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Robust Tracking Controller Backstepping Design for SISO Uncertain Nonlinear SystemË Yao Yu *. Yi-Sheng Zhong ** * Tsinghua National Laboratory for Information Science and Technology .Department of Automation, Tsinghua University, Beijing, 100084, P. R. China (e-mail: yu-y04@ mails.tsinghua.edu.cn). ** Tsinghua National Laboratory for Information Science and Technology. Department of Automation, Tsinghua University, Beijing, 100084, P. R. China (e-mail: [email protected]) Abstract: Controller design problem is dealt with for a class of plants with time-varying nonlinear uncertainties and unmodeled dynamics. A new method based on signal compensation is proposed to design a robust controller. A controller designed by this method consists of a nominal controller and a robust compensator. It is shown that robust tracking property and robust tracking transient performance can be achieved simultaneously. A salient feature of our results, shown in the present paper, is that the controller is linear and time-invariant one and we can tell the users how to tune on-line the parameters of the controller with the proposed robust and transient performance.

1. INTRODUCTION Recently, global stabilization of nonlinear strict-feedback form systems has attracted considerable attention. In this paper, our attention is focused on the problem of robust controller backstepping design for a class of SISO uncertain nonlinear strict-feedback form systems with time-varying uncertainties and stable dynamic uncertainties. Many researches have been done on this problem, but it is still not solved completely. Over the last decade, various robust stabilization techniques have been developed in the literature. Following the development of exact linearization techniques via smooth state feedback and diffeomorphic transformation ( Isidori, 1989 ), researches on certain triangular structure systems attracted considerable interest. One of the recent breakthroughs in nonlinear control is the introduction of backstepping algorithms for feedback linearizable systems. The relative-degree constraint, overparameterization, and growth condition are removed by allowing the controlled plant to be nonlinearly dependent on structure uncertainty, such as unknown parameters or unmodeled time-varying disturbances. Although the idea of integrator backstepping may be implicit in some earlier works, its use as a design tool was initiated by Tsinias(1989,1991), Byrnes and Isidori(1989), Sontag and Sussmann(1988), and Saberi et al.(1990). In Kanellakopoulos, Kokotovic, and Morse(1991), the cases with unknown parameters were investigated. In Freeman and Kokotovic (1992,1993), Marino and Tomei(1993), the cases with disturbance were considered. In Taylor, Kokotovic and kanellokopoulos(1989), Jiang and Mareels(1997), Jiang and Hill(1999), the cases with unmodeled Dynamics were treated.

In this paper, we consider robust control problem for a class of nonlinear strict-feedback form systems with uncertain parameters, bounded uncertainties and unmodeled dynamics. The uncertainties are norm-bounded, and the unmodeled dynamics is input-to-state stable as described in other papers. A new method is proposed based on signal compensation (Zhong, 2002) to design a robust controller. It will be shown that the states of the closed-loop system are ensured to be bounded and the global output tracking to a reference signal are achieved. And desired transient tracking property can also be assured The tracking error can be driven into an desired small neighborhood of the origin at an exponential rate. A controller designed by this method consists of a nominal controller and a robust compensator. The controller is linear and time invariant. This paper is organized as follows. In section 2, the plant description, the reference model, the assumptions on the uncertainties, reference input and unmodeled dynamics are presented. In section 3, controller design method and main results are stated. Section 4 gives the statement and the proof of the main results. An example is shown in section 5. Conclusions are stated in section 6. 2. PROBLEM DESCRIPTION Consider a nonlinear plant described by the following equations

ËThis work was supported by National Nature Science Foundation under Grant 60674017 and 60736024

978-3-902661-00-5/08/$20.00 © 2008 IFAC

14131

⎧ x1 ( t ) = x2 ( t ) + φ1 ( x,η , d , t ) ⎪ ⎪ x2 ( t ) = x3 ( t ) + φ2 ( x,η , d , t ) ⎪ ∑x ⎨ # ⎪ x t = x t + φ x,η , d , t ) ⎪ n −1 ( ) n ( ) n −1 ( ⎪ xn ( t ) = u ( t ) + φn ( x,η , d , t ) ⎩

(1)

10.3182/20080706-5-KR-1001.3269

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

∑η

η ( t ) = φη (η , x, d , t )

