then also the feedback system

x(t) = Ax(t) B[Cx(t), t] (4)ABSOLUTE STABILITY is exponentially stable.

Analysis of dynamical systems and circuits is mostly done un- The name of this result comes from its graphical interpre-der the assumption that the system is linear and time-invari- tation. The Nyquist plot, that is, the plot of G(i) in the com-ant. Powerful mathematical techniques are then available for plex plane as R, must not cross or circumscribe the circleanalysis of stability and performance of the system, for exam- centered on the real axis and passing through 1/ and 1/ple, superposition and frequency domain analysis. In fact, (Fig. 3).even if the system is nonlinear and time-varying, such as- An important aspect of the circle criterion is that it demon-sumptions can often be used to get a first estimate of the sys- strates how frequency domain properties can be used in atem properties. nonlinear setting. It is instructive to compare with the Ny-

The purpose of absolute stability theory is to carry the quist criterion, which states that the closed-loop system withanalysis one step further and get a bound on the possible in- linear feedback w(t) kv(t) is stable for all k [, ], pro-fluence of the nonlinear or time-varying components. The ap- vided that G(i) does not intersect the real axis outside theproach was suggested by the Russian mathematician Lure in interval [1/ , 1/]. The circle criterion replaces the inter-the 1940s and has, since then, developed into a cornerstone val condition with a circular disk. As a consequence, the sta-of nonlinear systems theory. bility assertion is extended from constant feedback to nonlin-

The basic setup is illustrated in Fig. 1, where the linear ear and time-varying feedback.time-invariant part is represented by the transfer function The proof of the circle criterion is based on a quadraticG(s) and the nonlinear parts of the system are represented by Lyapunov function of the forma feedback loop w (v, t). The analysis of the system isbased on conic bounds on the nonlinearity (Fig. 2): V (x) = xPx

where the matrix P is positive definite. It can be verified that (v, t)/v for all v = 0 (1)V(x) is decreasing along all possible trajectories of the system,provided that the frequency condition [Eq. (3)] holds. As aThis problem was studied in Russia during the 1950s. Inconsequence, the state must approach zero, regardless of theparticular, a conjecture by Aiserman was discussed, hopinginitial conditions.that the system would be stable for all continuous functions

in the cone [Eq. (1)], if and only if it was stable for all linearfunctions in the cone. This conjecture was finally proved to THE POPOV CRITERIONbe false, and it was not until the early 1960s that a majorbreakthrough was achieved by Popov (1). In the case that has no explicit time dependence, the circle

criterion can be improved. For simplicity, let 0 and hence

THE CIRCLE CRITERION 0 (v)/v for all v = 0 (5)

The Popov criterion can then be stated as follows.Popov and his colleagues made their problem statements interms of differential equations. The linear part then has the

Theorem 2 (Popov criterion). Suppose that : R R isformLipschitz continuous and satisfies Eq. (5). Suppose the sys-tem x Ax is exponentially stable and let G(i) C(iI A)1B. If there exists R such that

{x = Ax Bwv = Cx

(2)

and the corresponding transfer function isRe[(1 + i)G(i)] > 1

R (6)

then the systemG(s) = C(sI A)1Bx(t) = Ax(t) B[Cx(t)] (7)

Theorem 1 (Circle criterion). Suppose that the systemx Ax is exponentially stable and that : R2 R is Lipschitz is exponentially stable.continuous and satisfies Eq. (1). If

Note that the circle criterion is recovered with 0. Also thePopov criterion can be illustrated graphically. Introduce thePopov plot, where ImG(i) is plotted versus ReG(i). Then

0 < ReG(i) + 1G(i) + 1 R (3)

1J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

2 ABSOLUTE STABILITY

w

(, t)

G(s)

v

Figure 1. Absolute stability deals with linear time-invariant systemsin interconnection with nonlinear functions.

stability can be concluded from the Popov criterion if and onlyif there exists a straight line separating the plot from thepoint 1/. The slope of the line corresponds to the parame-ter . See the following example.

15

10

5

0

5

10

15

10 0Re

(a)

10

Im

1/1/

G(i )

Example. To apply the Popov criterion to the system

x1 = 5x1 4x2 + (x1)x2 = x1 2x2 + 21(x1)

let

G(s) = C(i A)1B = s + 82s2 + 7s + 6

The plot in Fig. 4 then shows that the Popov criterion is satis-fied for 1.

The theory of absolute stability had a strong developmentin the 1960s, and various improvements to the circle and Po-pov criteria were generated, for example, by Yakubovich.Many types of nonlinearities were considered and stronger

15

10

5

0

5

10

15

10 0Re

(b)

10

Im1/

G(i )

1/

criteria were obtained in several special cases (24). Impor-Figure 3. The circle criterion proves stability as long as the Nyquisttant aspects of the theory were summarized by Willems (5),plot does not cross or circumscribe the circle corresponding to theusing the notions of dissipativity and storage function.conic bounds on the nonlinearity. (a) 0 . (b) 0 .

GAIN AND PASSIVITY

In the results described so far, the results were stated interms of differential equations. A parallel theory was devel-oped by Zames (6) avoiding the state-space structure andstudying stability purely in terms of inputoutput rela-tions.

Im G

(i

)

12

10

8

6

4

2

0

2

15105Re G(i )

1/+

05

w= (v)

v

w

v

v

Figure 4. The Popov criterion can be applied when there exists astraight line separating the Popov plot from the point 1/.Figure 2. The nonlinearity is bounded by linear functions.

ABSOLUTE STABILITY 3

For this purpose, a dynamical system is viewed as a map The only property of saturation that would be exploited bythe Popov criterion is that 0 sat(v)/v 1 and, consequently,F from the input u to the output Fu. The map F is said to be

bounded if there exists C 0 such that

0 T

0w(t)[v(t) w(t)] dt T

0|Fu|2 dt C2

T0

|u|2 dtfor all w sat(v), T 0. However, the inequality will remainvalid even if some perturbation of amplitude smaller than onefor all T 0. The gain of F is denoted F and defined as theis added to the factor w in the product w(v w). One way tominimal such constant C. The map is said to be causal if twodo this is to introduce a function h(t) with the property

inputs that are identical until time T will generate outputsh(t)dt 1 and replace the previous expression by (w h that are also identical until time T.w) (v w), where h w is a convolution. The integral inequal-This makes it possible to state the following well-knownity then becomesresult:

Theorem 3 (Small gain theorem). Suppose that the input 0 T

0(w + h w)(v w) dt (9)

output maps F and G are bounded and casual. If

Using this inequality, the Popov criterion [Eq. (6)] can be re-F G < 1 (8)placed by the condition

then the feedback equations Re[(1 + i + H(i))(G(i) + 1)]] > 0 w R (10)

where H(i) e

i h(t)dt. The factor 1 i H(i) iscalled multiplier.

v = Gw + fw = Fv + e

The theory and applications of absolute stability have re-define a bounded causal map from the inputs (e, f ) to the out- cently had a revival since new computer algorithms make itputs (v, w). possible to optimize multipliers numerically and to address

applications of much higher complexity than previously. Theinequality [Eq. (9)] is a special case of what is called an inte-It is worthwhile to make a comparison with the circle crite-gral quadratic constraint, IQC. Such constraints have beenrion. Consider the case when . Then the condition [Eq.verified for a large number of different model components(3)] becomessuch as relays, various forms of hysteresis, time delays, timevariations, and rate limiters. In principle, all such constraints|G(i)| < 1 [0,]can be used computationally to improve the accuracy in sta-bility and performance analysis. A unifying theory for thisThis is the same as Eq. (8), since is the gain of andpurpose has been developed by Megretski and Rantzer (8),maxG(i) is the gain of the linear part [Eq. (2)].while several other authors have contributed with new IQCsAnother important notion closely related to gain is passiv-that are ready to be included in a computer library for sys-ity. The inputoutput map F is said to be passive iftem analysis.

