backstepping control of linear time-varying systems with known

1908 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 11, NOVEMBER 2003

Backstepping Control of Linear Time-VaryingSystems With Known and Unknown Parameters

Youping Zhang, Member, IEEE, BarısFidan, Student Member, IEEE, and Petros A. Ioannou, Fellow, IEEE

Abstract—The backstepping control design procedure has beenused to develop stabilizing controllers for time invariant plantsthat are linear or belong to some class of nonlinear systems. Theuse of such a procedure to design stabilizing controllers for plantswith time varying parameters has been an open problem. In thispaper we consider the backstepping design procedure for lineartime varying (LTV) plants with known and unknown parameters.We first show that a backstepping controller can be designedfor an LTV plant by following the same steps as in the lineartime-invariant (LTI) case and treating the plant parameters asconstants at each time . Its stability properties however cannot beestablished by using the same Lyapunov function and techniquesas in the LTI case. We then introduce a new parametrizationand filter structure that takes into account the plant parametervariations leading to a new backstepping controller. The newcontrol design guarantees exponential convergence of the trackingerror to zero if the plant parameters are exactly known. If theparameters are not precisely known but the time variations ofthe parameters associated with the system zeros are known,the appropriate choice of certain design parameters, withoutusing any adaptive law, leads to closed-loop stability and perfectregulation. This new control design is modified and supplementedwith an update law to be applicable to LTV plants with unknownparameters. In the adaptive control design, the notion of struc-tured parameter variations is used in order to include possiblea priori information about the plant parameter variations. Withthis formulation, only the unstructured plant parameters areestimated and are required to be slowly time varying, and thestructured plant parameters are allowed to have any finite speedof variation. The adaptive controller is shown to be robust withrespect to the unknown but slow parameter variations in theglobal sense. We derive performance bounds which can be used toselect certain design parameters for performance improvement.The properties of the proposed control scheme are demonstratedusing simulation results.

Index Terms—Adaptive control, backstepping, certainty equiv-alence, parametric robustness, structured parameter variations,time varying systems.

NOMENCLATURE

The following notation is used throughout this paper, unlessotherwise stated.

th element of vector .th coordinate column vector in .th row of matrix .

Manuscript received October 25, 2002; revised May 5, 2003. Recommendedby Associate Editor P. A. Iglesias. This work was supported by the NationalScience Foundation under grant ECS 9877193.

Y. Zhang is with Synopsys, Inc., Mountain View, CA 94043 USA (email:[email protected]).

B. Fidan and P. A. Ioannou are with the Department of Electrical Engi-neering, University of Southern California, Los Angeles, CA 90089 USA(email: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TAC.2003.819074

Matrix Frobenius norm of .Estimate of scalar or vector signal.Estimate error .shifted truncated -norm( -norm),

.

Set ,for a

given constant , where are somefinite constants, and is independent of .

for a given function ,where are some finite constants.

Mean-square-error (MSE) norm of function,.

Any exponentially decaying to zero signal.Any positive constant.( ) identity matrix.

I. INTRODUCTION

RESEARCH on adaptive nonlinear control has been accel-erated during the last decade, after introduction of a class

of controllers for a set of general classes of nonlinear systems[1]–[7]. These controllers are based on integrator backsteppingtogether with other nonlinear design tools such as nonlineardamping [1], [7], [8], tuning functions [7], [9], and andMTfilters [4], [7], [10], [11]. In the absence of modeling uncer-tainties, these controllers achieve global boundedness, asymp-totic tracking, passivity of the adaptation loop irrespective ofthe relative degree, and most importantly, systematic improve-ment of transient performance [7], [12]. These controllers havelater on been modified so that they can tolerate a class of mod-eling uncertainties, especially high frequency unmodeled dy-namics, in the global sense [13]–[16]. The set of systems whichcan be controlled by these controllers includes linear time-in-variant (LTI) systems. Moreover, for LTI systems, these con-trollers bear strong parametric robustness in the sense that globalstability can be achieved by choosing appropriate design param-eters without the precise knowledge of the values of the plant pa-rameters [6], [7], [17]. The corresponding adaptive controllerswhich deal with unknown but constant parameters [9], [7] canachieve arbitrarily improved transient performance [7], [12].

0018-9286/03$17.00 © 2003 IEEE

ZHANG et al.: BACKSTEPPING CONTROL OF LINEAR TIME-VARYING SYSTEMS 1909

The stability properties of this class of controllers are basedon the assumption that the plant parameters are time invariant(TI). In most applications, however, plant parameters may varywith time and therefore the properties of the controllers that aredesigned for LTI plants need to be evaluated in a time varying(TV) environment. The early attempts to design adaptive con-trollers for linear time-varying (LTV) systems are based on theuse of the certainty equivalence approach that combines a con-troller structure with a robust adaptive law [18]–[21]. These con-trollers use the notion that slow time variations of the plant pa-rameters act as a perturbation to the system in the same manneras unmodeled dynamics. Based on this notion, robust adaptivecontrol schemes for LTI systems are used to guarantee signalboundedness and small tracking error of the order of the timevariations of the plant parameters. Later, consideration of theTV nature of the plant and somea priori information about theparameter variations led to new adaptive model reference andpole placement control designs that allow the system to be fastTV [22], [23], [21]. These controllers bear the strong stabilityand robustness properties of their traditional counterparts forLTI systems. However, they can not guarantee good transientbehavior [24], [25], and generally can not be extended to non-linear time varying systems. In this paper, we fill this gap usingthe backstepping control design procedure.

