Robust control in uncertain nonlinear systems with

International Journal of Systems Science, 1999, volume 30, number 1, pages 19± 23

Robust control in uncertain nonlinear systems with exogenous signals

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S.-H. Lee² and J.-T. Lim² ³ This paper proposes a robust gain scheduling control law in nonlinear systems with exogenous signals and structural uncertainties. A matching condition is presented to cancel the unmodelled dynamics which appears in the linear term of the control input. Using the proposed control law, the closed-loop system which satis® es the given conditions provides the desired output error bound.

1.

Introduction

Gain scheduling control laws have been developed by many researchers (Huang and Rugh 1990, Rugh 1991). Using the extended linearization, Lawrence and Rugh (1990 ) and Khalil and Kokotovic (1991) proposed the stability results that deal with the response of a nonlinear system to slowly varying exogenous signals. The most recent work on the output regulation for a nonlinear system driven by an exogenous signal was done by Sureshbabu and Rugh (1995) and Lee and Lim (1997 b). They proposed a control law with derivative information on the exogenous signal and included nonlinear examples in order to illustrate enhanced regulation performance over existing controller design. Moreover, the gain scheduling in nonlinear systems with bounded uncertain time varying input was also proposed (Lee and Lim 1997a). Those controllers were designed and destined only for the systems with the exactly known plant dynamics. In practice, however, nonlinear plants are not exactly modelled owing to the unknown dynamics. Using those without any modi® cation causes undesirable performance or destabilizes the overall control system. The k th-order approximate equilibrium point was computed in uncertain systems with exogenous signals generated by the known exogenous system (Huang and Rugh, 1992, Huang 1995). Using the internal model principle, they proposed a k th-order robust control law. However, in most cases, it is di cult to obtain the exogenous system which generates the exogenous signals Accepted 3 April 1998. ² Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea. ³ To whom all correspondence should be addressed. 0020± 7721/99 $12.00

injected to the plant dynamics. Thus, this paper proposed a compensated state feedback controller to solve the regulation problems for the unknown exogenous system. Using the proposed control law, the system which satis® es a given matching condition can be controlled to provide the desired output error bound.

2.

Formulation

Consider the uncertain nonlinear system given by

Ç

x

f x, w

,

G x, w

D f x w

h x, w

y

,

D G x w u

,

,

1 2

where x is the n 1 state vector, u is the p 1 control input, w is the m 1 exogenous signal, y is the q 1 output, and D f and D G are unmodelled dynamics. The functions f , G and h are assumed to be continuously di erentiable. The following assumption is proposed for the nominal system. There exist an open neighbourhood C of m the origin in R and smooth functions x w and u w for each w C such that Assumption 1:

0

f x w ,w

rd

h x w ,w

where rd is the q

G x w ,w u w

,

,

1 reference input.

The nominal control input which achieves the control objective to minimize limt rd y t when D f D G 0 is given by (Rugh 1991) u

k x, w

u w

Kw x

x w

,

3

Ñ

where K w is determined so that the eigenvalues of ¶ f c x w , w /¶ x with (3) should have speci® ed values 1999 Taylor & Francis Ltd.

20

S.-H. L ee and J.-T. L im

with negative real parts for each w for f c x, w f x, w G x, w k x, w . Moreover, to minimize the e ect of the uncertainty, we add a compensator U x, w to the nominal controller k x, w . Then, the overall control input is given by k x, w

u

x, w . U

Using the assumptions 2 and 3, (6) can be rewritten as follows: Proof:

Ç

V t, e

W nominal

2w

4

f c x, w

Ç


e

f x, w D

G x, w Ac e

,

,

D f

Rc e

,

x, w

D Gk

Ç

q D G U

G

¶

^

R

p p

and

h

2

h

There exist functions n p : R R such that

1 :

m

,

G x, w

D G x w

h

x, w

1

h

¶

q w, w

Ç

R

n

n There exist a function q : R and constants g 1 and g 2 such that

1

x, w

x, w

q

,

,

g

D f x w

1

R

R

, h

2

R

g 2.

In order to analyse the error dynamics in the sense of Lyapunov stability, we de® ne a Lyapunov function T candidate V t, e t e t Q t e t . We notice that Q t is the well de® ned, continuously di erentiable T n n unique positive-de® nite solution of Ac t Q t Q t Ac t I . The solution of the Lyapunov equaiton 1 T T is obtained by vec Q t Ac t Ac t vec I 2 where the n -vector vec Q t is the vector composed of the columns of matrix Q t taken in order. Di erentiating V t, e t with respect to t and applying (5) to the result give

Ç

V t, e t

W nominal

2eT Q D f

T

2e Q D G U

6

where T

W nominal

e e

T

Ç

¶ q /¶

T

2e Q Rc e

e Qe

Ç

w w.

