Asymptotic Stabilization via Adaptive Fuzzy Control Yongping Pan∗, Rongjun Chen† ‡ , Hongzhou Tan† and Meng Joo Er∗ ∗ School
of Electrical and Electronic Engineering Nanyang Technological University, Singapore, 639798 Email: {yppan; emjer}@ntu.edu.sg † School of Information Science and Technology Sun Yat-sen University, Guangzhou 510006, China Email:
[email protected];
[email protected] ‡ Nanfang College, Sun Yat-sen University, Guangzhou 510970, China Abstract—This paper certifies that standard adaptive fuzzy control (AFC) can guarantee asymptotic stabilization performance rather than uniformly ultimately boundedness (UUB) even in the presence of fuzzy approximation errors (FAEs). Under a direct AFC scheme, the resulting optimal FAE is shown to be bounded by the norm of the plant state vector multiplying a globally invertible and nondecreasing function, which provides a pivotal property for asymptotic stability analysis. Without any additional control compensation, the closed-loop system is proved to be partially and asymptotically stable in the sense that all involved signals are UUB and the plant state variables converge to zero. The resulting control law is certainly continuous since it only contains an adaptive fuzzy system. Compared with previous adaptive approximation-based asymptotic stabilization approaches, the proposed approach not only simplifies control design, but also relaxes constraint conditions on the controlled plant. A simulation example of inverted pendulum control is provided to verify the discovery of this study. Keywords—Adaptive control, asymptotic stabilization, fuzzy approximation, uncertain nonlinear system.
I.
I NTRODUCTION
Recent years, adaptive approximation-based control (AAC) using fuzzy logic systems (FLSs) or neural networks (NNs) [1] has been attracting widespread concern due to its effectiveness of dealing with nonparametric uncertainties in nonlinear systems [2]–[18]. Yet, unlike classic adaptive control which guarantees asymptotic stability of the closed-loop systems, standard AAC only guarantees uniformly ultimately boundedness (UUB) due to the existence of inherent approximation errors [1]. To circumvent the problem involved in approximation errors, additional discontinuous control terms were widely applied to compensate for AAC systems [5]. Yet, the discontinuous control strategy is inapplicable in many practical applications since it requires infinite control bandwidth and causes chattering at the control input. Although some variations of the standard AAC with guaranteed partially asymptotic stabilization have been developed in [16]–[18], rigorous constraint conditions on the plants should be satisfied and tedious design should be done in all those approaches. This paper focuses on asymptotic stabilization using standard adaptive fuzzy control (AFC) in [19] for a class of single input single output (SISO) uncertain nonlinear systems in the general Brunovsky form in [18]. First, under a direct AFC scheme, the resulting optimal fuzzy approximation error (FAE) is shown to be bounded by the norm of the state vector
multiplying a globally invertible and nondecreasing function. Second, this bound result is applied to Lyapunov-based AFC design. Without any addition control compensation, the closedloop system is proved to be partially and asymptotically stable in the sense that all involved signals are UUB and the plant state variables converge to zero even in the presence of the FAE. A domain of attraction for initial state variables is also obtained during proof. The resulting control law is certainly continuous since it only contains an adaptive fuzzy system. Consequently, partially asymptotic stabilization is guaranteed by the standard AFC without any rigorous constraint condition and tedious design in [16]–[18]. The rest of this paper is organized as follows. The problem under consideration is formulated in Section II. The proof of the proposed argument is given in Section III. A simulation illustration is provided in Section IV. Conclusive remarks are summarized in Section V. Throughout this paper, R, R+ , Rn and Rn×m denote the spaces of real numbers, positive real numbers, real n-vectors and n × m matrixes, respectively, | · |, · and · ∞ denote the absolute value, 2-norm and ∞-norm, respectively, L∞ represent the space of essentially bounded functions, λmin (·) and λmax (·) represent the functions of minimal and maximal eigenvalues, respectively, min{·}, max{·} and sup(·) represent the functions of minimum, maximum and supremum, respectively, and C k represents the space of functions whose all k-order derivatives are existent and continuous, where n, m and k are non-negative integers. II.
P ROBLEM F ORMULATION
Consider the following SISO affine nonlinear system in the controllable canonical form, i.e. the Brunovsky form [18]: x˙ i = xi+1 (i = 1, 2, · · · , n − 1) x˙ n = f (x) + g(x)u (1) y = x1 where x(t) := [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is the measurable state vector, u(t) ∈ R and y(t) ∈ R are the control input and system output, respectively, f (x) : Rn → R satisfying f (0) = 0 is the class C 1 unknown nonlinear driving function, and g(x) : Rn → R is the class C 1 unknown control gain function (i.e. the affine term). Then, the following additional assumption is made accordingly. Assumption 1 [20]: There exist continuous functions f (x) and g(x), and a finite constant g0 such that |f (x)| ≤ f (x), and 0 < g0 ≤ g(x) ≤ g(x), ∀x ∈ Rn .
