Global adaptive neural dynamic surface control of strict-feedback systems

Author's Accepted Manuscript

Global Adaptive Neural Dynamic Surface Control of Strict-Feedback Systems Jeng-Tze Huang

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S0925-2312(15)00316-1 http://dx.doi.org/10.1016/j.neucom.2015.03.030 NEUCOM15255

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Neurocomputing

Received date: 21 August 2014 Revised date: 5 March 2015 Accepted date: 6 March 2015 Cite this article as: Jeng-Tze Huang, Global Adaptive Neural Dynamic Surface Control of Strict-Feedback Systems, Neurocomputing, http://dx.doi.org/10.1016/j. neucom.2015.03.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Global Adaptive Neural Dynamic Surface Control of Strict-Feedback Systems Jeng-Tze Huang Institute of Digital Mechatronic Techanology Chinese Culture University Taipei, Taiwan 11114 Email: [email protected]

Abstract By mainly activating an auxiliary robust control component for pulling back the transient escaped from the neural active region, the author has proposed a multi-switching robust neuroadaptive controller to achieve globally uniformly ultimately bounded (GUUB) tracking stability of strict-feedback systems recently. However, the so-called explosion of complexity problem, arising from the repeated differentiations of the virtual controllers in the backstepping procedure, has not been solved yet. The objective of this paper is to tackle this issue via using the dynamic surface control (DSC) technique. By a suitable modification to the DSC scheme, the complexity in the neuroadaptive control component can be successfully conquered. The modification is twofold, firstly, high-order instead of the common first-order filters are used, secondly, the filters’ inputs are space-modulated and proven to be bounded, which in turn ensures the bounded stability of the filter dynamics. However, the repeated differentiations in the auxiliary robust controller remains as the price of preserving the global stability. Nevertheless, since the neuroadaptive controller dominates most of the time, the overall controller still significantly improves the existing global designs in terms of computation and implementation. Simulation results are provided to demonstrate the validity of the proposed scheme. Keywords: Global stability, neural networks, dynamic surface control, high-order filter, smooth switching

Preprint submitted to Neurocomputing

March 20, 2015

1. Introduction It is known that the adaptive backstepping control design can be systematically formulated in a recursive manner to achieve the asymptotic stability for parametric strict-feedback systems [1]. However, in cases when the nonlinearities are totally unknown or nonlinearly parameterized, the adaptive control methodology cannot be directly applied. One popular approach for alleviating the above difficulty is to incorporate the neural networks (NNs) for resembling the dealt nonlinearities. By virtue of the universal approximation property, NNs are able to approximate any continuous function to an arbitrary accuracy in a compact subset of the input space by employing sufficient neurons [2, 3]. However, there are two major drawbacks associated with a general neuroadaptive backstepping design, first, due to the compactness of the neural active region, only semi-globally uniformly ultimately bounded (SGUUB) stability of the closed-loop system is ensured on the condition that the neural approximation remains valid for all time, which is difficult to verify beforehand; second, the complexity of the resulting controller grows rapidly with the system’s dimension due to the repeated differentiations of the virtual controllers, i.e., the so-called explosion of complexity problem. By using the multi-switching neuroadaptive backstepping approach, GUUB tracking stability of strict-feedback systems has been achieved recently in [4]. Based on such a scheme, a global direct adaptive neural controller was proposed in [5]. Despite these remarkable achievements, the second problem mentioned above has not been solved in these designs so far. First developed by Swaroop et. al. [6], the popular DSC scheme aims to solve such a problem by filtering instead of differentiating the virtual controller in each backstepping design step [7]. It is a powerful tool which has been incorporated with a wide variety of backstepping based controllers for solving such a problem. In [8], the DSC is incorporated with a sliding-mode controller while a nonlinear damping control is proposed in [6] for strictfeedback systems with uncertainties bounded by known functions. The adaptive DSC control is formulated to achieve the asymptotic instead of SGUUB tracking stability in cases when the uncertainty is linearly parameterized [9], [10]. It is then extended to the tracking control of a strict-feedback system with arbitrary unknown nonlinearities using NNs [11]. The so-called minimallearning-parameter technique is included to further alleviate the explosion of learning parameters problem in [12]. Along the line, the DSC scheme has been incorporated with various intelligent control methodologies for solving 2

versatile control problems such as the state/output feedback control of largescaled interconnected systems, nonlinear systems with time delays, and/or stochastic uncertainties, etc, [13]-[22]. However, as mentioned, only SGUUB stability is ensured in these designs. Besides, the corresponding stability analysis of the whole closed-loop system, consisting of the original system and the filter dynamics, is quite involved. As there are nonlinear functions of the full states and control gains, the boundedness of the filters’ inputs and hence the stability of the filter dynamics is particularly difficult to establish. In fact, so far there is still no systematic way to select the required control gains and the time constants of the filters for stabilizing a prescribed domain of attraction (DOA) [7]. Expectably, a direct incorporation of such a scheme with the previous design in [4] will destroy its global tracking stability if there are no proper modifications. In this regard, mainly by imposing certain constraints on the inputs and the velocities, the boundedness of the filter inputs and hence the GUUB tracking stability has been achieved for underactuated ship dynamics in [23]. Inspired by this, it is first noted that each virtual controller α in [4] has a form of α = mB αan + (1 − mB )αr , where αan is the neuroadaptive control component dominant in the neural active region, αr is the robust control component taking over the control authority outside the neural active region, and mB is the switching algorithm monitoring the exchange of the above mentioned two controllers. By incorporating a modified DSC scheme, where a high-order filter with mB αan as the input is constructed in each design step, the explosion of complexity in αan is annihilated. By virtue of the modulation of the switching function, the inputs to the filters, i.e., mB αan , are proven to be bounded for all time, which in turn ensures the bounded stability of the filter dynamics. However, the repeated differentiations of αr can not be avoided for preserving the GUUB stability. Nevertheless, since in practical applications the neuroadaptive control component dominates most of the time, the proposed controller is therefore more efficient and easier to implement than the earlier design in [4]. In summary, the main contribution of this paper includes: 1) the separation of the controller-filter pair has been achieved and hence it can be incorporated with any backstepping based control algorithms; 2) the explosion of complexity problem is solved while GUUB stability is preserved at the same time. To the best of our knowledge, this is the first time these objective have been simultaneously achieved for a general strict-feedback system in the literature. The remainder of the paper is organized as follows: the dealt problem 3

and the related achievements in the literature are briefly reviewed in Section 2. The central part of this paper, namely, the control design, is detailed in Section 3. Stability analysis of the resulting closed-loop system is provided in Section 4. To demonstrate its usefulness, simulation for a second-order strict-feedback system is conducted in Section 5. Finally, conclusion and future works are discussed in Section 6. 1.1. Preliminaries To facilitate the upcoming control design, the following definition is recalled for ease of reference. Definition 1. [1] Consider the following n0 th order nonlinear system x˙ = f (x, t)

