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6-9 August 2001 Montreal, CanadaA01-37085 AIAA 2001-4168

ROBUST ADAPTIVE SLIDING CONTROL FOR A CLASS OF MIMO NONLINEARSYSTEMS

Haojian Xu* and Maj Mirmirani**-University of Southern California, Department of Electrical Engineering, Los Angeles, CA 90089

t-California State University, Department of Mechanical Engineering, Los Angeles, CA 90032

ABSTRACTIn nonlinear adaptive control, a problem arises when

the identified gain matrix becomes singular. As theresult the adaptive law based on the inverse dynamicsof the identified model cannot be implemented. Whenthis occurs, even if the real system is alwayscontrollable, the closed loop system becomesuncontrollable. In this paper a method for generatingrobust adaptive control law for MIMO systems whichcircumvents this problem is presented. Assuming thatthe gain matrix of the unknown system satisfies acertain condition, estimation of the nonlinear gainmatrix is reduced to one of estimating a scalar nonzerovariable. Adaptive laws are designed to turn theunknown parameters on line as unknown functions areapproximated using one hidden layer network. Aspecific a-modification adaptive law is designed topreclude the possibility of closed-loop systeminstability. The resulting closed-loop system is provedto be globally stable with tracking error that convergesto a small residual set.

INTRODUCTIONControlling nonlinear multi-input multi-output

systems with partially or completely unknowndynamics is one of the challenging areas of research inadaptive control. Although there are some studiesfocusing on linear-in-parameters SISO systems,3'5'7there is no general methodology to treat MIMOnonlinear systems with unknown dynamics. The mostcommon approach is to approximate unknownnonlinear functions by neural network, fuzzy logic andother nonlinear approximation methods. Off-linetraining algorithms are used to train these "network"and the controller is designed based on the identifiedmodel. However, for real-time control, the designprocess becomes more complicated when learning andcontrol are attempted simultaneously. In addition, since

f Professor, Department of Mechanical EngineeringCopyright © 2001 The American Institute ofAeronautics and Astronautics Inc. All rights reserved.

nonlinear adaptive control laws are based on the inversedynamics, a problem arises when the identified gainmatrix becomes singular making the identified modeluncontrollable even when the real system is alwayscontrollable. This problem has not been solved in mostliteratures dealing with nonlinear unknownsystems. 1,2,5,10,14,16,17,19 Using gradient method withprojection to tune the unknown network parameters isnot a systematic design way and cannot ensure thesuccess of the method in all cases. A method wasproposed to circumvent this situation by directlyestimating the inverse of the gain function for a single-input single-output system.14 This method requiresknowledge of the inverse gain function, whichgenerally is not available. In addition, this methodcannot be readily extended to the case of multi-inputsystems. A small local convergence is achieved for oneclass of multi-input systems by limiting the learningrate and size of the initial errors of the "network"weights to a very small range such that theuncontrollable problem will not occur.1'2'10 The mostefficient method to deal with uncontrollability ofnonlinear adaptive systems is to use switchingmodification.7 An adaptive switching sliding controllerwas proposed by the authors which has proven effectivefor single-input single-output systems.20 In this paperwe will extend this method to a class of nonlinearMIMO systems. In this method, by assuming that theaverage of the sum of the gain matrix and its transposeis uniformly positive definite, one needs only toestimate a single nonlinear scalar function instead ofestimating every entry in the gain matrix. A switchingmodification function is included into the control law topreclude the possible uncontrollability of the identifiedsystems. The proposed controller is made of fourconstituents: a PD controller, two time-varying slidingmode controllers, two estimators and a switching law.An adaptive control law with a special a-modificationis used in the estimation of the output weights of thenetwork approximation of unknown functions. Theaddition of the a-modification guarantees that theclosed-loop system does not become unstable as resultof the network approximation errors and the switching