BIBLIOGRAPHY0

T0

u(t)y(t) dt for all T and all y = Fu

1. V. M. Popov, Absolute stability of nonlinear systems of automaticFor example, if the input is a voltage and the output is acontrol, Autom. Remote Control, 22: 857875, 1962. (Original incurrent, then passivity property means that the system onlyRussian, August, 1961.)can consume electrical power, not produce it.

2. V. A. Yakubovich, Absolute stability of nonlinear control systemsStability criteria can also be stated in terms of passivity.in critical cases, parts 13, Avtomaika i Telemechanika, 24 (3):For example, the circle criterion can be interpreted this way,293302; 24 (6): 717731, 1963; 25 (25): 601612, 1964. (Englishif 0 and is large.translation in Autom. Remote Control.)

3. V. A. Yakubovich, On an abstract theory of absolute stability ofnonlinear systems, Vestnik Leningrad Univ. Math., 10: 341361,MULTIPLIERS AND INTEGRAL QUADRATIC CONSTRAINTS1982. (Original in Russian, 1977.)

4. S. Lefschetz, Stability of Nonlinear Control Systems, New York:Less conservative stability criteria can often be obtained byAcademic Press, 1963.exploiting more information about the nonlinearity. One way

5. J. C. Willems, Dissipative dynamical systems, part 1, General the-to do this is to introduce so-called multipliers. Consider, forory; part 2, Linear systems with quadratic supply rates, Arch. Ra-example, a system with a saturation nonlinearity:tional Mech. Anal., 45 (5): 321393, 1972.

6. G. Zames, On the inputoutput stability of nonlinear time-varyingfeedback systemspart 1, Conditions derived using concepts ofloop gain; part 2, Conditions involving circles in the frequencyplane and sector nonlinearities, IEEE Trans. Autom. Control, 11:228238, 1966.

x = Ax Bsat(Cx) =

Ax + B if Cx 1Ax + BCx if |Cx| < 1Ax B if Cx 1

4 ABSTRACT DATA TYPES

7. G. Zames and P. L. Falb, Stability conditions for systems withmonotone and slope-restricted nonlinearities, SIAM J. Control, 6(1): 89108, 1968.

8. A. Megretski and A. Rantzer, System analysis via Integral Qua-dratic Constraints, IEEE Trans. Autom. Control, 47: 819830,1997.

ANDERS RANTZERLund Institute of Technology

ABSORBER. See ELECTROMAGNETIC FERRITE TILE ABSORBER.ABSORPTION MODULATION. See ELECTROAB-

SORPTION.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics EngineeringCopyright c 1999 John Wiley & Sons, Inc.

ADAPTIVE CONTROL

According to Websters dictionary, to adapt means to change (oneself) so that ones behavior will conform tonew or changed circumstances. The words adaptive systems and adaptive control were used as early as1950 (1). The design of autopilots for high-performance aircraft was one of the primary motivations for activeresearch on adaptive control in the early 1950s. Aircraft operate over a wide range of speeds and altitudes,and their dynamics are nonlinear and conceptually time-varying. For a given operating point, specified by theaircraft speed (Mach number) and altitude, the complex aircraft dynamics can be approximated by a linearmodel of the form (2)

where Ai, Bi, Ci, and Di are functions of the operating point i. As the aircraft goes through different flightconditions, the operating point changes, leading to different values for Ai, Bi, Ci, and Di. Because the outputresponse y(t) carries information about the state x as well as the parameters, one may argue that in principle,a sophisticated feedback controller should be able to learn about parameter changes by processing y(t) anduse the appropriate gains to accommodate them. This argument led to a feedback control structure on whichadaptive control is based. The controller structure consists of a feedback loop and a controller with adjustablegains as shown in Fig. 1. The way the controller parameters are adjusted on line leads to different adaptivecontrol schemes.

Gain SchedulingThe approach of gain scheduling is illustrated in Fig. 2. The gain scheduler consists of a lookup table

and the appropriate logic for detecting the operating point and choosing the corresponding value of the controlparameter vector . For example, let us consider the aircraft model in Eq. (1) where for each operating point i,i = 1, 2, . . ., N, the parameters Ai, Bi, Ci, and Di are known. For each operating point i, a feedback controllerwith constant gains, say i, is designed to meet the performance requirements for the corresponding linearmodel. This leads to a controller, say C(), with a set of gains {1, 2, . . ., i, . . ., N} covering N operating points.Once the operating point, say i, is detected, the controller gains can be changed to the appropriate value of iobtained from the precomputed gain set. Transitions between different operating points that lead to significantparameter changes may be handled by interpolation or by increasing the number of operating points. The twoelements that are essential in implementing this approach are a lookup table to store the values of i and theplant auxiliary measurements that correlate well with changes in the operating points.

Direct and Indirect Adaptive ControlA wide class of adaptive controllers is formed by combining an on-line parameter estimator, which provides

estimates of unknown parameters at each time instant, with a control law that is motivated from the known-parameter case. The way the parameter estimator, also referred to as the adaptive law, is combined with thecontrol law gives rise to two different approaches. In the first approach, referred to as indirect adaptive control

1

2 ADAPTIVE CONTROL

Fig. 1. Controller structure with adjustable controller gains.

Fig. 2. Gain scheduling.

(shown in Fig. 3), the plant parameters are estimated online and used to calculate the controller parameters.This approach has also been referred to as explicit adaptive control, because the design is based on an explicitplant model.

In the second approach, referred to as direct adaptive control (shown in Fig. 4), the plant model isparametrized in terms of the controller parameters, which are estimated directly without intermediate cal-culations involving plant parameter estimates. This approach has also been referred to as implicit adaptivecontrol, because the design is based on the estimation of an implicit plant model.

The principle behind the design of direct and indirect adaptive control shown in Figs. 3 and 4 is conceptu-ally simple. The design of C(c) treats the estimates c(t) (in the case of direct adaptive control) or the estimates(t) (in the case of indirect adaptive control) as if they were the true parameters. This design approach iscalled certainty equivalence and can be used to generate a wide class of adaptive control schemes by combiningdifferent on-line parameter estimators with different control laws.

The idea behind the certainty equivalence approach is that as the parameter estimates c(t) and (t)converge to the true ones c and , respectively, the performance of the adaptive controller C(c) tends to thatachieved by C(c) in the case of known parameters.

