IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 7, JULY 2012
Universal Neural Network Control of MIMO Uncertain Nonlinear Systems Qinmin Yang, Member, IEEE, Zaiyue Yang, Member, IEEE, and Youxian Sun Abstract— In this brief, a continuous tracking control law is proposed for a class of high-order multi-input–multi-output uncertain nonlinear dynamic systems with external disturbance and unknown varying control direction matrix. The proposed controller consists of high-gain feedback, Nussbaum gain matrix selector, online approximator (OLA) model and a robust term. The OLA model is represented by a two-layer neural network. The continuousness of the control signal is guaranteed to relax the requirement for the actuator bandwidth and avoid the incurred chattering effect. Asymptotic tracking performance is achieved theoretically by standard Lyapunov analysis. The control feasibility is also verified in simulation environment. Index Terms— Asymptotic convergence, neural networks, Nussbaum gain, online approximators.
I. I NTRODUCTION The adaptive control of nonlinear systems with unknown dynamics has received a great deal of attention and various approaches have been proposed in recent decades [1]–[5] including direct adaptive control [6] and indirect adaptive control [7] techniques. In order to achieve an asymptotic tracking, sliding mode controllers are also derived [8]. However, this controller design is discontinuous, and thus requires infinite actuator bandwidth and generates chattering [9]. Recently, due to their universal approximation properties [10], online approximation (OLA)-based control techniques have been utilized [11]–[14] extensively to parameterize the unknown plant nonlinearities. However, OLA-based control methodologies typically deliver uniformly ultimately bounded stability results due to their inherent functional reconstruction errors and external unknown disturbances [11]. In the recent literature, a significant effort is in place to achieve asymptotic stability, while also ensuring the continuousness of the control signal [9], [15]. Meanwhile, [15] removes the requirement of a priori knowledge of bounds on system dynamics and disturbance. However, they either utilize projection algorithm [9] or require persistence of excitation (PE) condition [13] to guarantee the boundedness of the neural network (NN) weights. The use of projection algorithm demands the selection of a predefined convex set so as to force the target NN weights [16] to lie within the set which is a challenge, and PE condition is Manuscript received July 31, 2011; revised April 17, 2012; accepted April 20, 2012. Date of publication May 22, 2012; date of current version June 8, 2012. This work was supported in part by the National Natural Science Foundation of China, under Grant 61004057 and Grant 61104008, and the Qianjiang Talent Program of Zhejiang Province, under Grant 2011R10024. The authors are with the State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310058, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2012.2197219
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generally not practical in most applications. Further, they all assume the control direction, which is the multiplier of the control term, to be known. The control direction is vital to the stability and performance of a control system, because it determines the direction of the system motion under any control. Solving the unknown control direction problem has drawn intense interests in recent decades. Several methods have been proposed, among which the Nussbaum gain are widely studied [17]. This method involves an adaptive mechanism to identify the unknown control direction, and can be readily integrated with other control design techniques, such as robust control [18], adaptive control [19], learning control [20], and backstepping design [21]. When a multi-input–multi-output (MIMO) system is studied, the unknown control direction is a matrix instead of a scalar. However, most previous studies only investigate the scalar unknown control direction, which is either positive or negative. Though the MIMO systems with unknown control directions are also considered in [22] and [23], the systems considered consist of n scalar unknown control directions rather than an n-by-n unknown control direction matrix. Thus, [22], [23] still utilize the conventional scalar Nussbaum gain. Because the unknown control direction matrix may be neither positive nor negative definite, e.g., [1, 0; 0, −1], the conventional scalar Nussbaum gain cannot be directly applied. In fact, a matrix selector has to be designed and then integrated with the Nussbaum gain to tackle the unknown varying control direction matrix. Therefore, the objective of this brief is to derive a continuous tracking control law for a general class of highorder MIMO uncertain nonlinear systems, which consists of both unknown system dynamics and unknown varying control direction matrix. The NN weights are tuned online without offline training phase. The knowledge of the bounds on system uncertainties and NN reconstruction errors is not required. Furthermore, the PE condition requirement to achieve stability is relaxed by adopting a novel NN updating law which also eliminates the need of choosing an appropriate convex set containing the target weights. The control problem is formulated in Section II. Section III presents the proposed control method, followed by the asymptotic convergence analysis in Section IV. The control performance is illustrated by simulations in Section V. Section VI draws the conclusion. II. P ROBLEM F ORMULATION Consider a class of high-order MIMO uncertain nonlinear systems x (m) = f (X) + B(X)u + d(t)
y=x
(1)
where t denotes the time horizon; x, y and u are n-dimensional state, output and control vectors. m denotes the order of the system. The term T X (t) ≡ x(t)T x(t) ∈ Rnm ˙ T . . . x (m−1) (t)T
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is the vector of all states and their derivatives, and d(t) ∈ Rn represents the unknown bounded disturbance vector. f (·) : Rnm → Rn and B(·) : Rnm → Rn×n are uncertain nonlinear smooth functions, (·)(i) (t) denotes the i th derivative with respect to time. The unknown control direction matrix B(X) is always nonsingular but has eigenvalues with unknown signs. The control objective is to drive the output to track a given trajectory yd , from an arbitrary initial condition y0 . Assumption 5: The disturbance d(t) is bounded above, i.e., d(t) ≤ d M , where d M ∈ R+ is an unknown positive constant with · denoting the standard Euclidean norm. III. C ONTROLLER M ETHODOLOGY
Thus, the ultimate control objective is to find an appropriate control law u such that y tracks yd eventually, namely, e → 0 when t → ∞. By recalling (5), a desired controller can be designed as
(7) u d = −B −1 (X) F( X¯ ) + kr with k being a positive constant. Substituting (7) into (5) yields r˙ = −kr + d.
(8)
Therefore, the selection of the control law ensures the convergence of the tracking error to zero if no disturbance is acting on the system. However, since both F( X¯ ) and B −1 (X) are unknown nonlinear functions, the feedback control u d cannot be implemented directly in practice.
A. Dynamics of Filtered Tracking Error First of all, it is reasonable to assume that yd is attainable by (1) when the disturbance is absent, i.e., the dynamics of yd can be written as follows: yd(m) = f (X d ) + B(X d )u d (X d )
(2)
where u d (Xd ) is the desired control law, which is unknown and * + T T X d ≡ x dT x˙dT . . . x d(m−1) . Hence, define the tracking error as e = y − yd .
(3)
Thereafter, the filtered tracking error can be defined as r ≡ λn−1 e(m−1) +λn−2 e(m−2) +· · ·+λ0 e =
m−1
λi e(i)
(4)
i=0
where λ0 , . . . , λn−1 are appropriately chosen constants such that λn−1 s m−1 + λn−2 s m−2 + · · · + λ0 is Hurwitz [15]. Consequently, e → 0 exponentially when r → 0. Without loss of generality, let λn−1 = 1. The dynamics in terms of the filtered tracking error can be readily obtained from (1)–(4) as r˙ = = =
m−1 i=0 m−2 i=0 m−2
λi e(i+1) =
m−1
λi (y − yd )(i+1)
i=0
λi (y − yd )(i+1) + (y − yd )(m) λi (y − yd )(i+1) + f (X) + B(X)u + d
i=0
− f (X d ) − B(X d )u d (X d ) = F( X¯ ) + B(X)u + d T where X¯ ≡ X T X dT ∈ R2 nm and ¯ = F( X)
m−2
(5)
λi (y − yd )(i+1) + f (X)
B. Unknown Varying Control Direction Matrix First, consider the varying and unknown control direction matrix B(X), and it is critical to build a negative feedback closed-loop. Because a matrix rather than a scalar is considered here, the matrix selector has to be introduced and combined with the Nussbaum gain to tackle B(X). Let B(X) consist of a nominal part B0 and a varying part (X) as follows: B (X) = B0 + (X) (9) where B0 ∈ Rn×n is nonsingular and constant, and (X) : Rnm → Rn×n is varying and could be singular. Both B0 and (X) are unknown. However, it is assumed that the norm of (X) is bounded by a constant to be specified later in Assumption 2, such that B(X) is always nonsingular. In order to deal with the unknown control direction matrix B(X), the following lemma has to be introduced first. Lemma 1 [24]: There exists a finite set of n-by-n invertible matrices {K 0 , …, K N−1 } so that, for any n-by-n invertible matrix D, there exists i ∈{0, …, N− 1} such that the spectrum of DK i is in the open right half of the complex plane, i.e., real (σ (DK i )) > 0. Remark 4: Assume that K i is normalized in the sense that ||K i || = 1; otherwise, one can always find a normalized matrix K i /||K i || satisfying real (σ (DK i /||K i ||)) > 0. The set given in Lemma 1 is called an unmixing set, which is known for a given dimension n. For example, for n = 1, the unmixing set is {1, −1}; for n = 2, the unmixing set is {ψ + (0), ψ + (2π/3), ψ + (4π/3), ψ − (0), ψ − (2π/3), ψ − (4π/3)}, where ψ + (ω) = [cos ω − sin ω; sin ω cos ω] and ψ − (ω) = [cos ω − sin ω; − sin ω − cos ω]. Remark 5: For our problem, let D = B0 , and then (X) can be treated as bounded nonlinear deviation from the nominal constant B0 . Therefore, there exists a matrix K i , such that the spectrum of B0 K i is in the open right half of the complex plane. Without loss of generality, let i = 0, i.e., real (σ (B0 K 0 )) > 0. As a result, the negative feedback closedloop is achievable by properly including the matrix K 0 into the control u. Since real (σ (B0 K 0 )) > 0, there exists a unique n-by-n positive definite matrix Q satisfying the Lyapunov equation
i=0
− f (X d ) − B(X d )u d (X d ).