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modification incorporated in the control law. Theclosed system is proved to be globally stable androbust. Unfortunately, there is a price to be paid forrobustness of the controller. The tracking error in thiscase converges to a small residual set instead ofconverging to zero.This paper is organized as follows: In section 2 thesliding mode control of MIMO linearizable systems ispresented. In section 3 a general method forapproximation of nonlinear functions is discussed. Insection 4 we present the proposed switching adaptivecontroller. In section 5, the theoretical results areapplied to the control of a generic hypersonic vehicle.Finally, section 6 includes the conclusions. Throughoutthis paper, | • | indicates the absolute value, and || • ||indicates the Euclidean vector norm.

imply ef(t) converges4'18

to zero with timesolutionsconstantA control law which forces the tracking errors to zerodefined in terms of these surfaces is called a slidingmode control solution for the dynamic system definedby (2.1). To obtain such a control law, the derivativesof the above set of sliding surfaces, £, ,-•• ,5^, areneeded and can be obtained by direct differentiation ofequations (2.2):

/,(*)/.(*)

blm(x)

where,(2.3)

SLIDING MODE CONTROL OF MIMQLINEARIZABLE SYSTEMS

As a first step, we introduce a sliding modecontroller for a multi-input multi-output nonlinearsystem whose dynamics can be described by thefollowing set of linear-in-the-input differentialequations:

(2.1)

y\ —x\^"'tym -xmIn (2.1), j te$K r , r = r, + "- + rm is the state, ul9-~,um

are control inputs, /,,•*•>/„, and biJ9i9j = !,-••, ware

smooth, and (-)(r) denotes the r-th time derivative ofthe function inside the bracket. The problem is todesign a set of control laws such that the outputs of thesystem y\,~-,ym track a set of desired trajectoriesdefined by yd , • • •, yd .The sliding mode solution of the above trackingproblem requires choosing a set of decoupled slidingsurfaces, S f , that describe the tracking error dynamics:

' ddl

(2.2)

where /I,, • • •, Am are positive constants that define thebandwidth of each decoupled error channel. The slidingsurfaces corresponding 5,.(0 = 0, / = 1,2, •••,#!represent a set of linear differential equations whose

\ (2.4)

and or,, j ,-",^,, /=!,...,w express the coefficients ofHurwitz binomial expansion of the correspondingsliding surfaces in (2.2).Equations (2.3) can be written in compact matrixformat as:S = f(x) + v(x) + B(x)u (2.5)with the appropriate matrix definitions:s = [s, ... sj/(*) = [/,(*) - fm(x)Y

(x)]T

bn(x)

In view of (2.5), the time derivative of the quadraticformat of sliding surfaces jSTS is:

(2.6)Therefore, asymptotic tracking is achieved if a controllaw u can be found for which the sliding condition

is satisfied.4'20

For example, from (2.5) it follows that if B(x) isinvertible, sliding condition required for asymptotictracking can be achieved by the control law.4

(2.7)

where k sgn(S) = [kl sgn(S, ),-,*„ sgn(Sw )]r, andkf > 0, i - 1, • • •, m .Similarly, if S T B(x)S ^ 0 the control law given by

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u -- (2.8)

can achieve the sliding condition sinceSTS = -&pD|S|<0. Here kPD is a positive gain. Inboth case, the dynamic of the system, f(x) and B(x) areassumed to be known.

In this paper, we will present an adaptive sliding controllaw for the class of systems defined by equations (2.1)when there is significant uncertainty in systemdynamics. We will focus on the extrem case wheny(jc)and B(x) are completely unknown functions. Before weproceed further, it is necessary to make an assumptionregarding B(x), prove a lemma, and describe in brief themethod for approximation of unknown nonlinearfunctions.

Assumption 1: Herein we assume thatj(B(x) + BT(x)J is uniformly positive definite over a

compact set Q e $R r , i.e.,

I > 6 > 0 , V j c e Q . (2.9)

where a( •) represents the minimal singular value ofthe matrix inside the bracket and b is its lower bound.

Lemma 1: For any square matrix B(x) satisfyingAssumption 1, there exists a continuous positivevariable ju(x) > b such that for any vector S the

quadratic form STB(x)S = p(x)\\S\\2 •Proof: The proof is presented in the Appendix.

