Composite Learning From Model Reference Adaptive ... - IEEE Xplore

Proceedings of 2015 International Conference on Fuzzy Theory and Its Applications (iFUZZY) The Evergreen Resort Hotel (Jiaosi), Yilan, Taiwan, Nov. 18-20, 2015

Composite Learning From Model Reference Adaptive Fuzzy Control Yongping Pan1 , Meng Joo Er2 , Lin Pan3 , and Haoyong Yu1 Abstract—Function approximation accuracy and computational cost are two major issues in approximation-based adaptive fuzzy control. In this paper, a model reference composite learning fuzzy control (MRCLFC) strategy is proposed for a class of affine nonlinear systems with functional uncertainties. In the MRCLFC, a modified modelling error that utilizes data recorded online is defined as the prediction error, a linear filter is applied to estimate the derivatives of plant states, and both the tracking error and the prediction error are exploited to update parametric estimates. It is proven that the closed-loop system achieves semiglobal practical exponential stability by an interval-excitation condition which is much weaker than a persistent-excitation condition. The proposed strategy can guarantee accurate function approximation under greatly reduced computational cost. The effectiveness of the proposed MRCLFC strategy has been verified by applying it to an control problem of aircraft wing rock.

dition in model reference adaptive control (MRAC) [14]. The difference between the concurrent learning and the composite adaptation lies in the construction of prediction errors. In the concurrent learning, a dynamic data stack composed of online recorded data is used in constructing prediction errors, and parameter convergence is guaranteed if an interval-excitation (IE) condition is satisfied such that sufficiently rich data are recorded in the data stack. Unfortunately, in this innovative design, exhaustive search should be applied to the data stack to maximize its singular value, and fixed-point smoothing must be applied to estimate the derivatives of all plant states. These deficiencies inevitably increase the computational cost of the concurrent learning algorithm. In this paper, a model reference composite learning fuzzy control (MRCLFC) strategy is proposed for a class of affine nonlinear systems with functional uncertainties. The design procedure of the proposed strategy is as follows: firstly, a fuzzy system is applied to approximate an plant uncertainty; secondly, a modified modelling error that utilizes online recorded data is defined as the prediction error; thirdly, a second-order linear filter is applied to estimate the derivative of a tracking error; and finally, both the tracking error and the prediction error are applied to update the fuzzy system. It is proven that the closed-loop system achieves semiglobal practical exponential stability under the IE condition which is much weaker than the PE condition. The proposed strategy guarantees accurate function approximation under greatly reduced computational cost compared with concurrent learning-based approaches. In this paper, N, R, R+ , Rn and Rn×m denote the spaces of natural numbers, real numbers, positive real numbers, real n-vectors, and n × m-matrixes, respectively,  ·  denotes the Euclidean norm, L∞ denotes the space of bounded signals, Ωc := {x|x ≤ c} denotes the ball of radius c, min{·}, max{·} and sup{·} denote the functions of minimum, maximum and supremum, respectively, λmin (A) and λmax (A) denote the minimal and maximal eigenvalues of A, respectively, and C k represents the space of functions whose k-order derivatives all exist and are continuous, in which c ∈ R+ , x ∈ Rn , A ∈ Rn×n , and n, m, k ∈ N.

I. I NTRODUCTION Adaptive fuzzy control (AFC) is motivated by the universal approximation property of fuzzy systems and their ability to incorporate linguistic information from a priori knowledge [1]. The effectiveness of AFC on tackling functional uncertainties of nonlinear systems has been demonstrated in many recent results, where some of them are reported in [2]–[13] and some references therein. Parameter convergence is desirable in AFC since it can bring several attractive features, including accurate function approximation, exponential tracking, and robustness against measurement noise and external perturbations [14]. However, the traditional AFC cannot ensure parameter convergence unless a persistent-excitation (PE) condition is satisfied. It is well known that the PE condition is very stringent and often infeasible in practical control systems [14]. Composite adaptive control (CAC) is an integrated direct and indirect adaptive control strategy which can achieve higher tracking accuracy and better parameter estimation by faster and smoother parameter adaptation [15]. Some recent results about fuzzy/neural network CAC are reported in [16]–[24]. However, parameter convergence still cannot be ensured in CAC without the PE condition. A concurrent learning technique was presented to achieve parameter convergence without the PE conThis work was supported in part by the National Research Foundation of Singapore under Grant no. NRF2014NRF-POC001-027, and in part by the Ministry of Education, Singapore (Tier 1 AcRF, RG29/15). 1 Y. Pan and H. Yu are with the Department of Biomedical Engineering, National University of Singapore, Singapore 117575, Singapore (e-mail: [email protected]; [email protected]). 2 M. J. Er is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]). 3 L. Pan is with the School of Electric and Electronic Engineering, Wuhan Polytechnic University, Wuhan 430023, China, and also with the Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Luxembourg (email: [email protected]).

