Jrl Syst Sci & Complexity (2007) 20: 545–553

ROBUST H∞ CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS∗ Weiping BI

Received: 19 April 2005 / Revised: 7 July 2006 c °2007 Springer Science + Business Media, LLC Abstract This paper studies the robust H∞ disturbance attenuation with internal stability for uncertain nonlinear control systems. By adding one power integrator technique, this paper designs a explicit smooth robust dynamic feedback law while rejecting the disturbance to any specified degree of accuracy. Further, the example and simulation results show the effectiveness of the proposed schemes. Key words Internal stability, robust dynamic feedback, specified degree.

1 Introduction ζ˙1 = u1 , ζ˙2 = u2 , ζ˙3 = ζ2 u1 ,

(1)

.. . ˙ζn = ζn−1 u1 was introduced by Murray and Sastry (1993) as a canonical representation for many nonlinear mechanical systems with nonholonomic constraints. Morin et al.(1998) addressed the parking problem for the mobile robot of unicycle type with uncertainties x˙ c = ν cos(θ + ε), y˙ c = ν sin(θ + ε),

(2)

θ˙ = ω, where ε is a small bias in orientation, which is the disturbance at the robot. Z. P. Jiang and I. Kanellakopoulos[1] dealt with the global output-feedback tracking for a benchmark nonlinear system. And robust control problems for nonlinear systems were considered in [2–8]. The purpose of this paper is to consider a perturbed version of the chained form (3). Using the power integrator technique, this paper constructs an explicit smooth robust dynamic feedback controller which attenuates the disturbance’s effect on the output to an arbitrary degree of Weiping BI College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China. Email: [email protected]. ∗ The research is supported by the Natural Science Foundation of Henan Province under Grant No. 2007120005.

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accuracy with internal stability. It complements the previous work in [9]. Further, the result obtained in this paper can recover the results in [9]. The paper is organized as follows. The systems description and problem statement are given in Section 2. Section 3 presents the main result and its proof. Section 4 proposes the example and simulation results. Finally, some concluding remarks are proposed in Section 5.

2 Problem Formulation This paper discusses a class of high-order nonlinear systems in the presence of both unknown time-varying disturbances and parametric uncertainties described by equations of the form: x˙ 1 = exp21 + f1 (x1 , t) + g1 (x1 , t)θ + φ1 (x1 , t)w, .. .

i x˙ i = expi+1 + fi (x1 , · · · , xi , t) + gi (x1 , · · · , xi , t)θ + φi (x1 , · · · , xi , t)w, .. .

(3)

x˙ n = eupn + fn (x1 , · · · , xn , t) + gn (x1 , · · · , xn , t)θ + φn (x1 , · · · , xn , t)w, y = h(x1 ), where x = (x1 , x2 , · · · , xn )T ∈ Rn represents the system state, u ∈ R control input, y ∈ R system output, w ∈ Rq disturbance signal, and θ ∈ Rs unknown constant vector. The mappings fi : Ri+1 → R, gi : Ri+1 → R1×s , and φi : Ri+1 → R1×q , i = 1, 2, · · · , n, are assumed to be smooth and vanishing at the origin. And e ∈ R+ , h(x1 ) is a known smooth function with h(0) = 0. The robust adaptive H∞ almost disturbance decoupling problem with internal stability (RAH∞ -ADD). If given any real number γ > 0, there is a smooth robust adaptive time-varying dynamic feedback law of the form ˙ ˆ t), θˆ = ε(x, θ,

ˆ t) u = u(x, θ,

ˆ t) = 0, u(0, θ, ˆ t) = 0 with ε(0, θ,

(4)

such that the closed-loop systems (3) and (4) satisfy the following: i) when w = 0, the origin of the closed-loop systems (3) and (4) is globally asymptotically stable; ii) ∀w(t) ∈ L2m , m = 1, 2, · · · , the response of the closed-loop systems (3) and (4) starting from the initial state x(0) = 0 is such that Z t Z t |y(s)|2mp1 ds ≤ γ 2 kw(s)k2m ds, ∀t ≥ 0. (5) 0

0

Let Lp =

½ Z z(t)|

∞

¾ kz(s)kp ds < ∞, p ≥ 1 .

