stepanyan_control of systems with slow actuators

8/12/2019 Stepanyan_Control of Systems With Slow Actuators

1/16

Control of Systems With Slow Actuators Using TimeScale Separation

Vahram Stepanyan

Mission Critical Technologies Inc, NASA Ames Research Center, Moffett Field, CA 94035 ,Nhan Nguyen

NASA Ames Research Center, Moffett Field, CA 94043

This paper addresses the problem of controlling a nonlinear plant with a slow actuatorusing singular perturbation method. For the known plant-actuator cascaded system theproposed scheme achieves tracking of a given reference model with considerably less controldemand than would otherwise result when using conventional design techniques. This isthe consequence of excluding the small parameter from the actuator dynamics via time

scale separation. The resulting tracking error is within the order of this small parameter.For the unknown system the adaptive counterpart is developed based on the predictionmodel, which is driven towards the reference model by the control design. It is proven thatthe prediction model tracks the reference model with an error proportional to the smallparameter, while the prediction error converges to zero. The resulting closed-loop systemwith all prediction models and adaptive laws remains stable. The benets of the approachare demonstrated in simulation studies and compared to conventional control approaches.

I. Introduction

In many applications, the plant to be controlled has much slower dynamics than the actuator throughwhich it is being controlled. That is, the cascaded plant-actuator system can be described as

x (t) = f (x (t )) + g (x(t ))y(t) z (t) = a (z(t )) + b(z(t ))u (t )

y(t) = c(z(t )) , (1)

where x(t) R n is the plants state, z (t) R m is the actuators state, y(t) R is the actuators output,u (t) R is the actuators input, and is a small constant. For these types of systems, the singular pertur-bations method has been successfully used for control design. 6 This approach requires some interconnectionconditions to be satised, which can be imposed directly on the systems under the consideration (See forexample 16 ) or in terms of Lyapunov functions and their derivatives (See for example 13 ). The developedcontrollers typically achieve local results (stability or asymptotic stability) of closed-loop systems (See forexample Refs. 2,6,8 just to mention a few of them). However, for some systems, global exponential stabilitycan be achieved. 1 As it is shown in Ref., 5 when the exponential stability is achieved, the actuators dynamicscan be neglected and the actuators output can be viewed as a control input for the plant. Therefore, the

control design can follow any known scheme that is suitable for the given plant dynamics.In some applications, the actuator dynamics can be substantially slower than the plants dynamics. Thatis, the cascaded plant-actuator system is now described by the equations

x (t) = f (x (t )) + g (x(t ))y(t) 1 z (t) = a (z(t )) + b(z(t ))u (t )

y(t) = c(z(t )) . (2)Senior Scientist, Mission Critical Technologies Inc., Senior Member AIAA, [email protected] Project Scientist, Integrated Resilient Aircraft Control Project, Intelligent Systems Division, Associate Fellow AIAA,

[email protected]


2/16

This is the case when the specic actuator, which is designed to control the plant in ordinary scenarios,fails, and the substitute is used to rescue the emergency situation. For example, the aircraft engine canbe used for the directional stability and control of the aircraft when the control surface deection systemfails. However, the engine dynamics is known to be much slower than the aircrafts rotational dynamics withthe rudder input. For these types of problems, the actuator dynamics cannot be ignored, and the actualcontrol input has to be designed to generate the necessary actuator output. In general, the actual controlinput depends is inversely proportional to the small parameter, thus resulting in high gain control signal.Therefore, to generate the necessary torque or force, the actuator may require a control signal exceeding thephysical limits.

The goal of this paper is to show that the application of the singular perturbation method based timescale separation can avoid demanding high magnitude control signals, and to develop a control algorithmthat guarantees the stable tracking of a given reference command in the presence of modeling uncertaintiesand external disturbances.

The rest of the paper is organized as follows. First, we give preliminaries for the singular perturbationsmethod, then we apply the method to the known plants and actuators dynamics. Afterwards, we presenttime scale separation based control design for the unknown system and provide stability analysis. Throughoutthe paper bold symbols are used for vectors, capital letters for matrices and small letters for scalars.

II. Preliminaries

Consider the singular perturbation model (1) with initial conditions x (0) = x 0 , z (o) = z 0 . We assumethat the functions f (x ) and g (x ) are continuously differentiable on some open connected set D x R n , andthe functions a (z ) and b(z ) are continuously differentiable on some open connected set D z R m . Let thecontrol input have the form u(t ) = (t, x , z ), where the function (t, x , z ) and its rst partial derivativeswith respect to ( t, x , z ) are continuous and bounded on any compact subset of D x D z .

The model is said to be in standard form if the algebraic equation

a (z ) + b(z )(t, x , z ) = 0 (3)

has isolated roots for all ( t, x ) [0, ) D x .Let z = h (t, x ) be an isolated root of (3). The system

x (t ) = f (x (t)) + g (x (t))h (t, x (t)) , x (0) = x 0 (4)

is called a reduced system. Denote the solution of (4) by x (t ).Introduce a change of variables = z h (t, x ) and t = . The system

d ( )d

= a ( + h (t, x )) + b( + h (t, x ))(t, x , + h (t, x )) , (0) = z 0 h (0, x 0 ) (5)

where ( t, x ) [0, ) D x are treated as xed parameters, is called a boundary layer system.In this paper, for the stability analysis of the closed-loop systems, we will use the following version of

Theorem 11.2 from Ref. 5

Theorem 1 Consider the singular perturbation problem for system (1) with initial conditions x (0) = x 0 ,z (o) = z 0 and control input u(t) = (t, x , z ). Let z = h (t, x ) be an isolated root of the algebraic equation (3). Assume that the following conditions are satised

The functions f (x ) and g(x ) are continuously differentiable on Dx ;

The function a (z ), b(z ) and c(z ) are continuously differentiable on D z ;

The function (t, x , z ) ant its rst partial derivatives with respect to its arguments are continuous and bounded on any compact subset of D x D z ;

The origin is an exponentially stable equilibrium point of the reduced system (4), R x is the region of attraction;

The origin is an exponentially stable equilibrium point of the boundary-layer system (5), uniformly in (t, x ), R z is the region of attraction.