We first consider the use of the backstepping controllers pro-posed in [6], [17] based on TI models to control LTV systemswith known parameters by treating the time varying parametersas constant at each time. We demonstrate that the quadraticLyapunov function-based analysis used in [6], [17] to show sta-bility and asymptotic tracking for LTI systems does not workfor LTV systems in general, even when the plant parametersare known exactly at each time. In addition, we establish thatsignal boundedness can only be guaranteed if the plant param-eters associated with the plant zeros vary slowly with time.

We, then, propose a new controller that guarantees stabilityand convergence of the tracking error to zero independent ofthe speed of variation of the plant parameters. The new con-troller uses integrator backstepping and nonlinear damping andexploits the fact that the TV plant parameters and their varia-tions are known exactly. The stability and performance of theproposed controller is examined in the presence of parametricuncertainty. The controller guarantees signal boundedness pro-vided that the time variations of the parametric uncertainty as-sociated with the plant zeros are small. In particular, if we knowthe time variations of these parameters exactly, then exponentialregulation can be achieved for zero reference input.

The new controller developed for the known parameter casebased on the LTV plant model is modified and combined withan adaptive law to deal with the case of unknown plant param-eters. The notion of structured parameter variations is used toincorporate any availablea priori information about the modesof variation of the plant parameters into the parameter estimates.The resulting adaptive backstepping controller has the followingadvantages as applied to LTV plants. First, only the unstructuredplant parameters are estimated and are required to be slowlyTV. The structured plant parameters are allowed to have any fi-nite speed of variation. Second, the performance bounds derivedcan be used to choose certain design parameters for improved

performance. Furthermore, we show that the proposed adaptivecontroller is robust with respect to unknown but slow parametervariations. Finally, we demonstrate the properties of the devel-oped controllers using simulations.

II. PROBLEM STATEMENT

A single-input–single-output (SISO) linear plant with param-eters that are smooth and bounded and have bounded deriva-tives which is strongly controllable and observable is topolog-ically equivalent to the following observable canonical form[21], [26], [27]:

(2.1)

(2.2)

where

......

...

...

...

The state–space model (2.1),(2.2) can also be represented in theinput–output form

(2.3)

where

are the right polynomial differential operators (PDOs) [19], [21]for the plant. Equivalently, (2.3) can also be represented usingthe right polynomial integral operator (PIO) as

We make the following assumptions about the plant.

Assumption 1 The PIO is exponentiallystable with a rate at least ,i.e., the transition matrix cor-responding to satisfies

for some constant.

Assumption 2 The PDO’s , are strongly co-prime with known orders , , re-spectively, and .

Assumption 3 The plant parameters , are timefunctions which are bounded and havebounded derivatives.

Assumption 4 The sign of the high frequency gainis known and constant, and there ex-ists a known constant such that

.


The control objective is to design an output feedback controllaw so that the closed-loop system is uniformly stable, and theplant output tracks as close as possible a bounded, continu-ously differentiable reference signal with measured boundedderivatives up to order .

III. B ACKSTEPPINGCONTROL: POINTWISE DESIGN

Let us assume that the plant parameters are known at eachtime and use the backstepping approach to design a controllaw that could meet the control objective if the plant parameterswere frozen in time at each point in time. We refer to this designas pointwise in time. In other words, we use the backsteppingdesign approach developed for LTI plants to a plant that is con-sidered for design purposes to be an LTI plant at each frozentime in the parameter space. Then we examine whether sucha design approach can lead to a controller that can handle theparameter time variations.

We consider the state dynamics (2.1) to construct a state es-timator. Selecting a design vectorsuch that the matrix (or the polynomial

) is Hurwitz, we can rewrite (2.1) in theLaplace domain (assuming frozen parameters at each time) as

Hence, we can use the following equation to derive a stateestimator:

(3.1)

Assuming that we have noa priori information about the statevector, we can set to zero and expand (3.1) as

Treating the plant as LTI (the plant coefficients as constant), weobtain

Denoting , (), and ( ) by , , and ,

respectively, we obtain a state estimation scheme which is verysimilar to [6], [7], and [17]

(3.2)

where

Noticing that for and ,we can generate ( ) and ( ) using thefollowing filters:

(3.3)

(3.4)

For LTI systems, is a constant vector. In our case, however,is a vector function of time. The TV nature ofdoes not affectthe form of the observer equation given by (3.2) which is thesame as that with constant. The observation error equation,however, is given by

where . It is clear that in the LTI case, whereis aconstant vector, as . Since is TV, i.e., ,

can no longer be guaranteed to go to zero in general.