Model Reference Adaptive ControlModel reference adaptive control (MRAC) is derived from the model-following problem or model reference

control (MRC) problem. The structure of an MRC scheme for a LTI, single-input single-output (SISO) plant isshown in Fig. 5. The transfer function Wm(s) of the reference model is designed so that for a given referenceinput signal r(t) the output ym(t) of the reference model represents the desired response the plant output y(t)should follow. The feedback controller, denoted by C(c), is designed so that all signals are bounded and theclosed-loop plant transfer function from r to y is equal to Wm(s). This transfer function matching guaranteesthat for any given reference input r(t), the tracking error e1 y ym, which represents the deviation of the

ADAPTIVE CONTROL 3

Fig. 3. Indirect adaptive control.

Fig. 4. Direct adaptive control.

plant output from the desired trajectory ym, converges to zero with time. The transfer function matching isachieved by canceling the zeros of the plant transfer function G(s) and replacing them with those of Wm(s)through the use of the feedback controller C(c). The cancellation of the plant zeros puts a restriction on theplant to be minimum-phase, that is, have stable zeros. If any plant zero is unstable, its cancellation may easilylead to unbounded signals.

The design of C(c) requires the knowledge of the coefficients of the plant transfer G(s). When isunknown we use the certainty equivalence approach to replace the unknown c in the control law with itsestimate c(t) obtained using the direct or the indirect approach. The resulting control schemes are knownas MRAC and can be classified as indirect MRAC of the structure shown in Fig. 3 and direct MRAC of thestructure shown in Fig. 4.

4 ADAPTIVE CONTROL

Fig. 5. Model reference control.

Adaptive Pole Placement ControlAdaptive pole placement control (APPC) is derived from the pole placement control (PPC) and regulation

problems used in the case of LTI plants with known parameters. In PPC, the performance requirements aretranslated into desired locations of the poles of the closed-loop plant. A feedback control law is then developedthat places the poles of the closed-loop plant at the desired locations. The structure of the controller C(c) andthe parameter vector c are chosen so that the poles of the closed-loop plant transfer function from r to y areequal to the desired ones. The vector c is usually calculated using an algebraic equation of the form

where is a vector with the coefficients of the plant transfer function G(s).As in the case of MRC, we can deal with the unknown-parameter case by using the certainty equivalence

approach to replace the unknown vector c with its estimate c(t). The resulting scheme is referred to asadaptive pole placement control (APPC). If c(t) is updated directly using an on-line parameter estimator, thescheme is referred to as direct APPC. If c(t) is calculated using the equation

where (t) is the estimate of generated by an on-line estimator, the scheme is referred to as indirect APPC.The structure of direct and indirect APPC is the same as that shown in Figs. 3 and 4, respectively, for thegeneral case. The design of APPC schemes is very flexible with respect to the choice of the form of the controllerC(c) and of the on-line parameter estimator.

Design of On-Line Parameter EstimatorsAs we mentioned in the previous sections, an adaptive controller may be considered as a combination of

an on-line parameter estimator with a control law that is derived from the known-parameter case. The way thiscombination occurs and the type of estimator and control law used give rise to a wide class of different adaptivecontrollers with different properties. In the literature of adaptive control the on-line parameter estimator hasoften been referred to as the adaptive law, update law, or adjustment mechanism.

Some of the basic methods used to design adaptive laws are

(1) Sensitivity methods(2) Positivity and Lyapunov design(3) Gradient method and least-squares methods based on estimation error cost criteria

The sensitivity method is one of the oldest methods used in the design of adaptive laws and is brieflyexplained below.

ADAPTIVE CONTROL 5

Sensitivity Method. This method became very popular in the 1960s (3), and it is still used in manyindustrial applications for controlling plants with uncertainties. In adaptive control, the sensitivity method isused to design the adaptive law so that the estimated parameters are adjusted in a direction that minimizes acertain performance function. The adaptive law is driven by the partial derivative of the performance functionwith respect to the estimated parameters multiplied by an error signal that characterizes the mismatch betweenthe actual and desired behavior. This derivative is called the sensitivity function, and if it can be generatedonline, then the adaptive law is implementable. In most earlier formulations of adaptive control, the sensitivityfunction cannot be generated online, and this constitutes one of the main drawbacks of the method. The useof approximate sensitivity functions that are implementable leads to adaptive control schemes whose stabilityproperties are either weak or cannot be established.

Positivity and Lyapunov Design. This method of developing adaptive laws is based on the directmethod of Lyapunov and its relationship to positive real functions. In this approach, the problem of designingan adaptive law is formulated as a stability problem where the differential equation of the adaptive law ischosen so that certain stability conditions based on Lyapunov theory are satisfied. The adaptive law developedis very similar to that based on the sensitivity method. The only difference is that the sensitivity functions inthe approach are replaced with ones that can be generated online. In addition, the Lyapunov-based adaptivecontrol schemes have none of the drawbacks of the MIT rule-based schemes.

The design of adaptive laws using Lyapunovs direct method was suggested by Grayson (4) and Parks(5) in the early 1960s. The method was subsequently advanced and generalized to a wider class of plants byPhillipson (6), Monopoli (7), and others (8,9,10,11).

Gradient and Least-Squares Methods Based on Estimation Error Cost Criteria. The main draw-back of the sensitivity methods used in the 1960s is that the minimization of the performance cost function ledto sensitivity functions that are not implementable. One way to avoid this drawback is to choose a cost functioncriterion that leads to sensitivity functions that are available for measurement. A class of such cost criteria isbased on an error, referred to as the estimation error (12), that provides a measure of the discrepancy betweenthe estimated and actual parameters. The relationship of the estimation error with the estimated parametersis chosen so that the cost function is convex, and its gradient with respect to the estimated parameters is im-plementable. Several different cost criteria may be used, and methods such as the gradient and least-squaresmay be adopted to generate the appropriate sensitivity functions.

On-Line Parameter Estimation

The purpose of this section is to present the design and analysis of a wide class of schemes that can be used foron-line parameter estimation. The essential idea behind on-line estimation is the comparison of the observedsystem response y(t) with the output of a parametrized model y(,t) whose structure is the same as that of theplant model. The parameter vector (t) is adjusted continuously so that y(,t) approaches y(t) as t increases. Theon-line estimation procedure involves three steps: In the first step, an appropriate parametrization of the plantmodel is selected. The second step involves the selection of the adjustment law, referred to as the adaptive law,for generating or updating (t). The third and final step is the design of the plant input so that the propertiesof the adaptive law imply that (t) approaches the unknown plant parameter vector as t .

We start by considering the SISO plant

6 ADAPTIVE CONTROL

where x Rn and only y, u are available for measurement. Note that the plant equation can also be written asan nth-order differential equation

The constants ai, bi are the plant parameters. We can express yn as

where

We can avoid the use of differentiators by filtering with our nth-order stable filter 1/(s) to obtain

where

In a similar way we can express the plant dynamics as follows:

where W(s) is an appropriate proper stable transfer function and , are defined similarly to , .In Eqs. (5), (6) the unknown vectors , appear linearly in equations where all the other terms are

known a priori or can be measured online. We use these parametric models to estimate the unknown vectors or by using the following approaches.