3/16

Then, for each compact set x Rx and z Rz there is a positive constant such that for all t 0,x 0 x , z 0 h (0, x 0) z and 0 < < , the singular perturbation problem (1) has a unique solution x (t, ), z (t, ) on [0, ), and

x (t, ) x (t ) = O() (6)

holds uniformly for all t 0. Moreover, there exists a time instance T () such that

z (t, ) h (t, x (t )) = O() (7)holds uniformly for all t T ().

III. Problem Formulation

Consider a single-input single-output system

x p (t ) = A p x p (t ) + b p [f p(x p (t )) + g p(x p (t ))ya (t ) + dP (t)]y p (t ) = c

p x p (t ) , (8)

with the initial conditions x ( 0) = x p0 , where x p R n is the plants state, f p(x p ) and g p (x p ) are continuouslydifferentiable unknown functions representing the modeling uncertainties, and ya (t ) R is the control effortgenerated by a slow actuator. It is reasonable to assume that the actuators dynamics are stable, minimumphase and of known relative degree r , and are described by the system

1 x a (t) = Aa x a (t ) + ba [u (t) + f a (ya (t), p a ) + da (t )]ya (t) = c

a x a (t) (9)

with initial conditions x a (0) = x a 0 . Here, x a R m is the actuators state, Aa is a Hurwitz matrix, thetransfer function Ga (s) = ca (s I Aa )

1 ba has stable zeros, is a small parameter, (0, 1] representsthe possible reduction in actuators control effectiveness, f a (ya , pa ) is a continuously differentiable functionrepresenting the modeling uncertainties, u(t) is the actuators control input, and time signals d p (t) and da (t )represent unknown external disturbances.

We notice that the actuators dynamics do not depend on the plants state to be controlled. However,the dynamics may depend on other external variables, which are lumped into the parameter p a . In general,the actuators state is not available for feedback. Therefore, the actuator related part of the overall controldesign must be considered in output feedback framework, assuming that the signal ya (t ) is available forfeedback.

To guarantee the controllability of the plant, we also assume that the pair ( A p , b p ) is controllable, thepair ( A p , c p ) is observable, and the function g p(x p ) never crosses zero. Without loss of generality, we takeg p (x p ) as a positive function, that is, there exists a positive constant g0 such that g p (x p) g0 > 0.

The control objective is to design an input signal u(t ) for the actuator such that the systems outputy p (t ) tracks the output of a given reference model

x m (t ) = Am x m (t ) + b

m yc (t)ym (t ) = c

x m (t ) , (10)

with initial conditions x m (0) = x m 0 , where yc (t ) is an external command that is bounded with its rsttime derivative. Here, Am is a Hurwitz matrix of the form Am = A p b p k for some control gain k , andbm = b p with = c p A

1m b p

1 . Since Am is Hurwitz, there exist a symmetric positive denite matrix P such that

Am P + P A m = Q (11)

for some symmetric positive denite matrix Q .


4/16

IV. Control design for known systems

In this section, we show that the magnitude of a control signal designed by the proposed time scaleseparation based algorithm is smaller than the magnitude of a control signal generated by the conventionalalgorithms by a term that is of the order of 1 . For comparison, we use backstepping design, 7 which is wellsuited for the cascaded system comprised of the dynamics in (8) and the actuator dynamics (9). To thisend we assume that the systems under consideration are known and disturbance free. Also, for the purposesof this section we assume that the functions f p(x p ), g p (x p ) and ga (ya ) are sufficiently smooth, and thereference command and its sufficiently many time derivatives are bounded. In this case, for convenience wewrite the actuator dynamics in the following normal form 10

1 x ca (t ) = A0x ca (t ) + b0[ k

a x a (t) + v(t )] 1

ca (t ) = A

c0

ca (t ) + b

ca x

ca (t )

ya (t ) = c0x ca (t) , (12)

where x ca R r is the portion of the actuators state corresponding to the chain of r integrators representedby the triplet ( c0 R r , A0 R r

r , b0 R r ) having the form

A0 = 0 I r 1

0 0

, c0 =1...0

, b0 =0...1

,

ca R m r is the actuators internal state, and the eigenvalues of Ac0 R (m

r ) (m r ) are the stable zerosof the actuators transfer function Ga (s). For the notational simplicity, we introduce a new control variablev(t ) = bu(t) + bga (ya (t), p a ), where b = 0 is the high frequency gain. Since the function ga (x a , p a ) andparameters and b are assumed to be known, u (t ) can be readily computed after designing v(t ).

Let the tracking error be e (t) = x p(t ) x m (t ). Its dynamics can be written as

e (t ) = A m e (t) + b p k x p(t ) + f p(x p (t )) + g p(x p (t ))ya (t ) y c (t ) . (13)

Since for the error dynamics (13) there exists a continuously differentiable feedback control law in the form

ya (t) = 1(t, x p ) = g 1

p (x p (t )) k

x p(t) f p (x p (t )) y c (t) , (14)

that guarantees that the derivative of the radially unbounded Lyapunov function

V 1 (e (t)) = e (t)P e (t) (15)

is negative denite for all e R n

V 1 (t ) = e (t )Qe (t ) < 0 , (16)

the linear block backstepping lemma (Lemma 2.23) from Ref. 7 can be applied. To be prepared for thatapplication, we introduce change of variables z c (t) = T c x ca (t ), where T c = diag(1 , , . . . , r

1). In these newvariables, the rst equation in (12) takes the form

z ci (t ) = zci +1 (t ), i = 1 , . . . , r 1 r z0r (t ) = k

a x a (t) + v(t )ya (t ) = zc1(t ) . (17)

Denoting i (t ) = z ci (t ) i (t, x ), i = 1 , . . . , r , where the stabilizing functions i (t, x p ) are dened as

