AN ADAPTIVE CONTROL SCHEME FOR OUTPUT FEEDBACK NONLINEAR SYSTEMS WITH ACTUATOR FAILURES

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

www.elsevier.com/locate/ifac

AN ADAPTIVE CONTROL SCHEME FOR OUTPUT FEEDBACK NONLINEAR SYSTEMS WITH ACTUATOR FAILURES Xidong Tang and Gang Tao * Suresh M. Joshi ** * Department of Electrical and Computer Engineering University of Virginia, Charlottesville, VA 22903

**Mail Stop 161, NASA Langley Research Center, Hampton, VA 23681

Abstract: An adaptive control scheme to achieve stability and output tracking is presented for the output-feedback nonlinear plant with unknown actuator failures. A state observer is designed for estimating the unavailable plant states, based on a chosen control strategy, in the presence of actuator failures with unknown failure values, time instants and pattern. An adaptive controller is developed by employing the backstepping technique, for which parameter update laws are derived to ensure asymptotic output tracking and signal boundedness of the closed-loop system, as shown by detailed stability analysis. Copyright © 2002 IFAC Keywords: Actuators, adaptive control, backstepping, failure compensation, nonlinear systems, tracking, stability. adaptive observer-based design (Wang and Daley, 1996), direct adaptive actuator failure compensation control of systems with known plant dynamics (Boskovic et al., 1998).

1. INTRODUCTION Actuator failures may lead to performance deterioration or dysfunction of control systems. For some critical systems such as flight control systems, actuator failures, if not compensated, may result in disasters. Some accidents caused by actuator failures could have been avoided, if it was successful to make use of the remaining working actuators which were actually enough for ensuring some desired system performance. To this end, effective controllers are needed to take action automatically, once actuator failures occur. However actuator failures are often unknown in terms of failure patterns, failure times and failure values, which not only introduce uncertainties into systems but also change the system structural properties. Hence, compensation schemes are expected to guarantee a specified control objective for any possible actuator failures. In order to handle failure uncertainties, an adaptive approach is a desirable one for designing control schemes with actuator failure compensation capacity.

Our research for the actuator failure compensation problem with direct adaptive designs was conducted in (Tao et al., 2001c) and (Tao et al., 2000) for linear systems, and some new results have also been presented in (Tao et al., 2 0 0 1 4 , (Tao et al., 2001b) and (Tao et al., 2001a) recently. In (Tang et al., 2001) an adaptive state feedback control scheme for nonlinear systems in the parametric strict-feedback form is proposed. Now a more complex case is investigated in this paper, that is, designing an adaptive control scheme with output feedback for nonlinear systems in the presence of unknown actuator failures. The nonlinear systems considered in the paper are in the output-feedback form, and the actuator failures are characterized by the pattern that some of the system inputs are stuck a t some fixed values not influenced by control action. In Section 2, we formulate the control problem we are investigating. In Section 3, an state observer for output feedback control design is proposed. With that state observer, an adaptive scheme based on the backstepping method is developed for asymptotic output tracking in spite of the actuator failures. The stability analysis for the closed-loop system is completed in Section 4.

Progresses have been made in the adaptive control of systems with actuator failures, including indirect adaptive LQ design (Ahmed-Zaid et al., 1991), indirect adaptive multiple model switching design (Boskovic and Mehra, 1999),

'

This research was supported in part by NASA Langley Research Center under grant NCC-1342 and by DARPA under grant F33615-01-C-1905.

415

2. PROBLEM STATEMENT

failed actuators, while p is changing at time instants t k , k = 1,2,. . .,q, such that the plant output y (t) asymptotically tracks a prescribed reference signal yT(t) (the pth derivative of yT(t), &'(t), is piecewise continuous) and the closed-loop signals are all bounded, despite the presence of unknown actuator failures and unknown plant parameters. It is clear that the key task is to design adaptive feedback control laws for v j ( t ) , j = 1 , 2 , . . . ,m, without knowing which of these u j ( t ) will have action on the plant dynamics, as there are arbitrary p failed actuators for V{jl,. . .,j,} c { 1,2,. . ., m}.