(6)
(B0 K 0 )T Q + Q (B0 K 0 ) = 2I
(10)
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,∞
where Q = 0 exp − (B0 K 0 )T τ exp (−B0 K 0 τ ) dτ and I is an n-by-n identity matrix. Let the maximum and minimum eigenvalue of Q be Q max and Q min , respectively. Obviously, Q max ≥ Q min > 0. Assumption 6: Assume ||Q (X)|| ≤ p < 1, where ||Q (X)|| ≡ σ max (Q (X)) is the largest singular value of Q (X). Since ||K 0 || = 1 and (B (X) K 0 )T Q + Q (B (X) K 0 ) = 2I + ( (X) K 0 )T Q +Q ( (X) K 0 ).
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Remark 6: Although a two-layer NN is utilized here as an illustration of the linearly parameterized OLA models, the results of this brief can be readily extended to more general OLA structures, which could be radial basis functions, splines, polynomials, fuzzy logic, etc. Remark 7: The NN reconstruction error bound ε M is considered to be unknown in this brief as opposed to traditional NN control methodologies [9], [13]. Thereafter, substituting (13) into (5) generates r˙ = Q −1 W T φ( X¯ ) + B(X)u + d + ε.
(15)
Assumption 2 insures that (B (X) K 0 ) Q + Q (B (X) K 0 ) is always nonsingular. T
D. Nussbaum Gain and Matrix Selector C. OLA ˆ X; ¯ Wˆ ) is employed to cope with Thereafter, an OLA F( ¯ ˆ X¯ ; Wˆ ) is the output of an the unknown function F( X ). F( q ˆ OLA model and W ∈ R is a vector of adjustable parameters. In this brief, we consider linearly parameterized approximators, which means the output can be expressed as T -ˆ X¯ ; Wˆ ) F( ˆ X¯ ; Wˆ ) = Z · Wˆ , Z (t) = F( (11) ∂ Wˆ -ˆ ˆ W =W (t )
where Z is the sensitivity function between the output of the approximator and the adjustable parameters or weights. When a two-layer NN [13] is utilized, (11) becomes ¯ ˆ X; ¯ Wˆ ) = Wˆ T φ(V T X), F(
Z = φ(V T X¯ )
(12)
where V ∈ R2nm×N and Wˆ ∈ R N×n are hidden and output layer weights respectively, φ(·) : R N → R N is the activation function in the hidden layer and N is the number of hidden layer nodes. On the other hand, based on the well-known universal approximation property of the NN [12], the smooth function Q · F( X¯ ) can be represented by the same NN structure as ¯ = W T φ(V T X) ¯ + ε = W T φ( X¯ ) + ε Q · F( X)
(13)
where Q is defined in (10) and constant vector W ∈ R N×n is the target values of Wˆ , such that it minimizes the L 2 norm ˆ X¯ ; Wˆ ) over (energy-norm) distance between Q · F( X¯ ) and F( 2 nm ¯ all X ∈ ⊂ R , provided is an arbitrary compact set. The weights of the hidden layer V are omitted for the purpose of simplification. It is reasonable to assume that W is bounded such that W ≤ W M [13], and W and W M are only “artificial” quantities required for analysis purposes. ε represents the functional reconstruction error satisfying ε ≤ ε M [11], with ε M being an unknown positive constant. It is demonstrated in [13] that if the hidden layer weights, V , are chosen initially at random and held constant while N is sufficiently large, the NN approximation error ε can be made arbitrarily small since the activation function vector forms a basis vector. Furthermore, from (12) and (13), the parameter estimation error is defined as W˜ = Wˆ − W.
(14)
Then, the Nussbaum gain and the matrix selector are designed below to tackle the unknown and varying control direction matrix. Firstly, define the control performance index θ that is used as the driving force for the adaptive schemes of Nussbaum gain and the matrix selector θ˙ = r T , where
θ (0) = θ0 = 0
= k g r + Wˆ φ( X¯ ) + T
Dˆ 2 r Dˆ r + e−t
(16) .
(17)
k g ∈ R+ is a feedback gain and Dˆ ∈ R+ is an adaptive gain whose expression will be given later by (22). Then, the Nussbaum gain is given as (18) g (θ ) = exp θ 2 sin (2πθ ) . Because g(θ ) can swing from positive infinite to negative infinite according to the control performance θ , it can correct inappropriate deviation caused by erroneous previous control. At last, the matrix selector is K (θ ) = (1 − mod (η, 1)) K η + mod (η, 1) K !η"
(19)
where mod(a, b) gives the residual of a over b, and the index η in (19) is given by rescaling θ as follows: ⎧ 1
⎪ if mod (ξ, 1) ∈ 0, 18 ⎨ ξ − 4 8 − mod(ξ, 1) , 1 7 η = ξ , if mod (ξ, 1) ∈ 8 , 8 ⎪
⎩ 7 ξ + 4 mod (ξ, 1) − 8 , if mod (ξ, 1) ∈ 78 , 1 ξ = mod (|θ | , N ). Functionally, the matrix selector K (θ ) picks a matrix K j from the unmixing set according to the value of θ as given by (16). According to [20], the selection process of K j includes the rescaling of θ and the proportional mix of two successive K j s. In this way, K (θ ) will continuously shift from K j to K j +1 . According to the value of θ , g(θ )K (θ ) can vary from K 0 to K N−1 with an arbitrary magnitude. In other words, g(θ )K (θ ) provides an opportunity to eventually build a negative feedback with an appropriate magnitude of feedback gain.