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II. P ROBLEM F ORMULATION Consider a class of nth-order single-input single-output (SISO) affine nonlinear systems as follows [25]:   x˙ = Λx + b f (x) + u (1) with Λ ∈ Rn×n and b := [0, · · · , 0, 1]T , where u(t) ∈ R is the control input, x(t) := [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is

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the vector of plant states, and f (x) : Rn →  R is an unknown C 1 function. A reference model is given by x˙ r = Ar xr + br r

denotes adaptive control, and ke ∈ Rn and kr ∈ Rn+1 are positive vectors of control gains. Substituting (6) into (1), one obtains the closed-loop tracking error dynamics as follows:   ˆ) e˙ = Ae + b f (x) − fˆ(x, W (7)

(2)

with br := [0, · · · , 0, br ]T ∈ Rn , where Ar ∈ Rn×n is a strictly Hurwitz matrix, xr (t) := [xr1 (t), xr2 (t), · · · , xrn (t)]T ∈ Ωcr ⊂ Rn is the vector of reference states, and r(t) ∈ R is a bounded command signal. Assume that x is measurable and (Λ, b) is controllable. The following definitions from [14] are used for facilitating control synthesis. Definition 1: A bounded signal Φ(t) ∈ RN is of IE over a time interval τd , Te ] if there exist Te , τd , σ ∈ R+  Te t ∈ [Te − T such that Te −τd Φ(τ )Φ (τ )dτ ≥ σI. N Definition 2: A bounded  t signal Φ(t)T ∈ R is of PE if there + exist σ, τd ∈ R so that t−τd Φ(τ )Φ (τ )dτ ≥ σI, ∀t ≥ 0. Let xre := [xTr , r]T denote an augmented reference signal, and e(t) := x(t)−xr (t) denote a tracking error. The objective of this study is to design a proper control law of the system (1) such that the closed-loop system achieves exponential stability under the IE rather than the PE condition. Remark 1: In this study, the SISO affine nonlinear system (1) is considered for clear illustration. As discussed in Remark 1 of [10], a class of multi-input multi-output (MIMO) affine nonlinear systems can be transformed into several SISO affine nonlinear systems in the form of (1) under certain conditions. Thus, the following results can to be extended to MIMO affine nonlinear systems. Moreover, by the backstepping technique [26], the following results are also possible to be extended to strict-feedback nonlinear systems.

with A := Λ − bkeT . The choice of kr satisfies bkrT xre = (Ar − Λ)xr + br r

and the choice of ke makes A strictly Hurwitz. Thus, for any matrix Q = QT > 0, a unique matrix solution P = P T > 0 exists for the following Lyapunov equation: AT P + P A = −Q. Noting the definition of εf , one obtains   T ˜ Φ(x) + εf (x) e˙ = Ae + b W

To facilitate the subsequent discussion, define t Φ(x(τ ))ΦT (x(τ ))dτ Θ(t) :=

in which τd ∈ R+ is an integral duration. Multiplying both sides of (7) by Φ(x), integrating the resulting equality at [t −τd , t] and after some transformations, one obtains t ˜ (t) = Θ(t)W Φ(x)(e˙ n − bAe − εf (x))dτ (12) t−τd

in which the time variable τ is omitted in the above integral part. Since e˙ n is unavailable, a second-order linear filter with unit gain is implemented as follows [28]:

eˆ˙ n = eˆn+1 (13) en+1 + ω 2 (en − eˆn ) eˆ˙ n+1 = −2ζωˆ

(3)