Choose a vector k = [k1 , k2 , · · · , kn ]T ∈ Rn such that h(s) := sn + kn sn−1 + · · ·+ k2 s+ k1 is a Hurwitz polynomial, where s is a complex variable. It follows form the result in [19] that the following ideal control law: (2) u∗ (x) := − f (x) − kT x g(x) makes the closed-loop system globally and exponentially stable. Yet, (2) is not enforceable since f (·) and g(·) are unknown here. Accordingly, the control objective of this study is to determine a standard AFC for the system in (1) under Assumption 1 such that the closed-loop system achieves asymptotic stabilization in the sense that limt→∞ x(t) = 0. Remark 1: For the simplification of discussion, the SISO affine nonlinear system in (1) with g > 0 is considered here. Yet, by the combination of the approaches in [21] and [22], the following results can be extended to a more general class of multi-input multi-output (MIMO) nonaffine nonlinear systems under certain conditions without much difficulty. In addition, from the discussion in [20], the following analysis can also be easily modified to the system in (1) with g < 0.
First, we derive the bound result of the optimal FAE w(·) in (4). From (2), one obtains u∗ (0) = − f (0) − kT 0 g(0) = 0. From the definition of θ ∗ in (5), one gets T
u(0, θ∗ ) = θ ∗ ξ(0) = 0. Applying u∗ (0) = 0 and u(0, θ∗ ) = 0 to (5), one obtains w(0) = u(0, θ∗ ) − u∗ (0) = 0. Since u∗ (·) in (2) and u(·) in (3) are of class C1 , w(x) in (4) is of class C1 . Combining w(x) ∈ C1 , w(0) = 0 and the Remark 3 of [23], there must exist a globally invertible and nondecreasing function ρ(x) : R+ → R+ such that w(x) ≤ ρ(x)x, ∀x ∈ Dx .
Then, subtracting and adding g(·)u∗ (·) on the second line of (1) and making some transformations, one obtains ˆ − u∗ (x)) (8) x˙ = Ax + b g(x)(u(x, θ) in which
III.
A SYMPTOTIC A DAPTIVE F UZZY C ONTROL
While f (·) and g(·) are unknown, one introduces a class C 1 linearly parameterized FLS [19]: T
ˆ =θ ˆ ξ(x) u(x, θ)
(3)
ˆ ∈ RM is the vector of to approximate u∗ (·) in (2), where θ n adjustable parameters, ξ(x) : R → RM satisfying ξ(·) ≤ φ is the vector of fuzzy basis functions, φ ∈ R+ is a finite constant determined by the fuzzy membership functions, and M is the number of fuzzy rules. Let compact sets Dx ˆ θ ˆ ≤ Mθ }, and Ωθ := := {x| x ≤ Mx }, Ωθ1 := {θ| 2 T ˆ ˆ {θ| θ ξ(0) = 0}, where Dx is regarded as a domain of fuzzy approximation, and Mx , Mθ ∈ R+ are user-defined finite constants. Then, define the optimal FAE w as follows: w(x) := u(x, θ ∗ ) − u∗ (x) where θ∗ is a vector of optimal parameters given by ˆ − u∗ (x)| . θ∗ = arg min sup |u(x, θ) θ∈Ωθ1 ∩Ωθ2
x∈Dx
(4)
(5)
Consider the SISO version of a pretreatment method in [16], [17]. By the Mean Value Theorem, f (·) is rewritten into f (x) = Δ(x)x
(6)
in which Δ(x) : Rn → Rn is a unknown function vector. Then, n NNs are applied to approximate Δ(·) such that the resulting approximation errors lie in a small gain-type norm bounded conic sector. This pretreatment method plays a key role in ensuring asymptotic stabilization of the plants in [16], [17]. Yet, it also significantly increases complexity of the control laws since n (rather than 1) NNs should be applied to indirectly approximate f (·). In the following derivation, we will show that such a pretreatment is not necessary and the asymptotic stabilization result can still be achieved without the increase of controller complexity.
(7)
⎡
⎢ A=⎢ ⎣
0 .. .
1 .. .
0 −k1
0 −k2
⎡ ⎤ ··· 0 .. ⎥ .. ⎢ . . ⎥,b = ⎢ ⎣ ⎦ ··· 1 · · · −kn
⎤ 0 .. ⎥ . ⎥. 0 ⎦ 1
From the selection of k, A is a stable matrix. Hence, for any given positive definite symmetric matrix Q ∈ Rn×n , there must exist a unique positive definite symmetric matrix solution P ∈ Rn×n for the Lyapunov equation: AT P + P A = −Q.