(1)

We say that the solution is GUUB if for all bounded initial state x(t0 ) ∈ Rn , there exists an  > 0 and a positive number T (, x(t0 )) such that |x(t)| <  for all t ≥ t0 + T . Secondly, the notion of input-to-state stability (ISS), i.e., bounded input implies bounded state, has been utilized especially for the control of nonlinear systems in recent years. The criterion for establishing the ISS property in linear systems will be used later in this paper and is cited below. Consider the linear system in a standard state-space form x˙ = Ax + Bu,

(2)

where x ∈ Rn is the state vector, u ∈ Rm is the input, A ∈ Rn×n , B ∈ Rn×m are proper constant matrix. Lemma 1. [24] The system (2) is ISS stable, i.e., the following inequality holds for all time, if A is Hurwitz. |x(t)| ≤ β(t)|x(0)| + γ|u|∞ where tA

β(t) = ||e || → 0

Z and γ = ||B||

(3)

∞

||eτ A ||dτ < ∞.

(4)

0

Moreover, for easy reference, we quote the following inequality here [25] η (5) 0 ≤ |η| − η tanh( ) ≤ cη δ, ∀η ∈ R, δ where cη = 0.2785 and δ is a positive constant. The benefit of the tanh(·) function over the sat(·) function used in the sliding-mode control design is its infinite differentiability which is indispensable in backstepping based designs. 4

2. Problem Formulation Consider a nonlinear system in a strict-feedback canonical form x˙ i = gi (¯ xi )xi+1 − fi (¯ xi ) x˙ n = gn (x)u − fn (x), 1 ≤ i ≤ n − 1 y = x1

(6)

where x = [x1 , · · · , xn ]T ∈ Rn , u ∈ R, y ∈ R are the states, input and output of the system, respectively; x¯i = [x1 , · · · , xi ]T ∈ Ri are the partial state vectors, fk (¯ xk ), gk (¯ xk ), k = 1, · · · , n are smooth nonlinear functions well-defined for all x ∈ Rn . The affine functions gk (¯ xk ) are assumed known and, without loss of generality, being positive for all time in the sequel. The following assumptions are essential for the subsequent control designs. A1) The reference output yd (t) is smooth and bounded. A2) The state vector x(t) is measurable. A3) The functions fk (¯ xk ), k = 1, · · · , n are unknown and bounded by |fk (¯ xk )| ≤ fku (¯ xk ),

∀¯ xk ∈ R k ,

(7)

where fku are known smooth functions. Assumption A1 is a necessity for establishing the boundedness of all the signals in the closed-loop system; A2 is made for the feasibility of state feedback; while A3 is indispensable for the applicability of the auxiliary robust controller embedded in the proposed control algorithm. The objective of this paper is to synthesize a control law u(t) such that the system output y(t) tracks the reference output yd (t) with high accuracy while all the closed-loop signals remain bounded for all time. Various nonlinear backstepping control methodologies have been formulated to attain such objectives. Among others, the neural adaptive approach is popular for its advantages of model independence, more efficient control efforts, without chattering behaviors, etc [3]. However, as mentioned, most existing designs ensure SGUUB of the tracking errors only. Moreover, the complexity of the resulting controller grows rapidly with the system dimension due to the

5

repeated differentiations of the virtual controllers required in the backstepping procedures. Mainly by adopting the multi-switching approach, GUUB tracking stability has been achieved in [4], [5]. However, the latter problem remains unsolved so far. We intend to tackle this issue in this paper. To further illustrate our motivation, let’s consider momentarily a simple second-order case of those in (6), i.e., x˙ 1 = g1 x2 − f1 x˙ 2 = g2 u − f2 ,

(8)

The standard backstepping procedure starts by defining the following two error states, z1 = x1 − yd z2 = x2 − α

(9)

where α is a virtual controller. By a direct differentiation of (9), it yields z˙1 = g1 (z2 + α) − y˙ d − f1 z˙2 = g2 u − f2 − α˙

(10)

For stabilizing the z1 subsystem, the virtual controller α of a typical neuroadaptive controller would be a function of x1 , y˙ d , and the estimated neural weight vector. On the other hand, the control input u has to cancel the signal α˙ first for stabilizing the z2 subsystem. As a result, the control input u will contain terms for cancelling the known parts of α˙ and some components for counteracting the unknown parts, e.g., f1 in this example. It can be expected that such control terms accumulate quickly with the order of system dimension, which is apparently unfavorable to practical implementation. The DSC scheme aims to conquer this drawback by filtering instead of differentiating the virtual controller itself. To this end, a first-order filter is constructed as follows τ q˙ + q = α (11) where τ > 0 is the filter time constant and q(t) is the output of the filter output. The solution to (11) is Z t 1 − 1 (t−ξ) − τ1 t + e τ α(ξ)dξ (12) q(t) = q(0)e 0 τ 6

1

Now τ1 e− τ (t−ξ) can be regarded as a delta function located at the instant t when τ → 0 [26]. It follows that 1

q(t) ≈ q(0)e− τ t + α(t) → α(t) as t → ∞.

(13)

It is then reasonable to assume that q(t) ˙ ≈ α(t) ˙ under such circumstances, which inspires the DSC technique to replace α˙ with q˙ for solving the explosion of complexity problem. Redefining the second error state as follows z2 = x2 − q

(14)

The corresponding z2 dynamics becomes z˙2 = g2 u − f2 − q˙

(15)

Apparently, q˙ is now cancellable and hence the above-mentioned difficulty can be alleviated. As can be seen, the two-step procedure involving in the DSC scheme is quite intuitive and simple. However, the resulting closed-loop system is generally interconnected in a complex way especially for high dimensional systems, which makes the corresponding stability analysis difficult. In fact, even though most existing DSC based control designs ensure SGUUB tracking stability, unfortunately, so far there are no specific ways for choosing the required control gains and filter time constants [7]. The paper aims to solve the above problem in an earlier global neuroadaptive control design, whose virtual controller α is in a form of [4] α = αan + αr ,

(16)

where αan is the neuroadaptive controller, αr is a robust controller for bringing the escaped transient back to the active region to ensure global stability. A direct inclusion of DSC scheme is expected to destroy the global tracking stability. Nevertheless, by a proper modification, i.e., a higher-order filter with a space-modulated input, the explosion of complexity in αan is avoided while those in αr remain as the price for preserving the global tracking stability of the closed-loop system. Since in practical applications the neuroadaptive controller dominates, i.e., mB = 1, most of the time, the complexity of the overall controller is effectively reduced and therefore is appealing for practical applications. Details are given in the upcoming derivation. 7