0

Next, assume that uncertain nonlinear system (3) satisfies the following basic conditions. Hypothesis 1

For i = 1, 2, · · · , n,

|fi (x1 , x2 , · · · ,xi , t)| ≤ (|x1 |pi + · · · + |xi |pi )βi (x1 , x2 , · · · ,xi ), kgi (x1 , x2 , · · · ,xi , t)k ≤ (|x1 |pi + · · · + |xi |pi )ηi (x1 , x2 , · · · ,xi ), kφi (x1 , x2 , · · · ,xi , t)k ≤ ϕi (x1 , x2 , · · · ,xi ),

ROBUST H∞ CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS

547

where βi (·), ηi (·), and ϕi (·) are known smooth nonnegative bounding functions. Hypothesis 2

p1 ≥ p2 ≥ · · · ≥ pn ≥ 1 are odd integers.

[5]

Lemma Let a, b, and ci , i = 1, 2, · · · , l, be real variables. Assume that dj : Rl+1 → R (j = 1, 2) are two smooth mappings. Then, for any positive integers m, n and real number N > 0, there are two nonnegative smooth functions d3 : Rl+2 → R, d4 : Rl+1 → R such that the following inequalities hold: |a|m+n i) |am [(b + ad1 (c1 , · · · , cl , a))n − (ad1 (c1 , · · · , cl , a))n ]| ≤ + |b|m+n d3 (c1 , · · · , cl , a, b); N m m ii) |bn (cm 1 + · · · + cl + b )d2 (c1 , · · · , cl , b)| ≤

|c1 |m+n + · · · + |cl |m+n + |b|m+n d4 (c1 , · · · , cl , b). N

3 Robust Adaptive Controller By means of the adding of a power integrator technique and the Lemma, we obtain a constructive solution to the RAH∞ -ADD problem of nonlinear system (3). Theorem 1 Under Hypothesis 1 and Hypothesis 2, there is a smooth robust adaptive dynamic controller of the form (4) that makes the RAH∞ -ADD problem solvable for nonlinear systems (3). Proof Utilizing adding a power integrator technique, we design recursively a smooth robust adaptive time-varying dynamic feedback law that solves the RAH∞ -ADD problem for (3). Step 1: Consider the x1 -subsystem of (3), and construct a Lyapunov function e = V1 (x1 , θ)

(2m−1)p +1

1 1 eT e x1 + θ θ, (2m − 1)p1 + 1 2λ

(6)

ˆ θˆ is a parameter estimate for θ, and λ a positive real number. There is a where θe = θ − θ, smooth function γ0 (x1 ) ≥ 0 such that for any β > 0, e + y 2mp1 − βkwk2m ≤ x(2m−1)p1 (exp1 + f1 (x1 , t) + g1 (x1 , t)θ) ˆ + x2mp1 γ0 (x1 ) V˙ 1 (x1 , θ) 1 2 1 +(τ1 (x1 , t) −

1 ˆ˙T e θ )θ + |x1 |(2m−1)p1 ϕ1 (x1 )kwk − βkwk2m , λ

(2m−1)p1

where τ1 (x1 , t) = x1

(7)

g1 (x1 , t).

By the Lemma, there are smooth functions γ1 (·) ≥ 0 and σ1 (·) ≥ 0, and (7) follows e + y 2mp1 − βkwk2m ≤ ex(2m−1)p1 (xp1 − x∗p1 ) + (τ1 (x1 , t) − 1 θˆ˙T )θe V˙ 1 (x1 , θ) 2 2 1 λ · µ ´ 1 ¶¸ ³ 1 ´³ ϕ2m (2m−1)p1 ∗p1 p1 1 (x1 ) 2m−1 ˆ , +x1 ex2 + x1 γ1 (x1 , θ) + γ0 (x1 ) + 1 − 2m 2mβ (8) 1 σ1 (x1 ). where |τ1 (x1 , t)| ≤ x2mp 1

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WEIPING BI

Clearly, the virtual smooth dynamic controller ˆ =− x∗2 (x1 , θ)

· ¸ p1 1 ³ ´³ ϕ2m (x ) ´ 2m−1 1 x1 1 1 ˆ + γ0 (x1 ) + 1 − 1 ne + γ1 (x1 , θ) e 2m 2mβ

ˆ = −x1 α1 (x1 , θ)

(9)

makes e + y 2mp1 − βkwk2m V˙ 1 (x1 , θ) (2m−1)p1

1 ≤ −nex2mp + ex1 1

³ 1 ˆ˙T ´ e 1 (xp21 − x∗p θ. 2 ) + τ1 (x1 , t) − θ λ

(10)

Inductive Step: Suppose at step k, there are the change of coordinates ˆ · · · , ξk = xk − x∗ (ξ1 , · · · , ξk−1 , θ) ˆ ξ1 = x1 , ξ2 = x2 − x∗2 (ξ1 , θ), k

(11)