2(t, x p ) = 11 (t ) + 1(t, x p ) e (t)P b pg p(x p (t ))

i (t, x p ) = i 1i 1(t) i 2(t ) + i 1(t, x p ) , i = 3 , . . . , r , (18)


5/16

and i > 0, i = 1 , . . . , r 1 are design parameters, the error system is written in the form

e (t)1 (t)

...r 1(t )

x

(t)

=

Am e (t) + b pg p(x p )1(t ) 11(t) + 2(t) e (t )P b pg p(x p )

... r 2(t ) r 1r 1(t)

f

(t, x

)

+

00...1

g

r (t )

r r (t) = k

a x a (t ) r r (t, x p ) + v(t) (19)

We dene the control law as

vb(t ) = k

a x a (t) r r (t ) r r 1(t ) +

r r (t, x p ) , (20)

where r > 0 is a design parameter. Then the closed-loop error system is reduced to

x (t) = f (t, x (t)) + gr (t) r r (t) = r r (t)

r r 1(t ) . (21)

Lemma 1 The control signal (20) guarantees the boundedness of all closed-loop signals and the exponential

convergence of the errors e(t) and i (t), i = 1 , . . . , r to zero.Proof. Consider the following candidate Lyapunov function

V 2 (e (t ), (t )) = V 1(e (t)) + 21 (t) + + 2r (t) . (22)

Its derivative can be readily computed to satisfy the inequality

V 2 (t ) = e (t )Qe (t ) 121 (t ) r 1

2r 1(t ) r

2r (t) < V 2(t) (23)

for all e (t) R n and z(t) R , where = 1max (P )min 1min (Q ), 1 , . . . , r 1 , r r with max (P ) denoting

the maximum eigenvalue of the matrix P . Since V 2 (t) is radially unbounded, the inequality (23) impliesthe global exponential stability of the system (21). Since the reference input yc (t) is bounded, from theboundedness of e (t ) and x m (t ) the boundedness of x (t) follows. Therefore, (t, x ) is bounded, implying the

boundedness of zc(t ). Therefore, the state x

ca (t) is bounded. Then, the boundedness of internal state

ca (t )follows from the input to state stability of the actuators internal dynamics.

Remark 1 We notice that the control signal u b(t ) is comprised of two parts. The rst part does not contain high gain and is designed from the perspective of retaining the slow mode of the actuators dynamics, since it cannot be altered without a high magnitude input signal. The second part is inversely proportional to the small parameter and is designed to cancel interconnecting terms. This part is a potential source of the high magnitude command that can exceed the actuators physical limits.

Next, we apply singular perturbation method to the cascaded error system (19), and introduce a new(slow) time variable = r t . Ignoring the actuators input to state stable internal dynamics and substituting results in the following system

rx ( )d = f (, x ) + gr ( )

dr ( )d

= k a x a ( ) + v( ) d r (, x p)

d . (24)

This is a singular perturbation problem. It is in the standard form if the algebraic equation

f (, x ) + gr ( ) = 0 . (25)

has isolated roots. Here we are concerned about the conditions of the existence of the isolated roots ingeneral. For our purposes it is sufficient to note that the equation (25) has a root x = [e = 0 , 1 =


6/16

0, . . . , r 2 = 0 , r 1 = 1r 1 r (t), which we denote by x

= h (t, r ). Introducing a change of variables ( ) = x ( ) h (t, r ), the boundary layer system is obtained from the rst equation in (24) as follows

d (t )dt

= f (t, (t)) . (26)

It is easy to see that the system (26) has an equilibrium at the origin. The reduced system takes the form

dr ( )

d

= k a x a ( ) + v( ) d r (, x p )

d e = h (, r ) . (27)

Unlike the previous case, here the control signal v( ) is designed to stabilize only the reduced system (27)and has the form

v p (t) = k

a x a (t) r r (t ) + r r (t, x p )|e = h ( t, r ) . (28)

Applying the control signal v p (t) results in the following closed-loop reduced systems in the real time scale t

r (t ) = r r r (t) , (29)

which has a globally exponentially stable equilibrium at the origin ( r = 0).

Lemma 2 Under the control action (28), the singular perturbation problem (24) with initial conditions e 0 , 1 (0) , . . . , r (0) has a unique solution e(t, ), 1 (t, ), . . . , r (t, ), which satises the relationship

r (t, ) r (t ) = O() (30)

for all t 0, where r (t) is the unique solution of the reduced system (27) with the initial condition r (o) =z cr (0) r (0, x 0). Further, there exists a time instance T

such that the relationship

x (t, ) = O() . (31)

holds for all t T . Moreover, all closed-loop signals are bounded.Proof. Consider the candidate Lyapunov function

V 3 (e (t ), (t)) = V 1 (e (t )) + 21 (t ) + + 2r 1(t ) . (32)

Its derivative along the trajectories of boundary layer system (26) can be easily computed to satisfy theinequality

V 3( (t ), z (t )) < V 3( (t ), z (t )) , (33)

with =

1max (P )min 1min (Q ), 1 , . . . , r 1 , which implies the global exponential stability of the boundary

layer system. The smoothness and boundedness conditions of Theorem 1 follow from the assumptionsimposed in this section. Therefore, from Theorem 1 it follows that (30) holds for all t 0, and there existsa time instance T () such that the relationship

x (t, ) h (t, r (t)) = O() (34)

holds for all t T (). From the exponential stability of (29) it follows that there exists a time instant T 1such that r (t) = O() for t T 1 . Then, from the denition of h (t, x (t)) it follows that the relationshiph (t, x (t )) = O() holds for t T 1 . Therefore (33) holds for t T , where T = max[ T (), T 1]. From theabove relationship and the boundedness of the reference model (10) it follows that the plants state x p (t) isbounded, implying the boundedness of the stabilizing functions i (t, x p ), i = 1 , . . . , r . Since (t) is bounded,the boundedness of z c (t ) follows, implying also the boundedness of the actuators state x ca (t). From theproperties of the actuators internal dynamics it follows that the internal state ca (t ) is bounded as well.