Thus, we propose a compensator given by U T

where w

x, w

q

k

1

w q

, w

7

e QG x, w . T

Under assumption 1± 3, if q < 1, then the closed-loop system (1) with (4) satis® es

Ç

W nominal

2 e Q

g

1

g 1

2

q

k

.

w

q

1 q

2q k k

2 e Q

T

2q w

g

1

2 w

w

2w

1k

T

h

1U

k

2

U

2

q

1

k q

g

2 e Q

g

g

1

U

g

1

2

1

2

1

w

k q

q

k

, h

where q < 1.

The compensator U x, w is chosen so as to cancel the e ect of uncertainty. To provide its physical meaning, we consider the case when QG 0 holds. Then, in order to achieve the control objective, x should track x w for each w C . Thus, for the single-input singleoutput case, U x, w decreases the absolute value of the nominal input k x, w when the uncertainty causes overtracking x > x w and increases it when under-tracking x < x w respectively by the amount q / 1 q k . For the case when QG < 0, a similar interpretation can be made in the reverse direction of the e ect of U x, w to the absolute value of the nominal input. Therefore, to implement this mechanism, U x, w is skilfully constructed as shown in (7). Then, U x, w restricts e on the switching surface w 0 from the e ect of uncertainty and it causes a chattering phenomenon. h x

D

^

If Ac x, w satis® es the L ipschitz condition for R , then there exists a ® nite constant L a such that

Lemma 2:

n

8

^

La e t 2

, ^

t

Using Ac q µe, w Ac q, w the de® nition of Rc t , we obtain L a /2 e . Proof:

0. L a µe and ^ Rc q, w, e

h

1: Suppose that ¶ f c x w , w /¶ x in the closed-loop system (5) is Hurwitz for each w C . From assumption 1, there exists a ® nite constant T such that ¶ q /¶ w T . Then, under assumption 2 and 3 there exist ® nite constants g 1 , g 2 and d from d , w t t 0, such that, for q < 1 and e 0 < eo , e t is bounded by eb as t where 1 2 µ 1 d M2 eo 2 g 2 g 4 / 1 q M1 D / /2 L a M1 , 1 2 µ 1 d M2 eb 2 g 2 g 4 / 1 q M1 D / /2 L a M1 2 2 and D µ 1 d M2 2 g 2 g 4 / 1 q M1 8d T 2 g 1 g 2 g 3 / 1 q L a M1 for 0 < µ < 1. Theorem

Lemma 1:

V

U

Rc t

,

2e QGU

2q k

h

U

T

2 e Q g

T

W nominal

2eT Q D G k

T

k

2

T

2w

Remark 1:

m

m

w

2

5

w .

2

Assumption 3:

h

U

W nominal

where Ac t : Ac q, w ¶ f c x, w^ /¶ x x q and Rc t : ^ 1 ^ µe, w Rc q, w , e Ac q, w dµ. The follow0 Ac q ing assumptions are proposed for the compensator design with a matching condition. Assumption 2:

2w

D G x w k x w

D G x w U

2eT Q h

T

W nominal

Letting q x w and e x q, we obtain the corresponding error dynamics such that

2eT Q D f

Ç

21

Robust control in uncertain nonlinear systems with exogenous signals

We note that there exist ® nite constants M1 and M2 such that Q t M1 and Q t M2 w t for t 0 respectively (Lee and Lim 1997 a, b). From lemma 1, we obtain Proof:

Ç

Ç

V t, e t

2

e

2 Q


g

¶

q w

Ç

2

1

Kw

e

g

1

, g 2,d

2 > 8 L a M1

g 2g

4

1 q

g 2g

3

1 q

M1

2

e

T

g

L a M1 e

3

Ç

h q, w

1

g 2g

3

1 q

g 2g

2

M2 d

1 /2

1

4

q

M1

;

h x, w

Ly q

x

,

Ly e

h

We notice that q < 1 is related to the stability margin for the input matrix perturbation in linear system case. Moreover, each value of max S g 1 , g 2 , d is proportional to 1 /q as shown in (9). Thus, the required robustness needs a best compromise between them. Under assumptions 1± 3, the closed-loop system (1) with (3) satis® es Remark 3:

W nominal

x2

x2

y

x1 ,

2 e Q g

1

q

G

g

2

k .

