), where B r and B rfi are linear wrt and fi respectively M = M 0 (μχ r 1 ;t)+m fi (μχ r 1 ;fi;t) in which M fi is linear wrt fi AIII4 The i -subsystem is bounded-input bounded-state (BIBS) stable wrt the input (μχ i 1 ;x i ) AIII5 There exist known functions ffi i (μχ i ;t) such that k i(χ; ; t)k» ffi i (μχ i ;t) i =1;:::;r (13) Remark 4 Assumption AIII1 is to make sure that the problem is well-posed in the sense that it guarantees that each subsystem has "relative degree" one from its input v i =μx i+1;m i to its output x i Assumption AIII is to capture the physical phenomenon that the inertia matrix M of a multi-dof mechanical system is always an spd matrix Assumption AIII4 is to assure that the internal dynamics of each subsystem in (1) is stable 5 For a matrix ffl, ffl is said to be linear wrt if all its elements are linear functions of 8

10 Assumption AIII5 is the semi-strict feedback structural assumption, which is similar to the parametric-strict feedback assumption in [1] However, this assumption is a much less restrictive structural assumption than the strict feedback assumption in [1] This is due to the fact that only the bounding functions of the uncertain nonlinearities i are required to be the function of μχ i and t, and i can contain bounded functions of χ j ;j >i and u In other words, bounded interactions among the i-th subsystem and the subsequent subsystems may be allowed, which violates the strict-feedback property } Remark 5 It is noted that the MIMO strict feedback form studied in [1] is a subset of the proposed semi-strict feedback form (1) By assuming that the input matrices of the first (r 1) subsystem are identity matrices (ie, B i = I m i mi ; 8i» r 1), the input matrix of the last subsystem assumes no parametric uncertainties (ie, B r is a function of χ only), the inertia matrix M is an identity matrix, no internal dynamics for each subsystem, and no uncertain nonlinearities (ie, assuming i =0; 8i), the proposed semi-strict feedback form (1) reduces to the MIMO strict feedback form in [1] It is thus clear that, aside from the control issues caused by the appearance of uncertain nonlinearities, the major difference between the proposed MIMO semi-strict feedback form and the MIMO strict feedback form in [1] is that the proposed form allows strong coupling and appearance of parametric uncertainties in the input matrix of each layer As a result, our previous SISO ARC design [0] cannot be generalized to solve the problem, which is qualitatively different from that in [1]; In [1], since input matrix assumes no parametric uncertainties, simple realizable input transformation like u = B r (χ) 1 v can be used to decouple the system, and the SISO backstepping adaptive design can thus be straightforwardly applied to each input-output pair to solve the problem } Remark 6 For SISO systems in the parametric-strict feedback form (ie, assuming m i = m =1, and i =0), the case of B i (μχ i ; ;t) being an unknown positive scalar b i is also studied in [1] However, over-parametrization about b i is used two parameter estimates are needed for one b i In addition, b i is assumed to be different from the unknown parameter set In contrast, our subsequent designs do not need over-parametrization for unknown parameters in the input matrices, the unknown parameters in the input matrices do not have to be different from, and the input gain function B i (μχ i ; ;t) does not have to be a constant scalar } Remark 7 In (1), in the absence of uncertain nonlinearities, in viewing the Assumption AIII3, the state equations of the first (r 1) subsystems can be linearly parametrized by the unknown parameter vector However, the state