Remark 2 Instead of the exponential tracking achieved by the backstepping control vb(t ) (20), the control law v p(t) (28) achieves only -tracking for the original error dynamics, with a transient time greater than T ]. However, the latter control signal has a smaller magnitude than the former one. To see this, we rst notice from the continuity of the function (t, x ) that (t, x ) (t, x )|e = h ( t,z ) = O (). Taking into account the relationship (80), we can write

vb(t) v p (t) = O() r r 1(t ) (35)

which implies that the two control signals differ by the high gain term r r 1(t), which is inversely propor-tional to r and comes from vb(t ) (20).


7/16

V. Control design for uncertain systems

In this section, we solve the same control problem assuming uncertain plants and actuators dynamicsthat are also subject to external disturbances d p (t) and da (t ) respectively, which have bounded derivatives.To be precise, we assume that the parameter , matrices A and A a , vector ba , and continuously differentiablefunctions f p (x p), g p (x p ), ga (ya ) are unknown. The control objective is to design a control law that uses theplants state x p (t ) and the actuators output ya (t) for feedback to achieve tracking of the reference model(10).

As in conventional adaptive control, using the universal approximation theorem, we approximate theunknown functions by radial basis functions networks on compact sets x and y a

f p(x p ) = w

f f (x p ) + f (x p )

g p(x p ) = w

g g (x p ) + g (x P )

ga (ya , pa ) = w

a a (ya ) + a (ya ) , (36)

where f (x p ), g (x p ) and a (ya ) are the approximation errors, uniformly bounded by positive constants f , g and

a respectively. Using the relationships (36), the error dynamics (13) can be written as

e (t) = A m e (t ) + b p k x p(t) + w

f f (x p) + w

g g (x p )ya (t ) + p(t, x p , ya ) y c (t ) , (37)

where p (t, x p , ya ) = f (x p (t))+ g (x p (t ))ya (t) + d p (t) is the combined disturbance term, which is uniformlybounded by a positive constant 1 as long as x p (t) x and ya (t ) y .

In this section, we assume that the actuators dynamics has a relative degree m r + 1 and is representedin the following observer canonical form

1 x a (t ) = A0x a (t ) qya (t) + ba [u(t ) + ga (ya (t ), pa ) + da (t)]ya (t ) = c

0 x a (t ) , (38)

where A 0 and c0 are dened similar to denitions in the previous section with r replaced by m , q R m is anunknown constant vector, and ba = [0 . . . ba,m r +1 . . . bam ] . Since the actuators dynamics is of relativedegree m r + 1, the ba,m r +1 is not zero. Let it be positive. Taking into account the approximation in(36), the dynamics of x a (t ) is written as

1 x a (t ) = A 0x a (t ) qya (t) + ba u (t ) + w

a a (ya ) + a (t, y a ) . (39)

where a (t, y a ) = a (ya ) + da (t) is the combined disturbance term, which is uniformly bounded by a positiveconstant a as long as ya y a .

To avoid generating high gain control signals we again apply the singular perturbation method. However,the method cannot be directly applied, since the exponential stability properties for the resulting boundarylayer and reduced systems cannot be establish in the adaptive control framework without parameter conver-gence. The latter can be guaranteed only for sufficiently rich regressor or input signals, which in general isdifficult to verify a priory. As an intermediate step, we follow the approach from Ref. 4 and design suitableprediction models for which the exponential stability properties can be guaranteed, and hence Tikhonovstheorem can be applied. The control objective is met when the closeness of the prediction models to thecorresponding dynamics is established. This will be done by the choice of adaptive laws independent of thecontrol design.

A. Prediction models

We consider the following prediction model for the plants tracking error dynamics

e (t) = A m e (t) + b k

(t)x (t ) + w f (t) f (x ) + w

g (t) g (x )ya (t) + p (t )sign( (t)) y c (t ) , (40)

where the initial conditions e (0) is chosen identical with e(0), (t) = [e (t ) e (t)] P b p , and all variableswith hat are the estimates of the corresponding variables without hat, and are available for the control


8/16

design. Let the prediction errors be e (t ) = e (t ) e (t ). Its dynamics can be written in the form

e (t ) = Am e (t) + b p k

(t )x p(t ) + w

f (t ) f (x p) + w

g (t ) g (x p )ya (t )

+ w g (t) g (x p)ya (t) + p (t, x p , ya ) p (t)sign (t) , (41)

where we introduce parameter estimation errors k (t ) = k k (t), w f (t) = w f w f (t), w g (t ) = w g w g (t), p (t ) = p p (t).

To be able to introduce a suitable prediction model for the actuator dynamics in (39), we re-parameterizethem following conventional adaptive observer-estimator scheme (see for example 9,15 ) and introduce lters

1 1 (t ) = ( A0 b0 k 0 ) 1(t ) + b

0ya (t), 1(0) = 0 1 2 (t ) = ( A0 b

0 k 0 ) 2(t ) + b0u(t), 2(0) = 0

1 3 (t ) = ( A0 b0 k 0 ) 3(t ) + b

0a (t, y a ), 3(0) = 0 1 (t ) = ( A0 b0 k

0 )( t) + b0

a (ya ), (0) = 0 , (42)

where b0 = [0 . . . 1 b0r +1 . . . b0m ] is chosen such that the pair ( A0 , b0) is controllable and the polynomial

s m r + b0r +1 s m r 1 + + b0m is Hurwitz, and the vector k0 is chosen such that the matrix A0 b

0k 0 isHurwitz. The actuators output ya (t ) is represented as

ya (t ) =

1

1(t) +

2

2(t ) + w

a

3(t) + f (t ) , (43)

where 1 R m and 1 = [bam . . . ba,m r +1 ] R r are unknown constant parameters to be estimatedonline, 1(t ) = 1(t), 2(t) = [ 21 (t) . . . 2r (t)]