ˆ = 0 (i = 2, 3, · · · , k), a positive definite and proper function with x∗1 = 0, x∗i (0, · · · , 0, θ) e = Vk (ξ1 , · · · , ξk , θ)

2mp −p +1 k X ξj 1 j 1 eT e + θ θ, 2mp1 − pj + 1 2λ j=1

(12)

and a virtual smooth dynamic feedback law · ξk ∗ ˆ xk+1 (ξ1 · · · , ξk , θ) = − (n − k + 1)e + γk (·) + γ˜k (·) + γˆk (·) e ³ ´ 1 ¸ p1k 1 ´³ 2m(p1 −pk ) Ψ 2m k (·) 2m−1 ˆ = −ξk αk (ξ1 , · · · , ξk , θ), + 1− ξk 2m 2mβ

(13)

with smooth nonnegative function αk (·) such that (x1 , x2 , · · · ,xk )-subsystem of (3) satisfies e + y 2mp1 − kβkwk2m ≤ −(n − k + 1)e(ξ 2mp1 + · · · + ξ 2mp1 ) V˙ k (ξ1 , · · · , ξk , θ) 1 k ³ ´ 1 ˆ˙T e k k ˆ ˆ +eξk2mp1 −pk (xpk+1 − x∗p (θ + ρk (ξ1 , · · · , ξk , θ)) k+1 ) + τk (ξ1 , · · · , ξk , θ, t) − θ λ and

ˆ kτk (·)k ≤ (ξ12mp1 + · · · + ξk2mp1 )σk (ξ1 · · · ξk , θ),

(14)

(15)

where smooth function σk (·) > 0. So, at Step k + 1, let ˆ ξk+1 = xk+1 − x∗k+1 (ξ1 , · · · , ξk , θ),

(16)

then, the xk+1 -equation of (1) becomes pk+1 ˆ t) + Gk+1 (x1 , x2 , · · · ,xk+1 , θ, ˆ t)θ ξ˙k+1 = xk+2 + Fk+1 (x1 , x2 , · · · ,xk+1 , θ, ∗ ˆ˙ ˆ t)w − ∂xk+1 θ, +Φk+1 (x1 , x2 , · · · ,xk+1 , θ, ˆ ∂θ

(17)

ROBUST H∞ CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS

549

where Fk+1 (·) = fk+1 (x1 , x2 , · · · ,xk+1 , t) − Gk+1 (·) = gk+1 (x1 , x2 , · · · ,xk+1 , t) − Φk+1 (·) = φk+1 (x1 , x2 , · · · ,xk+1 , t) −

n X ∂x∗k+1 j=1 n X j=1 n X j=1

∂xj

p

j + fj (x1 , x2 , · · · ,xj , t)), (xj+1

∂x∗k+1 gj (x1 , x2 , · · · ,xj , t), ∂xj ∂x∗k+1 φj (x1 , x2 , · · · ,xj , t). ∂xj

We can verify ˆ |Fk+1 (·)| ≤ (|ξ1 |pk+1 + · · · + |ξk+1 |pk+1 )β˜k+1 (ξ1 , · · · , ξk+1 , θ), ˆ kGk+1 (·)k ≤ (|ξ1 |pk+1 + · · · + |ξk+1 |pk+1 )˜ ηk+1 (ξ1 , · · · , ξk+1 , θ),

(18)

ˆ kΦk+1 (·)k ≤ Ψk+1 (ξ1 , · · · , ξk+1 , θ), for some smooth nonnegative functions β˜k+1 (·), η˜k+1 (·), and Ψk+1 (·), and consider a positive definite and proper function 2mp −p

e = Vk (ξ1 , · · · , ξk , θ) e + Vk+1 (ξ1 , · · · , ξk+1 , θ)

+1

ξk+1 1 k+1 . 2mp1 − pk+1 + 1

(19)

With the help of (14) and (17), we obtain e + y 2mp1 − (k + 1)βkwk2m V˙ k+1 (ξ1 , · · · , ξk+1 , θ) ∗pk ≤ −(n − k + 1)e(ξ12mp1 + · · · + ξk2mp1 ) + eξk2mp1 −pk [(ξk+1 + x∗k+1 )pk − xk+1 ] 2mp −pk+1

+ξk+1 1

pk+1 ˆ + ξ 2mp1 −pk+1 Φk+1 (·)w − βkwk2m (exk+2 + Fk+1 (·) + Gk+1 (·)θ) k+1

ˆ t)ρk+1 (ξ1 , · · · , ξk+1 , θ) ˆ +τk (·)ρk (·) − τk+1 (ξ1 , · · · , ξk+1 , θ, ³ ´ ˆ t) − 1 θˆ˙T (θe + ρk+1 (ξ1 , · · · , ξk+1 , θ)), ˆ + τk+1 (ξ1 , · · · , ξk+1 , θ, λ