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0 where y = r T tr W˜ T W˜ D˜ ∈ ⊂ Rn+2 and is an
E. Controller Design Let the control signal at time t be u(t) = g(θ )K (θ )R(r ) where the matrix
R(r ) =
(20)
arbitrary compact set containing y(t) = 0. U1 (y), U2 (y) can be defined as U1 (y) ≡ λ1 y2 ,
rr T r2
,
if r = 0
is used purely for the purpose of proof. Clearly, the control signal consists of the Nussbaum gain g(θ ), the matrix selector K (θ ), the high-gain filtered tracking error feedback k g r , the NN output Wˆ T φ( X¯ ), and a robust term ( Dˆ 2 r )/( Dˆ r + e−t ). It is noted that compared with traditional variable structure or sliding mode designs [8], a continuous robust term is utilized in (20) instead of signum functions to guarantee the continuousness of the control signal. Thus, the requirement of infinite bandwidth and chattering phenomenon in control input is avoided [9]. Finally, define the NN weight tuning law as W˙ˆ = kn φ( X¯ )r T − kr kn r Wˆ
1 1 2 2 1 1 1 1 1 1 λ1 ≡ min Q min , , , λ2 ≡ max Q max , , . 2 kn k D 2 kn k D (26) Note that from (23) and (24) we have kr 2 r W M − Dˆ r + r T Q (d + ε) + + D˜ r 4
≤ r − Dˆ + Dd + D˜ = 0. Thereafter, differentiating (23) yields
(21)
R+
are positive user design parameters. where kn , kr ∈ Further, the adaptive scalar Dˆ is written as ( t r (τ ) dτ Dˆ = k D (22) 0
where k D ∈ R+ is a positive constant. Remark 8: In comparison to [9], since the projection algorithm is not used, the challenge of selecting a predefined convex set is avoided. Furthermore, the boundedness of NN weights is proven simultaneously along with the asymptotic convergence of the tracking error in following section with no ¯ [13]. assumptions of PE condition on φ( X) Remark 9: In this control scheme, the continuous robust term ( Dˆ 2r )/( Dˆ r + e−t ) always exists and is employed to compensate the disturbance d(t) and the NN reconstruction error ε, the knowledge of whose bounds is not required to be known by designing Dˆ as an adaptive function of time instead of a constant as in [9] and [13]. IV. A SYMPTOTIC C ONVERGENCE The main theorem of the stability result for the proposed controller can now be stated. Theorem 2: Consider the closed-loop system consisting of the plant (1), the control given by (20), and the adaptive law (21), (22). Under Assumptions 1 and 2, the tracking error e → 0 asymptotically and closed-loop signals are all bounded. Proof: First, build a positive definite Lyapunov candidate 1 ˜2 1 ˜ ˜ T 1 tr W W + (23) U ≡ r T Qr + D , U0 =U (0). 2 2kn 2k D Meanwhile, it is important to observe that there exist two functions U1 (y) and U2 (y) such that U1 (y) ≤ U ≤ U2 (y)
(25)
where λ1 , λ2 ∈ R+ are defined as
if r = 0
I,
U2 (y) ≡ λ2 y2
(24)
U˙ = r T Q(F( X¯ ) + Bu + d) + tr W˜ r φ T ( X¯ )
−kr r tr W˜ Wˆ T + D˜ r
¯ )B(X)K (θ )R(r )+d +ε = r T Q Q −1 W T φ( X)+g(θ
¯ −kr r tr W˜ Wˆ T + D˜ r +tr W˜ r φ T ( X) = r T W Tφ( X¯ )+g(θ )r T Q B(X)K (θ )R(r )+ r T −r T
+r T Q (d +ε)+r T W˜ T φ( X¯ )−kr r tr W˜ Wˆ T + D˜ r
≤ r T W T + W˜ T φ( X¯ )+g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −r TWˆ Tφ( X¯ )−k g r 2 − Dˆ r +e−t +r T Q (d +ε)
−kr r tr W˜ T (W˜ + W ) + D˜ r ≤ g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t − Dˆ r + r T Q (d +ε)−kr r W˜ 2 +kr r W˜ W M + D˜ r
≤ g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t − Dˆ r '2 & 1 + r T Q (d + ε) − kr r W˜ − W M 2 kr 2 + r W M + D˜ r 4 ≤ g(θ )r T Q B(X)K (θ )R(r ) + θ˙ − k g r 2 + e−t '2 & 1 −kr r W˜ − W M 2 T ≤ g(θ )r Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t . (27)
Now consider g(θ )r T Q B(X)K (θ )R (r ) . If r = 0, we have g(θ )r T Q B(X)K (θ )R (r ) rr T r ˙ = g(θ )r T Q B(X)K (θ ) =r T Q B(X)K (θ ) g(θ )θ. r 2 r 2 (28)
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If K (θ ) = K 0 and from (10) we have
or equivalently
rT
r 1 = r T Q B(X)K (θ ) 2 2 r r
r × (Q (B0 + (X)) K 0 )T + Q (B0 + (X)) K 0 r
r 1 rT = 1+ . (29) (Q (X) K 0 )T + Q (X) K 0 r 2 r According to Assumption 2 and remember ||K j || = 1, we have
r 1 rT (Q (X) K 0 )T + Q (X) K 0 r 2 r ≤ Q (X) K 0 < p. Taking above inequality into (29) produces r r T Q B(X)K (θ ) = b (X) r 2 where b(X) ∈[1 − p, 1 + p] with 0 < p < 1. That is ˙ K (θ ) = K 0 . g(θ )r T Q B(X)K (θ )R (r ) = b (X) g(θ )θ,
(30)
Otherwise, if K (θ ) = K 0 r T Q B(X)K (θ )
r ≤L r 2
(31)
where L is the upper bound for ||QB(X)K (θ )||, which always exists because Q, B(X), and K (θ ) are all bounded. Thus, when K (θ ) = K 0 we have g(θ )r T Q B(X)K (θ )R (r ) ≤ L -g (θ ) θ˙
= Lsgn g (θ ) θ˙ g (θ ) θ˙ . (32) Then, we can write (30) and (32) into the compact form g(θ )r T Q B(X)K (θ )R (r ) ≤ ρ (θ ) g (θ ) θ˙ where ρ (θ ) is given below b (X),
ρ (θ ) = Lsgn g (θ ) θ˙ ,
if K (θ ) = K 0 if K (θ ) = K 0 .
(33)
(34)
If r = 0, we have g(θ )r T Q B(X)K (θ )R (r ) = 0, and (33) is still satisfied. Then, taking integration of (27) yields 0≤U
( t ( t
r 2 ds + r T g(θ )Q B(X)K (θ )R (r ) + θ˙ ds ≤ −k g 0 ( t0 −s + e ds + U0 0 ( t ( t 2 r ds + ≤ −k g (ρ (θ ) g (θ ) + 1) θ˙ ds + 1 + U0 0 0 ( θt ( t 2 ≤ −k g r ds + ρ (θ ) g (θ )dθ + θt + 1 + U0 (35) 0
0
which further gives ( θt ρ (θ ) g (θ )dθ + θt + 1 + U0 0≤ 0
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(
θt
− (θt + 1 + U0 ) ≤
ρ (θ ) g (θ )dθ
(36)
0
where θ t = θ (t). Note we have the following properties [20] ( 1 lim sup ρ (θ ) g (θ )dθ = +∞ (37) →∞ 0 and lim inf
→∞
1
(
ρ (θ ) g (θ )dθ = −∞.
(38)
0
Then, we can show that θ t is bounded by the method of contradiction. Suppose that θ t becomes divergent, then there are two possibilities: ,θ 1) if θt → +∞, (36) gives limθt →+∞ 1/θt 0 t ρ (θ ) g (θ )dθ ≥ −1, which contradicts with (38); , θ 2) if θt → −∞, (36) gives limθt →−∞ 1/θt 0 t ρ (θ ) g (θ )dθ ≤ −1, which contradicts with (37). ,θ As a result, θ t is bounded, and 0 t ρ (θ ) g (θ )dθ is also bounded; hence, we can find a positive constant C, such that , θt ρ g (θ ) (θ )dθ + θt + 1 + U0 ≤ C Q min . Then (23) and 0 (35) give ( t Q min r 2 ds + C Q min r 2 ≤ U ≤ −k g 0≤ 2 0 or equivalently 0≤
1 U 1 r 2 ≤ ≤ −k g 2 Q min Q min
as well as 0 ≤ kg
1 Q min
(
t
(
t
r 2 ds + C
(39)
0
r 2 ds ≤ C.
(40)
0
√ Since ||r || is continuous and r ≤ 2C from (39), 2 is Lipschitz continuous because |r 2 − r ∗ 2 | ≤ ||r || √ 2C |r − r ∗ | for any pair of r and r ∗ . Hence ||r ||2 is uniformly continuous, because very Lipschitz continuous function is uniformly continuous. Since ||r ||2 is also nonnegative and integrable, according to Barbalat’s lemma (40) implies ||r || → 0 as t → ∞, which further gives ||e|| → 0 as t → ∞. Then, we can conclude the asymptotic convergence of the tracking error. Furthermore, the boundedness of other signals also can be readily obtained. Equation (39) shows the uniform boundedness of U since the right hand side term is bounded. Therefore, W˜ and D˜ are bounded according to (23), which also implies the boundedness of Wˆ by definition. Dˆ is also bounded as Dd ˜ + Dd from (24). Thus, (17) and D˜ are bounded and Dˆ = D(t) and (20) show the boundedness of control signal. V. S IMULATION In order to demonstrate the feasibility of the theoretic results, the proposed adaptive NN controller is applied to the following third-order MIMO nonlinear system: (3) x1 (3) x = (3) = f (x) + B (x) u = f (x) + ( (x) + B0 ) u. x2
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(b) Fig. 3. System performance with the proposed controller. (a) Norm of NN ˆ weights. (b) Adaptive term D.