In view of assumption 1 and lemma 1 the control law(2.8) can be rewritten as:

(2.10)

where Se is a sliding unit vector defined bySe = S /\\S\\. Since ± (B(x) + B T (x)) is positivesymmetric, all of its eigenvectors are orthogonal and allof its eigenvalues are positive real. Therefore if oneexpands the error metric S as a linear combination ofthe eigenvectors of \(B(x) + BT(x)), it is readily seenthat ju(x) as a linear combination of all eigenvalues ofmatrix j(B(x) + B7 (x)) is positive real for all x. ju(x)herein called the gain function

APPROXIMATION OF UNKNOWNNONLINERA FUNCTIONS

The adaptive sliding control law presented in thispaper is in terms of the approximation of the unknownfunctions f(x) and //(x) using the general one layer"network"11'13'14'15. The one layer "network"approximation of an unknown continuous function A(JC),ha ( x ) , can be described in terms of some know basisfunctions gh(x^hj) by:

'*

y=iwhere the coefficients Oh.(i) and %h. are the outputand input weights respectively, and lh is the length ofthe nodes. The advantage of the one layer "network"approximation is that it can be easy realized bymassively parallel, analog computing framework. Withrespect to the control problem on hand, it is assumedthat the unknown functions / and B in (2.1) can beapproximated by the network (3.1) with any desireddegree of accuracy. Of course, using lemma 1, oneneeds to estimate the scalar function JL^X) instead ofestimating every entry in B(x).

Assumption 2: There exists a set of output weights #/ ' ,

/=!,...,TW, and 0? , such that the continuous functions/and ju can be approximated by (3.1) with any desired

>0 over any compact setaccuracy £f > 0 and

(3.2)

It is also assumed that the structure of the network,number of nodes, and input weights have already beenspecified by the designer, and the only unknown are theoutput weights. There are many functions of neuralnetwork that satisfy this universal approximationmodel.8'9'11'13'14'15. In the development below, it isassumed that every approximation function fja(x) orjua (x) , can use different basis functions. For example,one can use a radial basis neural network for ft

a(x]and a Volterra/Wiener neural network for jua (x) .Finally, denote the estimates of the approximation tothe unknown functions, ft

a (x) and jua (x) , by f° (x)and jua (x) respectively and define them by:



where 0f' (t) and 0f (t) are the estimates of 0f' and

0? respectively.Let0f'(t) = 0 j ' ( t ) - 0 f ' ( t ) 9 i'=l,...,/w (3.6)

Then the estimation errors can be expressed as:

7/flW = // f lW-// f lW = Z^ f/ l(Og/ l(^^/ /) (3.8)7 = 1ln

f i a ( x ) = £a(x)-â(x) = £0f(t)g^(x,^) (3.9)

THE ROBUSRT SWITCHINGADAPTIVE CONTROLLER

We now present the switching adaptive slidingcontrol law designed for the system (2,1) as:

u = -

In (4.1) above, /" (*) = [/,"(*) - /;(*)]r, andsgn(-) is a special sign function with hysteresis definedas:

sgn(jc) - -

1, if x>^

1, if --f- < x < -j- and sgn_(*) = 1

-1, if - -f- < * < -f- and sgn_(x) = -1

-1, if x < —~

(4.2)

where sgn_(*) = lim sgn(;c(/)) for r < t.In (4.1) and (4.2) 8 > 0 , a small constant, representsthe width for the switching function deadzone.<jv > 0 and a f > 0 are small constants, and kPD > 0is a gain chosen by the designer.Let a special sliding unit vector 5, be defined by:

(4'3)

In (4.3), O is a small positive constant.By taking 5, inside the bracket, it is noted that thecontrol law defined in (4.1) consists of 4 constituents: aswitching law J^sgn^"), a PD controller kPDS , twotime-varying sliding controllers crJWjtjILSj and

'f\\J v-vpi» an(l two estimators fa (x) and //a ( x ) .