(9)
To deal with the control singularity problem caused by the unknown control gain function g(·) in (1) [20], the following fuzzy approximation equation: ˆ − u∗ (x) = θ ˜ T Λξ(x) + c0 (x)w(x) g(x) u(x, θ) (10) ˜ := θ ˆ − θ∗ , c0 (x) : Rn → R+ is in [24] is introduced, where θ a unknown bounded function, Λ := diag(c1 , c2 , · · · , cM ), and ci ∈ R+ with i = 1, 2, · · · , M are unknown finite constants. From the proof in Appendix of [24], c0 (x) can be bounded by a finite constant c¯0 ∈ R+ , i.e. c¯0 := c0 (x)∞ , where the value of c¯0 can be affected by the maximal distance of fuzzy partitions. Substituting (10) into (8), one obtains T ˜ Λξ(x) + c0 (x)w(x) . x˙ = Ax + b θ (11) Choose a Lyapunov function candidate for (11) as follows: ˜ T Λθ/2γ ˜ V (z, t) = xT P x/2 + θ
(12)
T ˜T T
where z := [x , θ ] ∈ Rn+M , and γ ∈ R+ is a adaptive rate. Now, we establish the main result of this study. Theorem 1: For the system in (1) satisfying Assumption 1, select (3) with the following adaptive law: ⎧ ˆ < Mθ ⎪ −γxT P bξ(x) if θ ⎪ ⎪ ⎨ ˆ or (θ = Mθ and xT P bξ(x) ≤ 0) ˆθ˙ = (13) ˆθ ˆ T ξ(x) θ ˆ 2 ⎪ −γxT P bξ(x) + γxT P bθ ⎪ ⎪ ⎩ ˆ = Mθ and xT P bξ(x) > 0) if (θ
as the control law, where γ := γ0 /λmin (P ) with γ0 ∈ R+ to be a user-defined finite constant. Then the closed-loop system achieves partially asymptotic stability in the sense that all involving signals are UUB and limt→∞ x(t) = 0, ∀z ∈ S, where S is a domain of attraction given by (18).
Consequently, one obtains
Proof: Differentiating (12) along (11) with respect to time t and using (9) leads to
Remark 2: From (9) and the definition of A, one knows that a large k leads to a small P for a certain Q. Thus, according to (18), the domain of attraction S can be made large to be close to Dz0 by increasing Q and k concurrently. However, a large ˆ in (13), which degrades k reduces the learning speed of θ control performance or even destroys system stability. Here, the selection of γ to be equal to γ0 /λmin (P ) in the adaptive law of (13) completely resolves this problem. Moreover, since a large k in u∗ (·) only provides a nice property of the liner part of the closed-loop system in (12), the control performance is actually determined by the approximation performance of u(·) with respect to u∗ (·). Thus, the selection of γ0 plays the only key role in improving control performance.
˙ ˜ T Λˆ V˙ = −xT Qx/2 + θ θ/γ T ˜ Λξ(x) + c0 (x)w(e) + xT P b θ = −xT Qx/2 + xT P bc0 (x)w(x) ˙ ˜ T Λ xT P bξ(x) + ˆ θ/γ . +θ Substituting (13) into the above expression and using the result of the projection operator in [19], one obtains V˙ = −xT Qx/2 + xT P bc0 (x)w(x).
(14)
Applying (7) to (14) and noting c¯0 = c0 (x)∞ , one gets V˙ ≤ −λmin (Q)x2 /2 + c¯0 ρ(x)x2 P b = −x2 λmin (Q)/2 − c¯0 ρ(x)P b . Accordingly, one can state that c0 ρ(x)P b V˙ ≤ −λQ x2 /2, ∀λmin (Q) > 2¯
(15)
c0 ρ(x)P b ∈ R+ . with λQ := λmin (Q) − 2¯ Now, define continuous positive definite functions U1 and U2 , and a continuous positive semi-definite function U : U1 (z) := λ1 z2 , U2 (z) := λ2 z2 U (z) := λQ x2 /2 respectively, where λ1 := min{λmin (P )/2, λmin (Λ)/2γ}, and λ2 := max{λmax (P )/2, λmax (Λ)/2γ}. Then, one can show that V in (12) satisfies U1 (z) ≤ V (z, t) ≤ U2 (z) (16) V˙ (z, t) ≤ −U (z)
lim x(t) = 0, ∀ z(0) ∈ S.
t→∞
IV.