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Afterwards, a spike appears in the control signal when the controller realizes that the current Nussbaum gain matrix is not right and need search for the correct one. After all, from the simulation results, we can find that the design is capable of attaining satisfactory tracking performance while all other signals of the closed-loop system bounded.
-1500 -2000
Fig. 2.
Control input signal with the proposed controller.
0 . . 2 0 0 cos x 2 sin (x 2 ) + x 1 , (x) = 0.4 f (x) = and 2 sin x 1 0 . x1 + 0 sin x 2 1 −2 B0 = are unknown for control design. For this 2 −1 example, according to Remark 3 it can be readily found that 4π 0
. cos 4π 3 − sin 3 K0 = sin 4π cos (4π3) 3 can satisfy the condition of real (σ (B0 K 0 )) > 0. The objective of the control signal is to make the system output x follow the reference trajectory defined as yd = [− sin 2πt; cos 2πt]. The filtered tracking error is defined as r = e¨ + 4e˙ + 10e. The NN contains ten hidden neurons (i.e., N = 10) and traditional sigmoidal activation functions are employed. The hidden layer weights are selected initially at random and held constant. The other control parameters are: k g = 0.05, kn = 0.001, and k D = 0.01. A typical system response using the proposed controller is shown in Fig. 1 including the actual and desired system output trajectories. With unknown control direction matrix and the presence of disturbance and NN reconstruction error, the actual output can still asymptotically track the desired values. Fig. 2 illustrates the control signals and the norm of NN weights and the adaptive term Dˆ are also demonstrated in Fig. 3. During the initial phase, since the tracking error of x 2 is converging, the controller believes that the current Nussbaum gain matrix could be proper and thus remains low. Therefore, the output of the system is kept steady.
VI. C ONCLUSION In this technical note, a novel continuous control designs are proposed by incorporating with the Nussbaum gain matrix selector and the NN for a general class of MIMO high-order uncertain nonlinear systems with bounded disturbance and unknown state-dependent control direction matrix. Asymptotic tracking performance is obtained with the NN reconstruction error and external disturbance present, whose bounds are not required to be known. The risk brought by the projection algorithm and requirement of PE condition are also avoided. The theoretical analysis and the simulation results show the effectiveness of the proposed schemes. R EFERENCES [1] M. Chen, S. S. Ge, and B. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [2] Y. J. Liu, C. L. P. Chen, G. X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1162–1167, Jul. 2011. [3] A. K. Kostarigka and G. A. Rovithakis, “Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 22, no. 1, pp. 138–149, Jan. 2012. [4] Z. Hou and S. Jin, “Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 2173–2188, Dec. 2011. [5] A. K. Kostarigka and G. A. Rovithakis, “Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 22, no. 1, pp. 138–149, Jan. 2012. [6] J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [7] P. Ioannou and B. Fidan, Adaptive Control Tutorial (Advances in Design and Control). Philadelphia, PA: SIAM, 2006.
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[8] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [9] P. M. Patre, W. MacKunis, K. Kaiser, and W. E. Dixon, “Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and RISE feedback control structure,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2180–2185, Oct. 2008. [10] H. Lee and M. Tomizuka, “Robust adaptive control using a universal approximator for SISO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 1, pp. 95–106, Feb. 2000. [11] J. A. Farrell and M. M. Polycarpou, “Adaptive approximation based control: Unifying neural,” in Fuzzy and Traditional Adaptive Approximation Approaches. New York: Wiley, 2006. [12] D. A. White and D. A. Sofge, Eds., Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York: Van Nostrand, 1993. [13] F. L. Lewis, S. Jagannathan, and A. Yesilderik, Neural Network Control of Robot Manipulators and Nonlinear Systems. New York: Taylor & Francis, 1999. [14] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [15] Q. Yang and S. Jagannathan, “NN/RISE-based asymptotic tracking control of uncertain nonlinear systems,” in Proc. IEEE Int. Symp. Intell. Control, Sep. 2011, pp. 1361–1366. [16] W. T. Miller, R. S. Sutton, and P. J. Werbos, Eds., Neural Networks for Control. Cambridge, MA: MIT Press, 1990. [17] R. D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, no. 5, pp. 243–246, 1983. [18] L. Liu and J. Huang, “Global robust stabilization of cascade-connected systems with dynamic uncertainties without knowing the control direction,” IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1693–1699, Oct. 2006. [19] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1463–1469, Aug. 2003. [20] Z. Yang, S. C. P. Yam, L. K. Li, and Y. Wang, “Universal repetitive learning control for nonparametric uncertainty and unknown control gain matrix,” IEEE Trans. Autom. Control, vol. 55, no. 7, pp. 1710–1715, Jul. 2010. [21] Y. Zhang, C. Wen, and Y. C. Soh, “Adaptive backstepping control design for systems with unknown high-frequency gain,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2350–2354, Dec. 2000. [22] T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, no. 6, pp. 1021–1033, 2007. [23] T. P. Zhang and S. S. Ge, “Adaptive neural network tracking control of MIMO nonlinear systems with unknown dead zones and control directions,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 483–497, Mar. 2009. [24] A. Ilchmann, “High-gain adaptive stabilization of multivariable linear systems–revisited,” Syst. Control Lett., vol. 18, no. 5, pp. 355–364, 1992.
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Spectral Graph Optimization for Instance Reduction Konstantinos Nikolaidis, Student Member, IEEE, Eduardo Rodriguez-Martinez, John Yannis Goulermas, Senior Member, IEEE, and Q. H. Wu, Fellow, IEEE Abstract— The operation of instance-based learning algorithms is based on storing a large set of prototypes in the system’s database. However, such systems often experience issues with storage requirements, sensitivity to noise, and computational complexity, which result in high search and response times. In this brief, we introduce a novel framework that employs spectral graph theory to efficiently partition the dataset to border and internal instances. This is achieved by using a diverse set of border-discriminating features that capture the local friend and enemy profiles of the samples. The fused information from these features is then used via graph-cut modeling approach to generate the final dataset partitions of border and nonborder samples. The proposed method is referred to as the spectral instance reduction (SIR) algorithm. Experiments with a large number of datasets show that SIR performs competitively compared to many other reduction algorithms, in terms of both objectives of classification accuracy and data condensation. Index Terms— Graph Laplacian, instance instance-based learning, prototype reduction.
selection,
I. I NTRODUCTION In instance-based learning, algorithms suffer from two principal issues: first, the need to store the entire dataset in some type of memory, and second, the increased time complexity from having to search large portions of the stored prototypes in order to predict new queries. A third concern is the noisy instances present in the database that can degrade the performance of the system. In order to address these problems, instance reduction techniques which reduce the size of the stored datasets without compromising prediction accuracies, have attracted significant interest. One such variation, the editing algorithm, focuses only on improving classification accuracy by removing harmful instances. One of the simplest such examples is the edited nearest neighbor (ENN) [1], which is still widely used as a noise-filtering preprocessor within other methods [2], [3]. The majority of instance reduction techniques, however, deal not only with the harmful noisy instances but also with the redundant ones, in order to mitigate the aforementioned storage and search requirements without degrading prediction accuracies. Since instances that lie near the class boundaries hold the most valuable information to identify the decision surface and distinguish between different classes, many of the recent instance reduction algorithms employ various heuristics to retain border instances over Manuscript received January 27, 2012; revised April 2, 2012; accepted May 3, 2012. Date of publication May 30, 2012; date of current version June 8, 2012. This work was supported by a DTA Scholarship from the University of Liverpool. The authors are with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2012.2198832
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Universal Neural Network Control of MIMO Uncertain Nonlinear Systems Qinmin Yang, Member, IEEE, Zaiyue Yang, Member, IEEE, and Youxian Sun Abstract— In this brief, a continuous tracking control law is proposed for a class of high-order multi-input–multi-output uncertain nonlinear dynamic systems with external disturbance and unknown varying control direction matrix. The proposed controller consists of high-gain feedback, Nussbaum gain matrix selector, online approximator (OLA) model and a robust term. The OLA model is represented by a two-layer neural network. The continuousness of the control signal is guaranteed to relax the requirement for the actuator bandwidth and avoid the incurred chattering effect. Asymptotic tracking performance is achieved theoretically by standard Lyapunov analysis. The control feasibility is also verified in simulation environment. Index Terms— Asymptotic convergence, neural networks, Nussbaum gain, online approximators.