S} is the counterpart to the saturation function in theSISO case. Thus, by taking 5, inside the brackets in

(4.1), trJvOOlS, and ^l/^jc)^, are the

counterparts of the SISO sliding controllers for theMIMO case. The small linear boundary layer (£> is usedto smooth out the control discontinuity and avoidpossible singularity in calculating 5,. Note that the

switching law JAsgn(//a) is a hysteresis version of the

function ^ sgn(//fl), which switches between ±8^

whenever j u a ( x ) becomes zero. It is incorporated inthe denominator of (4.1) to avoid singularity of thecontrol law. Because

> 8U 12 > 0 , the control law in

(4.1) is exactly to exclude the possibility of asingularity. This design is critical since the potentialloss of controllability has been a main drawback ofmany nonlinear adaptive laws that are based on theinverse dynamics. Also, in the absence of thehysteresis, every switching represent a discontinuity inthe right-hand-side of the differential equations (2.1).The hysteresis modification of the pure sign function in(4.1) will guarantee desirable sliding motion andexistence and uniqueness of solutions of the differentialequations involved.12

Control law (4.1) can be rewritten in a compact formatas:11 = ^1* (4.4)

u = - 1sgn(//a

This presentation of (4.1) provides for an increasedinsight into the action of the controller. Theinterpretation of (4.4) is that control vector u expressedas a normalized directional vector 5, scaled by controleffort u. Sl represents the direction of the error metricS. Thus control vector u can be viewed as apportioningthe total control effort u in different directions. Thecomponents, which have large value in 5, , will obtainrelatively a larger control input than those componentswith smaller values. By doing so, the speed ofconvergence of every sliding surface will remain almostsame. This is consistent with the choices of the slidingsurfaces in (2.2) since these sliding surfaces areindependent.

The adaptive laws are defined as follows:



For S < O ,

(4.6)

For \\S\\>®,

/'« = */,$,£/; (*,#'),

p is a switch variable defined as:(4.7)

1,

0,

if / /" \<b-S -A

if

(4.8)where k^ > 0, / = 1, • • •, m and h^ > 0 are designparameters selected judiciously for best adaptation law,u is defined in (4.5), and A > 0 is a design parameterused to avoid discontinuity in p.

We now state the main result of this paper:Theorem'. Consider the system (2.1), the switchingcontrol law (4.1) and the adaptive laws (4.6) and (4.7).Assume that assumptions 1 and 2 hold. If the lowerbound of // satisfies the condition b > 2(S^ + s^) + A ,then given the small positive numbers O and sf , there

exist a positive constant <5* such that forc* c**

U U

PD and0*x , > ^v — . 0* ' f t c*<

< 5 ) 0 \ — O \-O

<j^>S*, all signals in the closed-loop system (2.1) arebounded and the tracking errors e^t) converge to asmall residual set bounded by /lp+lO, /=!,...,m.

Proof: Let us consider the following Lyapunov-likefunction:

V(t) =

2 \2kfi

In view of the adaptive laws (4.6) and (4.7), the firstderivative of V(f) can be obtained as:

= 0, if W<0 (4.10)

(4.11)V = STS+

The remaining of the proof is for \\S\\ > O .For simplicity, rewrite the control law (4.1) as:w = S> (4.12)

u = -^=^——l——————u (4.13)

I I A « I I (4'M)

First, analyze the first term of V in (4.11). Bysubstituting control laws (4.12) and (4.13) into (2.6),one has:

Tv(x) + /j(x)\\S\\u

° (x)) - ft" (x)}\S\uUsing (4.14),

(4.15)

Since,/(*)-/"(*)

= (/(*)-/'(*))-(/'= d f ( x ) - f a ( x )

(4.16)

(4.17)

where, /«(*) = |/,fl(jc) - /;(*)]Inserting (4.16) and (4.17),STS = -km \S\\2 - av ||v||5|| - af fS\

(4.18)

The last term in (4.18) represents the effect of theswitching and the approximation error. Let

(4.9)



6»+sl, (4.19) || - {(1 -

Then, similarly as shown in Ref. 20, the absolute valueof the last term in (4.18) can be expressed as:

Since

(4.20) can be further expressed as:

(4.20)

' (x) (4.21)

(4-22)

As a last step, using (4.22), the final STS can beexpressed in terms of an inequality as:

(4-23>

- Ji" (x)\\S\\u -STf° + pty° (x)\\S\\u\In reaching (4.23), the following inequalities were used:

< D s , for S>®

Now, inserting the adaptive law (4.7), the second termin (4.1 1) can be written as:

= SJ°(x) + - + sJ°(X) = STfa

Finally, in view of the adaptive law (4.7), the last termin V (4. 1 1) can be written as:

u\ (4.25)

. ThisHere, we use the identity /?//°(jc) = -/y\jua(x)result comes from following:f i a ( x ) = jua(x)-jua(x), Because p*Q only if

jua (x)\ < b - 6^ and \jua (x)\ > b - s^ as shown in theAppendix (7.4).Thus, //fl' (x) has negative sign.Combining (4.23), (4.24), and (4.25), V can beexpressed as:

V < -{(I -

(4.26)By properly choosing kPD , O, av , <J f , a ̂ such that thefollowing inequalities are satisfied:

k<S>>- \-s (4.27)

(4.28)

(4.29)

(4.30)

where k, > 0 is a constant.V can be made strictly negative:F<-t1 | |5r |<-Jfc10<0 (4.31)The result (4.31) is arrived at under Assumption 1 and2. Since Assumption 1 and 2 only hold on a compactset, i.e., jceQ, all states need to remain in this compactset for all t > 0 . Consider the set

(4.32)

where rQ > O is a constant.If rQ is chosen such that rQ > F(0) , together withV < 0 , then Q is an invariant set, and ;c(t) will remainin Q for all t > 0 if A:(0) e Q . Hence the Assumption 1and 2 will never be violated.Since V(f) is bounded from below and is non-increasingwith time, it has a limit, i.e., \irnV (t) = Vao. Then

/->oo

lim (™ |s(0|k/ = (V(Q) - VJ I k, < oo for \\S\\ > <S> . Thus/-^(X) -U 'I II 'I I'

5, 0f' (0 and 0f(t) are bounded for all / > 0 . Next wewill show the convergence property for the error metric\\S\\ . It is clear that V < 0 for all / > 0 and that V < 0when |iS| > O . Therefore, the total time during whichthe adaptation takes place is finite. This implies that thetotal time during which |s| stays outside of theellipsoid O is finite. If the error metric ||s| staysoutside the ellipsoid O only a finite around of time, itwill eventually converge to the ellipsoid, i.e., |s|| < O

as t -> oo . It follows that every surface \Sf \ < O whichin turn means that the tracking errors will converge to



the residual sets given by |e/(0| - >V+I (^,i = l,-,m.*'"'»QEDAs discussed in the SISO case, the trade-off forincorporating switching modification and theapproximation error of ^(x) into the control law is thatthe error metric |s||, or equivalently tracking errors,will converge to a small residual set defined by||5| < O instead of converging to zero.20

Remark 1: The system present in (2.1) can be derivedfrom the more general nonlinear systems described byequations:

has •. The width O will depend on ef ,

y. = hf(x) / = 1 , . - . , W

in which jc = [ jc , , - - - , j c / l ] 7 ' eSR" and p, g, h aresufficiently smooth functions. If the above system isfeedback linearizable, it can be reduced to the system(2.1).4 Let:

L';hm] (4.34)

(4.35)

where, the Lie Derivative expressions Lp and Lkp are

defined as:dh dhL (h) =< dh, p >= —— PI (x) + • • • + —— pn (x)dx{ dxn

L\ (h) = Lp (l*;1 (h)) =< dLkp{ (h\ p >

Here, ri is the equivalent linearizability index foroutput yi, i.e., one need to differentiate the outputchannel, yi, ri times until one of the control inputs is

different from zero. r = . indicates the relative

degree of the nonlinear system. When the relativedegree r < n , the internal dynamics of the system mustbe stable.