(2)

where F1 ( s ) is a robust filter of the form

y p ( t ) = x1 ( t )

F1 ( s ) =

where xi are the states, y p is the output, d is a bounded external disturbance vector, φi ( i = 1, 2," , n ) are regarded as nonlinear time-varying uncertainties and φη is a vector field describing dynamic uncertainties. There

are

known

ξi[i] = [ξi1 ξi 2 " ξii ] ςi

positive

φˆ1 = φ1 + y2 ( t ) = φ1 + x2 ( t ) + α1 y1 ( t ) + α1 ym ( t ) − β1r ( t ) − f1w1 ( t )

constant

vector

w1 ( t ) , only the states x1 ( t ) and r ( t ) are needed. If fact, w1 ( t ) = − F1 ( s )φˆ1 ( t )

and positive valued function

such that

(

β ⎛ α ⎞ = − ⎜1 + 1 ⎟ x1 ( t ) + 1 r ( t ) s ⎠ s ⎝

)

φi ( x,η , d , t ) ≤ ξi[i ] x[i ] ( t ) ∞ + ς i d ( t ) ∞ , η ( t0 ) where

and

It should be pointed out that, to get the robust control input

Assumption A A1)

Define

ς i ( 0, 0 ) = 0 ( i = 1, 2," , n ) 。

A2) The system given by (2) is bounded-inputs ( x and d ) bounded-state (η ) stable. It is required to design a controller which produces a control input u to drive the output y p of the plant to track a

y1 ( t ) = y p ( t ) − ym ( t ) = x1 ( t ) − ym ( t )

then one has

and

α1

and

β1 are given positive constants, r

(9)

(10)

(11)

(12)

From (1), (7), (8) and (10), one has

following reference model:

y m ( t ) = −α1 ym ( t ) + β1r ( t )

⎛ α ⎞ w1 ( t ) = − ⎜ 1 + 1 ⎟ y1 ( t ) s ⎠ ⎝

y1 ( t ) = −α1 y1 ( t ) + φˆ1 ( t ) + f1w1 ( t )

reference signal, denoted by ym , which is given by the

where

1 s + f1

y 2 ( t ) = x3 ( t ) + φ2 ( t )

(3) is a given

reference input.

(13)

where φ2 ( t ) = φ2 ( t ) − α12 y1 ( t ) + ( f1 + α1 ) ⎡⎣φˆ1 ( t ) + f1 w1 ( t ) ⎤⎦ + ρ m ( t )

ρ m ( t ) = α12 ym ( t ) + α1 β1r ( t ) − β1r ( t )

Assumption B

r ∈ C1 and there exist known and positive constants η r and η r such that

r ∞ ≤ ηr ,

r ∞ ≤ η r

(4)

3. CONTROLLER DESIGN

(14)

As the second step, consider the subsystem (13) with φ2 ( t )

as a disturbance and x3 ( t ) as a virtual control input and continue the design procedure. At the k th-step, consider the subsystem

y k ( t ) = xk +1 ( t ) + φk ( t )

Firstly, consider the subsystem

and regard xk +1 ( t ) as a virtual control input with the form

x1 ( t ) = x2 ( t ) + φ1 ( x1 ,η , d , t )

xˆk +1 ( t ) = uk ( t ) + f k wk ( t )

For the above system, a virtual controller is constructed as

xˆ2 ( t ) = u1 ( t ) + f1w1 ( t )

(5)

where u1 ( t ) is a nominal control input given by

u1 ( t ) = −α1 x1 ( t ) + β1r ( t )

where uk ( t ) is a nominal control input given by

uk ( t ) = −α k yk ( t ) (6)

w1 ( t ) is a robust control input to be designed, and f1 is a

where

The robust control input w1 ( t ) is given by

w1 ( t ) = − F1 ( s ) φˆ1 ( t )

is a positive constant. Let

yk +1 ( t ) = xk +1 ( t ) − xˆk +1 ( t )

positive constant to be determined. To perform backstep, apply the variable change

y2 ( t ) = x2 ( t ) − xˆ2 ( t ) = x2 ( t ) + α1 x1 ( t ) − β1r ( t ) − f1w1 ( t )