Let . Then, a plant parameterization to be used for con-trol design is obtained by differentiatingand using (3.2) asfollows:

where

Introducing the notation

we can write

(3.5)

The time variations affect the plant parametric (3.5) throughthe signal that also depends on the filtered values of, .Since is not considered to be known, it can only be treatedas a modeling error. The backstepping control design based on(3.5) with is given as follows:

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

where , are positive design constants and

The control law is

ifif

(3.13)


Let us analyze the stability properties of the controller (3.13)designed for when applied to the TV plant with byconsidering the following Lyapunov function:

which has been used in the LTI case. Its derivative can be com-puted using (3.5)–(3.11) as

As we can see, if , then , and will decay tozero exponentially fast, noting that is the generic notationfor exponentially decaying to zero signals and

for any constant , which can be chosenarbitrarily small. However when , due to the presence of

which depends on and , signal boundedness is not guaran-teed let alone asymptotic tracking unless the time variationsare sufficiently small or decay to zero with time.

The above analysis demonstrates that, in the presence of plantparameter variations, we can not prove stability with the controllaw based on the backstepping approach for LTI plants usingthe quadratic Lyapunov function based analysis of [6] and [17].In the following section, we modify the backstepping controldesign to take into account the time varying nature of the plantparameters.

IV. BACKSTEPPINGCONTROL: TIME VARYING DESIGN

In this section, we use the backstepping procedure for controldesign by taking into account the fact that the plant is TV. Asbefore, we assume that the plant parameters are known at eachtime .

A. Observer Design for the Time Varying Plant

The reason that the controller (3.13) can not guarantee per-fect tracking or even global stability is due to theterm in theparameterization (3.5). The signal, which acts as a perturba-tion to the closed-loop system, is due to the time variationof the parameter vector and depends on the closed-loop sig-nals , , and is therefore not guaranteed to be vanishing oreven bounded. However,can be constructed as follows ifisknown. Consider the filter

(4.1)

and define

(4.2)

It can be easily verified that with defined in (4.2)satisfies

and, therefore, converges to the true stateas . If isnot known then in (4.2) can be generated from

(4.3)

which follows from (4.1) by applying the linear swappinglemma [21], [28]. Combining (4.3) with (4.2), and denoting

and by and , respec-tively, we obtain

(4.4)

The signals , , and can be generated using the filters

(4.5)

(4.6)

(4.7)

The number of the filters can be reduced by defining, i.e., combining (4.5) and (4.6) as follows:

(4.8)

The filter (4.7) is used for backstepping design purposes. It iseasy to verify that the estimation error satisfies

(4.9)

which indicates that , and therefore exponentially.Using (4.7), (4.8), and (4.4), the following plant parameteriza-tion is obtained:

(4.10)

Note that the observer (4.4) incorporates the TV parametersin the filter design which gives us the desired observation error(4.9). In addition, only two filters are used, hence, this observerscheme has the potential of reducing the mathematical com-plexity of the control law. However, we also note that the numberof th order filters required for observer (4.4) is three, which isone more than that in the LTI case. This is for compensating forthe time variations of the plant parameters and achieve perfecttracking.

In the following section, we apply the backstepping proce-dure to design a controller for (2.1) and (2.2) based on observer(4.4) and parameterization (4.10) that are more suitable for LTVplants.

B. Backstepping Controller Design

Let us apply the backstepping controller design steps to theLTV plant given by (4.10).

Step 1)We treat as the first virtual control. We define

(4.11)

and choose

(4.12)

Step ) In each subsequent step, we individuallytreat as the virtual control and, therefore, the associ-ated error signals and stabilizing functions are recursivelydefined as

(4.13)


(4.14)

In the final step, when we differentiate , the controlappears in the form of . Therefore, we can design the controllaw as

ifif

(4.15)

where is the th stabilizing function bearing the same defi-nition as (4.14), which completes the design.

The stability of the control law (4.15) can be established byusing the Lyapunov function

(4.16)

whose derivative is given by

(4.17)

for any constant , which can be chosen arbitrarily small.From (4.16) and (4.17), it follows that asexponentially fast. Hence, the tracking error converges tozero exponentially, and the signalsand are uniformlybounded. To establish the boundedness of, we first see that

is bounded due to the exponential stability of .Using the boundedness of, we can recursively establish that

and finally are all bounded.We summarize our results using the following theorem.

Theorem 4.1:For the LTV plant (2.1) and (2.2) with As-sumptions 1–4, the controller given by (4.15) guarantees thatthe closed-loop system is internally stable, and the tracking errorconverges to zero exponentially fast.

We note that the traditional polynomial based model refer-ence controller scheme cannot guarantee perfect tracking whenthe TV plant parameters are completely known [19], and thata different filter structure has been proposed in [22] to resolvethis problem for the model reference control case. In our case,perfect tracking is achieved using twoth order filters. We alsonotice that by using only the signalsand instead of a seriesof ’s and ’s, we have significantly reduced the mathematical

complexity of the stabilizing functions. This is another advan-tage of the new controller.

The results of Theorem 4.1 are based on the assumption thatthe TV plant parameters are known for all . In many ap-plications, this assumption may not hold. Consequently, it is ofinterest to examine the robustness of the proposed controllerwhen only some nominal (TV) values of the plant parametersare known. In the following section, we address the robustnessof the controller with respect to parametric uncertainties.