10

1

exp

x2

w exp w

1

9

Remark 2:

Ç

w, w u,

exp

with the unmodelled dynamics D f 0. 1 cos x1 , 0 T and D G 0, 0. 4 sin w exp w . For rd 0, the nominal plant’ s operating points (or equilibrium traT jectory) are obtained by x w 0w and u w 1 w exp w . In order to place the linearized closed-loop system ¶ f c x w , w /¶ x eigenvalues both 2 ¸, ¸ 1 exp w , at we obtain Kw 1 2¸ exp w . Moreover, from assumptions 2 and 3 we obtain q 0. 4, g 1 0. 1 and g 2 0. Therefore, the overall control input is u

where L y is a Lipschitz constant.

V

x2

x1

Ç

2 3

2¸ exp w 1

1

Ç

y

0, q *

f or q

2

Consider the plant given by x1

M1 e

then it can be easily derived that both X > 0 and Y 0 are satis® ed for any g 1 , g 2 , d S . Thus, it guarantees 2 V t, e t X e for e t eb , eo . Finally, we conclude that e t is ultimately bounded by eb for e 0 < eo . Finally, if hc x, w satis® es the Lipschitz condition for n x D R , then we obtain rd

g

G q

q

T

2 R µ 1

d

2

Example:

where u w g 3 and K w g 4 for w C , X 1 µ 1 d M2 2 g 2 g 4 / 1 q M1 , and Y L a M1 e e eo e eb . In order to guarantee the existence of the positive range of eb , eo , both X > 0 and Y 0 should be satis® ed for given g 1 , g 2 and d . We de® ne S

g

1

as t , if limt G g 2. Moreover, we notice that eb in theorem 1 is monotonically increasing function with regard to the term g 2 / 1 q . Thus, the error bound can be reduced by adding the compensator U x, w for q < q *. h

2

e

Y,

2

X e

2

Rc

2 e Q

e u w

q

g

2 Q

w

M2

T

2

e

1 d

d

¶ g

1

1 2

Ç

Q

Ç

From (8) and (10), it is derived that there exists 0 q * < 1 such that

¸

x2

w exp w

2¸ exp w

2

1 exp w x1 w ¸

x2

w

2

1 exp w x1 w

w

.

Simulation is performed with/and without the proposed compensator U under the initial condition x0 0. 5 0. 5 . In ® gure 1( a), in order to guarantee output error below 0.05 in the closed-loop system with the constructed compensator U , we choose ¸ 5. Figure 1( b) shows that larger eigenvalues, that is ¸ 10, provide smaller error bound. Thus, we conclude that the compensated controller provides a smaller output error than the uncompensated controller. h

3.

Conclusions

In this paper, we propose a robust gain scheduling control law in the nonlienar system with unmodelled dynamics. In particular, a matching condition is presented so as to design the additional compensator. If the uncertain nonlinear system satis® es the given conditions, the proposed law reduces the output error bound to the desired bound.


22 S.-H. L ee and J.-T. L im

( a)

( b)

Figure 1.

Output for w(t) = sin (0 . 2t): (a )

k

= 5; (b) k = 10.

Robust control in uncertain nonlinear systems with exogenous signals References


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approximation method for the nonlinear servomechanism problem. IEEE Transactions on Automatic Control, 37, 1395± 1398. Kha lil , H. K. , and Kok otov ic , P. V ., 1991, On stability properties of nonlinear systems with slowly varying inputs. IEEE Transactions on Automatic Control, 36, 229. Law rence, D. A., and R ug h , W. J ., 1990, On a stability theorem for nonlinear systems with slowly varying inputs. IEEE Transactions on Automatic COntrol, 35, 860± 864.

23

Lee, S.-H. , and Lim, J. -T., 1997 a, Locally robustable gain scheduling in nonlinear systems with uncertain time varying inputs. International Journal of Systems Science, 28, 587± 593; 1997 b, Fast gain scheduling on tracking problems using derivative information. Automatica, 33, 2265± 2268. R u g h , W. J., 1991, Analytical framework for gain scheduling. IEEE Control Systems Magazine, 11, 79± 84. Su reshba bu , N., and R u gh , W. J ., 1995, Output regulation with derivative information. IEEE Transactions on Automatic Control, 40, 1755± 1766.