A S IMULATION E XAMPLE
Consider an inverted pendulum model in the form of (1) with n = 2 and [19]: ⎧ 2 4lp lp mp cos2 x1 ⎨ f (x) = gv sin x1 − mp lp x2 cos x1 sin x1 − mc +mp 3 mc +mp 4lp lp mp cos2 x1 ⎩ g(x) = cos x1 − mc +mp 3 mc +mp where x1 and x2 are the angular position and angular velocity of the pendulum, respectively, gv is the gravitational acceleration, mc is the mass of the cart, mp is the mass of the pendulum, and lp is the half-length of the pendulum. For simulation, select mc = 1kg, mp = 0.1kg, gv = 9.8m/s2 , lp = 0.5m, and five different initial state values x(0) as [π/6, π/6]T (Case 1), [π/12, π/3]T (Case 2), [0, −π/3]T (Case 3), [−π/12, 0]T (Case 4), and [−π/6, −π/3]T (Case 5). The control adjective is to make the plant state vector x converges to the equilibrium point zero.
(17)
The procedure of control design is as follows: first, to ˆ construct the FLS in (3), select θ(0) = [0, 0, · · · , 0]T , Mx = 2 2 1/2 ((π/6) + (π/3) ) , and the fuzzy membership functions (x + κ − (κ /2)(l − 1))2 i i i i , μAli = exp − i (κi /4)2
with Dz0 := Dx × Ωθ and ρ−1 (·) being the inverse function of ρ(·). The result in (16) implies that V (z, t) ∈ L∞ , ∀z ∈ Dz . ˜ θ ˆ ∈ L∞ , ∀z ∈ Dz . Using (3), one gets u ∈ L∞ , Hence, x, θ, ∀z ∈ Dz . Thus, all involved signals are UUB.
where Alii are linguistic variables of fuzzy rules, i = 1, 2, κ1 = π/6, κ2 = π/3, and li = 1, 2, · · · , 5; second, to determine the adaptive law in (13), let Mθ = 100, γ0 = 1000, k = [10, 25]T and Q = diag(10, 10), and obtain P = [15, 0.2; 0.2, 0.52] and λmin (P ) = 0.5172 by resolving (9).
∀ t > 0 and ∀ z ∈ Dz , where Dz is given by λ (Q) min Dz := z ∈ Dz0 |z < ρ−1 2¯ c0 P b
˜ ξ(x) ∈ L∞ , ∀z ∈ Dz , and c0 (x), w(x) ∈ Now, since x, θ, L∞ , ∀x ∈ Dx , (11) implies x˙ ∈ L∞ , ∀z ∈ Dz . From the definition of U (z) and x, x˙ ∈ L∞ , ∀z ∈ Dz , one obtains U˙ (z) ∈ L∞ , ∀z ∈ Dz , which is a sufficient condition for U (z) being uniformly continuous. Accordingly, from the Lemma 2 of [23], (15)−(17) implies that limt→∞ U(z) = 0, ∀ z(0) ∈ S where S is given by λ (Q) min . S := z ∈ Dz |U2 (z) < λ1 ρ−1 2¯ c0 P b
Simulation trajectories under various initial state values are shown in Fig. 1. One sees that both the angular position [see Fig. 1(a)] and the angular velocity [see Fig. 1(b)] rapidly converge to zero under fast parameter adaptations [see Fig. 1(c)] and smooth control inputs [see Fig. 1(d)], which verifies the correctness of the proposed argument. V.
(18)
C ONCLUSION
This paper has proved that the standard AFC can guarantee asymptotic stabilization rather than UUB even in the presence of the FAEs. Under a direct AFC scheme, the resulting optimal
0.6
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0.4
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Angular position
0.2 0 −0.2
Case 1 Case 2 Case 3 Case 4 Case 5
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−0.4 −1
−0.6 −0.8 0
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Norm of θ
40
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(c)
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(d)
Fig. 1. Simulation trajectories of the proposed approach. (a) Convergence of angular velocity. (b) Convergence of angular velocity. (c) Parameter adaptation. (d) Control input curve.
FAE is shown to be bounded by the norm of the plant state vector multiplying a globally invertible and nondecreasing function, which provides a nice property for asymptotic stability analysis. Without any additional control compensation, the closed-loop system is proved to be partially asymptotically stable in the sense that all involved signals are UUB and the plant state variables converge to zero. Compared with the previous AACs with guaranteed asymptotic stabilization, the proposed approach not only simplifies control design, but also relaxes constraint conditions on the plant. A simulation illustration of inverted pendulum control is provided to versify the conclusion of this study. Further work would focus on global asymptotic stabilization using standard AFC plus proportion differentiation (PD) control. ACKNOWLEDGMENT This work is supported in part by the Science and Engineering Research Council, Agency for Science, Technology and Research (A*STAR), Singapore under Grant no. 1122904016, and the National Natural Science Foundation of China-Guangdong Key Project under Grant no. U0935002.
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