V. PARAMETRIC ROBUSTNESS OF THEPROPOSEDCONTROLLER

In the previous sections, we assumed that the plant parameterswere known exactly. A natural question one may ask is: Whatif the plant parameters are not precisely known? That is, what ifthe actual plant dynamics are described by

(5.1)

instead of (2.1), and there exist errors ,between the actual parameters, and the pa-

rameters , used in the control design? This section an-swers this question.

Due to the parameter errorsand , the observer describedin (4.2) or (4.4) is no longer a true state observer. In fact, if wesubstitute and in (4.4), we obtain

Applying the swapping lemma, we have

where

The corresponding plant parameterization is

(5.2)

A lengthy analysis based on the plant parameterization (5.2)results in the following theorem, which establishes the robust-ness properties of the controller (4.15) with respect to the para-metric uncertainties.

Theorem 5.1:Assume that the parameter errorremains small for all time in the sense that

such that .Furthermore, select the design parameters, to satisfy

(5.3)


(5.4)

and

(5.5)

for , where is a state-space realization of the equation shown at the bottom of thepage. is the positive–definite symmetric solutionto the Lyapunov equation

, and is a constant positive–definite matrixsatisfying . Then, there exists asuch that , if the derivatives of the parameter errors

satisfy

for some , , then the closed loop system (5.1),(2.2), (4.15) is uniformly stable, and the tracking error is of theorder of in the mean square sense.

Proof: The proof is long and technical, and is presented inAppendix A.

Remark 5.1:Theorem 5.1 indicates that the uncertainty inthe parameters can be counteracted by increasing the valuesof the design parameters,, , , in particular. Notethat for sufficiently small, we can find design parameters,

such that (5.3)–(5.5) hold. As for theparameters, only thederivatives of the parameter errors have to be small, not nec-essarily the parameter errors themselves. This suggests that thebackstepping controller has strong parametric robustness as op-posed to the traditional ones. A special situation is the LTI case,where the time variations of the plant parameters or parametererrors are zero. Then, we reach the same conclusion as in [6]. Inthis case, if the reference input is zero, then exponential regula-tion is achieved.

In this section, we assume that the nominal values of the TVplant parameters are known. For stability, we require that theparametric uncertainty is small in the sense that the time varia-tion (first time derivative) of the parametric uncertainty is smallin the average sense, i.e., small most of the time. In the fol-lowing section, we combine the proposed controller designedfor LTV plants with known parameters with an appropriate pa-rameter estimation scheme to deal with the case of unknownplant parameters.

VI. A DAPTIVE BACKSTEPPINGCONTROL

In the previous sections, it is assumed that the plant parame-ters are known precisely or with some small error. In this sec-tion, we consider these parameters as unknown functions oftime which satisfy Assumptions 1–4. In order to incorporate anyavailablea priori information about the modes of variation ofthe plant parameters, we use the structured parameter variationsrepresentation [21], i.e., we assume that the plant parameters

, have the following known structure:

(6.1)

where , form the decompositionof which is a matrix of known time functions,

is the unstructured parameter vector that is un-known; is a known parameter vector which canbe decomposed to . Note that the leading

rows of , are zeros. Furthermore, we assumethe following about the leading nonzero term of and the un-structured parameters.

Assumption 5:The sign of is the same as the sign offor all . Moreover, the unstructured parameter vector

is differentiable with respect to time and satisfies

, i.e., the signals are bounded and

and for some and , a ”small” scalar.Assumption 5 requires the mean square value of the time vari-

ations to be of order , where will be required to be small.Next, we exploit the TV model based filter design of Section IVto construct a state estimator for the unknown parameter case.Using (6.1), we can rewrite (4.7) and (4.8) as

Applying the linear swapping lemma, we get


where , ,, and . Hence,

constructing the filters

(6.2)

(6.3)

(6.4)

(6.5)

where , issuch that is exponentially stable with stabilitymargin , i.e., has all eigenvalues nonpositive, andconsidering the “virtual observer”

(6.6)

it is straightforward to verify that the observation errorsatisfies

When , as . Hence, (6.6) is a true stateobserver for (2.1), (2.2) when the parameter vectoris knownand constant. If is not constant, then the observation error

is nonvanishing and is represented in the following transferfunction form:

(6.7)

Using (6.7), we can obtain the following plant parameterization:

(6.8)

where

Similar to the pointwise design of Section III, the parameter-ization (6.8) appears to be in the same form as the LTI case [7],[17] except for the term in , and is suitable for applyingthe adaptive backstepping design.

Remark 6.1:Note that (6.1) covers the general case in-cluding the fully structured, unstructured, and known parametercases. If the parameters are unstructured, then we simply have

. If the parameters are fully structured, thenis constant but unknown. The case where correspondsto the known parameter case.

Remark 6.2:Even though the filters (6.2) and (6.3) appearto be of high order since both , are matrices, theactual implementation of these two filters can be of lower order,depending on the elements of , . In general, if contains

linearly independent time functions, then thefilter can beobtained using th order filters. For example, suppose

where , ’s are constant matrices,are linearly independent scalar time functions, then it suffices toimplement

where . The matrix , can be realized usingas

Moreover, if , can be linearly represented by ,, respectively, say

where , are constant vectors of length, then the filtersignals and can be obtained through , as follows:

In this case the filters are of total order . In partic-ular, when the parameters are not structured, and

, then the total filter order is , which is similarto the LTI case [6], [7], [17].