2.1. Radial basis function neural networks Radial basis function neural networks (RBFNNs) are universal approximators for continuous functions, i.e., given an arbitrary continuous function, there is always an optimal neural weight vector (it may not be unique) with appropriate dimensions, such that the approximation errors are uniformly bounded by the given tolerance within the neural active region [2], [3]. Since the nonlinearities fi (¯ xi ) in (6) are unknown, it’s natural to construct appropriate neural networks for approximating them to render the adaptive control applicable. The outputs of the RBFNNs are in a form of θT φ(x), where θ ∈ Rnh is the tunable neural weight vector connecting the hidden layer to the output layer, nh is the number of the neurons in the hidden layer, and φ(x) = [φ1 (x), · · · , φnh (x)]T is the radial basis vector function. A typical basis Gaussian function is given by φi (x) = exp[−

(x − oi )T (x − oi ) ], d2i

i = 1, · · · , nh

(17)

where oi and di are the centers and the spreads of the Gaussian function, respectively. Within such a framework, the unknown functions fk (¯ xk ), k = 1, · · · , n in (6) can be re-parameterized in a form of fk (¯ xk ) = θk∗T φk (¯ xk ) + k (¯ xk ),

∀¯ xk ∈ Ωk

(18)

where θk∗ ∈ Rnk are the optimal weight vectors, k are the corresponding approximation errors, and Ωk are the neural active regions for f (¯ xk ), which, without losing generality, are chosen here as the hyper cubics defined by ∆

Ωk = {x||xj | ≤ rj , 1 ≤ j ≤ k}

(19)

There is a total of n subsets in a nested-supset relation Ω1 ⊃ Ω2 · · · ⊃ Ωn

(20)

Clearly, the reference output should lie within Ω1 for all time. The optimal weight vectors θk∗ ∈ Rnk are those that minimize dk (¯ xk ) over Ωk , respectively, i.e., ∆ θk∗ = arg minn { sup |fk (¯ xk ) − θkT φk (¯ xk )|}. (21) θk ∈R

k

x ¯k ∈Ωk

8

The approximation errors k (¯ xk ) are uniformly bounded by kk (¯ xk )k ≤ dk ,

∀¯ xk ∈ R k

(22)

where dk are unknown constants. For the tractability of the upcoming derivation, the following assumption is made here. A4) All the optimal weight vectors are bounded, i.e., ||θk∗ || ≤ θku , with θku , k = 1, · · · , n being unknown positive constants. 2.2. Switching function The proposed controller involves a set of multi-switching algorithms miB (¯ xi ), i = 1, · · · , n, which is in a form of [4] ∆

miB (¯ xi ) = Πik=1 Bk (xk ).

(23)

where Bk (xk ) is a basic one-dimensional switching function at disposal. It is noted that Bk (xk ) cannot be modeled by the conventional logic-based switching for the non-differentiability at the transition point. Even the differentiable switching functions in [4] are not adequate for the application here for their inability to completely shutdown the auxiliary robust controllers in Ωi . To solve the explosion of complexity problem, the switching algorithms have to let the neuroadaptive controllers completely in charge in the neuxi ) = 1, ∀¯ xi ∈ Ωi . In this respect, the following ral active regions, i.e., miB (¯ smooth function is a good candidate for that need [27] Z −1 −1 1 |xk | ∆ exp ( ) exp ( )dxk , Bk (xk ) = 1 − g rk xk − 1 2 − xk Z r¯k −1 −1 ) exp ( )dxk (24) g = exp ( xk − 1 2 − xk rk with rk , k = 1, · · · , i being the constants defining the boundaries of the compact subsets Ωi in (19), r¯k being the boundary on the righthand side of the transition regions. It is, however, not in a closed form and hence is not amenable to real implementation. In contrast, the following is a closed-form function and n-times differentiable, hence is more suitable for our needs [5]    1, ∀|xk | ≤ rk2, 2 ∆ x −r Bk (xk ) = cosn ( π2 sinn ( π2 r¯2k−r2k )), ∀rk ≤ |xk | ≤ r¯k , (25)   0, ∀|x | ≥ r¯ k. k k k 9

From viewing (23) and (25), it is easy to verify the sustenance of the following three equations which are very useful in the later stability analysis. miB (¯ xi )v(¯ xi , ρ(t)) ∈ L∞ ,

∀¯ xi ∈ R i

(26)

and (1 − miB (¯ xi ))v(¯ xi , ρ(t)) = 0, i ∂mB v(¯ xi , ρ(t)) = 0, ∀¯ xi ∈ Ωi , ∂xj

(27) 1 ≤ j ≤ i.

(28)

where ρ(t) is an arbitrary bounded vector function of time, and v(¯ xi , ρ(t)) is a continuous function well-defined in Ri . 3. Tracking Control Design In this section, a modified DSC scheme will be developed to solve the corresponding explosion of complexity problem in [4]. The design follows standard backstepping procedures. Particularly, a high-order filter with a space-modulated αan as the input is introduced in each design step. It’ll be shown that such an input function is actually bounded for all time, which in turn ensures the bounded stability of the filter dynamics. The GUUB stability of the overall closed-loop system is then easy to establish. Details are given in the following. Step 1: Consider first the case with i = 1 in (6), i.e., x˙ 1 = g1 x2 − f1 . Define the tracking error z1 = x1 − yd . By a direct differentiation, it yields z˙1 = g1 x2 − f1 − y˙ d

(29)

Treating x2 as the virtual controller for the z1 dynamics (29), the control objective can be attained via proper coupling effect. Denote the appropriate coupling by α2 . The proposed virtual controller is in a form of α2 = α2an + α2r ,

(30)

where α2an and α2r are respectively the virtual adaptive neural and robust controllers at disposal.

10

To avoid the complexity of the controller arising from the later repeated differentiations of the virtual controller, the popular DSC scheme will be incorporated herein. Let α2an pass through a (n − 1)’th-order filter as follows n−1 X

(k)

b2,k q2 = α2an ,

q2 (0) = α2an (0)

(31)

k=0

where q2 is the output of the filter, and b2,k , k = 0, · · · , n − 1 are positive numbers chosen to render the polynomial b2,n−1 sn−1 + · · · + b2,0 Hurwitz. The main reason for using a high-order instead of the commonly adopted first-order filter here is that the subsequent neuroadaptive controllers, i.e., (m) αian , i = 4, · · · , n depend on q2 , m > 1, which need be bounded to ensure the stability of the filter dynamics. Define the second state error as z2 (¯ x2 , q2 , t) = x2 − q2 − α2r

(32)

Note that the arguments entering z2 can be verified as referring to the later assignment of the virtual controller α2r in (35). By substituting (32) into (29), it yields z˙1 = g1 (q2 + z2 + α2r ) − f1 − y˙ d = g1 m1B q2 + g1 [α2r + (1 − m1B )q2 ] − [m1B + (1 − m1B )](f1 + y˙ d ) = −kz z1 + g1 z2 + g1 α2an + g1 [α2r + (1 − m1B )q2 ] −[m1B + (1 − m1B )](h1 + f1 ) + ζ2 (33) where h1 = −kz z1 + y˙ d and ζ2 is the filter error defined by ζ2 = g1 m1B q2 − g1 α2an .