, 3(t) = c0 (t ) and f (t ) = c

0 3 (t). Since the ltersare stable from the boundedness of the inputs the boundedness of the outputs follow. That is f (t) isbounded by some positive constant f since a (t, y a ) is bounded and the vector function 3(t ) is boundedsince the RBFs are bounded. The prediction model for the actuator dynamics in (39) is introduced as

ya (t) =

1 (t) 1(t ) +

2 (t ) 2(t) + w

a (t ) 3(t) + f (t)sign( (t)) , (44)

where again the variables with hat are the estimates of the corresponding variables without hat, andya (t ) = ya (t) ya (t) is the prediction error, which is given by the equation

ya (t ) =

1 (t ) 1(t) +

2 (t) 2(t ) + w

a (t ) 3(t ) + f (t ) f (t )sign( (t )) , (45)where 1(t) = 1 1(t ), 2(t ) = 2 2(t ) and w a (t ) = w a w a (t ).

Since g(x p ) is assumed to be positive, the neural network approximation (36) can be chosen such thatthe unknown weights w g are positive. 11 The RBFs can be chosen to be positive as well. Then the productw g (t) g (x p) is always positive. Therefore, it is reasonable to design the adaptive law such that all compo-nents of w g (t) are positive. This will be done by means of the projection operator. 12 In the same fashion,since 2 ,r > 0, the adaptive law for 2,r (t ) is designed to keep it positive. Having this in mind, we designthe adaptive laws for the estimates k (t ), w f (t), w g (t), p (t) as follows

p (t ) = |(t )|k (t ) = x p (t )(t)

w f (t ) = f (x )(t )w g (t ) = w g (t), ya (t ) g (x p )(t) , (46)

where > 0 is the adaptation rate, {, } denotes the projection operator, 12 introduced here to keep theestimates w g (t ) bounded away from zero.

The adaptive law for 2,r (t ) is dened by means of the projection operator.

2,r (t) = 2r (t ), ya (t)2r (t) + w

g (t ) g (x )(t )2r (t ) . (47)


9/16

The remaining adaptive laws are

1(t ) = ya (t) 1(t ) + w

g (t ) g (x )(t ) 1(t)2i (t ) = ya (t )2i (t ) + w

g (t ) g (x )(t )2i (t ) , i = 1 , . . . , r 1

w a (t ) = ya (t) 3(t ) + w

g (t ) g (x )(t ) 3(t)f (t ) = |ya (t)| + w

g (t ) g (x )|(t)| , (48)

where the projection operator ( , ) is introduced to keep (t) positive.The following lemma guarantees the closeness of the prediction models to the corresponding dynamics.

Lemma 3 The adaptive laws (46), (47) and (48) guarantee boundedness of the estimates k (t), w f (t ), w g (t ), p (t )as well as 1 (t ), w a (t ), 1 (t), f (t ) and the prediction errors e (t ) and ya (t ). Moreover, e (t) L2 and ya (t ) L2 .

Proof. Consider the following candidate Lyapunov function

V 4 (t ) = e (t)P e (t ) + 1 ( p p (t ))

2 + k

(t)k (t) + w f (t )w f (t ) + w

g (t)w g (t )

+

(t) 1(t) +

2 (t) 2(t ) + w

a (t)w a (t) + (

f f (t))2 . (49)

The derivative of V 4(t) is computed along the trajectories of systems (41), and (46) and (48).

V 4(t ) = e (t )[Am P + P A m ]e (t) + 2 e

(t )P b p k

(t )x p (t ) + w

f (t ) f (x p ) + w

g (t ) g (x p )ya (t )

+ w g (t ) g (x p )[

1 (t ) 1(t ) +

2 (t ) 2(t) + w

a (t ) 3(t) + f (t) f (t )sign( ya (t))] + p (t, x p , ya )

p (t)sign( (t )) + 2 1 ( p p (t)) p (t ) k

(t) k (t) w f (t ) w f (t ) w

g (t) w g (t )

1 (t) 1(t) 2(t ) 2 (t ) w

a (t) w a (t ) (

f f (t)) f (t)

= e (t)Q e (t) + 2 w g (t )[(t ) g (x p )ya (t ) 1 w g (t)] + 2 w

f (t )[(t) f (x p ) 1 w f (t)]

+ 2 k

(t)[(t )x p (t) 1 k (t )] + 2[(t) p (t, x p , ya ) p (t)|(t )|

1( p p(t)) p (t )]

+ 2

1 (t)[(t)w

g (t ) g (x p ) 1(t ) 1 1(t)] + 2 w

a (t)[(t )w

g (t) g (x p) 3(t ) 1 w a (t )]

+ 2 2(t )[(t)w g (t ) g (x p ) 2(t) 1 2(t )] + 2(t)w g (t) g (x p )f (t)

2w g (t ) g (x p )

f |(t)| + 2 w

g (t) g (x p)[

f f (t)]|(t)| 2 1(f f (t )) f (t )] . (50)

Substituting the adaptive laws and taking into account the properties of the projection operator 12

w f (t ) ya (t) g (x p)(t ) w g (t ), ya (t) g (x )(t) 0 (51)

2r (t) ya (t )2r (t ) + w g (t) g (x )(t)2r (t) 2r (t ), ya (t )2r (t ) + w

g g (x )(t)2r (t) 0 ,

we obtain

V 4(t ) e (t)Q e (t) + 2 (t) p (t, x p , ya ) 2 p (t)|(t)| 2ya (t )

1 (t ) 1(t ) +

2 (t) 2(t ) + w

a (t) 3 (t )

+ 2 (t )w g (t) g (x p)f (t ) 2w

g (t ) g (x p )

f |(t)| (

f f (t)) |ya (t )| . (52)

We notice that (t ) p (t, x p , ya ) p |(t )| 0, and (t )w

g (t) g (x p )f (t ) w

g (t) g (x p)

f |(t )| 0, sincew g (t) g (x p) is positive by construction. Moreover, from the adaptive law in (48) it follows that f (t ) 0as long as f (0) 0. Therefore the following inequality holds