A. Certainty Equivalence Control Law

The controller design follows the same procedure as in theknown parameter case. The idea is to recursively treatas a virtual control signal, and apply the backstepping procedureusing the certainty equivalence, i.e., replacing the unknown pa-rameter vector with its on-line estimate . The design stepsare as follows.

Step 1)

(6.9)

(6.10)

Step )

(6.11)


(6.12)

(6.13)

In Step ), the control appears in the form of

, therefore the control law can bechosen as

if

if

(6.14)where is either 0 or 1, the latter corresponding to the case

where appears explicitly in the control law. Note that for thecontrol law (6.14) to exist, the adaptive law must assure that

.With the control law (6.14), the corresponding error system

is given by

(6.15)where ,

......

.... . .

. . ....

. . .. . .

. . .

B. Adaptive Law With an Auxiliary Filter

The adaptive law for generating the parameter estimates usedin the control law (6.14) is based on the idea of introducing

an auxiliary filter to counteract the effect of term in theequation, therefore ending up with a new error system that is

suitable for synthesizing an adaptive law based on a Lyapunovfunction. We define the auxiliary filter

(6.16)

and the auxiliary error signal

Then, the error signal satisfies the equation

Since is not guaranteed to be bounded, we introduce the fol-lowing normalizing signal:

where and are design constants.Some important properties of are given by the followinglemma.

Lemma 6.1:We have

(6.17)

(6.18)

(6.19)

where is a positive constant.Proof: The state (2.1) can be rewritten as

from which we obtain

(6.20)

Observing (6.2)–(6.5) and (6.20), we see that, , , , andcan be represented as outputs of stable filters with inputsand

. Hence, the result (6.17) follows immediately. Similarly, (6.7)implies that , which together with(6.17) leads to (6.18). Finally, using the inequality

, we obtain

Now, define the normalized estimation error

Then, satisfies

(6.21)

By considering (6.21), (6.19), and the following Lyapunov-likefunction:

(6.22)


the following robust adaptive law can be chosen:

(6.23)

where is the nominal value of , i.e., is expected to beclose to the constant vector, is a positive definite symmetricgain matrix, is the projection operator to project along theboundary , which is definedas

if and

otherwise

and is the leakage coefficient [28] defined as

if

if (6.24)

is a known upper bound for , and is a smallconstant.

The stability properties of the adaptive law are described bythe following lemma.

Lemma 6.2:Assume that , then theadaptive law (6.23) guarantees the following.

i) , ,.

ii) .

iii) , , , , , . In particular,

if , then andas .

Proof: The properties in i) are direct consequences of theprojection and switching -modification, see [28]. To prove ii)and iii), let us consider the Lyapunov-like function (6.22). Forsimplicity and without loss of generality, we assume .The derivative of along the solution of (6.21), (6.23) can becomputed as

(6.25)

where

Using the inequality

we obtain

(6.26)

In view of (6.18), if ,hence, (6.26) implies that . In addition, integrating(6.25), we get that

Using and , it follows that

, and, consequently, using i) and ii),

it follows that .Due to the linearity of the stabilizing functions, depends

only on the parameter estimatesand is linear in , , ,

, , thus , and follows from(6.23) and . Using (6.16), satisfies

(6.27)

from which follows immediately.

If , then and all theproperties become properties, i.e., . Fi-

nally, using (6.21) and (6.27) we see that, which together with implies that

as .Having established the stability properties of the adaptive law,

we analyze the closed-loop stability properties of the adaptivecontrol scheme based on the error system (6.15) next. The fol-lowing theorem summarizes the results of this analysis, whichis presented in Appendix B in details.

Theorem 6.1:The adaptive controller described by (6.14)and (6.23), when applied to the LTV plant (2.1),(2.2), guaranteesthe existence of a constant such that , allthe closed-loop signals are uniformly bounded, and the trackingerror is of the order in the mean square sense, i.e.,

(6.28)

where and are finite positive constants. Moreover,canbe expressed as

where and is a finite positive constant indepen-dent of , , .

Proof: The proof is similar to that of Theorem 5.1 and ispresented in Appendix B.

Theorem 6.1 indicates that, using the adaptive controller(6.14) and (6.23), only the time variations of the unstructuredplant parameters are required to be small in the mean squaresense to guarantee closed-loop stability and tracking with small


MSE. The overall system is not necessarily restricted to beslowly TV. The requirement about the time variations of theunstructured plant parameters is necessary since it is not pos-sible to estimate unknown and arbitrarily fast TV parametersusing a general adaptive law with finite speed of adaptation[22], [28]. Once this requirement is satisfied, the mean squaretracking error is guaranteed to be of the order of the speed ofthe unstructured plant parameter variations.