equations of the last subsystem may not be linearly parametrized by any set of unknown parameters because of the appearance of M 1 (x I ;fi;t) in the state equations; although M is assumed to be linearly parametrized in terms of fi in AIII3, in general M 1 may not be linearly parametrized Introducing M greatly expands the applicability of the method since, as shown in [9], most mechanical systems, including robot manipulators, satisfy (1) but not the usual strict feedback forms [1], where linearly parametrizing state equations is required Remark 8 In Eq (1), the output vector y is partitioned into r blocks, and the outputs of the i-th block, y ib (empty if m i = m i 1 ), have a "relative degree" 6 of r i +1 In this way, we can have relative degrees } 6 The notion of "relative degree" should not be understood in the usual sense [31] since i are uncertain and may depend 9

11 ranging from 1 to r and solve the problem that different outputs of a MIMO system may have different relative degrees } III MIMO Semi-Strict Feedback Form II In the above MIMO semi-strict-feedback form, the output y is partitioned into r blocks to generate different "relative degrees" For some applications, it may be more natural to partition the input u into r blocks, which is shown in Fig and studied in the following Let the i-th and r-th subsystems have the same form as those in subsection III1 Instead of assuming 0 = m 0 = m 1» m» :::» m r = m as in subsection III1, here, it is assumed that m = m 1 m ::: m r 0 Partition u into r blocks as u =[u T 1b ;:::;ut rb ]T where u ib R m i mi+1 Now, connect the r subsystems in the following way The first m i m i+1 inputs of the i-th subsystem are u ib and the remaining m i+1 inputs are connected to the outputs of the i +1-th subsystem The output of the entire system is the output of the first subsystem, ie, y = x 1 The entire system is thus described by _x i = f i0 (μχ i ;t)+f i (μχ i ;t) + B i (μχ i ; ;t) 4 u ib x i+1 _ i = Φ i0 (μχ i ;t)+φ i1 (μχ i ;t) ; 1» i» r Di (μχ i ;t) i(χ; ; t) _x r = M 1 (μχ r 1 ;fi;t)[f r0 + F r + F fi (χ; t)fi + B r (χ; ; fi; t)u rb + D r (χ; t) r] _ r = Φ r (χ; ; fi; t) y = x 1 R m ; and u =[u T 1b ;:::;ut rb ]T (14) Inputs u 1b χ χ 1 1 Outputs u b r u rb r-th SUBSYSTEM x r -th SUBSYSTEM x 1-th SUBSYSTEM x 1 y Figure : MIMO Semi-strict Feedback Form II The same assumptions as in subsection III1 are made Details are omitted Remark 9 Due to the complexity associated with the strongly coupled uncertain MIMO nonlinear systems, even the presentation of the system in a meaningful manner becomes difficult The proposed two semi-strict feedback forms, (1) and (14), are quite natural and are in fact motivated by various practically significant on the control input The notion should be interpreted in the way introduced in [0], which is more natural 10

12 applications One practical application of the form (1) will be detailed in section VI, and a practical application of the form (14) will be a multi-axes mechanical device (eg, machine tools) driven by different types of actuators } IV Backstepping Designs via ARC Lyapunov Functions To systematically construct ARC Lyapunov functions for the two semi-strict feedback forms, we use the backstepping design procedure [1] Namely, we assume that an ARC Lyapunov function is known for an initial system and construct a new ARC Lyapunov function for an augmented system by adding another nonlinear system to the back of the initial system The main design difficulties here are due to the strong coupling and the appearance of parametric uncertainties in the input matrices Two backstepping designs are presented in this section The results are self-contained