αk

= xk +1 ( t ) + α k yk ( t ) − f k wk ( t ) Then

(7) where (8) 14132

y k ( t ) = −α k yk ( t ) + φˆk ( t ) + f k wk ( t )

φˆk ( t ) = φk ( t ) + yk +1 ( t )

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

The robust control input wk ( t ) is constructed as

⎡ y k ( t ) ⎤ ⎡ −α k ⎢ ⎥=⎢ ⎣ w k ( t ) ⎦ ⎣ 0 k = 1, 2," , n

wk ( t ) = − Fk ( s ) φˆk ( t ) Fk ( s ) =

1 s + fk

and the whole controller description

u ( t ) = un ( t ) + f n wn ( t )

where f k is a positive constant to be determined. Note that

y1 ( t ) = x1 ( t ) − ym ( t ) y2 ( t ) = x2 ( t ) + α1 y1 ( t ) − f1w1 ( t ) + α1 ym ( t ) − β1r ( t )

φˆk ( t ) = ( s + α k ) yk ( t ) − f k wk ( t )

yk ( t ) = xk ( t ) + α k −1 yk −1 ( t ) − f k −1 wk −1 ( t ) , k = 3, 4," , n

So the robust control input wk ( t ) can also be given by

u1 ( t ) = −α1 y1 ( t ) − α1 ym ( t ) + β1r ( t )

wk ( t ) = − Fk ( s ) φˆk ( t )

uk ( t ) = −α k yk ( t ) , k = 2,3," , n

⎛ α ⎞ = − ⎜ 1 + k ⎟ yk ( t ) s ⎠ ⎝

⎛ α ⎞ wk ( t ) = − ⎜1 + k ⎟ yk ( t ) , k = 1, 2," , n s ⎠ ⎝

Differentiating yk +1 ( t ) , one has

(17) One sees that the designed controller is a linear timeinvariant one.

y k +1 ( t ) = xk + 2 ( t ) + φk +1 ( t ) where

φk +1 ( t ) = φk +1 ( t ) − α k2 yk ( t ) + ( f k + α k ) ⎡⎣φˆk ( t ) + f k wk ( t ) ⎤⎦ Finally, one has

f k ⎤ ⎡ yk ( t ) ⎤ ⎡ 1 ⎤ ˆ φ (t ) , ⎢ ⎥+ − f k ⎦⎥ ⎣ wk ( t ) ⎦ ⎣⎢ −1⎦⎥ k

y n ( t ) = u ( t ) + φn ( t )

4. CLOSED-LOOP CONTROL PROPERTIES Let

x ( t ) = ⎡⎣ x1 ( t ) x2 ( t ) " xn ( t ) ⎤⎦

(15)

T

where

w ( t ) = ⎡⎣ w1 ( t ) w2 ( t ) " wn ( t ) ⎤⎦

φn ( t ) = φn ( t ) − α n2−1 yn −1 ( t ) + ( f n −1 + α n −1 ) ⎡⎣φˆn −1 ( t ) + f n −1wn −1 ( t ) ⎤⎦

τ ( t ) = ⎡⎣τ 1 ( t ) τ 2 ( t ) " τ n ( t ) ⎤⎦

yn ( t ) = xn ( t ) + α n −1 yn −1 ( t ) − f n −1wn −1 ( t )

The control input u ( t ) is constructed as

T

Assumption C

with the nominal control input un ( t ) is given by (16)

The parameters f i ( i = 1, 2," , n ) are sufficiently large,

fi +1 is much larger than fi ( i = 1, 2," , n − 1) and

and the robust control input wn ( t ) by

f1 max {β1 ,α i , ξij , i, j = 1, 2," , n, i ≥ j}

wn ( t ) = − Fn ( s ) φˆn ( t )

{

f1 max τ

1 Fn ( s ) = s + fn

∞

, ρm

∞

(

,ς j d

∞

}

)

, η ( t0 ) , j = 1, 2," , n

It will be shown that the conclusions stated in the following theorem hold.