I. I NTRODUCTION The adaptive control of nonlinear systems with unknown dynamics has received a great deal of attention and various approaches have been proposed in recent decades [1]–[5] including direct adaptive control [6] and indirect adaptive control [7] techniques. In order to achieve an asymptotic tracking, sliding mode controllers are also derived [8]. However, this controller design is discontinuous, and thus requires infinite actuator bandwidth and generates chattering [9]. Recently, due to their universal approximation properties [10], online approximation (OLA)-based control techniques have been utilized [11]–[14] extensively to parameterize the unknown plant nonlinearities. However, OLA-based control methodologies typically deliver uniformly ultimately bounded stability results due to their inherent functional reconstruction errors and external unknown disturbances [11]. In the recent literature, a significant effort is in place to achieve asymptotic stability, while also ensuring the continuousness of the control signal [9], [15]. Meanwhile, [15] removes the requirement of a priori knowledge of bounds on system dynamics and disturbance. However, they either utilize projection algorithm [9] or require persistence of excitation (PE) condition [13] to guarantee the boundedness of the neural network (NN) weights. The use of projection algorithm demands the selection of a predefined convex set so as to force the target NN weights [16] to lie within the set which is a challenge, and PE condition is Manuscript received July 31, 2011; revised April 17, 2012; accepted April 20, 2012. Date of publication May 22, 2012; date of current version June 8, 2012. This work was supported in part by the National Natural Science Foundation of China, under Grant 61004057 and Grant 61104008, and the Qianjiang Talent Program of Zhejiang Province, under Grant 2011R10024. The authors are with the State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310058, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2012.2197219
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generally not practical in most applications. Further, they all assume the control direction, which is the multiplier of the control term, to be known. The control direction is vital to the stability and performance of a control system, because it determines the direction of the system motion under any control. Solving the unknown control direction problem has drawn intense interests in recent decades. Several methods have been proposed, among which the Nussbaum gain are widely studied [17]. This method involves an adaptive mechanism to identify the unknown control direction, and can be readily integrated with other control design techniques, such as robust control [18], adaptive control [19], learning control [20], and backstepping design [21]. When a multi-input–multi-output (MIMO) system is studied, the unknown control direction is a matrix instead of a scalar. However, most previous studies only investigate the scalar unknown control direction, which is either positive or negative. Though the MIMO systems with unknown control directions are also considered in [22] and [23], the systems considered consist of n scalar unknown control directions rather than an n-by-n unknown control direction matrix. Thus, [22], [23] still utilize the conventional scalar Nussbaum gain. Because the unknown control direction matrix may be neither positive nor negative definite, e.g., [1, 0; 0, −1], the conventional scalar Nussbaum gain cannot be directly applied. In fact, a matrix selector has to be designed and then integrated with the Nussbaum gain to tackle the unknown varying control direction matrix. Therefore, the objective of this brief is to derive a continuous tracking control law for a general class of highorder MIMO uncertain nonlinear systems, which consists of both unknown system dynamics and unknown varying control direction matrix. The NN weights are tuned online without offline training phase. The knowledge of the bounds on system uncertainties and NN reconstruction errors is not required. Furthermore, the PE condition requirement to achieve stability is relaxed by adopting a novel NN updating law which also eliminates the need of choosing an appropriate convex set containing the target weights. The control problem is formulated in Section II. Section III presents the proposed control method, followed by the asymptotic convergence analysis in Section IV. The control performance is illustrated by simulations in Section V. Section VI draws the conclusion. II. P ROBLEM F ORMULATION Consider a class of high-order MIMO uncertain nonlinear systems x (m) = f (X) + B(X)u + d(t)
y=x
(1)
where t denotes the time horizon; x, y and u are n-dimensional state, output and control vectors. m denotes the order of the system. The term T X (t) ≡ x(t)T x(t) ∈ Rnm ˙ T . . . x (m−1) (t)T
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is the vector of all states and their derivatives, and d(t) ∈ Rn represents the unknown bounded disturbance vector. f (·) : Rnm → Rn and B(·) : Rnm → Rn×n are uncertain nonlinear smooth functions, (·)(i) (t) denotes the i th derivative with respect to time. The unknown control direction matrix B(X) is always nonsingular but has eigenvalues with unknown signs. The control objective is to drive the output to track a given trajectory yd , from an arbitrary initial condition y0 . Assumption 5: The disturbance d(t) is bounded above, i.e., d(t) ≤ d M , where d M ∈ R+ is an unknown positive constant with · denoting the standard Euclidean norm. III. C ONTROLLER M ETHODOLOGY
Thus, the ultimate control objective is to find an appropriate control law u such that y tracks yd eventually, namely, e → 0 when t → ∞. By recalling (5), a desired controller can be designed as
(7) u d = −B −1 (X) F( X¯ ) + kr with k being a positive constant. Substituting (7) into (5) yields r˙ = −kr + d.
(8)
Therefore, the selection of the control law ensures the convergence of the tracking error to zero if no disturbance is acting on the system. However, since both F( X¯ ) and B −1 (X) are unknown nonlinear functions, the feedback control u d cannot be implemented directly in practice.
A. Dynamics of Filtered Tracking Error First of all, it is reasonable to assume that yd is attainable by (1) when the disturbance is absent, i.e., the dynamics of yd can be written as follows: yd(m) = f (X d ) + B(X d )u d (X d )
(2)
where u d (Xd ) is the desired control law, which is unknown and * + T T X d ≡ x dT x˙dT . . . x d(m−1) . Hence, define the tracking error as e = y − yd .
(3)
Thereafter, the filtered tracking error can be defined as r ≡ λn−1 e(m−1) +λn−2 e(m−2) +· · ·+λ0 e =
m−1
λi e(i)
(4)
i=0
where λ0 , . . . , λn−1 are appropriately chosen constants such that λn−1 s m−1 + λn−2 s m−2 + · · · + λ0 is Hurwitz [15]. Consequently, e → 0 exponentially when r → 0. Without loss of generality, let λn−1 = 1. The dynamics in terms of the filtered tracking error can be readily obtained from (1)–(4) as r˙ = = =
m−1 i=0 m−2 i=0 m−2
λi e(i+1) =
m−1
λi (y − yd )(i+1)
i=0
λi (y − yd )(i+1) + (y − yd )(m) λi (y − yd )(i+1) + f (X) + B(X)u + d
i=0
− f (X d ) − B(X d )u d (X d ) = F( X¯ ) + B(X)u + d T where X¯ ≡ X T X dT ∈ R2 nm and ¯ = F( X)
m−2
(5)
λi (y − yd )(i+1) + f (X)
B. Unknown Varying Control Direction Matrix First, consider the varying and unknown control direction matrix B(X), and it is critical to build a negative feedback closed-loop. Because a matrix rather than a scalar is considered here, the matrix selector has to be introduced and combined with the Nussbaum gain to tackle B(X). Let B(X) consist of a nominal part B0 and a varying part (X) as follows: B (X) = B0 + (X) (9) where B0 ∈ Rn×n is nonsingular and constant, and (X) : Rnm → Rn×n is varying and could be singular. Both B0 and (X) are unknown. However, it is assumed that the norm of (X) is bounded by a constant to be specified later in Assumption 2, such that B(X) is always nonsingular. In order to deal with the unknown control direction matrix B(X), the following lemma has to be introduced first. Lemma 1 [24]: There exists a finite set of n-by-n invertible matrices {K 0 , …, K N−1 } so that, for any n-by-n invertible matrix D, there exists i ∈{0, …, N− 1} such that the spectrum of DK i is in the open right half of the complex plane, i.e., real (σ (DK i )) > 0. Remark 4: Assume that K i is normalized in the sense that ||K i || = 1; otherwise, one can always find a normalized matrix K i /||K i || satisfying real (σ (DK i /||K i ||)) > 0. The set given in Lemma 1 is called an unmixing set, which is known for a given dimension n. For example, for n = 1, the unmixing set is {1, −1}; for n = 2, the unmixing set is {ψ + (0), ψ + (2π/3), ψ + (4π/3), ψ − (0), ψ − (2π/3), ψ − (4π/3)}, where ψ + (ω) = [cos ω − sin ω; sin ω cos ω] and ψ − (ω) = [cos ω − sin ω; − sin ω − cos ω]. Remark 5: For our problem, let D = B0 , and then (X) can be treated as bounded nonlinear deviation from the nominal constant B0 . Therefore, there exists a matrix K i , such that the spectrum of B0 K i is in the open right half of the complex plane. Without loss of generality, let i = 0, i.e., real (σ (B0 K 0 )) > 0. As a result, the negative feedback closedloop is achievable by properly including the matrix K 0 into the control u. Since real (σ (B0 K 0 )) > 0, there exists a unique n-by-n positive definite matrix Q satisfying the Lyapunov equation
i=0
− f (X d ) − B(X d )u d (X d ).