and independent from the semi-strict feedback forms IV1 Initial MIMO Nonlinear Systems Consider the following initial MIMO system x_ I = f I0 (x I ;t)+f I (x I ;t) + B I (x I ; ;t)u I + D I (x I ;t) I y I = h I (x I ;t); u I ;y I R m I ; x I R n I; F I R n I p ; B I R n I mi (15) in which the norm of the vector of unknown nonlinear functions I is assumed to be bounded by a known function ffi I (x I ;t), ie, k Ik» ffi I (x I ;t) In addition, as in Assumption AIII3, B I (x I ; ;t) = B I0 (x I ;t)+ B I (x I ; ;t) where B I (x I ; ;t) is linear wrt The following notations will be used throughout the paper For a system matrix ffl, ^ffl is obtained by substituting the projected parameter estimates for the unknown parameters in ffl (eg, ^B I = B I (x I ; ^ ß ;t)) ~ffl refers to the estimation error offfl, ie, ~ffl = ^ffl ffl Since B I is linear wrt, there exist known matrices G Ir (x I ; ffl;t) and G Il (x I ; ffl;t), which are linear wrt ffl, such that B I (x I ; ;t)v r = G Ir (x I ;v r ;t) ; v T l B I (x I ; ;t)= T G Il (x I ;v l ;t); 8v r R m I 8v l R n I (16) G Ir and G Il are called the right and the left substitution matrices of B I wrt, respectively Since our intention is to use backstepping design procedure [1], we assume that there exists an ARC Lyapunov function V I (x I ; μ (l I ) ß ;t) for the system (15) with the associated ARC control law and adaptation function given by u I = ff I = ff Ia (x I ; μ (k I ) ß ;t)+ff Is (x I ; μ (k I ) ß ;t) and fi I (x I ; μ (k I ) ß ;ff I ;t) respectively By definition, V I satisfies Requirements R1-R3, in which Requirements R (4) and R3 (6) are rewritten as B1 I0 + F I + B I ff I I I0 + F I + B I ff I + D I W I + ~ T ß fi I I fi I» V I V I + c V I (t) I fi I (17) where V I and c VI can be freely adjusted by some controller parameters in a known form For convenience, 11

13 denote DV I I I0 + F I + B I ff I + D I I I fi I (18) IV Augmented MIMO Nonlinear Systems I Consider the following MIMO nonlinear system with the state vector χ e = [x T e ; T ] T, the input vector u e R m, and the output vector y e R m _x e = f e0 (x; t) +F e (x; t) + B e (x; ; t)u e + D e (x; t) e(x; ; u e ;t) _ = Φ 0 (x; t) +Φ (x; t) ; R n (19) y e = x e where x =[x T I ; xt e ; T ] T R n and n = n I + m + n In (19), the dynamics are allowed to depend on the states of the initial system: ie, the vectors or matrices f e0 ; F e ; B e ; D e ; Φ 0, and Φ can be functions of χ e as well as x I As in Assumptions AIII1-5, the following assumptions are made for the added system: AIV1 B e (x; ; t) is nonsingular for any Ω and can be linearly parametrized by, ie, B e = B e0 (x; t) + B e (x; ; t) where B e (x; ; t) is linear wrt AIV The - subsystem is bounded-input bounded-state (BIBS) stable wrt the input (x I ;x e ) AIV3 Uncertain nonlinearities e are bounded by k ek»ffi e (x; t) where ffi e (x; t) is known } Now, augment the system by connecting the first m I outputs of the system (19) to the inputs of the initial system (15): ie, u I = μx em I = U ex e, where U e =[I m I 0] R m I m The remaining outputs of the system (19), x e(m I +1);:::;x em,are combined with the outputs of the initial system to form the new outputs of the augmented system The inputs of (19) become the inputs of the augmented system The augmented system thus has the dimension n and is described by _x I = f I0 + F I + B I (x I ; ;t)μx em I + D I I _x e = f e0 (x; t) +F e + B e (x; ; t)u + D e e _ = Φ 0 (x; t) +Φ (x; t) (0) y = [y T I ; (N T e x e) T ] T where u R m, y R m, and N e =[0 I m m I ]T R m (m m I ) For a guaranteed transient performance, the following compatibility assumption has to be made with respect to the connection: AIV4 The initial output of the added system is compatible with the required initial ARC input of the original system, ie, μx em I (0) = ff I(0) } 1