with eˆn (0) = en (0) and eˆn+1 (0) = 0, in which ω ∈ R+ is the natural frequency, ζ ∈ R+ is the damping factor, and eˆn and eˆn+1 are estimates of en and e˙ n , respectively. Note that the integral in (12) is effective in reducing the influence of ˜ . Therefore, although measurement noise on calculating ΘW an additional filtering delay error is introduced by (13), the natural frequency ω in (13) can be made sufficiently small such that eˆn+1 ≈ e˙ n . The following lemmas give two results at the initialization stage t ∈ [0, Ta ) with Ta ∈ R+ . Lemma 1 [27]: For the system (1) under x(0) ∈ Ωcx0 driven by the control law (6), there exist cx > cx0 and Ta ∈ R+ such that x(t) ∈ Ωcx , ∀t ∈ [0, Ta ). Lemma 2 [28]: For the linear filter (13) on t ∈ [0, Ta ), given any μ ∈ R+ , there exists a sufficiently small ω ∈ R+ so that |ˆ en+1 − e˙ n | ≤ μ, ∀t ∈ [0, Ta ). Noting (11), the IE condition in Definition 1 can be rewritten as Θ(Te ) ≥ σ with Te , σ ∈ R+ , where σ can be regarded as an exciting strength. For a certain control problem with a given τd ∈ R+ , the epoch Te that satisfies the IE condition is usually not unique, and the corresponding σ can be time-varying. Let

ˆ := [w is applied to approximate f (x), where W ˆ1 , w ˆ2 , · · · , w ˆN T N ] ∈ Ωcw ⊂ R is the vector of adjustable parameters, Φ(x) := [φ1 (x), φ2 (x), · · · , φN (x)]T ∈ RN satisfying Φ ≤ ψ is the vector of fuzzy basis functions (FBFs), ψ ∈ R+ is a constant, and N is the number of fuzzy rules. An optimal approximation error εf is defined as follows: (4)

with W ∗ being a vector of optimal parameters given by    ˆ ) . (5) W ∗ = arg min sup f (x) − fˆ(x, W x∈Ωcx

The MRAC law of (1) is given as follows [25]: ˆ) u = −keT e +krT xre −fˆ(x, W       upd ure uad

(11)

t−τd

Since the C 1 function f (x) in (1) is unknown, a fuzzy system of the following form [1]:

ˆ ∈Ωc W w

(10)

B. Composite Learning Structure

A. Fuzzy Approximation-Based Control

εf (x) = f (x) − fˆ(x, W ∗ )

(9)

ˆ. ˜ := W ∗ − W in which W

III. C OMPOSITE F UZZY L EARNING C ONTROL

ˆ)=W ˆ T Φ(x) fˆ(x, W

(8)

(6)

where upd denotes proportional-derivative (PD) feedback control, ure denotes reference signal feedforward control, uad

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Te denote the first epoch that satisfies the IE condition, σc (t) := maxτ ∈[Te ,t] {σ(τ )} denote a current maximal exciting strength, and te (t) := arg maxτ ∈[Te ,t] {σ(τ )} denote an epoch corresponding to σc (t). To take advantage of the excitation information for parameter convergence, define the prediction error to be a modified modelling error as follows1 : ˜ (t) + ε(te ) (t) := Θ(te )W

C. Stability and Convergence Analysis Choose a Lyapunov function candidate for the closed-loop system composed of (7) and (10) as follows: ˜ /(2γ) ˜ TW V (z) = eT P e/2 + W

˜ T ]T ∈ Rn+N . The main result pertaining where z := [eT , W to stability and convergence is stated here. Theorem 1: Consider the system (1) under x(0) ∈ Ωcx0 driven by the control law (6) with (17), λmin (Q) > 1 and ˆ (0) ∈ Ωc , where the control gain kr is designed to satisfy W w (8), and the control gain ke is selected to make A in (7) strictly Hurwitz. If the IE condition Θ(t) ≥ σI is satisfied at t = Te with certain constants τd , σ ∈ R+ , then the closed-loop system achieves semiglobal practical exponential-like stability in the sense that all closed-loop signals are bounded on t ≥ 0 and ˜ (t) both the tracking error e(t) and the estimation error W converge to small neighbourhoods of 0 on t ≥ Te . Proof: Firstly, consider the control problem at t ∈ [0, ∞). The time derivative of V along (7) is as follows:   ˜ T eT P bΦ(x) − W/γ ˆ˙ V˙ = −eT Qe/2 + eT P bεf + W (19)

(14)

where ε(te ) is a lumped approximation error defined by te Φ(x)(e˙ n − eˆn+1 + εf (x))dτ. (15) ε(te ) := te −τd

Noting (12) and (15),  in (14) can be computed as follows: te Φ(x)(ˆ en+1 − bAe)dτ. (16) (t) = te −τd

ˆ as follows: Design a composite learning law of W   ˆ˙ = P W ˆ , γ(eT P bΦ(x) + kw ) W

(18)