where φˆn ( t ) = φn ( t ) ， α n and f n are positive constants . Then the following differential equation holds

w n ( t ) = − f n wn ( t ) − φˆn ( t )

and wn ( t ) can be implemented as

wn ( t ) = − Fn ( s ) φˆn ( t ) ⎛ α = − ⎜1 + n s ⎝

T

= ⎡⎣ ym ( t ) −α1 ym ( t ) + β1r ( t ) 0 " 0 ⎤⎦

u ( t ) = un ( t ) + f n wn ( t ) un ( t ) = −α n yn ( t )

T

⎞ ⎟ yn ( t ) ⎠

Theorem 1. For any given and bounded initial states x ( t0 ) and w ( t0 ) and for any given positive constant

find sufficiently large parameters which satisfy Assumption

C, such that the states x ( t ) , η ( t ) and w ( t ) are bounded and the following statements hold.

(1) If the initial states x ( t0 ) and w ( t0 ) are nonzero, then there is a T ≥ t0 such that

yk ( t ) ≤ ε ,

From (15) and (16) it follows that

y n ( t ) = −α n yn ( t ) + φˆn ( t ) + f n wn ( t )

Summarizing the design results, one has

ε , one can

wk ( t ) ≤ ε , t ≥ T , k = 1, 2," , n

(2) If the initial states x ( t0 ) and w ( t0 ) are zero, then

14133

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

yk ( t ) ≤ ε ,

wk ( t ) ≤ ε , t ≥ t0 , k = 1, 2," , n

Before showing the proof of the main results stated in Theorem 1, several preliminary lemmas are stated and proven.

0 0 ⎡ x1 ( t ) ⎤ ⎡ y1 ( t ) ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x2 ( t ) ⎥ = ⎢ y2 ( t ) ⎥ − ⎢ α1 y1 ( t ) ⎥ + ⎢ f1w1 ( t ) ⎥ + τ ( t ) ⎢ # ⎥ ⎢ # ⎥ ⎢ ⎥ ⎢ ⎥ # # ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ xn ( t ) ⎥⎦ ⎢⎣ yn ( t ) ⎥⎦ ⎣α n −1 yn −1 ( t ) ⎦ ⎣ f n −1wn −1 ( t ) ⎦

Lemma 1. Let

(18)

υ ( f ) = ⎡⎣υij ( f ) ⎤⎦ ⎡ 1 ⎢− f − α 1 ⎢ 1 ⎢ ⎢ ⎢ 0 ⎢⎣

then

So one has

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ 1⎥⎦

0 1 − f2 − α2

1 %

1 − f n −1 − α n −1

ξ1[1] x[1] ( t ) ≤ ξ11 y1 ( t ) + ξ11 τ 1 ( t )

−1

ξ k [ k ] x[ k ] ( t ) ≤ μ yk [k ] y[ k ] ( t ) + ξ k [ k ] τ [ k ] ( t ) + μ wk [k −1] ( f ) w[ k −1] ( t ) k = 2,3," , n

From Assumption A and inequalities (19), one sees that the conclusions of Lemma 2 hold.

υij ( f ) = 1, i = j

Let υ k 0 ( f

i −1

υij ( f ) = ∏ ( f k + α k ), i > j

Lemma

k= j

φˆk ( t ) ≤ yk +1 ( t ) ∞ + μˆ yk [k ] ( f ) y[ k ] ( t ) ∞

Let T

⎡ ξ k 1 + α1ξ k 2 ⎤ ⎢ ξ +α ξ ⎥ 2 k3 ⎢ k2 ⎥ ⎥ , # =⎢ ⎢ ⎥ ⎢ξ k ( k −1) + α k −1ξ kk ⎥ ⎢ ⎥ ξ kk ⎣ ⎦

+ μˆ wk [k −1] ( f ) w[ k −1] ( t ) ∞ + ςˆk ( f , t ) k = 2,3," , n − 1 + μˆ wn[n −1] ( f ) w[ n −1] ( t ) ∞ where

T

k

i= j

k −1

μˆ wkj ( f ) = υkj ( f ) f j + ∑υk ( i +1) ( f ) μ w( i +1) j ( f )

)

i= j

j = 1, 2," , k − 1 k

(

j =1

⎣

)⎦

+ υk 2 ( f ) ρ m ( t ) ， k = 2,3," , n

(21)

Proof. From the definition of φˆk ( t ) ( k = 1, 2," , n ) and

)

from Lemmas 1 and 2, one has that

k = 2,3," , n Proof.