14 (0) can be rewritten in the standard form (1) where f = 6 4 f I0 + F I + B I μx em I f e0 + F e Φ 0 +Φ ; B = B e ; D = 6 4 D I 0 0 D e ; = 4 I e 3 5 (1) IV3 Backstepping Design I In this subsection, by using backstepping design, an ARC Lyapunov function is constructed for the augmented system (0) based on the ARC Lyapunov function V I for the initial system (15) Since ff I is the ARC control law for the initial system (15) and the inputs of (15) are μx em I, the key point is to design an ARC control law for the system (19) such that μx em I tracks ff I and other outputs track their desired trajectories with a guaranteed transient performance To this end, define the tracking error z R m and V as z(x; μ (k I ) ß ;t)=x e μff(x I ; μ (k I ) ß ;t); μff = [ff T I ; (N T e y d (t)) T ] T V (x; μ (l V ) ß ;t)=v I (x I ; μ (l I ) ß ;t)+ 1 zt Ez () where E is any spd matrix and l V = maxfl I ;k I g The following three Lemmas show that V given by () satisfies all three requirements that are needed for V to be an ARC Lyapunov function for (0) The proofs of all three Lemmas are given in Appendix C Lemma 1 V in () satisfies Requirement R1 for the system (0) } Lemma Define L, ff ea, and ffi as where l ff = maxfl V ; k I +1g, L(x; μ (lff) ß ;t)= ^B e + B e (x; ff ea (x; μ (lff) ß ;t)=ff e0 ffi 0 ^ ß ffi 0 ffi(x; μ (l V ) ß ;u;t)=ffi 0 + G er (x; u; t) E B = 6 4 T z T EB e I1 ;t) z T EB e Im I I ;t) U T e E B T ;t) I U e (fi I ffi T 0 Ez) ff e0 = E I B I0U e + U [f I0 + B I0 μx em I ] f e0 ffi 0 = E 1 Ue T GT Il (x I I; U [F I + G Ir (x I ; μx em I ;t)] + F e (3) (4) and G er is the right substitution matrix of B e If L is nonsingular, by letting Ω u = fu s : and choosing u a as z T ELu s» 0g u a (x; μ (lff) ß ;t)=l 1 [ E 1 Qz + ff ea ] (5) 13

15 where Q is any spd matrix, Requirement R (5) is satisfied by V for the system (0) with the following adaptation function fi(x; μ (l V ) ß ;u;t)=fi I ffi T Ez (6) } Lemma 3 If u s Ω u is chosen such that z T [E(L ~ B e )u s + ff es ]» " e (t) (7) where " e is a design parameter and ff es = Ef ffi(x; μ (l V ) ß ;u a (x; μ ß (lff) ;t);t) ~ ß D I I + D e eg (8) then, Requirement R3 (6) is satisfied by V for the system (0) with V = minf V I ; min(q) max(e) and c V = c V I + " e } Remark 10 One solution to (7) can be found in the following way Let h(x; μ ß (lff) ;t) be a function satisfying h(x; μ ß (lff) ;t) sup Ω kff ; Ω es (x; μ ß (lff) ;u;t)k (9) For example, let h M keffi(x; μ (l V ) ß ;u a ;t)k + D Ikffi I + ked e kffi e (30) where M = k max min + " k Let ρ u be a positive scalar satisfying ρ u (x; μ (lff) ß ;t) sup Ω keb e (x; ~ ß ;t)l 1 E 1 k (31) which may not be difficult to calculate since B e is linear wrt ~ ß We assume that ρ u < 1 In the absence of input channel parametric uncertainties (ie, B e = 0), ρ u = 0 Therefore, as long as the input channel uncertainties are not so large, the assumption that ρ u < 1 can be satisfied Then, by choosing u s (x; μ (lff) ß ;t)= 1 4(1 ρu)"e h L 1 E 1 z (3) from (30) and (31), we have 1 LS of (7)» 4(1 ρu)"e h ( kzk + kzkkeb e (x; ~ ß ;t)l 1 E 1 kkzk) +kzkkff es k 1» ( p hkzk p (33) " "e e ) + " e» " e Thus, (7) is satisfied } Lemmas 1 to 3 lead to the following theorem: 14