(6)
(B0 K 0 )T Q + Q (B0 K 0 ) = 2I
(10)
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,∞
where Q = 0 exp − (B0 K 0 )T τ exp (−B0 K 0 τ ) dτ and I is an n-by-n identity matrix. Let the maximum and minimum eigenvalue of Q be Q max and Q min , respectively. Obviously, Q max ≥ Q min > 0. Assumption 6: Assume ||Q (X)|| ≤ p < 1, where ||Q (X)|| ≡ σ max (Q (X)) is the largest singular value of Q (X). Since ||K 0 || = 1 and (B (X) K 0 )T Q + Q (B (X) K 0 ) = 2I + ( (X) K 0 )T Q +Q ( (X) K 0 ).
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Remark 6: Although a two-layer NN is utilized here as an illustration of the linearly parameterized OLA models, the results of this brief can be readily extended to more general OLA structures, which could be radial basis functions, splines, polynomials, fuzzy logic, etc. Remark 7: The NN reconstruction error bound ε M is considered to be unknown in this brief as opposed to traditional NN control methodologies [9], [13]. Thereafter, substituting (13) into (5) generates r˙ = Q −1 W T φ( X¯ ) + B(X)u + d + ε.
(15)
Assumption 2 insures that (B (X) K 0 ) Q + Q (B (X) K 0 ) is always nonsingular. T
D. Nussbaum Gain and Matrix Selector C. OLA ˆ X; ¯ Wˆ ) is employed to cope with Thereafter, an OLA F( ¯ ˆ X¯ ; Wˆ ) is the output of an the unknown function F( X ). F( q ˆ OLA model and W ∈ R is a vector of adjustable parameters. In this brief, we consider linearly parameterized approximators, which means the output can be expressed as T -ˆ X¯ ; Wˆ ) F( ˆ X¯ ; Wˆ ) = Z · Wˆ , Z (t) = F( (11) ∂ Wˆ -ˆ ˆ W =W (t )
where Z is the sensitivity function between the output of the approximator and the adjustable parameters or weights. When a two-layer NN [13] is utilized, (11) becomes ¯ ˆ X; ¯ Wˆ ) = Wˆ T φ(V T X), F(
Z = φ(V T X¯ )
(12)
where V ∈ R2nm×N and Wˆ ∈ R N×n are hidden and output layer weights respectively, φ(·) : R N → R N is the activation function in the hidden layer and N is the number of hidden layer nodes. On the other hand, based on the well-known universal approximation property of the NN [12], the smooth function Q · F( X¯ ) can be represented by the same NN structure as ¯ = W T φ(V T X) ¯ + ε = W T φ( X¯ ) + ε Q · F( X)
(13)
where Q is defined in (10) and constant vector W ∈ R N×n is the target values of Wˆ , such that it minimizes the L 2 norm ˆ X¯ ; Wˆ ) over (energy-norm) distance between Q · F( X¯ ) and F( 2 nm ¯ all X ∈ ⊂ R , provided is an arbitrary compact set. The weights of the hidden layer V are omitted for the purpose of simplification. It is reasonable to assume that W is bounded such that W ≤ W M [13], and W and W M are only “artificial” quantities required for analysis purposes. ε represents the functional reconstruction error satisfying ε ≤ ε M [11], with ε M being an unknown positive constant. It is demonstrated in [13] that if the hidden layer weights, V , are chosen initially at random and held constant while N is sufficiently large, the NN approximation error ε can be made arbitrarily small since the activation function vector forms a basis vector. Furthermore, from (12) and (13), the parameter estimation error is defined as W˜ = Wˆ − W.
(14)
Then, the Nussbaum gain and the matrix selector are designed below to tackle the unknown and varying control direction matrix. Firstly, define the control performance index θ that is used as the driving force for the adaptive schemes of Nussbaum gain and the matrix selector θ˙ = r T , where
θ (0) = θ0 = 0
= k g r + Wˆ φ( X¯ ) + T
Dˆ 2 r Dˆ r + e−t
(16) .
(17)
k g ∈ R+ is a feedback gain and Dˆ ∈ R+ is an adaptive gain whose expression will be given later by (22). Then, the Nussbaum gain is given as (18) g (θ ) = exp θ 2 sin (2πθ ) . Because g(θ ) can swing from positive infinite to negative infinite according to the control performance θ , it can correct inappropriate deviation caused by erroneous previous control. At last, the matrix selector is K (θ ) = (1 − mod (η, 1)) K η + mod (η, 1) K !η"
(19)
where mod(a, b) gives the residual of a over b, and the index η in (19) is given by rescaling θ as follows: ⎧ 1
⎪ if mod (ξ, 1) ∈ 0, 18 ⎨ ξ − 4 8 − mod(ξ, 1) , 1 7 η = ξ , if mod (ξ, 1) ∈ 8 , 8 ⎪
⎩ 7 ξ + 4 mod (ξ, 1) − 8 , if mod (ξ, 1) ∈ 78 , 1 ξ = mod (|θ | , N ). Functionally, the matrix selector K (θ ) picks a matrix K j from the unmixing set according to the value of θ as given by (16). According to [20], the selection process of K j includes the rescaling of θ and the proportional mix of two successive K j s. In this way, K (θ ) will continuously shift from K j to K j +1 . According to the value of θ , g(θ )K (θ ) can vary from K 0 to K N−1 with an arbitrary magnitude. In other words, g(θ )K (θ ) provides an opportunity to eventually build a negative feedback with an appropriate magnitude of feedback gain.