(34)

For stabilizing the equilibrium z1 = 0, we propose that α2an (x1 , θˆ1 , t) = m1B g1−1 (h1 + θˆ1T φ1 ), α2r (x1 , q2 , t) = (1 − m1B ){−q2 + g1−1 [h1 − f1u tanh(

z1 f1u )]}. w0

(35)

The explicit dependence on t of α2an and α2r comes from the reference trajectory yd and its derivative. Since yd is smooth and bounded according to A1, therefore it can be concluded that α2an ∈ L∞ if θˆ1 ∈ L∞ based on (26). Such a reasoning will be repeatedly used in the upcoming derivations. 11

The update algorithm for θˆ1 is given as follows ˙ θˆ1 = −σ1 [z1 m1B φ1 (z1 ) + σ ¯1 θˆ1 ],

(36)

where σ1 , σ ¯1 > 0 are tuning gains. Step 2: By a direct differentiation of z2 in (32), it yields z˙2 = x˙ 2 − q˙2 − α˙ 2r ∂α2r ∂α2r ∂α2r = h2 + g2 x3 − q˙2 − [ (g1 x2 − f1 ) + q˙2 + ] ∂x1 ∂q2 ∂t −h2 − f2

(37)

where h2 = −kz z2 − g1 z1 .

(38)

Based on (28), it can be seen that those terms resulting from α˙ 2r in (37) are nonzero only outside Ω1 and therefore will be handled by pure robust control components. In backstepping design, x3 in (37) will act as a virtual controller to stabilize the z2 dynamics via invoking proper coupling effect. Denote the ideal coupling by α3 , which is in a form of α3 = α3an + α3r ,

(39)

where α3an is the neuroadaptive controller dominating in the active region, α3r is the robust controller being in charge outside the active region. Let α3an pass through the following (n − 2)’th-order filter n−2 X

(k)

b3,k q3 = α3an ,

q3 (0) = α3an (0)

(40)

k=0

where q3 is the output of the filter and b3,k , k = 0, · · · , n − 2 are positive numbers chosen to render the polynomial b3,n−2 sn−2 + · · · + b3,0 Hurwitz. Again, to introduce the virtual controller into the z2 -dynamics in (37), define the third error state as (1)

z3 (¯ x2 , q3 , q¯2 , t) = x3 − q3 − α3r . 12

(41)

Likewise, the arguments entering z3 can easily be verified as referring to the later assignment of the virtual controller α3r in (43). By substituting (41) into (37), it yields z˙2 = −kz z2 − g1 z1 + g2 (z3 + q3 + α3r ) − q˙2 ∂αr ∂α2r ∂α2r q˙2 + ] − (h2 + f2 ) −[ 2 (g1 x2 − f1 ) + ∂x1 ∂q2 ∂t = −kz z2 − g1 z1 + g2 z3 + g2 α3an + g2 [α3r + (1 − m2B )q3 ] ∂αr ∂α2r ∂α2r −[ 2 (g1 x2 − f1 ) + q˙2 + ] ∂x1 ∂q2 ∂t −[m2B + (1 − m2B )](q˙2 + h2 + f2 ) + ζ3

(42)

where ζ3 = g2 m2B q3 − g2 α3an is the filter error state for α3an . For stabilizing the equilibrium z2 = 0, we propose that (1) α3an (¯ x2 , q¯2 , θˆ2 , t) = m2B g2−1 (q˙2 + h2 + θˆ2T φ2 ), ∂αr ∂αr (1) α3r (¯ x2 , q3 , q¯2 , t) = g2−1 { 2 [g1 x2 − f1u tanh(w0−1 z2 2 f1u )] ∂x1 ∂x1 ∂α2r ∂α2r + } + (1 − m2B ){−q3 q˙2 + ∂q2 ∂t z2 f u +g2−1 [q˙2 + h2 − f2u tanh( 2 )]}, w0

(43)

(1)

where q¯2 = [q2 , q˙2 ]T . The update algorithms for θˆ2 is given by ˙ θˆ2 = −σ2 [z2 m2B φ2 (¯ x2 ) + σ ¯2 θˆ2 ]

(44)

where σ2 , σ ¯2 > 0 are tuning gains. Step i (3 ≤ i ≤ n − 1): Denote the ideal coupling for xi by αi , which is in a form of αi = αian + αir ,

(45)

where αian , αir are respectively the virtual adaptive and robust controllers at disposal. Let αian pass through the following (n − i + 1)’th-order filter n−i+1 X

(k)

bi,k qi

= αian ,

k=0

13

qi (0) = αian (0)

(46)

where qi is the output of the filter and bi,k , k = 0, · · · , n − i + 1 are positive numbers chosen to render the polynomial bi,n−i+1 sn−i+1 + · · · + bi,0 Hurwitz. On the other hand, it is noted that the robust virtual controller αir , due to the iterative design procedure and the corresponding repeated differenti(ξ ) (ξ ) ations, would be a function of x¯i−1 , q¯λii = [qλi , · · · , qλii ]T , 2 ≤ λi ≤ i, ξi = i − λi , and t explicitly. Define the error state (ξ )

zi (¯ xi , q¯λii , t) = xi − qi − αir .

(47)

As explained previously, the time t entering the arguments of the variables on both sides of (47) indicates a summary dependence on yd and its derivatives (ξ ) up to certain orders, which are all bounded by A1. If q¯λii can be shown to xi )zi will be bounded by (26). be bounded, then miB (¯ By a direct differentiation, the zi -dynamics becomes z˙i = x˙ i − q˙i − α˙ ir = hi + gi xi+1 − q˙i − [

i−1 X ∂αr i

k=1

+

ξi i X X ∂αir

(c+1)

q (c) λi

λi =2 c=0

∂qλi

+

∂xk

(gk xk+1 − fk )

∂αir ] − hi − fi , ∂t

(48)

where hi = −kz zi − gi−1 zi−1

(49)

Denote the virtual controller for xi+1 by αi+1 . It is also in a form of an r , αi+1 = αi+1 + αi+1

(50)

Likewise, the following (n − i)’th-order filter is constructed to introduce the DSC scheme into the zi -subsystem in (48), n−i X

(k)

an bi+1,k qi+1 = αi+1 ,

an qi+1 (0) = αi+1 (0)

(51)

k=0

where qi+1 is the output of the filter and bi+1,k , k = 0, · · · , n − i are positive numbers chosen to render the polynomial bi+1,n−i sn−i + · · · + bi,0 Hurwitz. 14

Define the (i + 1)’th error state vectors and the corresponding filter errors as r zi+1 = xi+1 − qi+1 − αi+1 , i an ζi+1 = gi mB qi+1 − gi αi+1 .