16 Theorem If L is nonsingular and (7) is satisfied, V defined by () is an ARC Lyapunov function for the augmented system (0) with the control functions u a given by (5) and u s determined from (7) The adaptation function fi is given by (6) 4 Remark 11 Since Ω^ can be chosen arbitrarily close to Ω and B e is nonsingular for any Ω, ^B e is nonsingular Noting that the last two terms of L in (3) are linear wrt, nonsingularity of L can thus be guaranteed by using a small adaptation rate } IV4 Augmented MIMO Nonlinear Systems II In the previous subsection, we constructed an ARC Lyapunov function for the augmented system (0) The state equations of the added system (19) have the same form as those in the first (r 1) subsystem in (1), which are required to be linearly parametrized by the unknown parameter vector when e = 0 In this subsection, the added system will take the form of the r-th subsystem in (1) so that the results will be applicable for most mechanical systems as mentioned in Remark 7 The main difficulty here is that the state equations of the added system may not be linearly parametrized by any group set of unknown parameters Let us consider the following augmented system x_ I = _x e = f I0 (x I ;t)+f I (x I ;t) + B I (x I ; ;t)μx em I + D I(x I ;t) I M 1 (x I ;fi;t)[f e0 + F e (x; t) + F fi (x; t)fi + B e (x; ; fi; t)u + D e (x; t) e] _ = Φ (x; ; fi; t); R n y = [y T I ; (N T e x e ) T ] T (34) In viewing Assumptions AIII1-5, e satisfies Assumption AIV3, the -subsystem satisfies Assumption AIV, and AIV5 B e is nonsingular and B e (x; ; fi; t) =B e0 (x; t) +B e (x; ; t) +B efi (x; fi; t) where B e (x; ; t) and B efi (x; fi; t) are linear wrt and fi respectively AIV6 M is an spd matrix and there exist positive scalars k m and k M such that k m I m» M» k M I m AIV7 M(x I ;fi;t)=m 0 (x I ;t)+m fi (x I ;fi;t) in which M fi is linear wrt fi } We assume that the compatibility assumption AIV4 is satisfied and proceed to construct an ARC Lyapunov function for (34) as in subsection IV3 Define z as in () and V as V (x; μ (l V ) ß ;fi;t)=v I (x I ; μ (l I ) ß ;t)+ 1 zt M(x I ;fi;t)z (35) Similar to (16), let G Mr (x; ffl;t) and G Ml (x; ffl;t) denote the right and left substitution matrices of the matrix M fi (x; fi; t) in terms of fi Let G r (x; ffl;t) and G l (x; ffl;t) denote the right and left substitution matrices of the matrix B e (x; ; t) (in terms of ) respectively, and G fir (x; ffl;t) and G fil (x; ffl;t) for B efi (x; fi; t) (in terms of fi) 15

17 From (34), each component of _M has the following form _M ij ij I0 + F I + B I0 μx em I + G Ir(x I ; μx em I ;t) ij I I (36) Since M(x I ;fi;t) can be linearly parametrized in terms of fi, so ij ij I = 0, from (36), _M(x I ;fi;t) can be linearly parametrized in terms of the augmented parameter set e =[ T ;fi T ;# T ] T, where # =[fi 1 T ;fi T ;:::;fi lfi T ] T R l fip Thus, there exist known vector d M (x I ; ffl;t) and matrix D M (x I ; ffl;t)=[d M ; D Mfi ; D M# ] such that 1 νt _M(x I ;fi;t)ν = ν T [d M (x I ;ν;t)+d M (x I ;ν;t) e + ~ M(x; ν; e ;t)]; 8ν R m (37) where ~ M linearly depends on I and can be bounded by a known function ffi M, ie, k ~ M (x; ν; e ;t)k»ffi M (x; ν; t) (38) Since M fi is linear wrt fi, there exists a known matrix D p# (x; t) such that M fi (x I ;fi;t)u T e [F I + B I (x I ; ;t)μx em I ]=D p#(x; t)# (39) Since Ω and fi Ω fi, we have # Ω #, where Ω # is a known bounded set having the form Ω # = f# : # min <#<# max g Thus, we can define ^# ß = ß # ( ^#), the projection of ^#, in the same way as^ ß for ^ Corresponding to Lemmas 1-3, we have the following three Lemmas The proofs of the three Lemmas are given in Appendix D Lemma 4 V in (35) satisfies Requirement R1 for the system (34) } Lemma 5 If L(x; μ (lff) ß ; ^fi ß ;t) is nonsingular, by letting Ω u = fu s : z T Lu s» 0g and choosing u a as u a (x; μ ß (lff) ; ^fi ß ; ^# ß ;t)=l 1 [ Qz + ff ea (x; μ ß (lff) ; ^fi ß ; ^# ß ;t)] (40) V of (35) satisfies Requirement R (4) for the system (34) in terms of the augmented unknown parameter vector e =[ T ;fi T ;# T ] T The corresponding adaptation function fi e (x; μ (l V ) ß ;u;t)=[fi T ; fi fi T ; fi # T ]T is given by fi (x; μ (l V ) ß ;u;t)=fi I (x; μ (k I ) ß ;ff I ;t) ffi T (x; μ (l V ) ;u;t)z fi fi (x; μ (l V ) ß ;u;t)= ffi T fi (x; μ (l V ) ß ;u;t)z (41) fi # (x; μ (k I ) ß ;t)= ffi T z #0 ß 16