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. /
0 where y = r T tr W˜ T W˜ D˜ ∈ ⊂ Rn+2 and is an
E. Controller Design Let the control signal at time t be u(t) = g(θ )K (θ )R(r ) where the matrix
R(r ) =
(20)
arbitrary compact set containing y(t) = 0. U1 (y), U2 (y) can be defined as U1 (y) ≡ λ1 y2 ,
rr T r2
,
if r = 0
is used purely for the purpose of proof. Clearly, the control signal consists of the Nussbaum gain g(θ ), the matrix selector K (θ ), the high-gain filtered tracking error feedback k g r , the NN output Wˆ T φ( X¯ ), and a robust term ( Dˆ 2 r )/( Dˆ r + e−t ). It is noted that compared with traditional variable structure or sliding mode designs [8], a continuous robust term is utilized in (20) instead of signum functions to guarantee the continuousness of the control signal. Thus, the requirement of infinite bandwidth and chattering phenomenon in control input is avoided [9]. Finally, define the NN weight tuning law as W˙ˆ = kn φ( X¯ )r T − kr kn r Wˆ
1 1 2 2 1 1 1 1 1 1 λ1 ≡ min Q min , , , λ2 ≡ max Q max , , . 2 kn k D 2 kn k D (26) Note that from (23) and (24) we have kr 2 r W M − Dˆ r + r T Q (d + ε) + + D˜ r 4
≤ r − Dˆ + Dd + D˜ = 0. Thereafter, differentiating (23) yields
(21)
R+
are positive user design parameters. where kn , kr ∈ Further, the adaptive scalar Dˆ is written as ( t r (τ ) dτ Dˆ = k D (22) 0
where k D ∈ R+ is a positive constant. Remark 8: In comparison to [9], since the projection algorithm is not used, the challenge of selecting a predefined convex set is avoided. Furthermore, the boundedness of NN weights is proven simultaneously along with the asymptotic convergence of the tracking error in following section with no ¯ [13]. assumptions of PE condition on φ( X) Remark 9: In this control scheme, the continuous robust term ( Dˆ 2r )/( Dˆ r + e−t ) always exists and is employed to compensate the disturbance d(t) and the NN reconstruction error ε, the knowledge of whose bounds is not required to be known by designing Dˆ as an adaptive function of time instead of a constant as in [9] and [13]. IV. A SYMPTOTIC C ONVERGENCE The main theorem of the stability result for the proposed controller can now be stated. Theorem 2: Consider the closed-loop system consisting of the plant (1), the control given by (20), and the adaptive law (21), (22). Under Assumptions 1 and 2, the tracking error e → 0 asymptotically and closed-loop signals are all bounded. Proof: First, build a positive definite Lyapunov candidate 1 ˜2 1 ˜ ˜ T 1 tr W W + (23) U ≡ r T Qr + D , U0 =U (0). 2 2kn 2k D Meanwhile, it is important to observe that there exist two functions U1 (y) and U2 (y) such that U1 (y) ≤ U ≤ U2 (y)
(25)
where λ1 , λ2 ∈ R+ are defined as
if r = 0
I,
U2 (y) ≡ λ2 y2
(24)
U˙ = r T Q(F( X¯ ) + Bu + d) + tr W˜ r φ T ( X¯ )
−kr r tr W˜ Wˆ T + D˜ r
¯ )B(X)K (θ )R(r )+d +ε = r T Q Q −1 W T φ( X)+g(θ
¯ −kr r tr W˜ Wˆ T + D˜ r +tr W˜ r φ T ( X) = r T W Tφ( X¯ )+g(θ )r T Q B(X)K (θ )R(r )+ r T −r T
+r T Q (d +ε)+r T W˜ T φ( X¯ )−kr r tr W˜ Wˆ T + D˜ r
≤ r T W T + W˜ T φ( X¯ )+g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −r TWˆ Tφ( X¯ )−k g r 2 − Dˆ r +e−t +r T Q (d +ε)
−kr r tr W˜ T (W˜ + W ) + D˜ r ≤ g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t − Dˆ r + r T Q (d +ε)−kr r W˜ 2 +kr r W˜ W M + D˜ r
≤ g(θ )r T Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t − Dˆ r '2 & 1 + r T Q (d + ε) − kr r W˜ − W M 2 kr 2 + r W M + D˜ r 4 ≤ g(θ )r T Q B(X)K (θ )R(r ) + θ˙ − k g r 2 + e−t '2 & 1 −kr r W˜ − W M 2 T ≤ g(θ )r Q B(X)K (θ )R(r )+ θ˙ −k g r 2 +e−t . (27)
Now consider g(θ )r T Q B(X)K (θ )R (r ) . If r = 0, we have g(θ )r T Q B(X)K (θ )R (r ) rr T r ˙ = g(θ )r T Q B(X)K (θ ) =r T Q B(X)K (θ ) g(θ )θ. r 2 r 2 (28)
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 7, JULY 2012
If K (θ ) = K 0 and from (10) we have
or equivalently
rT
r 1 = r T Q B(X)K (θ ) 2 2 r r
r × (Q (B0 + (X)) K 0 )T + Q (B0 + (X)) K 0 r
r 1 rT = 1+ . (29) (Q (X) K 0 )T + Q (X) K 0 r 2 r According to Assumption 2 and remember ||K j || = 1, we have
r 1 rT (Q (X) K 0 )T + Q (X) K 0 r 2 r ≤ Q (X) K 0 < p. Taking above inequality into (29) produces r r T Q B(X)K (θ ) = b (X) r 2 where b(X) ∈[1 − p, 1 + p] with 0 < p < 1. That is ˙ K (θ ) = K 0 . g(θ )r T Q B(X)K (θ )R (r ) = b (X) g(θ )θ,
(30)
Otherwise, if K (θ ) = K 0 r T Q B(X)K (θ )
r ≤L r 2
(31)
where L is the upper bound for ||QB(X)K (θ )||, which always exists because Q, B(X), and K (θ ) are all bounded. Thus, when K (θ ) = K 0 we have g(θ )r T Q B(X)K (θ )R (r ) ≤ L -g (θ ) θ˙
= Lsgn g (θ ) θ˙ g (θ ) θ˙ . (32) Then, we can write (30) and (32) into the compact form g(θ )r T Q B(X)K (θ )R (r ) ≤ ρ (θ ) g (θ ) θ˙ where ρ (θ ) is given below b (X),
ρ (θ ) = Lsgn g (θ ) θ˙ ,
if K (θ ) = K 0 if K (θ ) = K 0 .
(33)
(34)
If r = 0, we have g(θ )r T Q B(X)K (θ )R (r ) = 0, and (33) is still satisfied. Then, taking integration of (27) yields 0≤U
( t ( t
r 2 ds + r T g(θ )Q B(X)K (θ )R (r ) + θ˙ ds ≤ −k g 0 ( t0 −s + e ds + U0 0 ( t ( t 2 r ds + ≤ −k g (ρ (θ ) g (θ ) + 1) θ˙ ds + 1 + U0 0 0 ( θt ( t 2 ≤ −k g r ds + ρ (θ ) g (θ )dθ + θt + 1 + U0 (35) 0
0
which further gives ( θt ρ (θ ) g (θ )dθ + θt + 1 + U0 0≤ 0
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(
θt
− (θt + 1 + U0 ) ≤
ρ (θ ) g (θ )dθ
(36)
0
where θ t = θ (t). Note we have the following properties [20] ( 1 lim sup ρ (θ ) g (θ )dθ = +∞ (37) →∞ 0 and lim inf
→∞
1
(
ρ (θ ) g (θ )dθ = −∞.
(38)
0
Then, we can show that θ t is bounded by the method of contradiction. Suppose that θ t becomes divergent, then there are two possibilities: ,θ 1) if θt → +∞, (36) gives limθt →+∞ 1/θt 0 t ρ (θ ) g (θ )dθ ≥ −1, which contradicts with (38); , θ 2) if θt → −∞, (36) gives limθt →−∞ 1/θt 0 t ρ (θ ) g (θ )dθ ≤ −1, which contradicts with (37). ,θ As a result, θ t is bounded, and 0 t ρ (θ ) g (θ )dθ is also bounded; hence, we can find a positive constant C, such that , θt ρ g (θ ) (θ )dθ + θt + 1 + U0 ≤ C Q min . Then (23) and 0 (35) give ( t Q min r 2 ds + C Q min r 2 ≤ U ≤ −k g 0≤ 2 0 or equivalently 0≤
1 U 1 r 2 ≤ ≤ −k g 2 Q min Q min
as well as 0 ≤ kg
1 Q min
(
t
(
t
r 2 ds + C
(39)
0
r 2 ds ≤ C.
(40)
0
√ Since ||r || is continuous and r ≤ 2C from (39), 2 is Lipschitz continuous because |r 2 − r ∗ 2 | ≤ ||r || √ 2C |r − r ∗ | for any pair of r and r ∗ . Hence ||r ||2 is uniformly continuous, because very Lipschitz continuous function is uniformly continuous. Since ||r ||2 is also nonnegative and integrable, according to Barbalat’s lemma (40) implies ||r || → 0 as t → ∞, which further gives ||e|| → 0 as t → ∞. Then, we can conclude the asymptotic convergence of the tracking error. Furthermore, the boundedness of other signals also can be readily obtained. Equation (39) shows the uniform boundedness of U since the right hand side term is bounded. Therefore, W˜ and D˜ are bounded according to (23), which also implies the boundedness of Wˆ by definition. Dˆ is also bounded as Dd ˜ + Dd from (24). Thus, (17) and D˜ are bounded and Dˆ = D(t) and (20) show the boundedness of control signal. V. S IMULATION In order to demonstrate the feasibility of the theoretic results, the proposed adaptive NN controller is applied to the following third-order MIMO nonlinear system: (3) x1 (3) x = (3) = f (x) + B (x) u = f (x) + ( (x) + B0 ) u. x2
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 7, JULY 2012
-3
Desired & Actual Output Trajectory 2
2 x1 r1
1.5 Weights
x1 & r1
1 0 -1 -2
Norm of NN Weights
x 10
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(a) 2 0.015
0.01
0
D hat
x2 & r2
Addaptive Term D hat
x2 r2
1
-1 -2
Fig. 1.