(

ςˆk ( f , t ) = ∑υkj ( f ) ⎡ξ j[ j ] τ [ j ] ( t ) ∞ + ς k d ( t ) ∞ , η ( t0 ) ⎤

+ μ wk [k −1] ( f ) w[ k −1] ( t ) ∞ + ς k d ( t ) ∞ , η ( t0 )

)

μˆ ykj ( f ) = υk ( j −1) ( f ) − υk ( j +1) ( f ) α 2j + ∑υki ( f ) μ yij

φk ( x,η , d , t ) ≤ μ yk [k ] y[ k ] ( t ) ∞ + ξk[k ] τ [k ] ( t ) ∞

(

j = 1, 2," , k

φ1 ( x,η , d , t ) ≤ ξ11 y1 ( t ) ∞ + ξ11 τ 1 ( t ) ∞

(

μˆ y11 = ξ11 ,

ςˆ1 ( t ) = ξ11 τ 1 ( t ) ∞ + ς 1 d ( t ) ∞ , η ( t0 )

satisfy that

+ ς 1 d ( t ) ∞ , η ( t0 )

(20)

φˆn ( t ) ≤ μˆ yn[n] ( f ) y[ n ] ( t ) ∞ + ςˆn ( f , t )

⎡ f1ξ k 2 ⎤ ⎢ fξ ⎥ μ wk [k −1] ( f ) = ⎢ 2 k 3 ⎥ , k = 2,3," , n ⎢ # ⎥ ⎢ ⎥ ⎣ f k −1ξ kk ⎦

φk ( k = 1, 2," , n )

satisfy the following

φˆ1 ( t ) ≤ y2 ( t ) ∞ + μˆ y11 y1 ( t ) ∞ + ςˆ1 ( t )

Proof. The proof is omitted since it is straightforward.

Lemma 2.

) = 0, k = 2,3," , n . 3. φˆk ( t ) ( k = 1, 2," , n )

inequalities.

υij ( f ) = 0, i < j

μ y11 = ξ11 , μ yk [k ]

(19)

From the definition of yk ( k = 1, 2," , n ) it

follows that 14134

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

⎡ φˆ1 ( t ) ⎤ ⎡ y2 ( t ) + φ1 ( t ) ⎤ ⎢ ⎥ ⎢ ⎥ 2 ⎢ φˆ2 ( t ) ⎥ − + + + + y t α y t φ t f α f w t ρ t ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 1 1 m ⎢ 3 ⎥ ⎢ ˆ ⎥ 2 ⎢ y4 ( t ) − α 2 y2 ( t ) + φ3 ( t ) + ( f 2 + α 2 ) f 2 w2 ( t ) ⎥ ⎢ φ3 ( t ) ⎥ ⎥ ⎢ # ⎥ = υ ( f ) ⎢# ⎢ ⎥ ⎢ ⎥ ⎢ y ( t ) − α 2 y (t ) + φ ( t ) + ( f + α ) f w (t )⎥ ⎢φˆn−1 ( t ) ⎥ n n − 2 n− 2 n −1 n−2 n− 2 n− 2 n − 2 ⎢ ⎥ ⎢ ⎥ ⎢⎣ −α n2−1 yn−1 ( t ) + φn ( t ) + ( f n−1 + α n−1 ) f n−1wn−1 ( t ) ⎦⎥ ⎢⎣ φˆn ( t ) ⎥⎦

Together with Assumption A and Lemmas 1 and 2, the conclusions of Lemma 3 hold. Lemma 4.

For any given positive constant

εφ

k −1 ⎡ k +1 ⎤ ≤ ε k ⎢ ∑ yi ( t ) ∞ + ∑ wi ( t ) ∞ + 1⎥ i =1 ⎣ i =1 ⎦ ≤ ε k ⎡ k + 1 y[ k +1] ( t ) + 1 + k − 1 w[ k −1] ( t ) ⎤ ∞ ∞⎦ ⎣ If choose ε k such that ε k k + 1 ≤ ε φ , then (23) holds.