18 In the above equations, L and ff ea in (40), and ffi e =[ffi T ;ffit fi ;ffit #0 ]T in (41) are given by L(x; μ ß (lff) ; ^fi ß ;t)= ^B T e + B e I ;t) ^MU T e Z B ff ea = ff e0 (x; μ (l V ) ß ;t) ^MUe T ffi (x; μ (l V ) ß ;u;t)=ffi 0 + G r (x; u; t) ffi fi (x; μ (l V ) ß ;u;t)=ffi fi0 + G Mr (x I [fi I(x; μ (k I ) ß ;ff I ;t) ffi T 0 z] ffi 0 U T ffi #0 (x; μ (k I ) ß ;t)=d M# (x I ;z;t) D p# (x; t) fi (x; μ (l V ) ß ;u;t) I ^ ffie0 eß ;t)+g fir (x; u; t) (4) where ffi e0 =[ffi T 0 ; ffit fi0 ; ffit #0 ]T and Z B (x; μ ß (lff) ;t)= 6 4 T z T B e I1 ;t) z T B e Im I T ;t) ff e0 I B I0U μff fe0 + M + M 0Ue I0 + B I0 μx em I ] d M (x I ;z;t) ffi 0 (x; μ (l V ) ß ;t)=ue T G T Il (x I I; ;t)+fe + D M (x I ;z;t) M 0 Ue I I + G Ir (x I ; μx em I ffi fi0 (x; μ (k I ) ß ;t)=f fi G Mr (x I ; U [f I0 + B I0 μx em μff ]+ Mfi (x I ;z;t) (43) } Lemma 6 If u s Ω u is chosen such that z T [(L ~ L u )u s + ff es ]» " e (t) (44) where ~L u (x; μ ß (lff) ; ^fi ß ;t)=b e (x; ~ ß ;t)+b efi (x; fi ~ ß ;t) M fi (x; fi ~ ß ;t)ue T Z B ff es (x; μ ß (lff) ; ^fi ß ; ^# ß ;u;t)= ffi e (x; μ (l V ) ß ;u a ;t) ~ eß + ~ (45) then, Requirement R3 (6) is satisfied by V for (34) with V = minf V ; min(q) I g and c km V = c V I + " e } Remark 1 One solution to (44) can be found in the same way as in Remark 10 except that h and ρ u are required to satisfy h em kffi e (x; μ (l V ) ß ;u a ;t)k + k M kue T ρ u (x; μ (lff) ß where em = k emax emin + " e k ;t) sup Ω k ~ L u L 1 (x; μ (lff) Ikffi I + kd e kffi e + ffi M (x; z; t) ;t)k (46) } Lemmas 4 to 6 lead to the following theorem: Theorem 3 If L of (4) is nonsingular and (44) is satisfied, V defined by (35) is an ARC Lyapunov function for the augmented system (34) The control function u a is given by (40), u s is determined from (44), and the adaptation function fi e for the augmented parameter set e is given by (41) 4 17

19 Remark 13 Although the state equations of the system (34) cannot be linearly parametrized, by introducing M as a weighting matrix in forming V in (35), _V can be linearly re-parametrized, which makes the utilization of adaptive control possible } V ARC Of MIMO Nonlinear Systems In Semi-Strict Feedback Forms In this section, the two backstepping designs presented in Section IV will be applied recursively to solve the ARC of nonlinear systems transformable to the two semi-strict feedback forms in section III V1 ARC For MIMO Semi-Strict Feedback Form I In order to recursively apply the two backstepping designs presented in section IV to solve the ARC problem for the system (1), it is necessary to make sure all assumptions in section IV are met As shown later, all assumptions are straightforwardly met except the compatibility Assumption AIV4 on the connections To overcome this problem, an additional tool, trajectory initialization [1], will be used Namely, instead of tracking the desired outputs y d (t) = [yd1b T ;:::;yt drb ]T directly, the controller will be designed to track the filtered outputs y t (t) =[yt1b T ;:::;yt trb ]T, in which the i-th block of outputs, y tib R m i mi 1, are generated by the following (r i +1)-th order stable system (y (r i+1) tib y (r i+1) dib )+fi i1 (y (r i) tib y (r i) dib )+:::+ fi i(r i+1) (y tib y dib )=0 (47) The initial conditions y tib (0);:::;y (r i) tib (0) for (47) will be specified later to make sure that the compatibility assumption