(f f (t )| ya (t )| f (t)ya (t ) f (t )sign( (t )) ya (t ) . (53)

Taking into account the inequalities above and the error denition (45) we obtain

V 4(t) e (t )Q e (t ) y2a (t ) , (54)


10/16

which implies that the error signals e (t), k (t), w f (t), w g (t ), p p (t ), 1(t ), w a (t), 2(t) and

f f (t )are bounded. Then, the estimates k (t ), w f (t), w g (t), p (t), 1(t), w a (t ), 2 (t ) and f (t ) are bounded aswell. At this moment we cannot conclude anything about the boundedness of e (t), e (t), ya (t ), ya (t ) or ya (t).However, we can integrate the inequality (54) and obtain

t

0[e ( )Q e ( ) + y2a ( )]d V 4 (0) V 4 (t ) . (55)

From Lemma 2 it follows that V 4 (t ) is bounded, therefore e (t) L2 and ya (t) L2 .

B. Adaptive control design

We are interested in designing a control signal u (t ) for the cascaded system

e (t) = Am e (t ) + b k

(t)x (t) + w f (t ) f (x ) + w

g (t) g (x )ya (t ) + p(t)sign( (t)) y c (t )

1 2(t) = ( A0 b0k 0 ) 2(t) + b

0u (t) , (56)

where ya (t) = q (t) 1(t) + ba (t) 2(t) + w

a (t ) a (t) + f (t)sign( (t )), such that e (t) 0 as t . Wenotice that the second system in (56) is of relative degree r , stable and minimum phase. Therefore, it canbe represented in the normal form

1 2(t) = A0 2(t ) + b0[ k

a 2(t ) + u(t )] 1

ca (t) = A

c0

ca (t) + b

ca 2(t) , (57)

where the triplet ( c0 R r , A 0 R r r , b0 R r ) is the normal form of representation of chain of r integrators,

and the eigenvalues of Ac0 R (m r ) (m r ) are the stable zeros of the minimum phase transfer function

G (s) = c0 (s I A 0 + b0k 0 )b

0 . The control design besides the tracking task must guarantee also theboundedness of 2(t). Then the boundedness of ca (t) will follow from the input-to-state stability of theinternal dynamics.

To this end consider a change of variables z o (t ) = T o 2(t ), where T o = diag(1 , , . . . , r 1). In these newvariables, the second equation in (56) takes the form

zoi (t ) = zoi +1 (t ), i = 1 , . . . , r 1

r zor (t ) = k 0 T 1o z 0(t) + u(t ) . (58)

We notice that

w g (t ) g (x )ya (t) = w

g (t ) g (x ) q (t) 1(t) + (t)z

o1 (t ) + w

a (t ) 3(t) + f (t)sign( (t)) ,

where the inequality (t ) > 0 is guaranteed by the adaptive law (48). Therefore, we design the stabilizingfunction z o1 (t) = 1(t, x p ) for the tracking error prediction model as follows

1(t, x p) = k

(t)x p (t ) w

f (t) f (x p ) p (t)sign( (t )) + y c (t)2r (t)w

g (t) g (x p )

q (t) 1(t) + w

a (t ) 3 (t ) + f (t )sign( (t ))2r (t) , (59)

If we introduce the variable 1(t ) as in the previous section, that is 1(t) = z o1 (t ) 1(t, x p ), the errorprediction model takes the convenient form

e (t) = Am e (t ) + bw

g (t) g (x p )1(t) . (60)

However, the dynamics of 1 (t ) involves the derivative of 1 (t, x p), which is not available in this case becauseof unknown terms in the plants dynamics. One remedy for the situation is the command ltered approach


11/16

presented in. 3 Following Ref., 3 we rst lter the stabilizing function 1(t, x p ) through a second order stablelter

z11 (t ) = z12 (t )z12 (t ) = 2 z12 (t ) [z11 (t ) 1(t, x p )] , (61)

then use the error variable 1 (t ) = z o1 (t) z11 (t ) instead of 1(t). The tracking error prediction model nowcan be written as

e (t ) = Am e (t) + b p w

g (t) g (x p )1 (t ) + b p w

g (t ) g (x p )[z11 (t ) 1(t, x p )] , (62)

The compensated error e (t ) = e (t ) (t ), where (t) is dened according to equation

(t) = A m (t ) + b p w

g (t ) g (x p )[z11 (t ) 1(t, x p ) + a 1(t)] , (63)

satises the equation

e (t ) = A m e (t ) + b p w

g (t ) g (x p )1 (t ) , (64)

where 1(t ) = 1(t) a 1 (t ), a 1 (t) is to be dened in the next step.Following the block-backstepping 7 and command ltering 3 procedures, the stabilizing functions i , i =

2, . . . , r are dened as follows.

2(t, x p ) = 1 1 (t ) + z12 (t ) (e (t) (t)) P b p w

g (t) g (x p) i (t, x p ) = i 1 i 1(t) i 2(t ) + zi 1,2 (t ) , i = 3 , . . . , r , (65)

where i (t) = zoi (t) zi 1(t), z i 1(t ) is generated through the stable command lter

zi 1 (t) = zi 2(t )zi 2 (t) = 2 zi 2(t ) [zi 1(t ) i (t, x p )] . (66)

The prediction models (56) in terms of error signals 1(t ), . . . , r (t ) can be written in the following compactform

e (t )1 (t )

...r 1(t )

x ( t )=

Am e (t) + b p w

g (t) g (x p)1 (t) + b p w

g (t ) g (x p )[z11 (t ) 1(t, x p )] 1 1(t) + 2 (t ) (e (t ) (t )) P b p w

g (t) g (x p) + z21 (t ) 2(t, x p )...

r 2(t ) r 1 r 1(t ) + zr 1(t) r (t, x p )

f ( t, x )+

00...1

gr (t)

r r (t ) = r zr 1(t) + u(t) k

0 T 1o z

o (t ) (67)