(20)

where 2mp −pk+1

1 ˆ t) = τk (ξ1 , · · · , ξk , θ, ˆ t) + ξ τk+1 (ξ1 , · · · , ξk+1 , θ, k+1

2mp −pk+1

1 ˆ = ρk (ξ1 , · · · , ξk , θ) ˆ + λξ ρk+1 (ξ1 , · · · , ξk+1 , θ) k+1

and

Gk+1 (·),

³ ∂x∗

k+1

∂ θˆ

2mp1 ˆ )σk+1 (ξ1 , · · · , ξk+1 , θ). kτk+1 (·)k ≤ (ξ12mp1 + · · · + ξk+1

By means of (18), (21), (22), and the Lemma, we can show

´T

,

(21)

(22)

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WEIPING BI

e + y 2mp1 − (k + 1)βkwk2m V˙ k+1 (ξ1 , · · · , ξk+1 , θ) 2mp −pk+1

≤ −(n − k)e(ξ12mp1 + · · · + eξk2mp1 ) + eξk+1 1 2mp −pk+1

+ξk+1 1

h

p

∗p

k+1 k+1 (xk+2 − xk+2 )

³ ∗pk+1 pk+1 exk+2 + ξk+1 γk+1 (·) + γ˜k+1 (·) + γˆk+1 (·)

1 2m ³ ´i (·) ´ 2m−1 1 ´³ 2m(p1 −pk+1 ) Ψk+1 1˙ + 1− ξk+1 + (τk+1 (·) − θˆT )(θe + ρk+1 (·)). 2m 2mβ λ

(23)

Obviously, the smooth virtual controller h ˆ = − ξk+1 (n − k)e + γk+1 (·) + γ˜k+1 (·) + γˆk+1 (·) x∗k+2 (ξ1 , · · · , ξk+1 , θ) e 1 2m ³ ip 1 (·) ´ 2m−1 1 ´³ 2m(p1 −pk+1 ) Ψk+1 k+1 ˆ + 1− ξk+1 = −ξk+1 αk+1 (ξ1 , · · · , ξk+1 , θ) 2m 2mβ

(24)

with αk+1 (·) > 0. Thus, (23) follows e + y 2mp1 − (k + 1)βkwk2m V˙ k+1 (ξ1 , · · · , ξk+1 , θ) 2mp −pk+1

2mp1 ≤ −(n − k)e(ξ12mp1 + · · · + ξk+1 ) + eξk+1 1

p

∗p

k+1 k+1 (xk+2 − xk+2 )

³ 1˙ ´ + τk+1 (·) − θˆT (θe + ρk+1 (·)). λ

(25)

So, the inductive argument is completed. Step n: Repeatedly using the inductive step, one can verify that there exist a change of coordinates (ξ1 , ξ2 , · · · , ξn ) of the form (11), a positive definite and proper Lyapunov function e = Vn−1 (ξ1 , · · · , ξn−1 , θ) ˜ + Vn (ξ1 , · · · , ξn , θ)

ξn2mp1 −pn +1 2mp1 − pn + 1

and a smooth dynamic feedback control law ˆ u∗ = x∗n+1 (ξ1 , · · · , ξn , θ) 1 ¸ pn · 2m(p −p ) 1 ³ ξn 1 ´³ ξn 1 n Ψn2m (·) ´ 2m−1 =− e + γn (·) + γ˜n (·) + γˆn (·) + 1 − e 2m 2bβ ˆ = −ξn αn (ξ1 , · · · , ξn , θ)

(26)

such that e + y 2mp1 − nβkwk2m V˙ n (ξ1 , · · · , ξn , θ) ∗pn n ≤ −e(ξ12mp1 + · · · + ξn2mp1 ) + eξn2mp1 −pn (xpn+1 − xn+1 )

³ ´ ˆ t) − 1 θˆ˙T (θe + ρn (ξ1 , · · · , ξn , θ)). ˆ + τn (ξ1 , · · · , ξn , θ, λ

(27)

ROBUST H∞ CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS

551

Then, the robust adaptive controller ˙ ˆ t), θˆ = λτnT (ξ1 , · · · , ξn , θ,

ˆ = −ξn αn (ξ1 , · · · , ξn , θ), ˆ u = u∗ (ξ1 , · · · , ξn , θ)

(28)

and let nβ = γ 2 , yields e + y 2mp1 − γ 2 kwk2m ≤ −e(ξ 2mp1 + · · · + ξ 2mp1 ). V˙ n (ξ1 , · · · , ξn , θ) n 1

(29)

Clearly, the system (3) is globally asymptotically stabilized by the smooth robust dynamic feedback control (28) when w = 0. And, by means of Vn (·) positive definite with Vn (0) = 0, we can derive from (29) Z

t

Z |y(s)|2mp1 ds ≤ γ 2

0

t

kw(s)k2m ds,

∀ t ≥ 0.