Now we are ready to prove Theorem 1. For let

⎡1 1 ⎤ Pk = ⎢ ⎥ ⎣1 2 ⎦

, if

Assumption C is satisfied, then

φˆ1 ( t ) f1

φˆk ( t ) fk

≤ ε φ ⎡ y[2] ( t ) ⎣

+ 1⎤ ∞ ⎦

(22)

Consider the following positive function n

V = ∑ vk k =1

≤ ε φ ⎡ y[k +1] ( t ) ⎣

k = 2,3," , n − 1 φˆn ( t ) ≤ ε φ ⎡ y[n] ( t ) ⎣ fn

∞

∞

+ w[ k −1] ( t )

)

+ 1⎤ ⎦

+ 1⎤ ⎦

⎡ y (t ) ⎤ vk = ⎡⎣ yk ( t ) wk ( t ) ⎤⎦ Pk ⎢ k ⎥ ⎣ wk ( t ) ⎦ The derivative of V along the trajectories of the closed-loop

(24)

system is given by

and μˆ wkj ( f

)，

εk

, if

∞

∞

one sees that for any given positive constant

( j = 1, 2," , n )

where (23)

+ w[ n −1] ( t )

Proof. From the definition of μˆ ykj ( f

fj

n

V = ∑ vk k =1

n

= −2∑ ⎡⎣α k yk2 ( t ) + α k yk ( t ) wk ( t ) + f k wk2 ( t ) + wk ( t ) φˆk ( t ) ⎤⎦ k =1

are sufficiently large and satisfy

n

= −∑ ⎡⎣α k vk + α k yk2 ( t ) + 2 ( f k − α k ) wk2 ( t ) + 2wk ( t ) φˆk ( t ) ⎤⎦

Assumption C, then

k =1

1 ≤ ε k , k = 1, 2," , n fk

μˆ ykj ( f ) fk

μˆ wkj ( f ) fk

2 ⎧ ⎡ φˆk ( t ) ⎤ ⎫ ⎪ 2 2 ⎥ ⎪⎬ ≤ −∑ ⎨α k vk + α k yk ( t ) + ( f k − 2α k ) wk ( t ) − ⎢ ⎢ fk ⎥ ⎪ k =1 ⎪ ⎣ ⎦ ⎭ ⎩ n

≤ ε k , k = 1, 2," , n;

j = 1, 2," , k

≤ ε k , k = 2,3," , n;

j = 1, 2," , k − 1

ςˆk ( f , t ) ∞ fk

⎡ φˆk ( t ) ⎤ ⎥ ≤ −α 0V + ∑ ⎢ fk ⎥ k =1 ⎢ ⎣ ⎦

where α 0 = min {α k , k = 1, 2," , n} . So one has 2

fk

≤ ε k , k = 1, 2," , n

μˆ yk [k ] ( f ) 1 yk +1 ( t ) ∞ + y[ k ] ( t ) ∞ fk fk +

⎡ φˆk ( t ) ⎤ 1 −α 0 ( t − t0 ) ∞ ⎥ (26) V (t ) ≤ e V ( t0 ) + ∑ ⎢ α 0 k =1 ⎢ fk ⎥ ⎣ ⎦ n

We only show the proof of (23). The proof of (22) and (24) can be performed in a similar way. For k = 2,3," , n − 1 , from (20) and (25), one has

≤

2

n

(25)

φˆk ( t )

k = 1, 2," , n ,

μˆ wk [k −1] ( f ) fk

w[ k −1] ( t ) ∞ +

From Lemma 4, it follows that for any given positive constant ε φ , if Assumption C is satisfied, one has 2

⎡ φˆk ( t ) ⎤ ∞ ⎥ ⎢ ≤ 2ε φ2 ⎡ y[2] ( t ) ∑ ⎢⎣ ⎢ ⎥ fk k =1 ⎣ ⎦ n

2 ∞

+ 1⎤ ⎥⎦

n −1

+ ∑ 3ε φ2 ⎡ y[k +1] ( t ) ⎣⎢ k =2

1 ςˆk ( f , t ) fk

+ 3ε φ2 ⎡ y[n] ( t ) ⎢⎣

14135

2 ∞

2 ∞

+ w[ k −1] ( t )

+ w[ n −1] ( t )