(52)

By substituting (50)-(52) into (48), after some manipulation, it yields r an + (1 − miB )qi+1 ] + gi [αi+1 z˙i = −kz zi − gi−1 zi−1 + gi zi+1 + gi αi+1

−[

i−1 X ∂αr i

k=1 −[miB

∂xk

(gk xk+1 − fk ) +

ξi i X X ∂αir

(c+1)

q (c) λi

λi =2 c=0

+

∂qλi

∂αir ] ∂t

+ (1 − miB )](q˙i + hi + fi ) + ζi+1

(53)

For stabilizing the equilibrium zi = 0, we propose that (ξ )

an αi+1 (¯ xi , q˙i , q¯λii , θˆi , t) = miB gi−1 (q˙i + hi + θˆiT φi ), (ξ +1) r αi+1 (¯ xi , qi+1 , q¯λii , t)

=

gi−1 {

i−1 X ∂αr i

k=1 ξi i

+

∂xk

[gk xk+1 − fku tanh(w0−1 zi

X X ∂αr i

(c+1)

q (c) λi

λi =2 c=0 ∂qλi +(1 − miB ){qi+1 + zi f u −fiu tanh( i )]}. w0

+

∂αir u f )] ∂xk k

∂αir } ∂t

gi−1 [q˙i + hi (54)

The update algorithms for θˆi are given by ˙ θˆi = −σi [zi miB φi + σ ¯i θˆi ]

(55)

where σi , σ ¯i > 0 are tuning gains. r Remark 1. The terms on the right-hand side of αi+1 in (54) can be divided r into two parts, one for cancelling α˙ i and the other for dealing with the term (1 − miB )(q˙i + hi + fi ), which is nonzero only when x¯i ∈ / Ωi . From the iterative characteristics, it is not hard to see that those terms for cancelling α˙ ir are either being multiplied by (1 − mjB ) or by ∂mjB /∂xk 1 ≤ j ≤ i − 1, 1 ≤ k ≤ j, which are zeros for all x¯j ∈ Ωj by (27) and (28). Consequently, it can be an r concluded that αi+1 is in charge when x¯i ∈ Ωi while αi+1 is activated outside Ωi .

15

Step n: We finally arrive at the subsystem where the actual control input appears. Denote the ideal coupling for xn by αn , which is in a form of αn = αnan + αnr ,

(56)

where αnan and αnr are respectively the virtual adaptive and robust controllers at disposal. Let αnan pass through the following first-order filter 1 X

bn,k qn(k) = αnan ,

qn (0) = αnan (0)

(57)

k=0

where qn is the output of the filter and bn,k , k = 0, 1 are positive numbers chosen to render the polynomial bn,1 s + bn,0 Hurwitz. By a direct differentiation of the following zn error state, (ξ )

zn (x, q¯λnn , t) = xn − qn − αnr ,

2 ≤ λn ≤ n,

ξn = n − λn ,

(58)

it yields z˙n = x˙ n − q˙n − α˙ nr ξn n−1 n X X X ∂αnr ∂αnr (c+1) = hn + gn u − [ (gk xk+1 − fk ) + q (c) λn ∂x k ∂q k=1 λ =2 c=0 λ n

n

∂αr + n ] − [mnB + (1 − mnB )](q˙n + hn + fn ), ∂t

(59)

where hn = −kz zn − gn−1 zn−1

(60)

For stabilizing the equilibrium zn = 0, we propose that u = uan + ur

(61)

where (ξ ) uan (x, q˙n , q¯λnn , θˆn , t) = mnB gn−1 (q˙n + hn + θˆnT φn ),

u

r

(ξ +1) (x, q¯λnn , t)

=

gn−1 {

n−1 X ∂αr

n

k=1

∂xk 16

[gk xk+1 − fku tanh(w0−1 zn

∂αnr u f )] ∂xk k

+

ξn n X X ∂αnr

(c+1)

q (c) λn

λn =2 c=0

+

∂qλn

∂αnr ∂t

+(1 − mnB )[q˙n + hn − fnu tanh(

zn fnu )]}. w0

(62)

The update algorithms for θˆn is given by ˙ ¯n θˆn ] θˆn = −σn [zn mnB φn (x) + σ

(63)

where σn , σ ¯n > 0 are tuning gains. 4. Stability Analysis Comparing to the design in [4], we now have two closely coupled subsystems - the original error dynamics and the filter dynamics, as the result of introducing the DSC algorithms into the proposed controller. The stability of the whole closed-loop system is established in two consecutive steps. First, it’ll be shown that the inputs to the filters are all bounded signals, and hence the boundedness of the filter outputs follows immediately. After that, the original closed-loop error system can be regarded as a strict-feedback system subject to bounded disturbances and hence the corresponding stability analysis becomes relatively easy. By substituting the virtual and the actual controllers in (35), (43), (54), and (62) into the subsystems (33), (42), (53), and (59) respectively, after some straightforward calculations, it yields the following closed-loop z-system z˙1 = −kz z1 + g1 z2 + m1B (θ˜1T φ1 + d1 ) z1 +(1 − m1B )[−f1u tanh( ) − f1 ] + ζ2 w0 .. . z˙i = −kz zi − gi−1 zi−1 + gi zi+1 + miB (θ˜iT φi + di ) i−1 X ∂αir ∂αr + [−fku tanh(w0−1 zi i fku ) − fk ] ∂xk ∂xk k=1 +(1 − miB )[−fiu tanh(

zi fiu ) − fi ] + ζi+1 w0

.. . 17

z˙n = −kz zn − gn−1 zn−1 + mnB (θ˜nT φn + dn ) n−1 X ∂αr ∂αnr [−fku tanh(w0−1 zn n fku ) − fk ] + ∂xk ∂xk k=1 +(1 − mnB )[−fnu tanh(

zn fnu ) − fn ], w0

i = 2, · · · , n − 1,

(64)

where θ˜i = θˆi − θi∗ . Define the following three constants ∆

kv = min[2kz − 2, σ1 σ ¯1 , · · · , σn σ ¯n ], n 1X ∆ [ (¯ σi θi∗T θi∗ + d2i ) + dζ + n(n + 1)cη δ], dv = 2 i=1 r 2dv zb = , kv