Consider a output-feedback form nonlinear plant

xi = Xi+l

+ Cpi(?J), i = 1 , 2 , .. . , p - 1 W

j=1

...

m

j=1 W

j=l

y=x1,

3. ADAPTIVE COMPENSATION CONTROL

(2.1)

where u j E R, j = 1,2,.. .,m, are the inputs whose actuators may fail during operation, x = [ X I , 2 2 , . . . , xnIT E R" is the unmeasured state vector, y E R is the output, cpi(y), i = 1 , 2 , . . . , n , and Pj(y), j = 1,2,..., m, are known smooth nonlinear functions, and b,J, r = 0,1, . . . ,y = n-p, j = 1,2,. . . ,m, are unknown constant parameters. The actuator failures of interest are modeled as uj(t) = t i j , t 2 t j , j E {1,2,..., m } (2.2) where the failure value f i j and failure time instant t j are unknown, and so is the failure index j. The basic assumption for the actuator failure compensation problem is that (Al) the plant (2.1) is so designed that for any up to m - 1 actuator failures, the remaining actuators can still achieve a desired control objective, implemented with the knowledge of the plant parameters and failure parameters. Suppose that there are pk actuators failing at a time instant t k , k = 1,2,. . .,q, and to < t l < tz < . . .
For the plant (2.4) in the output-feedback form with p failed actuators a t each time interval (tk,tl;+l), k = O , l , . . .,q, an adaptive control scheme will be developed in this section, which will be proved to achieve the control objective proposed above for Vt > to in the next section. Since the failure pattern uj(t) = t i j , j = j1 , . . . ,j,, 0 5 p 5 m - 1, is assumed to be unknown in this problem, a desirable adaptive control design is expected to achieve the control objective for any possible failure pattern. For a fixed failure pattern, there is a set of failed actuators, while there is another set of working actuators. For this set of working actuators, there is a resulting zero dynamics pattern for the plant (2.4). For closed-loop stabilization and tracking, the zero dynamic system of the plant (2.4) needs to be stable. As such zero dynamics depend on the pattern of working actuators whose number may range from 1 to m, which leads to many possible characterizations of required stable zero dynamics conditions related to different possible control designs, we choose to specify the following one with which a stable adaptive control scheme can be developed to achieve the control objective.

For the plant (2.1), we assume that

xj#jl,,,,,jp

(A2) the polynomials sign[b,,j]Bj(s) are all stable ones, V { j l , . . . ,j,} C { 1 , 2 , . . . ,m } , Vp E {0,1,. . . , m - l}, where Bj(s)=b,,jsY

+bl,js

+ b,-i,j~,-' + ...

+boj, j

= 1,2,. . . ,m. (3.1)

Remark 3.1. Assumption (A2) implies that the plant (2.1) is minimum phase for all possible actuator failure cases under a special control strategy (see (3.2) below). The plant (2.4) has relative degree p, and under the control strategy (3.2), it has a linear zero dynamics system = A,q+~$(y), where the eigenvalues of A, E R T x y are the roots of the polynomial Cjfj, , , , , , j , sign[b,,j]Bj(~), which is stable from Assumption (A2). Note that while the zero dynamics depend on the actuator failures, that is, A,

The control objective is to design an output feedback control scheme for the plant (2.4) with p

416

is determined by the failure pattern, I#&) depends on both the failure pattern and failure values. 0

With E=x-%, it follows from (3.5) and (3.7) that i = Aoq lim E(t) = 0 exponentially. (3.9) t+cc

For an adaptive control design, another assumption is required:

To construct an adaptive observer, we denote kl,, and k 2 , r j as the adaptive estimates of kl,, and k 2 , T j r j = 1 , 2 , . . . , m , r = 0,1,. . . ,y.Then, an adaptive estimate of x is

is known, j = 1 , 2 , . . . ,m.

(A3) the sign of b,,

To develop a solution to the stated control problem, we choose the control strategy: vj

= sign[b,,j]-vo,

1

~

j = 1 , 2 , . . . ,m, (3.2)

PAY)

. . ,x,]

cp(v)=[cp1 ( Y ) l c p 2 ( Y ) l . .

kl,r=

The backstepping technique (KrstiC et al., 1995) is now applied to derive a stable adaptive control scheme for the system (3.5), with a design procedure of p steps.

1

C ~ign[bY,jlbr,j,~ = 0 , 1 .i. .,7,

j#jl

r=O

3.2 Backstepping Design

T

.l % ( d l T

Y

for which the update laws for k l , r and k 2 , T j are developed next together with a failure compensation control law for vo(t) in (3.2) to ensure closed-loop stability and output tracking.