SPRLyapunov Design Approach. We start by rewriting Eq. (6) as follows (for simplicity we dropthe subscript ):

where = L 1(s)and L(s) is chosen so that L 1(s) is a proper transfer function and W(s)L(s) is a proper strictlypositive real (SPR) transfer function. Let (t) denote the estimate of at time t. Then the estimated value of z

ADAPTIVE CONTROL 7

based on (t) is given by

and the estimation error is given by

Let

denote the normalized estimation error, where n2s is the normalizing signal, which we design to satisfy

Typical choices for ns that satisfy this condition are n2s = TP for any P = PT > 0, and the like. When L ,the condition is satisfied with m = 1, that is, ns = 0, in which case = 1. We express in terms of the parametererror = :

For simplicity, let us assume that L(s) is chosen so that WL is strictly proper and consider the following statespace representation of Eq. (8):

where Ac, Bc, and Cc are the matrices associated with a state-space representation that has a transfer functionW(s)L(s) = CTc(sI Ac) 1Bc.

Let us now consider the following Lyapunov-like function for the differential equation (9):

where = T > 0 is a constant matrix and Pc = PTc > 0 satisfies the algebraic equations

for some vector q, matrix Lc = LTc > 0, and small constant > 0. The existence of Pc = PTc > 0 satisfying theabove equations is guaranteed by the SPR property (12) of W(s)L(s) = CTc(sI Ac) 1Bc. The time derivative of

8 ADAPTIVE CONTROL

V is given by

We now need to choose = as a function of signals that can be measured so that the indefinite terms in V arecanceled out. Because e is not available for measurement, cannot depend on e explicitly.

We know that PcBc = Cc, which implies that eTPcBc = eTCc = . Therefore

The choice for = to make V 0 is now obvious, namely, for

we have

Using the above inequality, we can prove the following theorem.

Theorem 1. The adaptive law in Eq. (4) guarantees that:

(i) , L .(ii) , ns, L 2, independent of the boundedness of .

(iii) If ns, , L , and is perstistently exciting (PE)that is, there exist positive constant 1, 0, T0 suchthat

then (t) exponentially fast.

The proof of the theorem can be found in Ref. 12.Gradient Method. In this method, we consider the parametric model in Eq. (5). Similar to the previous

subsection, we define (t) to be the on line estimate of and the normalized estimation error as

where z= T(t) and m2 = 1 + n2s and n2s is chosen so that /m L . The adaptive law is designed to minimizethe performance index J(), i.e.,

ADAPTIVE CONTROL 9

which gives

Different choices for the performance index lead to different adaptive laws.Let us consider the simple quadratic cost function (instantaneous cost function)

Applying the gradient method, the minimizing trajectory (t) is generated by the differential equation

where = T > 0 is a scaling matrix that we refer to as the adaptive gain. We have

The following theorem holds:

Theorem 2. The adaptive law in Eq. (18) guarantees that:

(i) , ns, L .(ii) , ns, L 2, independent of the boundedness of .

(iii) If ns, L and is PE, then (t) exponentially fast.

The proof of the theorem can be found in Ref. 12.Least Squares. Let (t), , z be defined as above, and let m2 = 1 + n2s, (t) be the estimate of at time

t, and m satisfy /m L . We consider the following cost function:

where Q0 = QT0 > 0, 0, 0 = (0), which includes discounting of past data and a penalty on the initial estimate0 of . Because z/m, /m L , we have that J() is a convex function of over Rn at each time t. Hence, anylocal minimum is also global and satisfies

which yields the so-called nonrecursive least-squares algorithm

10 ADAPTIVE CONTROL

where

We can show that P, satisfy the differential equations

We refer to Eqs. (22) and (23) as the continuous-time recursive least-squares algorithm with forgetting factor.The stability properties of the least-squares algorithm depend on the value of the forgetting factor .

In the identification literature, Eqs. (22) and (23) with = 0 are referred to as the pure least-squaresalgorithm and have a very similar form to the Kalman filter. For this reason, the matrix P is usually calledthe covariance matrix. The pure least-squares algorithm has the unique property of guaranteeing parameterconvergence to constant values as described by the following theorem:

Theorem 3. The pure least-squares algorithm guarantees that:

(i) , ns, , , P L .(ii) , ns, L 2.

(iii) limt (t) = , where is a constant vector.(iv) If ns, L and is PE, then (t) converges to as t .

The proof of the theorem can be found in 12.Bilinear Parametric Model. As will be shown in the next sections, a certain class of plants can be

parametrized in terms of their desired controller parameters, which are related to the plant parameters viaa Diophantine equation. Such parametrizations and their related estimation problem arise in direct adaptivecontrol and, in particular, direct MRAC, which is discussed in the next section.

In these cases appears in the form

where is an unknown constant, z, , z0 are signals that can be measured, and W(s) is a known proper transferfunction with stable poles. Because the unknown parameters , appear in a special bilinear form, we referto Eq. (24) as the bilinear parametric model.

Known Sign of . The SPRLyapunov design approach and the gradient method with an instantaneouscost function discussed in the linear parametric case extend to the bilinear one in a rather straightforwardmanner.

Let us start with the SPRLyapunov design approach. We rewrite Eq. (24) in the form

ADAPTIVE CONTROL 11

where z1 = L 1(s)z0, = L 1(s), and L(s) is chosen so that L 1(s) is proper and stable and WL is proper andSPR. The estimate z of z and the normalized estimation error are generated as

where ns is designed to satisfy

and (t), (t) are the estimates of , at time t, respectively. Letting , , it follows fromEqs. (25) to (27) that

Now T T = T T + T T = T T, and therefore,

By choosing

we can see that the following theorem holds.

Theorem 4. The adaptive law in Eq. (30) guarantees that:

(i) , , L .(ii) , ns, , L 2.

(iii) If , L , is PE, and L 2, then (t) converges to as t .(iv) If L 2, the estimate converges to a constant independent of the properties of .

The proof of the theorem can be found in Ref. 12. The case where the sign of is unknown is also givenin Ref. 12.

Model Reference Adaptive Control

Problem Statement. Consider the SISO LTI plant described by

12 ADAPTIVE CONTROL

where Gp(s) is expressed in the form

where Zp, Rp are monic polynomials and kp is a constant referred to as the high-frequency gain.The reference model, selected by the designer to describe the desired characteristics of the plant, is

described by

which is expressed in the same form as Eq. (32), that is,

where Zm(s), Rm(s) are monic polynomials and km is a constant.The MRC objective is to determine the plant input up so that all signals are bounded and the plant output

yp tracks the reference model output ym as close as possible for any given reference input r(t) of the class definedabove. We refer to the problem of finding the desired up to meet the control objective as the MRC problem. Inorder to meet the MRC objective with a control law that is implementable (i.e., a control law that is free ofdifferentiators and uses only measurable signals), we assume that the plant and reference model satisfy thefollowing assumptions:

Plant Assumptions.

P1. Zp(s) is a monic Hurwitz polynomial of degree mp.P2. An upper bound n on the degree np of Rp(s) is known.P3. The relative degree n= np mp of Gp(s) is known.P4. The sign of the high-frequency gain kp is known.

Reference Model Assumptions.

M1. Zm(s), Rm(s) are monic Hurwitz polynomials of degree qm, pm, respectively, where pm n.M2. The relative degree nm = pm qm of Wm(s) is the same as that of Gp(s), that is, nm = n.

MRC Schemes: Known Plant Parameters. In addition to assumptions P1 to P4 and M1, M2, let usalso assume that the plant parameters, that is, the coefficients of Gp(s), are known exactly. Because the plantis LTI and known, the design of the MRC scheme is achieved using linear system theory.