The compensated error is dened as i (t ) = i (t ) ai (t ), where ai (t ) is generated by the dynamics

ai (t ) = i ai (t) + zi +1 ,1(t ) i (t, x p ) + a,i +1 (t ), i = 1 , r 1 a,r = 0 . (68)

From the above constructions it follows that r (t ) = r (t ). The dynamics of the compensated error can bewritten as

1 (t ) = 1 1 (t ) + 2(t) e (t )P b p w

g (t) g (x p )i (t ) = i 1(t) i i (t ) + i +1 (t), i = 2 , . . . , r 1

r r (t ) = r zr 1(t ) + u(t) k

0 T 1o z

o (t ) . (69)


12/16

We apply the singular perturbation method to the cascaded system

e (t )1 (t )

...r 1(t )

x ( t )

=

Am e (t ) + b p w

g (t ) g (x p )1 (t ) 1 1(t) + 2 (t) e (t )P b p w

g (t) g (x p)...

r 2(t ) r 1 r 1(t)

f ( t, x )

+

00...1

g

r (t )

r r (t) = r zr 1(t) + u(t ) k

0 T 1o z

o (t ) (70)

by introducing slow time variable = r t . In the slow time the systems are written as

rdx ( )

d = f (t, x ) + g r (t)

dr (t )d

= dz r 1( )

d + u(t) k 0 T

1o z

0(t) (71)

This system is in the form of the standard singular perturbation model, if the algebraic equation

f (t, x ) + g r (t ) = 0 (72)

has isolated roots. We are not concerned about nding all the roots of the equation (72), but we are interestedin the root x = [ e = 0 1 = 0 . . . r 2 = 0 r 1 = 1r 1 r (t)]

, which we denote by x = h (t, r ).Introducing a change of variables ( ) = x ( ) h (t, r ), the boundary layer system is obtained from therst equation in (71) as follows

d (t)dt

= f (t, + h (t, r ) + g r (t) = f (t, ) . (73)

The system (73) has its equilibrium at the origin. The following lemma shows that it is globally exponentiallystable.

Lemma 4 The boundary layer system (73) is globally exponentially stable.

Proof. Consider a candidate Lyapunov function

V 5(t ) = e

(t )P e (t) + 21 (t) + +

2r 1(t ) . (74)

It is straightforward to compute its derivative along the solutions of system (73). After some algebra weobtain

V 5(t) = e (t)Q e (t) 1 21 (t) r 1

2r 1(t) . (75)

Denoting = 1max (P )min 1min (Q ), 1 , . . . , r 1 the following inequality can be readily derived

V 5(t ) v5(t ) , (76)

which along with the radial unboundedness of V 5 (t ) implies the global exponential stability of (73). The reduced model is derived from the second equation in (71) by substituting x ( ) = h (, r ) and = 0.

Since the only term k0 T 1o z 0(t ) does not explicitly depend on x , we can write the reduced system in the

formdr ( )

d =

dz r 1( )d

+ v(t) k 0 T 1o z

0 (t ) . (77)

The control signal v( ) is designed in the actual time scale as

u (t ) = r r (t ) + k

0 T 1o z

0(t) + r zr 2(t ) , (78)

which in the actual time scale results in the following closed-loop reduced system

r (t ) = r r r ( ) . (79)


13/16

Obviously the reduced system has a globally exponentially stable equilibrium at the origin. The rest of theconditions of Theorem 1 follow from the smoothness and boundedness assumptions imposed on the referenceinput, as well as on the plants and actuatorss dynamics. Therefore, for all t 0 the singular perturbationproblem (71) has a unique solution x (t, ), r (t, ) with initial conditions x (0) and r (0)) respectively, andthe relationship

r (t, ) r (t ) = O() (80)

holds, where r (t ) is the unique solution of the reduced system with initial condition r (0) = zor (0) zr 1(0).Moreover, there exists a time instance T () such that the relationship

x (t, ) h (t, r (t)) = O() (81)

holds for all t T (). From the exponential stability of (79) it follows that there exists time instant T 1 suchthat r (t) = O() for t T 1 . Then, from the denition of h (t, r (t)) it can be seen that the relationshiph (t, r (t )) = O() holds for t T 1 . Therefore,

x (t, ) = O() , (82)

holds for t max[T (), T 1]. Next, we apply the result of Ref. 3 to the error systems (19) and (67). Sincex (t, ) = O() and r (t ) = O() for all t max[T (), T 1], it can be shown that

x (t, ) = O() + O( 1 )r (t, ) = O() + O(

1 ) (83)

for all t max[T (), T 1], where [x (t, ) r (t, )] is the solution of the error prediction system (67).From (83) it follows that e (t ) and (t ) are bounded. Then, from Lemma 2 it follows that e (t) is bounded.

Since the reference command yc (t) is bounded, x m (t) is bounded, implying the boundedness of x p (t). SinceLemma 2 guarantees the boundedness of the parameter estimates, it follows that the stabilizing functions i (t, x p ), i = 1 , . . . , r are bounded, implying also boundedness of ( zi 1 (t ), z i 2(t)) , i = 1 , . . . , r . Then from thedenition of error signal (t) it follows that z o (t ) is bounded, implying the boundedness of ya (t) and 2(t).Then the control signal u (t ) is bounded as well. From the the denition of ya (t) (44) it follows that 1(t) isbounded, hence 1(t) is bounded. Then from the lter equation (42) it follows that ya (t) is bounded. Thatis, all closed loop signals are bounded. Therefore, e (t) and ya (t ) are bounded. From the Barabalats 14 itfollows that e (t) 0 and ya (t) 0 as t .

The obtained results are formulated in the form of the following theorem.

Theorem 2 Consider the uncertain plant (8) and slow actuator (9). The adaptive controller given by (28),along with the stabilizing functions (14) and (18), auxiliary lters (61) and (66), the prediction models (40)and (44), and the adaptive laws (46), (47) and (48), guarantees that the plants output tracks the output of the given reference model (10) with the error directly proportional to the small parameter in the actuators dynamics and inversely proportional to the frequency of the lters (61).