To express the plant (2.4) with the control strategy (3.2), we define '

n

r=O j=1

where vo is a control signal derived from a backstepping design to be given in Section 3.2.

x=[x1,x2,

r

(3.3)

,..., j ,

k2,rj=br,jiij, r=O,1 ,..., y,j = j l , . . . , j P ,

Define vectors of unknown constants

k2,rj=01 r=O, 1 , .. . ,y,j f j , , . . . , j p

k1=[k1,01

(3.4)

m

T

,

k2=[k201r...ik90rnik%11,.. .rk%lrni...ik%ym]Tr(3.11)

and rewrite the plant (2.4) in a compact form as Y

k l , l , . . . l k1,Y-ll

and vectors of signals

2

w = [ ~ o , i , c l l , i , .. . , ~ ~ - - l , ii ]=~1,, 2,..., 71,

.. < l l , i , . . ., < ~ r n , i >. ., . < Y m , i I T , i = 1 , 2 , . . . 71, pT,i and crj,il i = 1,2,. . .,n, are

~i=[bl,it.

-

0 10 00 1 A= .. . 0 o... -0 o . . .

... 0 .'.O .. . 0 0 0 0

1

0 0

_ _

t
(3.12)

the ith where variable of p T , < r j l r=O, 1,.. ., y- 1, j=1,2,. . .,m.

1

0 ERnxn,c= : ERnX1. (3.6) 1 0 0 0

:-

Step 1: With the output tracking error

= Y - Yr,

z1

(3.13)

where yr is a reference signal with the piecewise continuous pth derivative @), it follows from (3.5) -( 3.8) that i1=E2

-

+ z2 + 91 (y) -

GT

T -624- k13&,2+ W 2 kl+ &;k2+

(2

+Cpl(9 ) - i r . (3.14)

Choose the auxiliary error signal 22

^.

(3.15)

= P7,2 - K y r - a1 3

where R is an estimate of

al=k(-clzl - d l Z l =R(-cizi- dizi-

-

22

K

= $-,

and

- $01(y))

~21-

~ T k 2 -t 2 -

cpi(y)). (3.16)

Substituting (3.15) into (3.14) results in

z1=-c1zs -d1%1+kl,Yz2+%2 - i =-c1z1 -dlzl

+62

+W~ICl+€32 Y

r

n

r=O j = 1

Y

2 - kl,yi(GT+a1)

+ kl,yz2 + k 1 , 7 ( ~ 7 , -RGr 2

-ks,Yi(Gr+a*),

~-

, . -

pal)

(3.17)

where kl,? = k l , y - k ~ , y kl , = kl - k s , k2 = k 2 - k 2 , i= K - R, and c1 , dl are some positive constants.

r=O

417

The time-derivative of the partial Lyapunov candidate function V1 = f z ? + %k2, where X1 > 0 is a chosen constant gain, and k ~is positive , ~ due to Assumption (A2), is derived as V1=-c1z1

r,T kl -

2

2

- dlz,

+ .Ti2

+

ZlE2

+

+ VIil,-( + + a 1 ) k- b k k

il,yZ1z2

- zlkl,,(y,

where ci and d i , i = 2 , 3 , . . .,p-1, are some positive constants to be chosen, and vi, ri and ui, i = 2 , 3 , . . .,p - 1, are the tuning functions given as

A1

dlz:+

=-c,z:-

Z1€2+ 12.1,9122+ v1i1,,+rp1+ v 3 2

with the choice of the adaptive law for R:

R = -A121 (yr where v1, T I a n d functions for kl,,,

al),

dY

(3.18)

(3.23)

which are the first tuning and k2, are given as

kl

,

Q1), 7 1 =~1w2 8 4 = ~ 1 & 2 .

Introducing the partial Lyapunov candidate function = V,-l ~ Z Z with , the definition zi+l = pLy,i+1 - R Y P - ai, we derive the time-derivative of Vi from (3.21) and (3.22) as

+

(3.19)

Step i = 2 , 3 , . . . , p - 1: According to (3.15), define z i , i = 2 , 3 , . . . , p - 1, in a similar way: Zi

dY

~ 1 ,

kyT-

?I1=ZI (&,2-

+

aai-1

vi=vi-1- pLy,2zi,Ti =ri-1- w2 22,

,.