We consider the feedback control law

ADAPTIVE CONTROL 13

where

c0, 3 R1, 1, 2 Rn 1 are constant parameters to be designed, and (s) is an arbitrary monic Hurwitzpolynomial of degree n 1 that contains Zm(s) as a factor, i.e., (s) = 0(s)Zm(s), which implies that 0(s) ismonic, Hurwitz, and of degree n0 = n 1 qm. The controller parameter vector

is to be chosen so that the transfer function from r to yp is equal to Wm(s).The inputoutput properties of the closed-loop plant are described by the transfer function equation

where

We can now meet the control objective if we select the controller parameters 1, 2, 3, c0 so that theclosed-loop poles are stable and the closed-loop transfer function Gc(s) = Wm(s) is satisfied for all s C. Choosing

and using (s) = 0(s)Zm(s), the matching equation Gc(s) = Wm(s) becomes

Equating the coefficients of the powers of s on both sides of Eq. (38), we can express it in terms of the algebraicequation

where = [T1, T2, 3]T; S is an (n + np 1) (2n 1) matrix that depends on the coefficients of Rp, kpZp,and and p is an n + np 1 vector with the coefficients of Rp Zp0Rm. The existence of to satisfy Eq. (39)and, therefore, Eq. (38) will very much depend on the properties of the matrix S. For example, if n > np, morethan one will satisfy Eq. (39), whereas if n = np and S is nonsingular, Eq. (39) will have only one solution.Lemma 1. Let the degrees of Rp, Zp, , 0 and Rm be as specified in Eq. (34). Then

(i) The solution of Eq. (38) or (39) always exists.(ii) In addition, if Rp, Zp are coprime and n = np, then the solution is unique.

14 ADAPTIVE CONTROL

The proof of the lemma can be found in Ref. 12.MRAC for SISO Plants. The design of MRAC schemes for the plant in Eq. (31) with unknown parame-

ters is based on the certainty equivalence approach and is conceptually simple. With this approach, we developa wide class of MRAC schemes by combining the MRC law, where is replaced by its estimate (t), withdifferent adaptive laws for generating (t) online. We design the adaptive laws by first developing appropriateparametric models for , which we then use to pick up the adaptive law of our choice from the precedingsection.

Let us start with the control law

whose state-space realization is given by

where = [T1, T2, 3, c0]T and = [T1, T2, yp, r]T, and search for an adaptive law to generate (t), the estimateof the desired parameter vector .

It can be seen that under the above control law, the tracking error satisfies

where = 1/c0, = [T1, T2, 3, c0]T. The above parametric model may be developed by using the matchingEq. (38) to substitute for the unknown plant polynomial Rp(s) in the plant equation and by canceling the Hurwitzpolynomial Zp(s). The parametric model in Eq. (42) holds for any relative degree of the plant transfer function.

A linear parametric model for may also be developed from Eq. (42). Such a model takes the form

where

The main equations of several MRAC schemes formed by combining Eq. (41) with an adaptive law basedon Eq. (42) or (43). The following theorem gives the stability properties of the MRAC scheme.

Theorem 5. The closed-loop MRAC scheme in Eq. (41), with (t) adjusted with any adaptive law with nor-malization based on the model in Eq. (42) or (43) as described in the preceding section, has the followingproperties:

(i) All signals are uniformly bounded.(ii) The tracking error e1 = yp ym converges to zero as t .

ADAPTIVE CONTROL 15

(iii) If the reference input signal r is sufficiently rich of order 2n, r L , and Rp, Zp are coprime, then thetracking error e1 and parameter error = converge to zero for the adaptive law with known sgn(kp).

The proof of the theorem can be found in Ref. 12.

Adaptive Pole Placement Control

In the preceding section we considered the design of a wide class of MRAC schemes for LTI plants with stablezeros. The assumption that the plant is minimum-phase, that is, it has stable zeros, is rather restrictive inmany applications. A class of control schemes that is popular in the known-parameter case are those thatchange the poles of the plant and do not involve plant zeropole cancellations. These schemes are referredto as pole placement schemes and are applicable to both minimum- and nonminimum-phase LTI plants. Thecombination of a pole placement control law with a parameter estimator or an adaptive law leads to an adaptivepole placement control (APPC) scheme that can be used to control a wide class of LTI plants with unknownparameters.

Problem Statement. Consider the SISO LTI plant

where Gp(s) is proper and Rp(s) is a monic polynomial. The control objective is to choose the plant input up sothat the closed-loop poles are assigned to those of a given monic Hurwitz polynomial A(s). The polynomialA(s), referred to as the desired closed-loop characteristic polynomial, is chosen according to the closed-loopperformance requirements. To meet the control objective, we make the following assumptions about the plant:

P1. Rp(s) is a monic polynomial whose degreen n is known.P2. Zp(s), Rp(s) are coprime, and degree(Zp) < n.

Assumptions P1 and P2 allow Zp, Rp to be non-Hurwitz, in contrast to the MRC case, where Zp is requiredto be Hurwitz. If, however, Zp is Hurwitz, the MRC problem is a special case of the general pole placementproblem defined above with A(s) restricted to have Zp as a factor.

In general, by assigning the closed-loop poles to those of A(s), we can guarantee closed-loop stability andconvergence of the plant output yp to zero provided there is no external input. We can also extend the PPCobjective to include tracking, where yp is required to follow a certain class of reference signals ym, by using theinternal model principle as fol8lows: The reference signal ym L is assumed to satisfy

where Qm(s), the internal model of ym, is a known monic polynomial of degree q with nonrepeated roots on thej axis and satisfies

P3. Qm(s), Zp(s) are coprime.

For example, if yp is required to track the reference signal ym = 2 + sin 2t, then Qm(s) = s(s2 + 4) andtherefore, according to assumption P3, Zp(s) should not have s or s2 + 4 as a factor.

16 ADAPTIVE CONTROL

In addition to assumptions P1 to P3, let us also assume that the coefficients of Zp(s), Rp(s), i.e., the plantparameters are known exactly.

We consider the control law

where P(s), L(s), M(s) are polynomials [with L(s) monic] of degree q + n 1, n 1, q + n 1, respectively, tobe found, and Qm(s) satisfies Eq. (45) and assumption P3.

Applying Eq. (46) to the plant in Eq. (44), we obtain the closed-loop plant equation

whose characteristic equation

has order 2n + q 1. The objective now is to choose P, L such that

is satisfied for a given monic Hurwitz polynomial A(s) of degree 2n + q 1. It can be seen that assumptionsP2 and P3 guarantee that L, P satisfying Eq. (49) exist and are unique. The solution for the coefficients of L(s),P(s) of Eq. (49) may be obtained by solving the algebraic equation

where Sl is the Sylvester matrix of QmRp, Zp of dimension 2(n + q) 2(n + q),

and l, p, a are the vectors whose entries are the coefficients of L(s), P(s) and A(s), respectively. The coprimenessof QmRp, Zp guarantees that Sl is nonsingular; therefore, the coefficients of L(s), P(s) may be computed fromthe equation

The tracking error e1 = yp ym is given by

ADAPTIVE CONTROL 17

For zero tracking error, Eq. (51) suggests the choice of M(s) = P(s) to null the first term. The second term in Eq.(51) is nulled by using Qmym = 0. Therefore, the pole placement and tracking objective are achieved by usingthe control law

which is implemented using n + q 1 integrators to realize C(s) = P(s)/Qm(s)L(s). Because L(s) is not necessarilyHurwitz, the realization of Eq. (52) with n + q 1 integrators may have a transfer function, namely C(s), withpoles outside C . An alternative realization of Eq. (52) is obtained by rewriting it as

where is any monic Hurwitz polynomial of degree n + q 1. The control law (53) is implemented using 2(n+ q 1) integrators to realize the proper stable transfer functions ( LQm)/, P/.