Remark 2:In view of (4.26), the dead zone width O is anecessary feature of the proposed robust adaptivecontroller. When the magnitude of the tracking errorcan be compared with switching width andapproximation error, the adaptive estimation is stoppedto avoid the parameter drift that is typical in adaptivecontrol. Since the tracking error converges to theresidual set \et(t)\ < /£7r/+lO , i = \, — ,m as ||S||Ô,the accuracy of the tracking depends on the width O ofthe dead zone. However, from (4.26) and (4.27), one

8 and &PD . To obtain close tracking, one needs a largegain, k?D , small sf and S. The value of S if we

rewritten as 8 = —— - —— - —— =• which is a function of

the quotient of (Sb + sb ) / b . Let (Sb + sb ) / b becalled the "Switching and Approximation Error Ratio(SAER)". Since (4.27), (4.28) and (4.29) require S< 1 ,

i.e., ————— <1, SAER cannot be larger than 0.5.1 — uA±^l\

Furthermore, generally it is desirable to keep crv , crf

less than unity. This requirement will be satisfied bykeeping SAER less than 1/3. Another way to reduce thetracking error is to increase the gain, kPD . However, alarge gain is undesirable, since it will require a largecontrol input. Thus, there is a tradeoff between thetracking error, SAER, the control gain and theapproximation error, s f . The choices of the SAER andthe gain, kPD , depend on the maximum control effortthat the actuator can provide.

Remark 3: The a-modification term/*//{j//|5|||M|g//(jc,^/) has been incorporated inadaptive law (4.7) by design to ensure stability androbustness. It is activated only when the estimate //fl

approaches zero. In fact, it guarantees the convergenceof the tracking error even if jua approaches zero sinceit appears as a negative term in the derivative ofLyapunov-like function. For system (2.1) even ifj(B + BT) is negative definite, the control law (4.1)can be made valid by changing the a-modification to:-fkô-^\\S\\u\g^(x9^). Thereby this special a-modification ensures robustness whereas the classicala-modification is to avoid the estimate of the parameterunbounded.6

Remark 4: The time-varying terms av |v(jc)| and

crv /*(*) appearing in control law (4.1) areincorporated to have a dual role. First, as time-varyingsliding gain, they ensure overall system stability as seenin equation (4.26). Secondly, it can be viewed as speciala-modifications in the control law used to avoid |v(jc)|

and |/fl(jc)| unbounded.



SIMULATION RESULTSNomenclatureaC

speed of sound, ft/sD Drag coefficient

CL lift coefficientCM (q) moment coefficient due to pitch rateCM (a) moment coefficient due to angle of attackCM (Se) moment coefficient due to elevator deflectionCT

cDh

LMM y

mqR

TVaftySe

//p

thrust coefficientreference length, 80 ftdrag, Ibfaltitude, ftmoment of inertia, 7,000,000 slug-ft2

lift, IbfMach number

pitching moment, Ibf-ftmass, 9375 slugspitch rate, rad/sradius of the Earth, 20,903,500 ftradial distance from Earth's center, ftreference area, 3603 ft2

thrust, Ibfvelocity, ft/sangle of attack, radthrottle settingflight-path angle, radelevator deflection, radgravitational constantdensity of air, slugs/ft3

In this section, simulation results are used to illustratethe theoretical results. The plant used in this simulationis a rigid-body longitudinal model of a generichypersonic vehicle cruising at Mach 15 and at analtitude of 110,000 ft.21'22 The dynamics of thehypersonic vehicle can be written by following thedifferential equations:

• _ Tcosa-D //sin/m r2

. L + Tsina (//-F2r)cos/mV

h = V sin ya = q-y

Vr2

(6.1)

where,

Myy = \r = h + RE

with

CL = #(0.493 + 1.9 1/M)CD = 0.0082(17k*2 + -0.054M

CT = *(l + 0.15)/7 if /7<1.15/7) if J3>\

CM (q) = (c 1 2V]q(- 0.025M + 1 ,3?)x (- 6.83tf 2 + 0.303^ - 0.23)