Besides establishing stability and tracking properties, thetheorem provides guidelines for performance improvement aswell. It shows that theMSEperformance can be improved byamplifying , , and possibly for small enough to satisfythe stability conditions. Unlike [12], arbitrary performanceimprovement is only assured in terms of the normalizedtracking error . The bound on dependson . However, although might increase byincreasing , , , this can be counteracted by reducingthe normalization coefficient . Hence, the performance of theadaptive backstepping controller can be improved by adjustingthe design parameters, , , and . We demonstratethis fact via simulations in Section VII.

C. Fully Structured Parameter Variations

The case of fully structured parameter variations correspondsto being constant. We generalize it to the situation that

. For this special class of LTVplant, the proposed adaptive controller (6.23) and (6.14) hasthe following properties.

Corollary 6.1: If the speed of parameter variations satisfy

, then the adaptive controller(6.14) and (6.23) guarantees that all the closed-loop signals areuniformly bounded, and the tracking error converges to zeroasymptotically.

Proof: This is a direct consequence of Theorem 6.1 andLemma 6.2 iii).

Due to the transformation (6.1), the parameter vectormaynot reflect the plant parameters themselves, and can containmore or less than elements, which corresponds tothe overparameterized and the underparameterized case, respec-tively. Corollary 6.1 indicates that when full knowledge of theparameter variations is available, then regardless of the speed ofthe parameter variations of the plant, global stability is guaran-teed, and asymptotic tracking is achieved.

In addition, in the case of fully structured parameter varia-tions, is exponentially vanishing. Therefore, in this case, wecan apply the tuning design given in [6], [17] instead of the cer-tainty equivalence approach using parameterization (6.8). Theadvantage is a guaranteed performance improvement, as in theTI case [7], [12].

VII. SIMULATION RESULTS

Let us consider a simple unstable second-order LTV plantwhose state-space representation is

(7.1)

(7.2)

(a)

(b)

Fig. 1. Response using the pointwise design and exact knowledge ofg , g ,g . (a)c = d = c = d = 1. (b) c = c = 5, d = d = 1.

where , , and . Itis required to design a controller so that the outputtracks thereference signal .

Let us first apply the pointwise design, i.e., the controlscheme (3.6)–(3.13) together with the estimation filters (3.3)and (3.4), assuming that , , are all known exactly. Notingthat , , for the plant, the filter parameters arechosen as , and the design parameters are chosenas . Fig. 1(a) shows the result. Althoughthe output signal is bounded, tracking performance is not thatsuccessful. Next, we increase the values of the design constants

and to 5. Tracking performance is enhanced as shown inFig. 1(b). However, asymptotic tracking is not achieved.

Then, we repeat the same simulations with the LTV designi.e., the control scheme (4.11)–(4.15) together with the estima-tion filters (4.5)–(4.7). Tracking is perfect as shown in Fig. 2.Next, we consider some parametric uncertainty. We assume thatour plant model is a little bit erroneous, e.g., models of the ac-tual functions , , of the plant are ,

. Choosing the design parameters as ,, we can see from Fig. 3 that the system is

stabilized, and a relatively small tracking error (smaller thanthat of controller (3.13) with known parameters) is obtained.


(a)

(b)

Fig. 2. Response using the LTV design and exact knowledge ofg , g , g .(a) c = d = c = d = 1. (b) c = c = 5, d = d = 1.

Fig. 3. Response using the LTV design in presence of parametric uncertainty(g (t) = 1, g (t) = g (t) = 2, c = c = 5, d = d = 1).

Note that the parametric uncertainty has amplitude 1, however,its derivative has a much smaller amplitude 0.1. This demon-strates that the time variation of the uncertain parameter, notthe size of uncertainty itself, determines the system stability andperformance.

(a)

(b)

Fig. 4. Adaptive control with different choices forc , c , � (d = d = 1,� = 2, � = 1). (a) Tracking. (b) Parameter estimation.

Finally, we consider the unknown parameter case assumingthat the plant structure (7.1)–(7.2) is known but the functions,

, are unknown. In order to build up an adaptive controller,we first write the plant parameter in structured parameter varia-tions form as follows:

where

Noting that and hence are zero, the estimation filtersare implemented using (6.2), (6.4), and (6.5). The following


(a)

(b)

Fig. 5. (a) Response using the adaptive controller (c = c = 5, d = d =

1, � = 10, � = 2, � = 1). (b) Adaptive tracking with different choices for� (c = c = 5, d = d = 1, � = 10, � = 2).

control law is designed based on the steps in Section VI se-lecting :

The adaptive law and the associated auxiliary signal are de-fined as

where , , , , , , are design constants, is theswitching- coefficient defined in (6.24).

Fig. 4 shows the tracking error and the parameterestimate for simulations with different choices of the designparameters and the adaptive gainparameters. The switchingand normalization parameters are set as , ,

, in all of these simulations. The response for, , is redrawn in Fig. 5(a)

in order to make it comparable with the results for the caseswith known and unknown parameters. As can be seen in thesefigures, the system is stabilized, and the tracking error remainsin a neighborhood of 0, for all design parameter choices.

As can be seen in Fig. 4, increasing the value of the designparameters improves the tracking performance as in theknown parameter case. By increasing the adaptive gain, not onlyparameter estimation gets faster but the tracking performance isimproved further as well.