(17)

where γ ∈ R+ is a learning rate, kw ∈ R+ is a weight factor, ˆ , •) is a projection operator given by and P(W  ˆ  < cw or W ˆ  = cw & W ˆ T• ≤ 0 •, if W ˆ T P(W , •) = . ˆ ˆ W •− W ˆ 2 •, otherwise W

in which (9) is used in obtaining (19). From the projection modification results of [25], the learning law (17) guarantees ˆ (t) ∈ Ωc as W ˆ (0) ∈ Ωc , and that W w w   ˙ T T ˜ T . ˜ e P bΦ(x) − W/γ ˆ W ≤ −kw W

The following lemma is useful in control analysis. Lemma 3: There exist constants ε¯f , ε¯ ∈ R+ such that εf (x) and ε(te ) defined in (4) and (15) satisfy |εf (x)| ≤ ε¯f and ε(te ) ≤ ε¯, respectively, ∀x ∈ Ωcx and ∀te ∈ [0, Ta ), where ε¯f can be arbitrarily small by increasing the number N in (3), and ε¯ can be arbitrarily small by increasing the natural frequency ω in (13) and the number N in (3). Proof : Noting the universal approximation property of fuzzy systems [1], there exists a constant ε¯f ∈ R+ so that |εf (x)| ≤ ε¯f , ∀x ∈ Ωcx , where ε¯f can be made arbitrarily small by increasing N in (3). Noting Lemmas 1 and 2, one obtains |e˙ n −ˆ en+1 | ≤ μ, ∀x ∈ Ωcx and ∀t ∈ [0, Ta ), where μ can be made arbitrarily small by increasing ω in (13). Now, using the definition of ε(te ) in (14) and noting Φ ≤ ψ, there exists a constant ε¯ := ψτd (μ + ε¯f ) ∈ R+ such that ε(te ) ≤ ε¯, ∀x ∈ Ωcx and ∀te ∈ [0, Ta ), where ε¯ can be made arbitrarily small by increasing ω in (12) and N in (3).  Remark 2: In the concurrent learning, fixed-point smoothing should be applied to estimate x˙ n so that the prediction error can be computed, and singular value maximization should be applied to a data stack such that the IE condition can be satisfied to achieve parameter convergence. These deficiencies inevitably increase the computational cost of the concurrent learning algorithm. In the proposed composite learning, e˙ n is estimated by the filter (13) such that fixed-point smoothing is not needed, and the novel prediction error  in (14) is defined such that singular value maximization is avoided.

Applying the foregoing inequality to (19) and noting  = ˜ + ε(te ) in (14), one obtains Θ(te )W   ˜ T Θ(te )W ˜ + ε(te ) . V˙ ≤ −eT Qe/2 + eT P bεf − kw W Noting Lemma 3, one obtains |ε(x)| ≤ ε¯f and ε(te ) ≤ ε¯, ˆ , W ∗ ∈ Ωc , one ∀x ∈ Ωcx and ∀te ∈ [0, Ta ). Noting W w ˜  ≤ 2cw . Applying these results to the foregoing obtains W inequality, one obtains V˙ ≤ − eT Qe/2 + eP b¯ εf T ˜ ˜ − kw W Θ(te )W + 2kw cw ε¯.

(20)

Secondly, it follows from (20) with Θe ≥ 0 that V˙ ≤ −(λmin (Q) − 1)e2 /2 + 2kw cw ε¯ + P b2 ε¯2f /2 for all x ∈ Ωcx and t ∈ [0, Ta ). Noting (18), one obtains V˙ ≤ −ks V + 2ks c2w /γ + 2kw cw ε¯ + P b2 ε¯2f /2 = −ks V /2 − ks (V − η(ks , γ, μ))/2 where ks := (λmin (Q) − 1)/λmax (P ) ∈ R+ and η(ks , γ, μ) := 4c2w /γ + 4kw cw ε¯/ks + P b2 ε¯2f ∈ R+ . Let Ωcz0 := Ωcx0 ∩ Ωcr × Ωcw and Ωcz := Ωcx ∩ Ωcr × Ωcw so that Ωcz0 ⊂ Ωcz . It is implied from the above inequality that V˙ (t) ≤ −ks V (t)/2, ∀V (t) ≥ η

(21)

over z(t) ∈ Ωcz and t ∈ [0, Ta ). From the definition of V in (18), there exist constants λa := min{1/2, 1/(2γ)} ∈ R+ and λb := max{1/2, 1/(2γ)} ∈ R+ such that

1 The special definition is based on a consideration that Θ(t )W ∗ is not e computable by (12) due to the unmeasurable e˙ n .

λa z2 ≤ V (z) ≤ λb z2 .