APPC. The APPC scheme that meets the control objective for the unknown plant is formed by com-bining the control law in Eq. (53) with an adaptive law based on the parametric model in Eq. (5). The adap-tive law generates on-line estimates a, b of the coefficient vectors a of Rp(s) = sn + Tan 1(s) and bof Zp(s) = Tbn 1(s), respectively, to form the estimated plant polynomials Rp(s, t) = sn + Tan 1(s), Zp(s,t) = Tbn 1(s). The estimated plant polynomials are used to compute the estimated controller polynomials L(s,t), P(s, t) by solving the Diophantine equation

for L, P pointwise in time, or the algebraic equation

for l, where Sl is the Sylvester matrix of RpQm, Zp; l contains the coefficients of L, P; and l contains thecoefficients of A(s). The control law in the unknown-parameter case is then formed as

Because different adaptive laws may be picked up from the section On-Line Parameter Estimation above, awide class of APPC schemes may be developed.

The implementation of the APPC scheme requires that the solution of the polynomial Eq. (54) for L, P orof the algebraic Eq. (55) for l exists at each time. The existence of this solution is guaranteed provided thatRp(s, t)Qm(s), Zp(s, t) are coprime at each time t, that is, the Sylvester matrix Sl(t) is nonsingular at each time t.

Theorem 6. Assume that the estimated plant polynomials RpQm, Zp are strongly coprime at each time t. Thenall the signals in the closed-loop APPC scheme are u.b., and the tracking error converges to zero asymptoticallywith time.

The proof of the theorem can be found in Ref. 12. The assumption that the estimated polynomials arestrongly coprime at each time t is restrictive and cannot be guaranteed by the adaptive law. Methods that relaxthis assumption are given in Ref. 12.

18 ADAPTIVE CONTROL

Adaptive Control of Nonlinear Systems

In the previous sections we dealt with the problem of designing controllers for linear time-invariant systems.In this section, we show how the techniques of adaptive control of linear systems can be extended or modifiedfor nonlinear systems. Although the techniques presented can be applied to a variety of nonlinear systems, wewill concentrate our attention on adaptive state feedback control of SISO feedback-linearizable systems.

Feedback-Linearizable Systems in Canonical Form. We start with an nth-order SISO feedback-linearizable system in canonical form, whose dynamics are as follows:

where y, u R are the scalar system output and input, respectively, f , g are smooth vector fields, and x [x1, x2, . . ., xn]T is the state vector of the system. In order for the system in Eq. (57) to be controllable andfeedback-linearizable we assume that

A1. A lower bound for g(x) [i.e., |g(x)| > > 0 x Rn] and the sign of g(x) are known.

The control objective is to find the control input u that guarantees signal boundedness and forces y tofollow the output ym of the reference model

where A is a Hurwitz n n matrix, r L , and therefore xm L . In order to have a well-posed problem, it isassumed that the relative degree of the reference model is equal to n. If e ym y is the tracking error, thenits nth time derivative satisfies

Let h(s) = sn + k1sn 1 + + kn be a Hurwitz polynomial (here s denotes the d/dt operator). Also let [e,e, . . ., e(n 1)]T. Under assumption A1, the system Eq. (57) is a feedback-linearizable system. Therefore, if weknow the vector fields f and g, we can apply the static feedback

where k [kn, kn 1, . . ., k1]T. Then the error system in Eq. (59) becomes

which implies that e, L and therefore all closed-loop signals are bounded, and limt e(t) = 0.

ADAPTIVE CONTROL 19

In many cases, the vector fields f and g are not completely known and thus adaptive versions of thefeedback law (60) have to be applied. For instance, using the usual assumption of linear parametrization, if thevector fields f and g are of the form

where i, i = 1, 2, are vectors with unknown constant parameters, one may replace the feedback law in Eq. (60)with the certainty-equivalent one [the certainty-equivalent feedback-linearizing (CEFL) controller]

where i, i = 1, 2, are the estimates of the unknown parameter vectors i, i = 1, 2. These estimates are generatedby an on-line adaptive law. We next propose the following adaptive laws for updating i:

where i, i = 1, 2, are symmetric positive definite matrices and 1 = bc f , 2 = ubcg, bc = [0, . . ., 0, 1]. Thenext theorem summarizes the properties of the proposed control law.

Theorem 7. Consider the system in Eq. (57) and the feedback control law in Eqs. (62) and (63). Let assumptionA1 hold. Then, if T2(t)g(x(t)) = 0 for all t, all the closed-loop signals are bounded and the tracking errorconverges to asymptotically to zero.

Parametric-Pure-Feedback Systems. Let us now try to extend the results of the previous section tononlinear systems that take the form

where u, zi R, f ij, gnj are smooth functions, and Rp is the vector of constant but unknown system parameters.Let us rewrite Eq. (64) as

where f i1() = f i0() zi+1. Systems of the form in Eq. (65) are called parametric-pure-feedback (PPF) systems(13,14). Note that the above class of systems includes as a special case the system in Eq. (57) of the previoussection.

20 ADAPTIVE CONTROL

The control objective is to force the system output y to asymptotically track a reference signal ym. Weassume that the first n 1 time derivatives of ym are known. Also it will be assumed that ym and its first n 1 time derivatives are bounded and smooth signals.

Let us now assume that the parameter vector is known and construct a control law that meets thecontrol objectives. Before we design the feedback law, we will transform the system in Eq. (64) into a suitableform. The procedure we will follow is based on the backstepping integrator principle (13).

Step 0. Let 1 z1 ym. Let also c1, . . ., cn be positive constants to be chosen later.Step 1. Using the chain-of-integrators method, we see that, if z2 were the control input in the z1 part of Eq.

(65) and were known, then the control law

would result in a globally asymptotically stable tracking, since such a control law would transform the z1part of Eq. (65) as follows:

However, the state z2 is not the control. We therefore define 2 to be the difference between the actual z2and its desired expression in Eq. (66):

Step 2. Using the above definition of 2, the definition of 1, and the z1 part of Eq. (65), we find that

Step 2. Using the above definitions of 1, 2, we have that

where (2) is a (p + p2)-dimensional vector that consists of all elements that are either of the form 2,i orof the form 2,ij, where by 2,i we denote the ith entry of the vector . In the system (69) we will think ofz3 as our control input. Therefore, as in step 1, we define the new state 3 as

ADAPTIVE CONTROL 21

Substituting Eq. (70) into Eq. (69) yields

Step i (2 < i n 1). Using the definitions of 1, . . ., i and working as in the previous steps, we may expressthe derivative of i as

where the vector (i) contains all the terms of the form i1i2 i with 1 j i. Defining now i+1 as

we obtain that

Step n.Using the definitions of 1, . . ., n 1 and working as in the previous steps, we may express the derivativeof n as follows:

where the vector contains all the terms of the form i1i2 ij with 1 j n, Ym [ym, ym, ym, . . .,y(n 1)m]T, and [0(z1, . . ., zn) + T1(z1, . . ., zn)] is given by

Using the definitions of 1, . . ., n, and rearranging terms, we may rewrite Eq. (75) as follows:

22 ADAPTIVE CONTROL

Using the above methodology, we have therefore transformed the system in Eq. (65) into the followingone:

The above system is feedback-linearizable if the following assumption holds.