Later, fixing , , , ,, , the effect of the normalization coefficient

is tested. The results in terms of the tracking errors are shownin Fig. 5(b). As seen in this figure decreasinghas a similareffect with increasing on enhancement of tracking.

In the aforementioned simulations, we see that the param-eter estimates adapt to the parameter changes. We have alsoobserved that the control effort remains within a reasonablebound. Since the only unknown TV parameter is slowlytime varying, stability is guaranteed.

VIII. C ONCLUSION

In this paper, we introduced a new backstepping controllerfor LTV systems with known and unknown parameters. Thecontroller guarantees exponential tracking when the plant pa-rameters are known exactly. When the plant parameters are notknown exactly but their time variations are small enough, re-gardless of the size of the parameter errors (except for the highfrequency gain), global stability can be guaranteed by choosingcertain design parameters properly. Hence, the proposed con-troller has strong parametric robustness properties which mostof the traditional model reference controllers do not have.

When the plant parameters are unknown, the proposed con-troller is combined with an online parameter estimator to form anew adaptive controller. This new adaptive controller guaranteesthe following. All the closed-loop signals are globally uniformlybounded. The tracking error remains small and of the order ofthe speed of the unstructured plant parameter variations, whichis required to be small in the mean square sense. If the plantparameter variations are fully structured, the tracking error con-verges asymptotically to zero. The performance bounds for thetracking error developed can be used to select certain designparameters for performance improvement. The expected perfor-mance of the proposed controller and the effects of design pa-rameter selections on the transient performance are illustratedby simulation results.

The proposed controller is suitable for use in many applica-tions where the plant parameters are time varying. An exampleof such application is the control of aerospace systems where theparameters of the system vary with time and/or flight conditions


[29]–[31]. Application of the proposed backstepping scheme toflight control of high performance aircrafts and hypersonic airbreathing vehicles is currently under investigation.

APPENDIX IPROOF OFTHEOREM 5.1

Using (5.2), (4.11), and (4.14), we obtain the following errorequation:

(A.1)

where

......

.... . .

. .....

. . .. . .

. ..

We first consider the term . We have

(A.2)

Let the th order monic polynomial and the th ordermonic polynomial be a decomposition of , i.e.,

. Then, the operator in of (A.2) canbe written as

which is the sum of two proper and exponentially stable I/Ooperators. Hence, the operator in of (A.2), which we denoteas , is a proper and exponentially stable I/O operator.Using (A.2), (4.11), and the definition of , we can write

where . Suppose thatis a state-space realization of

, then

(A.3)

(A.4)

Note that and are independent of the designparameters . Since is an exponentially stable matrix,there exists a positive-definite matrix such that

Define , then we substitute (A.4) into theerror equation of and get

(A.5)

Next, we consider the term , which we write in the followingstate-space form:

(A.6)

Augmenting (A.6) with (3.3), we get

(A.7)

Finally, we augment the error systems (A.1) and (A.3) with(A.7) using (A.5) and get

......

(A.8)

where

......

......

. . ....

. . .. . .

.... . .

. . .. . .

.... . .

. . .. . .

.... . .

. . .. . .


We first analyze the homogeneous part of (A.8) by individu-ally considering the following two partial Lyapunov functions:

where are constants to be chosen andis a constant matrix satisfying . The

derivatives of , along the solution of (A.8) are computedas

Using the inequalities

we get

(A.9)

(A.10)

Let us choose

where is an arbitrary constant, and consider theLyapunov function

where is another constant to be chosen. Using (A.9) and(A.10), we have the following:

We first pick

where , are arbitrary constants. With these choices,if

where are arbitrary constants, then

where . Since , , , ,are arbitrary, the existence of , , is guaranteed pro-

vided that (5.3)–(5.5) are satisfied. Hence, if, , and for


are chosen to satisfy (5.3)–(5.5), then the homoge-neous part of (A.8) is exponentially stable. Now, let us suppose

, , and for are chosen as such, and go back to(A.8). Since , the corresponding output is alsobounded. Therefore it suffices to consider the subsystem where

, , are all zero.Next, we define a fictitious normalizing signal

(A.11)

where is a constant suchthat , are exponentiallystable. The normalizing property of can be described bythe following lemma.

Lemma 8.1:Regardless of the boundedness of any closed-loop signal, we have

(A.12)

for all and some .Proof: Using the inequality

(A.13)

which follows directly from the definition of given by(A.11), we have that for all for whichis exponentially stable

(A.14)

where is the impulse response of and. Note that is finite since is

exponentially stable. Using Holder’s inequality, we have thatand

(A.15)

The result follows directly from (A.14) and (A.15).Now, define the following normalized errors:

and consider the Lyapunov function

where , , are chosen as before. Using (A.13) and conti-nuity of

(A.16)Using (A.8), (A.16), and the fact that

for an arbitrary function , the derivative of can be computedas

Therefore, , all the normalized signals arebounded and small in the order ofin the mean square sense. Inparticular, . On the other hand, using (3.6)–(3.12) and(4.11)–(4.14), we can represent the control law (3.13) or (4.15)in the form