VI. Simulation Example

We demonstrate the benets of the approach on a rst order unstable system

x p (t) = x2 p (t ) + xa (t ) , (84)

controlled by the actuator

xa (t) = 0.1(x a (t) u(t )) . (85)

For this simulation, the control objective is to track a reference command yc (t) = 1. That is e(t ) = x p (t) 1.The control signal u b(t) takes the form

(t) = xa (t ) + am e(t ) + x2 p (t ) (86)

ub(t) = ka (t ) + xa (t ) 1e(t ) 1 a 2m e(t) a m (t) 2x p (t ) x

2 p (t) + xa (t) ,


14/16

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Plant output

Feedback linearizationBacksteppingSingular perturbations

Figure 1. Output tracking performance for different control strategies.

whereas the signal u p (t ) has form

u p(t ) = ka (t ) + xa (t ) + 1 2

1am

(t ) + 13

+ xa (t ) 1a m

(t ) + 1 .

Figure 1 displays the tracking performance for three controllers: feedback linearization, backsteppingdesign and proposed design with time scale reparation. For the backstepping design and proposed designthe control gains are the same: am = 3 , ka = 1. For the feedback linearization the control gains are chosensuch that the eigenvalues of the resulting error systems are the same. As it can be seen from the plots thetracking is almost the same. The actuators output performance is displayed in Fig. 2 and the control signals

0 5 10 151

0.8

0.6

0.4

0.2

0

0.2

0.4Actuator output


Figure 2. Actuators output performance for different control strategies.

are displayed in Fig. 3. As it can be seen the magnitude of the control signal generated by the singularperturbations method based approach is substantially smaller the that of generated by feedback linearizationand backstepping design.

We notice that with the control signal u p (t ), the error system that is comprised of the boundary layersystem and reduced system has eigenvalues 3 and 0.1. Next we run the simulation for the feedbacklinearization and backstepping approach with the gains chosen such that the error system has poles at the


15/16

0 5 10 156

4

2

0

2

4

6

8

10

12

14Control signal


Figure 3. Control signals for different control strategies.

same location. The tracking performance and the corresponding control signals are displayed in Fig. andFig. respectively. As it can be seen from the gures the tracking is sluggish for both designs, as it is expected,though the control signals have smaller magnitude.

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Plant output

Feedback linearizationBackstepping

Figure 4. Output tracking performance for backstepping and feedback linearization with poles at 3 and 0.1.

VII. Conclusion

We considered a problem of controlling a plant with a slow actuator. It has been shown that thesingular perturbations method can be applied for this problem. The resulting control signal has substantiallysmaller magnitude than one generated by conventional design technique, although the tracking error can beguaranteed to be only bounded with the bound proportional to the small parameter in actuators dynamics.

VIII. Acknowledgment

The authors wish to thank Dr. Eugene Lavretsky of The Boeing Company for fruitful discussions.


16/16

0 5 10 151

0.5

0

0.5

1

1.5

2

2.5

3Control signal

Feedback linearizationBackstepping

Figure 5. Control signals for backstepping and feedback linearization with poles at 3 and 0.1.

References1 C.-C. Chen. Global Exponential Stabilization for Nonlinear Singularly Perturbed Systems. IEE Ptoc. - Control Theory

Appl. , 45(4):377382, July 1998.2 P. D. Christodes and A. R. Teel. Singular Perturbations and Input-to-State Stability. IEEE Trans. Autom. Contr. ,

41(11):16451650, Nov 1996.3 J. A. Farrell, M. Polycarpou, M. Sharma, and W. Dong. Command Filtered Backstepping. IEEE Trans. Autom. Contr. ,

54(6):13911395, June 2009.4 N. Hovakimyan, E. Lavretsky, and C. Cao. Adaptive Dynamic Inversion via Time-Scale Seperation. IEEE Trans. Neural

Networks , 19(10):17021711, 2008.5 H.K. Khalil. Nonlinear Systems, Third Edition . Prentice Hall, New Jersey, 2002.6 P. V. Kokotovic, H. K. Khalil, and L. OReilly. Singular Perturbations Methods in Control: Analysis and Design .

Academic Press, New York, 1986.7 M. Krstic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear and Adaptive Control Design . John Wiley & Sons, New

York, 1995.8 C. L. Lin and B. S. Chen. On the Design of Stabilizing controllers for Singularly Perturbed Systems. IEEE Trans.

Autom. Contr. , 37(11):18281834, 1992.9 K.S. Narendra and A.M. Annaswamy. Stable Adaptive Control . Prentice Hall, 1989.10 p. Sannuti and A. Saberi. Special Coordinate Basis for Multivariable Linear Systems - Finite and Innite Zero Structure,

Squaring Down and Decoupling. The International Journal of Control , 45:16551704, 1987.11 J. Park and I. Sandberg. Universal Approximation Using Radial Basis Function Networks. Neural Computation , 3:246

257, 1991.12 J. B. Pomet and L. Praly. Adaptive Nonlinear Regulation: Estimation from the Lyapunov Equation. IEEE Trans.

Autom. Contr. , 37(6):729740, 1992.13 A. Saberi and H. Khalil. Qudratic-type Lyapunov Functions for Singularly Perturbed Systems. IEEE Trans. Autom.

Contr. , AC-29:542550, 1984.14 S. S. Sastry and M. Bodson. Adaptive Control: Stability, Convergence and Robustness . Prentice Hall, 1989.15 J.J. Slotine and W. Li. Applied Nonlinear Control . Prentice Hall, New Jersey, 1991.16 A. N. Tikhonov. On the Dependence of the Solutions of Differential Equations on a Small Parameter. Mat. Sborni. N.

S. 22(64) (in Russian) , pages 193204, 1948.

stepanyan_control of systems with slow actuators

Documents