(65)

where dζ is defined later in (67). We can now state our main results as follows. Theorem 1. Consider the closed-loop system consisting of the plant (6), the controller (61)-(62), and the NN weight tuning algorithms (36), (44), (55), and (63). If the control and the parameter tuning gains are selected to fulfill kz > 1 and σ, σ ¯ > 0, then all the signals in the closed-loop system are uniformly ultimately bounded. Moreover, the output tracking error z1 = y−yd converges uniformly to the following set, whose radius can be made arbitrarily small by using sufficiently large gains kz , σk , σ ¯k , k = 1, · · · , n. ∆

Ωz1 = {z1 | |z1 | ≤ zb }

(66)

Proof 1. The proof proceeds in two consecutive steps, first, the boundedness of the signals αian , which in turn guarantees the bounded stability of the constructed filter dynamics, is proven. The boundedness of the error states ζi+1 , i = 1, · · · , n − 1 then follows immediately by definition. Second, by treating the above error states as the bounded disturbance to the z-dynamics (64), the GUUB stability of the whole closed-loop system is then proven. S1: First, it is easy to see that m1B z1 (x1 , t)φ1 (x1 ) is bounded based on (26), which in turn ensures the boundedness of θˆ1 (t) in (36) from Lemma 1. The signal α2an (x1 , θˆ1 , t) in (35) is then bounded based on (26), which in turn im(k) plies the boundedness of q2 , k = 0, · · · , n − 2 in (31), again based on Lemma 18

1. On the other hand, it is apparent that m1B g1 and g1 α2an are bounded, which, together with the boundedness of q2 , guarantee the boundedness of the error signal ζ2 in (34) by definition. Similarly, m2B z2 (¯ x2 , q2 , t)φ2 (¯ x2 ) is bounded ˆ based on (26), which in turn ensures the boundedness of θ2 (t) in (44). The (1) x2 , q¯2 , θˆ2 , t) follows immediately. It implies the boundboundedness of α3an (¯ (k) edness of q3 , k = 0, · · · , n − 2 in (51), again based on Lemma 1. Since ζ3 = g2 m2B q3 − g2 α3an and the boundedness of g2 m2B and g2 α3an is apparent, together with the boundedness of q3 , therefore the boundedness of ζ3 can be attained. The rest proof will be proceeded via induction. For an arbitrary integer i, 3 ≤ i < n, we assume that αkan , 2 ≤ k ≤ i (m) are all bounded. Then qk , m = 0, · · · , n − i + 1 are all bounded by Lemma (ξ ) 1. Based on (26), it is easy to see that the signals miB zi (¯ xi , q¯λii , t)φi are bounded. Considering miB (¯ xi )zi φi as the bounded input to the θˆi (t) dynamics in (55), the boundedness of the signal θˆi (t) can be concluded from Lemma 1. an in (54), and hence the boundedness of The boundedness of the signals αi+1 an the filters’ outputs qi+1 and gi αi+1 follows immediately. The boundedness of the signals ζi+1 can then be easily obtained. Since the cases with i = 2, 3 are established earlier, the proof for the rest can be concluded by induction. Define the filter error vector ζ = [ζ2 , · · · , ζn ]T . It is now obvious that there exists some positive constant dζ , such that max ||ζ||2 ≤ dζ . t≥0

(67)

By viewing the error vector ζ as the additive bounded disturbance to the closed-loop z-dynamics in (64), the corresponding stability analysis can be expected to be much simpler than the existing DSC based backstepping control designs [6], [11]. Consider the following Lyapunov function n

V

X 1 T (z z + σi−1 θ˜iT θ˜i ). = 2 i=1

Differentiating V along the system (64) yields

V˙

T

= z z˙ +

n X

σi−1 θ˜iT θ˜˙ i

i=1

19

(68)

T

= −kz z z + −fi ]} +

n X

u

zi f zi {miB (θ˜iT φi + i ) + (1 − miB )[−fiu tanh( i ) w0

i=1 n i−1 XX i=2 k=1

+

n−1 X

zi ζi+1 +

i=1

∂αir ∂αr [−zi fku tanh(w0−1 zi i fku ) − zi fk ] ∂xk ∂xk

n X

σi−1 θ˜iT θ˜˙ i

i=1

n n X i−1 X X 1 i 2 T 2 i ≤ −kz z z + [ mB (zi + di ) + (1 − mB )cη δ] + cη δ 2 i=1 i=2 k=1 n

X 1 + (z T z + ζ T ζ) − σ ¯i θ˜iT θˆi 2 i=1 n

1 X ˜T ˜ ≤ −(kz − 1)z z − σ ¯i θi θi + dv 2 i=1 T

≤ −kv V (t) + dv

(69)

It can be easily deduced from (69) that 1 dv dv |z(t)|2 ≤ V (t) ≤ (V (0) − )−kv t + , 2 kv kv

∀|z(0)| ≥ zb ,

t ≥ 0. (70)

By taking square root on both sides of the inequality (69), it yields r p dv |z(t)| ≤ 2V (t) ≤ 2[V (0) − ]e−kv t + zb2 kv

(71)

Since the first term within the square root on the right-hand side of (71) decreases monotonically to zero, therefore given an arbitrary constant zc > zb , we can always find a positive constant T such that |z(t)| ≤ zc for any t > T . The uniform ultimate boundedness of z(t) then follows by Definition 1. In particular, z1 (t) will converge eventually to the set Ωz1 , whose size can be made arbitrarily small by choosing kz , σk , σ ¯k , k = 1, · · · , n sufficiently large. Remark 2. As mentioned, most existing DSC based control designs ensure SGUUB stability of the closed-loop system. However, the corresponding stability criteria for the control gains and the filter time constants to fulfill rely on the prior knowledge of the maximal value of a complex function in a prescribed DOA, which is difficult to estimate. In contrast, by virtue of the 20

bounded stability of the filter dynamics built in the first place, it is noted that the resulting closed-loop system in (64) has exactly the same structure as the previous one in [4], with merely an additional bounded disturbance term ζ arising from the introduced DSC scheme. After that, the rest of stability analysis and the final rule of gain selection are basically the same as in [4]. Remark 3. The major advantages of the design here over the existing similar schemes are especially conspicuous when the states run out of the neural active regions. There are at least three scenarios for this to happen, 1) poor transient behaviors; 2) exogenous disturbances; 3) the DOA does not actually cover the whole neural active region. Instability or unacceptable tracking performances may occur in the existing similar schemes since no adequate control actions are involved outside the neural active regions. In contrast, the proposed design prohibits their occurrence by invoking the auxiliary robust controllers under such circumstances. Remark 4. The virtual controllers αi+1 in (54) and the actual controller u in (62) may seem a little bit complicated, nevertheless, as explained in remark 1 and by the nested-supset relation in (20), all robust control components αkr , 2 ≤ k ≤ i + 1 will be shut down while only the neuroadaptive control components αkan will be activated when x¯i ∈ Ωi . Since the system state stays in the neural active region most of the time, therefore all the robust control components αkr , 2 ≤ k ≤ n and ur will be shut down under such circumstances. Consequently, the computational complexity in the proposed controller reduces to a general DSC based neuroadaptive control algorithm for most of the operation. 5. Simulation To illustrate the validity of the proposed design, simulation on a secondorder system is conducted in this section. Example 1: Consider the following system x˙ 1 = g1 (x1 )x2 − f1 (x1 ) x˙ 2 = g2 (x)u − f2 (x) y = x1

(72)

g1 = 1.0, g2 = 1.0, f1 = −(x1 + x31 ), f2 = −(x1 x2 + x21 + x22 ).