(A.17)

for some bounded functions , , , .Substituting

in (A.17), we get

where , is a Hurwitz polynomial oforder , ,

. Hence, applying [30, Th.(22), p. 113], we obtain

where and are impulse responses ofand , respectively. Since

and are both exponentiallystable and , , , are bounded smooth functions of

which is a bounded function of time, the supremum termsmentioned before are all finite. Hence, using , we get


or

where is some constant. Applying the Bellman–Gronwall Lemma, we have

Now, if

where , then and .Once is bounded, is bounded. Then, we apply the same

argument as before, and conclude that all the closed-loop signalsare uniformly bounded, and the closed-loop system is internallystable. In addition, all the error signals satisfy

. That is, all the error signals, including the tracking errorare of the order of in the mean square sense, i.e.,

for any and some constants independent of.

APPENDIX IIPROOF OFTHEOREM 6.1

Following equations (6.9)–(6.13), it can be easily shown thatthe control law (6.14) can be represented as

(B.1)

for some continuous functions , , , . On the otherhand, from (2.3), we get

(B.2)

Substituting (B.2) into (B.1), we obtain

where so that is some Hurwitzpolynomial of order , ,and

. Let us define a fictitious normalizingsignal

where is a constant such thatis an exponentially stable operator. Applying the arguments wehave used in the proof of Theorem 5.1 again, we derive that

where is some finite positive constant. Since

and

we have

where , i.e., suchthat

Applying the Bellman–Gronwall lemma, we obtain

Let , then , we have .Since bounds which bounds all the closed-loop signals,it follows that all signals are uniformly bounded. In addition, thetracking error satisfies .

In order to get some quantitative results, we first derive someperformance bounds for the estimation error. We start by cal-culating the bound of the Lyapunov function. Noting thatis assumed to be zero for simplicity and without loss of gener-ality, (6.26) yields

(B.3)From (B.3), we notice that is a function which decreaseswith increasing , , . Therefore, the bound on ,is decreasing with the increase of, , . Using the fact

that , and , weintegrate (6.25) and get

(B.4)

(B.5)

where is some positive constant independent of.


Next, we derive some performance bounds for the auxiliarysignal . Considering the quadratic function

it follows directly from (6.27) and the definition of that

Therefore

Since , , , depend only on , , , , and is non-increasing with the increase of , using (6.23) and(B.5), we obtain

(B.6)

where is some positive constant independent of, , . Fi-nally, combining (B.4) and (B.6) we obtain (6.28).

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Youping Zhang (M’02) received the B.S. degreefrom the University of Science and Technology ofChina, Hefei, Anhui, R.O.C. in 1992, and the M.S.and Ph.D. degrees in electrical engineering from theUniversity of Southern California, Los Angeles, CAin 1994 and 1996, respectively.

From 1996 to 1999, he was a Research Engineerwith the United Technologies Research Center, EastHartford, CT. He joined Numerical Technologies,Inc., San Jose, CA, in July 1999 and stayed fornearly four years until it was acquired by Synopsys

in February 2003. At Numerical Technologies, he was in several differentpositions including software product development, technology research, andtechnical marketing for resolution enhancement technologies. He is currentlya Technical Marketing Manager for Mask Synthesis at Synopsys. His researchinterests are in the areas of optimizations, numerical methods, computer aideddesigns, and intelligent systems.


Barıs Fidan (S’02) received the B.S. degrees inelectrical engineering and mathematics from MiddleEast Technical University, Ankara, Turkey in 1996,and the M.S. degree in electrical engineering fromBilkent University, Ankara, Turkey in 1998. He iscurrently working on the Ph.D. degree in ElectricalEngineering-Systems at the University of SouthernCalifornia, Los Angeles.

His research interests include adaptive andnonlinear control, switching and hybrid systems,robotics, high performance and hypersonic flight

control, semiconductor manufacturing process control, and disk-drive servosystems.

Petros A. Ioannou (S’80–M’83–SM’89–F’94)received the B.Sc. degree (first class honors) fromUniversity College, London, U.K., and the M.S.and Ph.D. degrees from the University of Illinois,Urbana, in 1978, 1980, and 1982, respectively.

In 1982, he joined the Department of ElectricalEngineering-Systems, University of SouthernCalifornia, Los Angeles, California, where he iscurrently a Professor and the Director of the Centerof Advanced Transportation Technologies. Hisresearch interests are in the areas of adaptive control,

neural networks, nonlinear systems, vehicle dynamics and control, intelligenttransportation systems, and marine transportation. He was a Visiting Professorat the University of Newcastle, NSW, Australia and the Australian NationalUniversity, Canberra, in fall 1988, the Technical University of Crete in summer1992 and fall 2001, and served as the Dean of the School of Pure and AppliedScience at the University of Cyprus in 1995. He is the author/coauthor of fivebooks and over 150 research papers in the area of controls, neural networks,nonlinear dynamical systems, and intelligent transportation systems.

Dr. Ioannou was a recipient of the Outstanding Transactions Paper Award in1984, and the recipient of a 1985 Presidential Young Investigator Award.

backstepping control of linear time-varying systems with known

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