The altitude h is required to track the sinusoidalfunction hd =hQ +100sin(0.l*;r*f) with a steady statetracking error of less than 1ft. Simultaneously, thevelocity V is required to follow a step function withmagnitude of 50ft/s and a steady error of less than 1ft/s. Both of these are taken about a trimmed conditionof h0= 110,000 ft and F0=15mach. The model forthis vehicle has the same format of equations (4.33)with the throttle setting J3com and the elevatordeflection^ as the control inputs. The linearizabilityindices of h is 4 and V is 3 respectively. Thus, therelative order of the system is r=n=l. Therefore, thelinearizable system has no zero dynamics and is givenby:

(6.2)

The linearized model (6.2) is exactly in the format ofthe general nonlinear system (2.1) with

ft com

/2(4)

.ftcom

com

8,

where,



and

The detailed development of (6.2) and the expressionsof F0 , /*<4) can be found in Ref. 24.Consider the workspace { \V - F0 1 < 1 00 ft/s;

|A-A 0 |<300f t ; |ar|<10°; \y\ <30° }, it is easy to showthat the average of B(x) and its transpose is uniformlypositive definite in this workspace. Assuming acompletely unknown nonlinear plant, a one layer radialbased neural network with basis functionsg(x9<£) = exp(-;r<j2||jc-^| ) was used to approximatethe unknown functions / and //. The mean £, is thecenter of the radial basis function, and a2 is thevariance representing a measure of the width of thebasis function. The simulation results are shown inFigure 1. The altitude h converges to the desiredtrajectory after 25 seconds, whereas the velocityconverges to the neighbor of the desired state in 20seconds.

N«gral Ad»p1iv« Control for Hypersonic Aircraft

I "I

JO 40 50 60

3£ 0.05

Figure 1. Adaptive Control for Hypersonic VehicleResponse to a 50ft/s step-velocity and 100ft sinusoidal

altitude commandsThe dot line '- -' indicates the desired trajectory and

solid line '—' indicates the real trajectory.

CONCLUSIONSA new switching adaptive sliding controller for a

class of MIMO systems is presented. The analysis forthe MIMO systems is considerably more complex ascompared with the SISO case reported in previous

work. The design of the controller requires estimatingthe input gain matrix. By imposing only a mildcondition, the problem of calculating the estimate of theentire elements of the gain matrix is reduced to one ofobtaining the estimate of the value of a single scalarfunction of the state. A hysteresis type switchingconstituent is added to the control law that precludesthe closed loop system loses controllability. Thepossible system instability which can arise when theestimate of the gain function approaches zero is solvedby including a set of special a-modifications. Theclosed system is proved to be globally stable androbust. The tracking error is shown to converge to asmall residual set. The size of the residual set isdetermined by the switching width and the magnitudeof the approximation error. The above controlmethodology can be applied to neural and fuzzyadaptive control, and other adaptive control wherenonlinear approximation is used for unknown functions.

APPENDIXProof of Lemma 1Since any matrix can be expressed as the sum of asymmetric matrix and a skew-symmetric matrix, B(x)can be written as:

BT(x) B(x)-BT(x)

\S, since theThus, STB(x)S =

quadratic form associated with a skew-symmetricmatrix is always zero.Also for any vector S and the unit vector Se along thatvector 5r = 5 5 ' . Thus

Since j(B(x) + BT (x)) is positive symmetric definite,all eigenvectors are orthogonal and all eigenvalues arereal. Let e{ , • • • , em indicate a set of unit eigenvectors ofthe span of 9em are

orthonormal and form a set of basis in y\m . Thus theunit vector Se can be expressed a linear combination

where c{9-",cm are corresponding coefficients and

I

Letcl+ — + c2

m =1.2. m

O"l9o"29"-9c7m indicates the eigenvalues of



+ c |s|2

Where,

> min(cr, , a2 '",(7m)>b

Since j(B(x) + BT (x)) is symmetric, all eigenvaluesare real. ju(x) is linear combination of all eigenvaluesof matrix ±(B(x) + BT ' (x)) and is real for all x.

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