A1. 0(z) + T1(z) = 0 for all z.

Note now that in the case where (and thus ) is known, a controller that meets the control objective isthe controller of the form

Under the above control law, the closed-loop dynamics become

It can be shown that the matrix A0 is a stability matrix, provided that ci > 2.

Theorem 8. The control law in Eq. (79) guarantees that all the closed-loop signals are bounded and that thetracking error converges to zero exponentially fast, provided that the design constants ci satisfy ci > 2.

In the case where the vector is not known, the certainty-equivalence principle can be employed in orderto design an adaptive controller for the system. However, the problem of designing parameter estimators forthe unknown parameters is not as easy as it was in the linear case. This can be seen from the fact thatthe states i, i > 1, are not available for measurement, since they depend on the unknown parameters. Toovercome this problem a recursive design approach similar to the approach above can be constructed. Thedifference between this approach [called adaptive backstepping (13)] and the approach presented above is thefollowing: in the approach presented above the states i, i > 1, depend on the unknown vector of parameters; in the new approach they are appropriately redefined so they depend on the parameter estimate vector .

ADAPTIVE CONTROL 23

Then the derivatives of i, i > 1, depend on the derivatives of the parameter estimates . In order to overcomethis problem, the adaptive backstepping approach makes use of the so-called tuning functions (13).

Next we present the adaptive controller that results from applying the adaptive backstepping procedureto the system in Eq. (65) for the case where

A2. Tf i2() are independent of zi+1 and Tgn2 = 0.

Also for simplicity, and without loss of generality, we will assume that f i1() = 0. The case where assumptionA2 is not valid will be treated in the next subsection. The adaptive controller that results from applying theadaptive backstepping procedure is recursively defined as follows:

Control law:

Parameter update law:

Tuning functions:

Regressor vectors:

Here ci > 2, i are positive design constants, and = T > 0 is a positive definite design matrix. The nexttheorem summarizes the properties of the above control law.

Theorem 9. Suppose that assumptions A1, A2 hold. Then the above adaptive control law guarantees that allthe closed-loop signals are bounded and that the tracking error converges asymptotically to zero.

The proof of the theorem can be found in Ref. 13.

24 ADAPTIVE CONTROL

A Control Law That Overcomes the Loss-Of-Controllability Problem. A significant problem thatarises in adaptive control of linear-in-the-parameters feedback-linearizable systems is the computation ofthe feedback control law when the identification model becomes uncontrollable although the actual systemis controllable; so far, there is no known solution to this problem. For instance, for the case of the systemin Eq. (57) the parameter estimation techniques used in adaptive control cannot guarantee, in general, that|2(t)Tg(x(t))|> 0 for each time t, that is, they cannot guarantee that the identification model is controllable.Also, for the case of PPF systems presented in the previous subsection, the adaptive backstepping techniquesguarantee global stability only in the case where assumption A2 is valid. Such restrictions are made becausethe computation of the adaptive control law depends on the existence of the inverse of the matrix that consistsof the estimated input vector fields (or the Lie derivatives of the output functions along those vector fields).Even in the case of known parameters where the inverse of the corresponding matrix exists (this is triviallysatisfied for feedback-linearizable systems), the inverse of the estimate of this matrix might not exist at eachtime due to insufficiently rich regressor signals, large initial parameter estimation errors, and so on.

We next show how one can overcome the problem where the estimation model becomes uncontrollable, byappropriately using switching adaptive control. We will apply the switching adaptive control methodology tothe PPF system of the previous subsection, by removing assumption A2.

Consider the Lyapunov function for the PPF system of the previous subsection.

By differentiating V with respect to time, we obtain that

Let us define

Note now that, using the definition of i, we can rewrite the is as follows:

where i and wi are appropriately defined known functions. Therefore, we have that

where is defined to be the vector whose entries are the elements ij, and 0, 1 areappropriately definedknown functions.

We are now ready to present the proposed controller. The control input is chosen as follows:

ADAPTIVE CONTROL 25

where:

One has

where denotes the estimate of . k() is a positive design function satisfying k(1, ) = 0 iff 1 = 0 and

is a continuous-switching signal that is used to switch from control u1 to control u2 and vice versa:

s is hysteresis-switching variable defined as follows:

where s (t) limt s(), where t.

The parameter estimates are updated using the following smooth projection update law (15)

where is a symmetric positive definite design matrix and PC is defined as follows (15):

26 ADAPTIVE CONTROL

where

where 0 < < 1, q 2, and j are positive design constants.The following theorem summarizes the properties of the control law in Eq. (83, 84, 85, 86, 87, 88, 89).

Theorem 10. Consider the system in Eq. (65) and the control law in Eqs. (83, 84, 85, 86, 87, 88, 89). Letassumption A1 hold. Moreover assume that the following hold:

C1. K > 1; k() satisfies Eq. (85).C2. j are sufficiently small. Moreover, (0) C, where

Then for any compact X0 Rn and for any positive constant c the following holds: there exist a positiveconstant K such that, for any initial state x0 X0, the control law in Eqs. (83, 84, 85, 86, 87, 88, 89) with K > Kguarantees that all the closed-loop signals are bounded and, moreover, that the tracking error 1 converges infinite time to the residual set

The idea of using switching adaptive controllers of the form presented above was first introduced in Ref.16, where the proposed methodology was applied to systems of the form in Eq. (57). The controller of Ref. 16was extended in Ref. 17 for PPF systems of the form in Eq. (65).

Acknowledgment

This article was supported by NASA grant NAGW-4103.

BIBLIOGRAPHY

1. J. A. Aseltine, A. R. Manchini, and C. W. Sartune, A survey of adaptive control systems, IRE Trans. Automat. Control,6 (3): 1958.

2. D. McRuer, I. Ashkenas, D. Graham, Aircraft Dynamics and Automatic Control, Princeton, NJ: Princeton Univ. Press,1973.

3. J. B. Cruz, Jr., System Sensitivity Analysis, Stroudsburg, PA: Dowden, Hutchinson & Ross, 1973.4. L. P. Grayson, Design via Lyapunov second method, Proc. 4th JACC, 1963.5. P. C. Parks, Lyapunov redesign of model reference control systems, IEEE Trans. Autom. Control, 11: 1966.6. P. H. Phillipson, Design methods for model reference adaptive systems, Proc. Inst. Mech. Eng., 183 (35):, 695706, 1968.7. R. V. Monopoli, Lyapunovs method for adaptive control design, IEEE Trans. Autom. Control, 3: 1967.8. I. D. Landau, Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, 1979.