(73)

where

21

The reference output is yd = sin(t). The number of neurons for approximating f1 and f2 are 15 and 225, respectively. The corresponding neural active regions are given by [−1.4, 1.4] and [−1.4, 1.4] × [−2.8, 2.8], respectively. Note that a larger number of neurons generally leads to a higher accuracy of tracking. However, poorer transient may result due to the necessity of simultaneous update of a large number of estimated neural weights. Therefore, a trade-off might beneeded regarding this matter in real applications. The filter is constructed as b0 q˙ + q = α2an , b0 = 0.001. Contrary to most existing DSC based designs, it is reminded that the value of b0 here is irrelevant to the stability of the closed-loop system. However, it is beneficial to choose a small time constant when taking (13) into account. The bounding functions for the unknown f1 and f2 are q q q u u 2 6 f1 = 2.0( x1 + 1 + x1 + 1), f2 = 2.0( x21 x22 + 1 + x21 + x22 ).(74) The rest of the numerical values used in the simulation are kz1 = 10.0, kz2 = 5.0, σ1 = σ2 = 5.0, w0 = 0.05, r1 = 1.4, r2 = 2.8, r¯1 = 1.5, r¯2 = 2.9. The corresponding controllers are given by α2an = m1B g1−1 (h1 + θˆ1T φ1 ), α2r = (1 − m1B ){−q2 + g1−1 [h1 − f1u tanh(

z1 f1u )]}, w0

uan = m2B g2−1 (q˙2 + h2 + θˆ2T φ2 ), ∂α2r ∂α2r ∂αr ∂αr ur = g2−1 { 2 [x2 − f1u tanh(w0−1 z2 2 f1u )] + q˙2 + ∂x1 ∂x1 ∂q2 ∂t u z2 f +(1 − m2B )[q˙2 + h2 − f2u tanh( 2 )]}. w0

(75)

The first case study is with an initial condition of [x1 (0), x2 (0)]T = [0.3, 0.2]T , while the estimated neural weights are assigned randomly within [0, 0.1]. The system starts within Ω2 , and hence it will behave the same under either the proposed design or the existing DSC-based schemes as long as the trajectory stays within Ω2 for all time. The output tracking result has a high accuracy as shown in Fig. 1. It is reminded that the proposed scheme, just like other similar designs, does not guarantee the invariance of the neural active regions. As evidenced by Fig. 2, the x2 state, though remaining bounded, escapes from Ω2 for a short period of time. This corresponds to the first scenario outlined in remark 3. During that period of time, the auxiliary 22

robust controller ur will be activated to bring the escaped transients back, as shown by the switching signals in Fig. 3. It justifies the necessity of including the extra robust controllers to ensure the GUUB stability of the closed-loop system. The price is the increase of control complexity due to the activation of ur when m2B 6= 0. Nevertheless, the computational burden is still greatly reduced when compared to the original design in [4] since both switching signals m1B and m2B are equal to one most of the time. On the other hand, the effectiveness of the neural compensation is evidenced by the on-line identification of the unknown functions f1 and f2 shown in Figs. 4-5. Note that the accuracy of identification depends on various factors such as the number of neurons, centers and spreads of the neural gaussian functions, and the so-called persistence of excitation [3, 27]. Therefore, it may lead to different levels of identification accuracy when unknown functions f1 , f2 and/or reference trajectories are different. To further highlight the contribution of the proposed design, a second case study with the initial lying outside Ω1 , i.e., [x1 (0), x2 (0)]T = [2.0, 2.5]T , is conducted next. Such an initial may happen in real applications due to big exogenous disturbances, i.e., the second scenario outlined in remark 3. The reference trajectory is changed to 0.5(sin t + sin 0.5t) in this case. The output trajectories without and with switching are depicted respectively in Figs. 6-7. It is observed that instability happens in the former case while the robust controller ur is activated to bring the transient back in the latter case. The tracking accuracy is as good once the transient is brought back into the neural active regions. The corresponding switching signals are shown in Fig. 8. As can be seen, the neuroadaptive controller still dominates and therefore the computation burden is significantly reduced when compared to the original design. Last, a third case study with the initial close to the edge of Ω1 , i.e., [x1 (0), x2 (0)]T = [1.4, 2.0]T , is conducted here to demonstrate the validity of the proposed design in handling the third scenario outlined in remark 3. The output trajectories without and with switching are depicted respectively in Figs. 9-10. The instability depicted in Fig. 9 is effectively annihilated by the proposed design as can be seen in Fig. 10. The reason for such a discrepancy is the activation of the auxiliary robust controller ur outside the neural active region, as evidence by the switching signals in Fig. 11. It is reminded that, due to the difficulty of the estimation of the DOA, such a phenomenon is likely to happen in real applications. Example 2: Next we’ll apply our design to the Brusselator model, which 23

describes the oscillatory behaviors of certain types of chemical reactors and serves as a paradigm for the research of chaos. The controlled Brusselator model is in a form of (72) with [4, 28] g1 = x21 , g2 = 2 + cos(x1 ), f1 = (p2 + 1)x1 − p1 , f2 = x21 x2 − p2 x1 . where p1 , p2 > 0 are the system parameters. The bounding functions for the unknown functions f1 and f2 are q u f1 = 2.0[p1 + (p2 + 1) x21 + 1], q q f2u = 3.0[x21 x22 + 1 + p2 x21 + 1]

(76)

(77)