0

So, the proof of the Theorem is completed. Remark When gi (·) ≡ 0, i = 1, 2, · · · , n or θ ≡ 0, the result of this paper recovers the Theorem 1 and Theorem 2 in [9].

4 Example and Simulation This section presents one example and its simulation to illustrate the main features of our schemes. Example x˙1 = x32 + x31 + x31 θ + x1 ω, x˙2 = u, (30) y = x1 . e = First, we choose the Lyapunov function as V1 (x1 , θ)

x41 4

e + 21 θeT θ,

h ³ 2 ´i e + y 6 − ω 2 ≤ x3 (x3 − x∗3 ) + x3 x∗3 + x3 5 + θˆT θˆ + x1 . V˙1 (x1 , θ) 1 2 2 1 2 1 4 ˆ = −x1 (5 + θˆT θˆ + Let x∗2 (x1 , θ)

x21 1 3 4 )

ˆ then = −x1 α1 (x1 , θ),

e + y 6 − ω 2 ≤ −2x6 + x3 (x3 − x∗3 ) + (x6 − θˆ˙T )θ. e V˙1 (x1 , θ) 1 1 2 2 1 ˆ we have Let ξ2 = x2 − x∗2 (x1 , θ), ∗ ∗ ∗ ∂x∗ ˆ − ∂x2 x3 θe − ∂x2 x1 ω − ∂x2 θ. ˆ˙ ξ˙2 = u − 2 (x32 + x31 + x31 θ) 1 ˆ ∂x1 ∂x1 ∂x1 ∂θ

e = V1 (x1 , θ) e + Let V2 (x1 , ξ2 , θ)

ξ26 6 ,

we have ∗

5 5 ∂x2 ˆ V˙2 + y 6 − 2ω 2 ≤ −2x61 + x31 [(ξ2 + x∗2 )3 − x∗3 (x3 + x31 + x31 θ)) 2 ] + ξ2 u − ξ2 ∂x1 2 ∂x∗

+

ξ210 x21 ( ∂x21 )2 4

´ ³ ³ ∂x∗ ∂x∗ ´³ ∂x∗ ´T ˙ + x61 − ξ25 2 x31 − θˆT (θe + ρ2 ) − x61 − ξ25 2 x31 ξ25 2 . ∂x1 ∂x1 ∂ θˆ

552

WEIPING BI

Finally, the robust adaptive controller ∂x∗

³ u = −ξ2 1 + γ2 (·) + γ˜2 (·) + γˆ2 (·) + is such that V˙2 + y − 2βω ≤ 6

2

−x61

−

ξ26 ,

ξ24 x21 ( ∂x21 )2 ´ 4

where

1 5 · 36 12 + 4 · 34 α13 + α1 , 9 4 3 i 23 3 4 ³ ∂x∗2 ´2 2 2 h ∂x∗2 2 ˆ γ˜2 (·) = ξ2 (1 + θ) + ξ2 (3ξ2 + x1 α1 ) α13 , 2 ∂x1 3 ∂x1

γ2 (·) =

2 h³ 4 ∂x∗ ´³ ∂x∗ ´T i6 x1 − ξ25 2 x1 ξ2 2 , 3 ∂x1 ∂ θˆ ³ 2 ´1 ˆ = 5 + θˆT θˆ + x1 3 , α1 (x1 , θ) 4 γˆ2 (·) =

when ω = 0, initial value (x1 , x2 , θ)|t=0 = [2, −5, 1.2], the simulation result is shown as follows. 2

1

0

−1

−2

−3

−4

−5

0

50

100

150

5 Conclusion The contributions of this paper are to improve the adding power integrator technique and solve the robust H∞ disturbance attenuation with internal stability problem for a class of timevarying and uncertain nonlinear systems (3). And, the result of this paper can recover the results in [9]. Therefore, this paper extends earlier work and exploits a new application of adding one power integrator technique for a wider class of uncertain nonlinear systems which can not be controlled by existing methods.

ROBUST H∞ CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS

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