are met To accomplish this task, it is necessary to know the explicit dependence of the control law on the filtered desired trajectory y t and its derivatives Therefore, in the following, these known time functions will appear as independent variables in the expression of the control law as opposed to appearing implicitly in the control law as in section IV The following notations are used Similar to (16), G ir (μχ i ; ffl;t) and G il (μχ i ; ffl;t) denote the right and the left substitution matrices of the matrix B i (μχ i ; ;t), G r (χ; ffl;t) and G l (χ; ffl;t) for the matrix B r (χ; ; t), G fir (χ; ffl;t) and G fil (χ; ffl;t) for the matrix B rfi (χ; fi; t), and G Mr (μχ r 1 ; ffl;t) and G Ml (μχ r 1 ; ffl;t) for the matrix M fi (μχ r 1 ;fi;t) 8k, define μy (k) tib as μy(k) tib =[yt tib ; :::;(y(k) tib )T ] T Recursively define y ju by y 1u = y t1b ;:::;y ju = [y (1)T j 1u ; yt tjb ]T = [y (j 1)T t1b ;:::;ytjb T ]T R m j, and L ju by L 1u = L 1 ;:::;L ju = [Uj T L j 1u N j ]L j The design proceeds in the following steps: V11 Step 1 The first system is defined as the first subsystem of (1), the first two equations of (1) for i = 1, whose inputs are μx ;m1 and outputs are μy 1 = y 1b = x 1 Noting Assumptions AIII1-5, the first system can be considered as a special case of the augmented MIMO nonlinear system I (0) with m I = 0 x 1 ; 1 ; μx ;m1 ; f 10 ; F 1 ; B 1 ; D 1, and 1 in (1) correspond to x e ; ; u; f e0 ; F e ;B e ; D e ; and e in (0), respectively Therefore, the backstepping design results in section IV3 can be used to construct an ARC 18

20 Lyapunov function V 1 for the first system V 1 is given by V 1 (x 1 ;y t1b )= 1 zt 1 E 1z 1 ; z 1 (x 1 ;y t1b )=x 1 ff 0 (y t1b ) (48) where ff 0 (y t1b )=y t1b (t) The associated control functions can be obtained through Eqs (5) and (7) For simplicity, in the following, the robust control term u s is chosen according to (3) in Remark 10 After some lengthy substitutions and calculations, the control function ff 1 (χ 1 ; ^ ß ; μy (1) t1b ;t) and the adaptation function fi 1 (χ 1 ; ^ ß ; μy (1) t1b ;t) are obtained as ff 1 = ff 1a + ff 1s ; ff 1a = L 1 1 [y(1) t1b f 10 ffi ^ 10 ß E 1 1 Q 1z 1 ] 1 ff 1s = 4(1 ρu1)"e1 h 1 L 1 1 E 1 1 z 1 (49) fi 1 = (ffi 10 + G 1r (χ 1 ;ff 1 ;t)) T E 1 z 1 where ffi 10 (χ 1 ;t)=f 1 L 1 (χ 1 ; ^ ß ;t)= ^B 1 = B 1 (χ 1 ; ^ ß ;t) h 1 (χ 1 ; ^ ß ; μy (1) t1b ;t) MkE 1 [ffi 10 + G 1r (χ 1 ; ff 1a ;t)k + ke 1 D 1 kffi 1 (χ 1 ;t) ρ u1 (χ 1 ; ^ ß ;t) sup ke Ω 1B 1 (χ 1 ; ~ ß ;t)l 1 1 E 1 1 k It can be verified that the control law (49) has the following structure ff 1 = L 1 1u y(1) t1b + ff 1y(χ 1 ; ^ ß ; μy (1) t1b ;t) ff 1p(χ 1 ; ^ ß ;y t1b ;t) ff 1y = ff 1s L 1 1u E 1 1 Q 1z 1 ff 1p = L 1 1u [f 10 + ffi ^ 10 ß ] (50) (51) with the property that every element of ff 1y contains z 1 as a factor V1 Step i In this subsection, mathematical induction will be used to prove the general results for all intermediate design steps For this purpose, all recursive formula will be described first with all proofs given in Appendix E 8i» r 1, the control function ff i (μχ i ; μ ß (i) ; μy (i) t1b ;:::;μy(1) tib ;t) and adaptation function fi i(μχ i ; μ ß (i) ; μy (i) t1b ;:::;μy(1) tib ;t) are calculated based on the control functions ff j and adaptation functions fi j of all previous steps as ff i = ff ia + ff is ff ia = L 1 i n E 1 i i 1 j j0 + Ui T i Q i z i E 1 i Ui T BT i 10 E i 1z i 1 + Ui T " Pi 1 j=1 P i j k=0 P i 1 1 y (k) tjb i tjb ffi i0 h^ ß P i i j=1 j )T U j+1 E j+1 z j+1 Ui T 1 ff is = 4(1 ρui)"ei h i L 1 i E 1 i z i # i 1 j0 + B j0 μx j+1;m j ) + N i y (1) tib f 1 (fi i 1 ffi T i0 E iz i ) o (5) fi i = fi i 1 (ffi i0 + G ir (μχ i ;ff i ;t)) T E i z i 19