To ensure the applicability of the backstepping tool, the affine function g1 in (76) has to be nonzero for all time, which in turn implies that the state x1 has to nonzero for all time. The reference trajectory is 3 + 1.5 cos(t)/(1 + sin2 t). It is actually a biased one-dimensional lemniscate curve with width 1.5. The initial condition is [x1 (0), x2 (0)]T = [2.0, 0.5]T . The neural active regions are given by Ω1 = [1.6, 4.4] and Ω2 = [1.6, 4.4] × [−2.8, 2.8], which are those in Example 1 with their centers shifted to [3.0, 0]T . Note that the switching functions in (25) have to be shifted simultaneously. The control gains are kz1 = 10.0, kz2 = 10.0,, and the remaining numerical parameters are identical to those in Example 1. For comparisons, the output trajectories without and with switching are depicted respectively in Figs. 12-13. Due to the periodical traverses the boundary of Ω1 by the system’s trajectory, the pure adaptive DSC neural controller exhibits poor tracking performances at the start and around the peak of yd . In contrast, by properly activating the auxiliary robust controller the proposed design improves such a drawback significantly as shown in Fig. 13. The same discrepancy appears in the x2 trajectories in Figs. 14-15. The switching signals in Figs. 16 provide evidences for the above reasoning. The simulation results on a practical system model here further highlight the advantages of the proposed design over the existing similar designs in the literature. 6. CONCLUSION The paper aims to solve the so-called explosion of complexity problem in a previous global neuroadaptive control design for strict-feedback systems. The 24

popular DSC approach can not be directly applied to this issue as it only ensures the SGUUB stability in general. By suitably modifying the DSC scheme and restricting it to the neural active region, the above-mentioned problem is solved for the neuroadaptive control component, while the complexity of the auxiliary robust controller remains unchanged for preserving the achieved global stability. The modification is two-fold, as high-order instead of the common first-order filters are adopted, and second, the inputs to those filters are proven to be bounded, which in turn ensures the boundedness of the filter states. The bounded stability of the filter dynamics is a very desirable property as it alleviates the difficulty in the stability analysis and gain selections. Compared to the previous design [4], the improvement is significant in terms of computation since the neuroadaptive controller dominates most of the time. It is also superior to most existing DSC-based designs for achieving global instead of semiglobal stability. By virtue of the achieved modularity, the proposed design is capable of incorporating with any backstepping based designs for solving the explosion of complexity problem. Among others, its extension to mobile robots [29][30] or strict-feedback systems in a networked control architecture, which might suffer from communication-induced time delays, packet dropouts, and measurement or quantization errors [31]-[33], is particularly of our interests. It can be expected that, via incorporating the modified DSC algorithm here to the above-mentioned systems, the two common drawbacks of most existing designs, i.e., semiglobal stability only and the difficulty of deciding the required control gains and time constants of the constructed filters, can be effectively eliminated. Finally, it is believed that implementation on real systems is always a better way for demonstrating the validity of a control design. Regarding this, application of the design here to a mobile robot in [30] with implementation will soon be undertaken in the near future. Acknowledgment The author would like to thank the Associate Editor and the reviewers for their comments which greatly improve the presentation of the paper. This work is supported by the Ministry of Science and Technology, with granted number 102-2221-E-034-014. [1] M., Krstic, I., Kanellakopoulos, and P.V., Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. 25

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29

Figure Caption • Fig. 1 Output trajectory of the first case study in Example 1. • Fig. 2 x2 trajectory of the first case study in Example 1. • Fig. 3 m1B , m2B of the first case study in Example 1. • Fig. 4 Estimated f1 function of the first case study in Example 1. • Fig. 5 Estimated f2 function of the first case study in Example 1. • Fig. 6 Output trajectory of the second case study without switching in Example 1. • Fig. 7 Output trajectory of the second case study with switching in Example 1. • Fig. 8 m1B , m2B of the second case study with switching in Example 1. • Fig. 9 Output trajectory of the third case study in Example 1 without switching. • Fig. 10 Output trajectory of the third case study in Example 1 with switching. • Fig. 11 m1B , m2B of the third case study with switching in Example 1. • Fig. 12 Output trajectory of the case without switching in Example 2. • Fig. 13 Output trajectory of the case with switching in Example 2. • Fig. 14 m1B , m2B of the case with switching in Example 2.

30

1.5 y yd

1

Output

0.5 0 −0.5 −1 −1.5 0

5

10

15 20 Time (sec)

25

31 Figure 1: Output trajectory of the first case study in Example 1.

30

4

2

x2

0

−2

−4

−6 0

5

10

15 Time (sec)

32

20

Figure 2: x2 trajectory of the first case study in Example 1.

25

30

1

mB

0.8 0.6 0.4 m1 B

0.2

m2 B 0 0

5

10

33

15 Time (sec)

20

Figure 3: m1B , m2B of the first case study in Example 1.

25

30

4 Estimated f1

3

f1

Estimated f1

2 1 0 −1 −2 −3 0

5

10

15 Time (sec)

34

20

Figure 4: Estimated f1 function of the first case study in Example 1.

25

30

16 Estimated f2

14

f2

Estimated f2

12 10 8 6 4 2 0 0

5

10

35

15 Time (sec)

20

Figure 5: Estimated f2 function of the first case study in Example 1.

25

30

4

2

x 10

y yd

Output

1.5

1

0.5

0 0

0.2

0.4 Time (sec)

0.6

36 Figure 6: Output trajectory of the second case study without switching in Example 1.

0.8

2.5 2.5

2

y yd

2 1.5 1

1.5

0.5

Output

0

1

0

0.5

1

0.5 0 −0.5 −1 0

5

10

15 Time (sec)

20

25

37 Figure 7: Output trajectory of the second case study with switching in Example 1.

30

1.2 1 0.8

mB

1 0.6 0.4

0.5 0 0

0.5

1 m1 B

0.2 0 0

m2 B 5

10

15 Time (sec)

38

20

25

Figure 8: m1B , m2B of the second case study with switching in Example 1.

30

20000

Output

15000

y yd

10000

5000

0

−5000 0

0.02

0.04

0.06 0.08 TIme (sec)

0.1

39 Figure 9: Output trajectory of the third case study in Example 1 without switching.

0.12

1 y yd

Output

0.5

0

−0.5

−1

−1.5 0

5

10

15 Time (sec)

20

25

40 Figure 10: Output trajectory of the third case study in Example 1 with switching.

30

1

mB

0.8 0.6 0.4

1 0.5 m1 B

0 0

0.5

m2 B

1

0.2 0 0

5

10

15 Time (sec)

41

20

25

Figure 11: m1B , m2B of the third case study with switching in Example 1.

30

8 10

y yd

7 5

Output

6 0 0

5

0.5

1

4 3 2 1 0

10

20 30 Time (sec)

40

42 Figure 12: Output trajectory of the case without switching in Example 2.

50

5 4.5

Output

4 3.5 3 2.5 2 1.5 0

10

20 30 Time (sec)

43 Figure 13: Output trajectory of the case with switching in Example 2.

40

50

1.5 m1 B m2 B

mB

1

0.5

0 0

10

20 30 Time (sec)

44

Figure 14: m1B , m2B of the case with switching in Example 2.

40

50