2 ∞

2 ∞

+ 1⎤ ⎥⎦

+ 1⎤ ⎦⎥

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

≤ 3ε φ2 ≤ where

{( n + 1) ⎡⎣ y (t )

3ε φ2 ( n + 1)

λ0

V (t )

(

∞

2 ∞

}

2 + w(t ) ∞ ⎤ + n ⎦

(27)

where α = 10, β = 10 . It is required that For the simulation, we take

g = 9.8 , m = k /10 = g −2 , d ( t ) = sin ( 3t ) , L ( t ) = g + 3sin ( 2t )

+ 3ε φ2 n

)

λ0 = λmin ( Pk ) = 3 − 5 / 2 . From (26) and (27) it

follows that

V (t )

∞

≤ V ( t0 ) +

3ε φ2 ( n + 1)

λ0α 0

V (t )

+ ∞

3ε φ2 n

⎛

⎞ 1 η ( t ) ⎟⎟ + d ( t ) ⎝ L (t ) m ⎠

φ ( t ) = L ( t ) mg sin ⎜⎜

The initial values are given by x ( 0 ) = 0,η ( 0 ) = 0 . The

α0

robust controller designed by the method shown in the last section is as follows:

If

λ0α 0

εφ <

(28)

3 ( n + 1)

u ( t ) = −α x ( t ) + β r ( t ) + f1w1 ( t )

where

then

3ε φ2 n ⎤ 1⎡ V ( t ) ∞ ≤ ⎢V ( t0 ) + (29) ⎥ π ⎢⎣ α 0 ⎥⎦ 3ε φ2 ( n + 1) . From (26), (27) and (29) one where π = 1 − λ0α 0 has n

e ∑ ⎡⎣ y ( t ) + w ( t )⎤⎦ ≤ k =1

2 k

2 k

−α 0 ( t − t0 )

λ0

Choose

εφ < εφ <

V ( t0 )

3ε φ2 ⎪⎧ ( n + 1) ⎡ 3ε φ2 n ⎤ ⎪⎫ + + V t ( ) ⎢ ⎥ + n⎬ ⎨ 0 λ0α 0 ⎩⎪ λ0π ⎣⎢ α 0 ⎦⎥ ⎭⎪ It can be shown (see Lemma A in Appendix A) that if

εφ

n

e ∑ ⎡⎣ y ( t ) + w ( t )⎤⎦ ≤ k =1

2 k

2 k

εφ

so that

λ0α 0

3 ( n + 1)

≈ 0.2523

λ02α 0ε ≈ 0.0779 3 ⎡⎣( n + 1) V ( t0 ) + nλ0 + ( n + 1) λ0ε ⎤⎦ 2

⎛ gm ⎞ ≈ 158.2622 f1 ≥ ⎜ ⎜ ε k ⎟⎟ ⎝ φ ⎠

is

2

λ02α 0ε 3 ⎡⎣( n + 1) V ( t0 ) + nλ0 + ( n + 1) λ0ε ⎤⎦

then

⎛ α⎞ w1 ( t ) = − ⎜1 + ⎟ y1 ( t ) , w1 ( 0 ) = 0 s⎠ ⎝

and choose f1 to satisfy

sufficiently small so that

εφ <

ε = 0.03 .

−α 0 ( t − t0 )

λ0

(30)

⎛ gm ⎞ 1 g f1 ≥ ⎜ ym ( t ) ∞ + η ( 0 ) + d ( t ) ⎟ ≈ 646.0341 ⎜ε k ⎟ εφ εφ ⎝ φ ⎠ So we set f1 = 700 . Simulation results are shown in Figs 1 through 3. We can see that the state x , unmeasured unmodeled dynamics η are globally bounded. The tracking error is driven into the desired small neighborhood of the origin with desired transient performance.