= p~y,i- K y ,(i-1)

- ai-1.

(3.20)

Differentiating (3.20), we obtain

. - K y ,( i ) - K:Y r( 2 - 1 ) - ai-1

&=fiy,i

~

=pLy,i+1-4py,1

docl -dY

(E2+kl,Lypy,2+&1

ail,?

+E32+&+(PI(Y))

(3.24)

aaiPl :

_-

- ~ y p - ~ yaai-l ya )i -R. -

h,Ly--

aai-l k:l - -aaiPl : k2. aka ai2

Design the stabilizing function

ai

(3.21)

as

Design the control signal vo(t) for (3.2) as

vo = ap+ ky?), where apis constructed as

418

(3.25)

In summary, we have developed an adaptive actuator failure compensation scheme for the system (3.5), which consists of the controller law wj=sign[b,,j] ?Jo=ap

1

-vO,

/% (Y1

j = 1,2, . . . ,m,

+ r2yy

(3.31)

and the adaptive laws for updating the controller parameters, I;=-X121($,

kl,y=XZ?ip,

ly-2 1 -2 v = v - 2-1 A+ k2x1 + -k1,? + -2I-,k , rl-1-kl 2x2 -1-

- E T ~ E l

Now we prove that with the adaptive control design proposed in Section 3 , the closed-loop signal boundedness and asymptotic output tracking are guaranteed for Vt > t o , so that the stated actuator failure compensation objective is achieved.

(3.27)

i= 1

where P , which is positive definite and symmetric, satisfies the Lyapunov equation PA,+A;fP = - I , and X2 > 0, rl = I'T > 0 , r2= I'g > 0. The timederivative of V is

Theorem 4.1. The adaptive output feedback control scheme consisting of the controller (3.31) and the filters (3.7) along with the parameter update laws (3.32) applied to the system (3.5), based on Assumptions (A2) -(A3), ensures global boundedness of all closed-loop signals and global asymptotic output tracking: limt-tm(y(t) - y,(t)) = 0. Proof: For each time interval ( t k , t k + l ) , k = 0,1,. . .,q, a Lyapunov function V such as (3.27) can be constructed. Starting from the first time interval, it can be seen from (3.30) that V ( t )5 V ( t o ) and V 5 0 for Vt E [to,t 1) . Hence we conclude il, i 2 and E are bounded for that z , k , il,y, t E [ t o , t l ) .From the boundedness of y, and 2 1 , y is bounded. It follows from (3.7) that J and ( , j , r = 0,1,. . .,y,j = 1,2,. . .,m, are bounded. Also from (3.7), we obtain that

(3.28)

With the choice of the update laws k l , y = X 2 ~ p ,k.1 = F I T p r

k2=r2up,

kl =rlTP, k2 =r2Vpr (3.32)

4. STABILITY ANALYSIS

LK

1-7p 1 +p2 r2 k2 + x 2di

Q1)

where a p ,up, I - ~ and up are derived from the recursive backstepping procedure.

in which cp>O and dp>O are designing constants. Consider the Lyapunov candidate function p-

+

(3.29)

p,,i = eT(sI - A,)-le,-,vo

we have

(4.1)

for 0 5 T 5 7, 1 5 i 5 n, where ei is the ith coordinate vector in Rn. Express the plant (3.5) in the differential equation form n

419

Y

failures is constructed and used for an adaptive control design. Based on the state observer, a backstepping method is applied to develop an adaptive compensation scheme to ensure that the plant output tracks a reference output asymptotically and that all closed-loop signals are bounded, in spite of the unknown actuator failures in addition to unknown plant parameters.

As in (KrstiC et al., 1995), rewrite (4.2) in the input-output form with transfer function G(s) = 1 kl~,s~+...+k,,ls+kl,o 7 “input” n

,,y(nL

~

r

n

x&-z)(Y)-y

Q2TJ&)(Y),

(4.3)

r=O ~ = 1

z=1

and “output” signal

210:

VO =G( S)[a].