0.005 0
0.2
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1 t (s)
1.2
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2
0
Actual and desired system output with the proposed controller.
0
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1 t (s)
1.2
(b) Fig. 3. System performance with the proposed controller. (a) Norm of NN ˆ weights. (b) Adaptive term D.
500 0 u1
-500 -1000 -1500 -2000
0
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1 t (s)
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500 0 u2
-500 -1000
Afterwards, a spike appears in the control signal when the controller realizes that the current Nussbaum gain matrix is not right and need search for the correct one. After all, from the simulation results, we can find that the design is capable of attaining satisfactory tracking performance while all other signals of the closed-loop system bounded.
-1500 -2000
Fig. 2.
Control input signal with the proposed controller.
0 . . 2 0 0 cos x 2 sin (x 2 ) + x 1 , (x) = 0.4 f (x) = and 2 sin x 1 0 . x1 + 0 sin x 2 1 −2 B0 = are unknown for control design. For this 2 −1 example, according to Remark 3 it can be readily found that 4π 0
. cos 4π 3 − sin 3 K0 = sin 4π cos (4π3) 3 can satisfy the condition of real (σ (B0 K 0 )) > 0. The objective of the control signal is to make the system output x follow the reference trajectory defined as yd = [− sin 2πt; cos 2πt]. The filtered tracking error is defined as r = e¨ + 4e˙ + 10e. The NN contains ten hidden neurons (i.e., N = 10) and traditional sigmoidal activation functions are employed. The hidden layer weights are selected initially at random and held constant. The other control parameters are: k g = 0.05, kn = 0.001, and k D = 0.01. A typical system response using the proposed controller is shown in Fig. 1 including the actual and desired system output trajectories. With unknown control direction matrix and the presence of disturbance and NN reconstruction error, the actual output can still asymptotically track the desired values. Fig. 2 illustrates the control signals and the norm of NN weights and the adaptive term Dˆ are also demonstrated in Fig. 3. During the initial phase, since the tracking error of x 2 is converging, the controller believes that the current Nussbaum gain matrix could be proper and thus remains low. Therefore, the output of the system is kept steady.
VI. C ONCLUSION In this technical note, a novel continuous control designs are proposed by incorporating with the Nussbaum gain matrix selector and the NN for a general class of MIMO high-order uncertain nonlinear systems with bounded disturbance and unknown state-dependent control direction matrix. Asymptotic tracking performance is obtained with the NN reconstruction error and external disturbance present, whose bounds are not required to be known. The risk brought by the projection algorithm and requirement of PE condition are also avoided. The theoretical analysis and the simulation results show the effectiveness of the proposed schemes. R EFERENCES [1] M. Chen, S. S. Ge, and B. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [2] Y. J. Liu, C. L. P. Chen, G. X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1162–1167, Jul. 2011. [3] A. K. Kostarigka and G. A. Rovithakis, “Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 22, no. 1, pp. 138–149, Jan. 2012. [4] Z. Hou and S. Jin, “Data-driven model-free adaptive control for a class of MIMO nonlinear discrete-time systems,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 2173–2188, Dec. 2011. [5] A. K. Kostarigka and G. A. Rovithakis, “Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 22, no. 1, pp. 138–149, Jan. 2012. [6] J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [7] P. Ioannou and B. Fidan, Adaptive Control Tutorial (Advances in Design and Control). Philadelphia, PA: SIAM, 2006.
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[8] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [9] P. M. Patre, W. MacKunis, K. Kaiser, and W. E. Dixon, “Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and RISE feedback control structure,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2180–2185, Oct. 2008. [10] H. Lee and M. Tomizuka, “Robust adaptive control using a universal approximator for SISO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 1, pp. 95–106, Feb. 2000. [11] J. A. Farrell and M. M. Polycarpou, “Adaptive approximation based control: Unifying neural,” in Fuzzy and Traditional Adaptive Approximation Approaches. New York: Wiley, 2006. [12] D. A. White and D. A. Sofge, Eds., Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York: Van Nostrand, 1993. [13] F. L. Lewis, S. Jagannathan, and A. Yesilderik, Neural Network Control of Robot Manipulators and Nonlinear Systems. New York: Taylor & Francis, 1999. [14] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [15] Q. Yang and S. Jagannathan, “NN/RISE-based asymptotic tracking control of uncertain nonlinear systems,” in Proc. IEEE Int. Symp. Intell. Control, Sep. 2011, pp. 1361–1366. [16] W. T. Miller, R. S. Sutton, and P. J. Werbos, Eds., Neural Networks for Control. Cambridge, MA: MIT Press, 1990. [17] R. D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, no. 5, pp. 243–246, 1983. [18] L. Liu and J. Huang, “Global robust stabilization of cascade-connected systems with dynamic uncertainties without knowing the control direction,” IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1693–1699, Oct. 2006. [19] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1463–1469, Aug. 2003. [20] Z. Yang, S. C. P. Yam, L. K. Li, and Y. Wang, “Universal repetitive learning control for nonparametric uncertainty and unknown control gain matrix,” IEEE Trans. Autom. Control, vol. 55, no. 7, pp. 1710–1715, Jul. 2010. [21] Y. Zhang, C. Wen, and Y. C. Soh, “Adaptive backstepping control design for systems with unknown high-frequency gain,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2350–2354, Dec. 2000. [22] T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, no. 6, pp. 1021–1033, 2007. [23] T. P. Zhang and S. S. Ge, “Adaptive neural network tracking control of MIMO nonlinear systems with unknown dead zones and control directions,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 483–497, Mar. 2009. [24] A. Ilchmann, “High-gain adaptive stabilization of multivariable linear systems–revisited,” Syst. Control Lett., vol. 18, no. 5, pp. 355–364, 1992.
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Spectral Graph Optimization for Instance Reduction Konstantinos Nikolaidis, Student Member, IEEE, Eduardo Rodriguez-Martinez, John Yannis Goulermas, Senior Member, IEEE, and Q. H. Wu, Fellow, IEEE Abstract— The operation of instance-based learning algorithms is based on storing a large set of prototypes in the system’s database. However, such systems often experience issues with storage requirements, sensitivity to noise, and computational complexity, which result in high search and response times. In this brief, we introduce a novel framework that employs spectral graph theory to efficiently partition the dataset to border and internal instances. This is achieved by using a diverse set of border-discriminating features that capture the local friend and enemy profiles of the samples. The fused information from these features is then used via graph-cut modeling approach to generate the final dataset partitions of border and nonborder samples. The proposed method is referred to as the spectral instance reduction (SIR) algorithm. Experiments with a large number of datasets show that SIR performs competitively compared to many other reduction algorithms, in terms of both objectives of classification accuracy and data condensation. Index Terms— Graph Laplacian, instance instance-based learning, prototype reduction.
selection,
I. I NTRODUCTION In instance-based learning, algorithms suffer from two principal issues: first, the need to store the entire dataset in some type of memory, and second, the increased time complexity from having to search large portions of the stored prototypes in order to predict new queries. A third concern is the noisy instances present in the database that can degrade the performance of the system. In order to address these problems, instance reduction techniques which reduce the size of the stored datasets without compromising prediction accuracies, have attracted significant interest. One such variation, the editing algorithm, focuses only on improving classification accuracy by removing harmful instances. One of the simplest such examples is the edited nearest neighbor (ENN) [1], which is still widely used as a noise-filtering preprocessor within other methods [2], [3]. The majority of instance reduction techniques, however, deal not only with the harmful noisy instances but also with the redundant ones, in order to mitigate the aforementioned storage and search requirements without degrading prediction accuracies. Since instances that lie near the class boundaries hold the most valuable information to identify the decision surface and distinguish between different classes, many of the recent instance reduction algorithms employ various heuristics to retain border instances over Manuscript received January 27, 2012; revised April 2, 2012; accepted May 3, 2012. Date of publication May 30, 2012; date of current version June 8, 2012. This work was supported by a DTA Scholarship from the University of Liverpool. The authors are with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2012.2198832
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