V ( t0 ) + ε

The inequality above implies the conclusions of Theorem 1. 5. NUMERICAL EXAMPLE Consider the following plant as a numerical example (Jiang and Hill, 1999).

x ( t ) = u ( t ) + φ ( t ) k m y p (t ) = x (t )

η ( t ) = − η ( t ) + x ( t ) where u is the torque input, η is the unmeasured unmodeled dynamics, x is the state, y p is the output. The reference model is

y m ( t ) = −α ym ( t ) + β r ( t ) , r ( t ) = 1, t ≥ 0 14136

Fig. 1. Plot of tracking error

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

Fig. 2. Plot of robust control input

Fig. 2. Plot of unmodeled dynamics 6. CONCLUSIONS For strict-feedback systems with time-varying nonlinear uncertainties and unmodeled dynamics, a new method is proposed of designing robust controller. A nominal controller is designed to get exact output tracking for the nominal plant; a robust compensator is added to achieve robust properties. Under Assumptions A, B and C, robust tracking property and robust tracking transient performance are achieved. The robust controller is a linear and timeinvariant one, so it can be realized easily.

uncertainties. IEEE Transactions on Automatic Control, 42, 292-308 Jiang, Z.P. and Hill, D.J. (1999). A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics. IEEE Transactions on Automatic Control, 44, 1705-1711. Kanellakopoulos, I., Kokotovic, P.V. and Morse, A.S. (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Transactions on Automatic Control, 36, 1241-1253. Marino, R. and Tomei, P. (1993). Robust stabilization of nonlinear feedback linearizable time-varying uncertain systems. Automatica, 29, 181-189 Saberi, A., Kokotovic, P.V. and Sussmann, H.J. (1990). Global Stabilization of partially linear composite systems. SIAM Journal of Control and Optimization, 28, 1491-1503. Sontag, E.D. and Sussmann, H. J. (1988). Further comments on the stabilizability on the angular velocity of a rigid body. Systems and Control Letters, 12, 437-442. Taylor, D.G., Kokotovic, P. V. and Kanellokopoulos, I. (1989). Adaptive regulation of nonlinear systems with unmodeled dynamics. IEEE Transactions on Automatic Control, 34, 405-412 Tsinias, J. (1989). Sufficient Lyapunov-like conditions for stabilization. Mathematics of Control, Signals, and Systems, 2, 343-357. Tsinias, J. (1991). Existence of control Lyapunov functions and applications to state feedback stabilizability of nonlinear systems. SIAM Journal of Control and Optimization, 29, 457-473. Zhong, Y.S. (2002). Robust output tracking control of SISO plants with multiple operating points and with parametric and unstructured uncertainties. Int. Journal of Control, 75, 219 -241 Appendix A. Lemma A If

εφ

is sufficiently small so that (28) holds and

that

λ02α 0ε εφ < 3 ⎡⎣( n + 1) V ( t0 ) + nλ0 + ( n + 1) λ0ε ⎤⎦

(A.1)

then REFERENCES Byrnes, C.I. and Isidori, A. (1989). New results and examples in nonlinear feedback stabilization. Systems and Control Letters, 12, 437-442. Freeman, R.A. and Kokotovic, P.V. (1993). Design of softer robust nonlinear control laws. Automatica, 29, 14251437. Freeman, R.A. and Kokotovic, P.V. (1992). Backstepping design of robust controllers for a class nonlinear systems. Preprints of second IFAC nonlinear control systems design symposium, Bordeaux, France, 307-312 Isidori, A. (1989). Nonlinear Control Systems, 105-203, Springer-Verlag, Berlin. Jiang, Z.P. and Mareels, M.Y. (1997). A small-gain control method for nonlinear cascaded systems with dynamic

3ε φ2 ⎧⎪ ( n + 1) ⎡ 3ε φ2 n ⎤ ⎫⎪ (A.2) ⎢V ( t ) + ⎥ + n⎬ < ε ⎨ λ0α 0 ⎪⎩ λ0π ⎣⎢ 0 α 0 ⎦⎥ ⎪⎭ 2 2 Proof. If (28) holds, then λ0 α 0 − 3ε φ ( n + 1) λ0 > 0 . In this case, from (A.1) it follows that

3ε φ2 ⎡⎣( n + 1) V ( t0 ) + ( nλ0 ) ⎤⎦

λ02α 0 − 3ε φ2 ( n + 1) λ0

<ε

which implies that

3ε φ2 ⎡⎣( n + 1) α 0V ( t0 ) + 3ε φ2 ( n + 1) n + λ0π nα 0 ⎤⎦

λ02πα 02

So (A.2) holds.

14137

<ε

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