REFERENCES

(4.4)

Ahmed-Zaid, F., P. Ioannou, K. Gousman and R. Rooney (1991). Accommodation of failures in the f-16 aircraft using adaptive control. IEEE Control Systems Magazine 11, 73-78. Boskovic, J. D. and R. K. Mehra (1999). Stable multiple model adaptive flight control for accommodation of a large class of control effector failures. Proceedings of the 1999 American Control Conference pp. 1920-1924. Boskovic, J . D., S.-H. Yu and R. K. Mehra (1998). A stable scheme for automatic control reconfiguration in the presence of actuator failures. Proceedings of the 1998 American Control Conference pp. 2455-2459. KrstiC, M., I. Kanellakopoulos and P. V. KokotoviC (1995). Nonlinear and Adaptive Control Design. Wiley-Intersciense. New York, NY. Tang, X. D., G. Tao and S. M. Joshi (2001). Adaptive actuator failure compensation control of parametric strict-feedback systems with zero dynamics. Proceedings of the 40th IEEE Conference on Decision and Control pp. 20312036. Tao, G., S. H. Chen and S. M. Joshi (2001a). An adaptive actuator failure compensation controller using output feedback. Proceedings of the 2001 American Control Conference pp. 3085-3090. Tao, G., S. H. Chen and S. M. Joshi (2001b). An adaptive control scheme for systems with unknown actuator failures. Proceedings of the 2001 American Control Conference pp. 11151120. Tao, G., S . M. Joshi and X. L. Ma (2001~).Adaptive state feedback and tracking control of systems with actuator failures. IEEE Transactions on Automatic Control 46, 78-95. Tao, G., X. D. Tang and S. M. Joshi (20014. Output tracking actuator failure compensation control. Proceedings of the 2001 American Control Conference pp. 1821-1826. Tao, G., X. L. Ma and S. M. Joshi (2000). Adaptive output tracking control of systems with actuator failures. Proceedings of the 2000 American Control Conference pp. 2654-2658. Wang, H. and S. Daley (1996). Actuator fault diagnosis: an adaptive observer-based technique. IEEE Transactions on Automatic Control 41,1073-1078.

Substituting (4.4) into (4.1), we have

pT,2=eT(sI- A,)-’e,-,G(s)[~I,

(4.5)

which results in the boundedness of pT+, r = 0,1,. . .,y,i = 1,2,. ..,n, because y is bounded, cpz(.), i = 1 , 2,..., n, and p J ( . ) ,j = 1,2,..., m, are smooth, and the matrix A, and the polynomial kl,,sY + . . . + kl,l s + kl,o are stable. It is in turn implied from (3.8) and the boundedness of 6 that z is bounded. According to (4.4), it can also be seen that vo is a bounded signal. Since ,f?,(y) # 0 for Vy 6 R, the boundedness of vJ is guaranteed too, j = 1,2,. . .,m. Therefore, all closed-loop signals are bounded for t E [to,t l ) . At time t = t l , there occur pl actuator failures, which result in the abrupt change of K , kl,-,, k l and k 2 . Since the change of values of these parameters are finite and z is continuous, it can be concluded from V S O that V ( t ) < V ( t f ) = V ( t ; ) + VI 5 V ( t o ) V1 for t E ( t l , t 2 ) with a positive constant Vl. By repeating the argument above, the boundedness of all the signals are proved for the time interval ( t l ,t 2 ) . Continuing in the same way, finally we have that V ( t )< V ( t $ )= V ( t ; ) + Vq V ( t o ) + C ; = , y k for t E ( t q , m )with some positive constants V k , k = 1,2,. . .,q. Due to the finite times of actuator failures, it can be obtained that V ( t )is bounded for Vt > t o , and so are all the closed-loop signals.

+

<

To prove output tracking, considering the last time interval ( t q ,m) with a positive initial V ( t : ) , we see that it follows from (3.30) that z E L2 and E E L 2 . On the other hand, we can conclude that Z E L- and i E Lc0 from the boundedness of the closed-loop signals. In turn it follows that over the time interval ( t q ,m), limL+m21 ( t )= 0, which means that the output tracking error y ( t ) - y r ( t ) V is such that limt+co(y(t)-yT(t))= 0. 5. CONCLUDING REMARKS An adaptive output feedback actuator failure compensation control scheme for a canonical class of nonlinear systems: those in the output-feedback form, is presented as a further development of the backstepping technique. In order to estimate the unmeasured states of the controlled plant, an adaptive observer for the uncertain plant with an equivalent unknown disturbance due to actuator

420