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(22)

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the definition of σc under Lemma 2 is applied in the above expression. It follows from (18) and (21) that

Based on the results of (21) and (22), the  UUB Theorem [29, Th. 4.5] is invoked to conclude that if η < cz0 /λb , then Ωcz0 is positively invariant, and the closed-loop system has UUB stability for every z(0) ∈ Ωcz0 in the sense of e(t) ∈ Ωce and ˆ (t) ∈ Ωc , ∀t ≥ 0 implying Ta = ∞. From the definition W w of η, for any given cx0 ∈ R+ , there are sufficiently  large ke in (6), γ in (17) and ω in (13) to guarantee η < cz0 /λb . Using ˆ (t) ∈ Ωc , ∀t ≥ 0, one gets x(t), u(t), (t) ∈ e(t) ∈ Ωce , W w L∞ , ∀t ≥ 0. Thus, all closed-loop signals are bounded, ∀t ≥ 0. Moreover, since cx0 can be arbitrarily increased to include all possible x(0), the stability is semiglobal. Thirdly, consider the control problem at t ∈ [Te , ∞). Since there exist constants σ, τd ∈ R+ such that Θ(Te ) ≥ σI, it is derived from (20) that on t ≥ Te , one has

V˙ (t) ≤ −ke V (t) + ρ(ks , γ, μ), ∀t ≥ Te with ke := min{ks , 2γkw σc } ∈ R+ , which implies that the closed-loop system has semiglobal exponential-like stability in ˜ (t) converge to small neighthe sense that both e(t) and W bourhoods of 0 dominated by ke , γ and ω on t ≥ Te .  IV. A N I LLUSTRATIVE E XAMPLE Consider the following wing rock dynamics model [14]:     0 1 0 x˙ = x+ (f (x) + bu) 0 0 1 with f (x) = [1, x1 , x2 , |x1 |x2 , |x2 |x2 , x31 ]T α, where x1 is the aircraft roll angle (rad), x2 is the roll rate (rad/s), u is the aileron command (rad), b ∈ R+ is a control gain, and α ∈ R6

˜ TW ˜ + ρ(ks , γ, μ) V˙ ≤ −eT Qe/2 − kw σc W with ρ(ks , γ, μ) := 2kw cw ε¯ + P b2 ε¯2f /2 ∈ R+ , in which

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in (13). The conventional model reference AFC (MRAFC) is selected as a baseline controller, where parameter settings of the two controllers are the same for fair comparison. Simulation studies are carried out in MATLAB running on Windows 7, where the solver is chosen to be fixed-step ode 5, the step size is set to be 0.001 s, and other settings are kept at their default values. Simulation results of the MRAFC and the MRCLFC without measurement noise are shown in Figs. 1 and 2, respectively. For the control performance, one observes that under the same parameter settings, the MRCLFC achieves better tracking accuracy with much less oscillations at x2 and u than the MRAFC. For the learning performance, one observes that the MRAFC exhibits serious oscillations at both ˆ and fˆ, and approximation of f by fˆ is unclear, whereas W the MRCLFC achieves accurate approximation of f by fˆ with ˆ and fˆ. Simulation results of the reduced oscillations at both W MRAFC and the MRCLFC under 45dB Gaussian white noise

is a vector of unknown coefficients. For simulation studies, set b = 3, α = [0.8, 0.2314, 0.6918, −0.6245, 0.0095, 0.0214]T , x(0) = [68π/180, −57 π/180]T , and     0 1 0 x˙ r = xr + r −1 −1 1 in which r = 57π/180 at t ∈ [15, 17] s, r = −57π/180 at t ∈ [25, 27] s and r = 0 for other time [14]. The design procedure of the proposed MRCLFC approach is as follows: firstly, to construct the FBFs in (3) with N = 52 = 25, let Ωcx = [−1, 1] × [−1, 1] and choose 5 Gaussian functions with standard deviation equaling to 1/3 for each state xi with i = 1 and 2 to evenly cover Ωcx ; secondly, solve (8) to obtain kr = [−1, −1, 1]T ; thirdly, select ke = [1, 1]T so that A is strictly Hurwitz; fourthly, solve (9) with Q = 10I to obtain P ; fifthly, set τd = 5 s in (11); sixthly, set γ = 50, kw = 1 and cw = 10 in (17); and finally, set ω = 100 and ζ = 0.7

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measurement are shown in Figs. 3 and 4, respectively, where qualitative analysis of these results is the same as the previous case except chattering at the control inputs u occur for both the controllers, which verifies robustness against measurement noise of the proposed control strategy.

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