ADAPTIVE CONTROL 27

9. K. S. Narendra, A. M. Annaswamy, Stable Adaptive Systems, Englewood Cliffs, NJ: Prentice-Hall, 1989.10. S. Sastry, M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Englewood Cliffs, NJ: Prentice-Hall,

1989.11. K. J. Astrom, B. Wittenmark, Adaptive Control, Reading, MA: Addison-Wesley, 1989.12. P. A. Ioannou, J. Sun, Robust Adaptive Control, Upper Saddle River, NJ: Prentice-Hall, 1996.13. M. Krstic, M. I. Kanellakopoulos, and R. Kokotovic, Nonlinear and Adaptive Control Design, New York: Wiley, 1995.14. D. Seto, A. M. Annaswamy, J. Baillieul, Adaptive control of nonlinear systems with a triangular structure, IEEE Trans.

Autom. Control, 7: 1994.15. J.-B. Pomet, L. Praly, Adaptive nonlinear stabilization: Estimation from the Lyapunov equation, IEEE Trans. Autom.

Control, 37: 729740, 1992.16. E. B. Kosmatopoulos and P. A. Ioannou, A switching adaptive controller for feedback linearizable systems, IEEE Trans.

Autom. Control, to be published.17. E. B. Kosmatopoulos and P. A. Ioannou, Robust switching adaptive control using control Lyapunov functions, Proc.

IEEE Conf. Decision and Control, 1997.

PETROS A. IOANNOUELIAS B. KOSMATOPOULOSUniversity of Southern California

BILINEAR SYSTEMS

An electrical circuit or other engineering system often com-municates with its external environment by input signalsthat control its behavior and output signals; it is then calleda control system. If the components of a control system allobey Ohms Law or one of its analogs, such as Hookes Law,the system is called linear. In linear control systems, theeffect of the controls is additive and the output measure-ment is linear. They are discussed in the article in thisencyclopedia on MULTIVARIABLE SYSTEMS.

What distinguishes a bilinear system (BLS) is that al-though it is linear in its state variables, some control signalu(t) exerts its effect multiplicatively. BLS may be given asmathematical models of circuits or plants or may be cho-sen by a designer to obtain better system response than ispossible with a linear system. Their study is a first step to-ward nonlinear control theory. Industrial process control,economics, and biology provide examples of BLS with mul-tiplicative controls such as valve settings, interest rates,and neural signals, respectively. This topic of research be-gan in the early 1960s with independent work in the USSRand in the USA; see the surveys of Bruni et al. (1) andMohler (2, 3) for historical development and reviews of theearly literature.

Notation in This Article. The symbol R means the realnumbers and Rn the n-dimensional real linear space; Cmeans the complex plane, with R(s) the real part of s C.The bold symbols ah, x, z will represent elements (col-umn vectors) of Rn; x (x transposed) is a row vector; x.is the complex conjugate transpose of x. Given a vec-tor function x(t) = col[x1(t), . . . , xn(t)], its time derivative isx(t)def= dx(t)/dt. Capital letters A, B, F, X are square matri-ces and I the identity matrix diag (1, 1, . . . , 1). The trace ofmatrix A is the sum of its diagonal elements, written tr(A);det(A) is its determinant. in are integers; r, s, t are realscalars, as are lowercase Greek letter quantities. Germantype g, sl, . . . will be used for Lie algebras.

Control Systems: Facts and Terminology

The following discussion is a brief reminder of state-spacemethods; see Sontag (4). The state variables in an electricalcircuit are currents through inductors and voltages acrosscapacitors; in mechanics, they are generalized positionsand momenta; and in chemistry, they are concentrationsof molecules. The state variables for a given plant consti-tute a vector function depending on time x(t). Knowledge ofan initial state x(0), of future external inputs, and the first-order vector differential equation that describes the plantdetermine the trajectory {x(t), t 0}. For the moment, wewill suppose that the plant elements, such as capacitors,inductors, and resistors, are linear (Ohms Law) and con-stant in value. The circuit equations can usually be com-bined into a single first-order vector differential equation,

with interaction matrix F and a linear output function.Here is a single-input single-output example, in which thecoefficient vector g describes the control transducer, theoutput transducer is described by the row vector h, andv = v(t) is a control signal:

x = Fx + vg, y = hx (1)or written out in full,

dxi

dt=

nj=1

(Fi jx j) + vgi, i = 1, . . . , n; y =n

i=1hixi (1)

(It is customary to suppress in the notation the time-dependence of x, y and often the control v.)

As written, equation 1 has constant coefficients, andsuch a control system is called time-invariant, whichmeans that its behavior does not depend on where wechoose the origin of the independent variable t; the systemcan be initialized and control v exerted at any time. Whenthe coefficients are not constant, that is, made explicit inthe notation, e.g.,

x = F (t)x + vg(t), y = h(t)xwhich is called a time-variant linear system.

For both linear and bilinear systems, we will need thesolution of x = Fx, which for a given initial condition x(0)is x(t) = eFtx(0). The matrix exponential function is definedby

eFt = I + Ft + F2t2

2+ . . . + F

ktk

k!+ . . . (2)

it is the inverse Laplace transform of the matrix (sI F )1and can be computed by numerical methods described byGolub and Van Loan (5). Its most familiar use in electricalengineering is to solve equation 1

x(t) = eFtx(0) + t

0

e(tr)Fv(r)gdr

The polynomial

PF (s)def= det(sI F ) = (s 1)(s 2) (s n)

is called the characteristic polynomial of F, and its rootsare called the eigenvalues of F. The entries in the matrixeFt are linear combinations of terms like

tmi eit , i = 1, . . . , nIf the eigenvalues are distinct, the integers mi van-ish, but if i is a multiple eigenvalue, mi may bepositive. For a given F, the matrices {exp(Ft), t R}are a group under matrix multiplication: exp((r + t)F ) =exp(rF )exp(tF ), (exp(tF ))1 = exp(tF ).

For different applications, different restrictions may beplaced on the class U of admissible control signals. In thisarticle, U will usually be the class of piecewise constant(PWC) signals. The value of a control at an instant of tran-sition between pieces need not be defined; it makes no dif-ference. If there is a single number > 0 that is an upperbound for all admissible controls, call the class U as a re-minder.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright 2007 John Wiley & Sons, Inc.

2 Bilinear Systems

A control system, linear or not, is said to be controllableon its state space if for any two states x, x, a timeT > 0 exists and a control in U for which the trajectorystarting at x(0) = |x ends at x(T ) = x. For time-invariantlinear systems, there is a simple and useful conditionnecessary for controllability. Stated for equation 1, thisKalman rank condition is

rank(g, Fg, . . . , Fn1g) = nIf it is satisfied, the matrix-vector pair {F, g} is called acontrollable pair. If the controls are not bounded, this con-dition is sufficient for controllability, but if they are in someU, the control may be small compared with Fx for large xand have an insufficient effect.

BILINEAR SYSTEMS: WHAT, WHY, WHERE?

Precisely what are BLS and why should one use bilinearsystems in control engineering? This section will give someanswers to those questions, starting with a formal defini-tion, and give some examples.

Definition. A bilinear system is a first-order vector dif-ferential equation x = p(x, u) in which x is a state vector;u is a control signal (scalar or vector); and the componentsof p(x, u) are polynomials in (x, u) that are linear in x, useparately, but jointly quadratic, with constant real coeffi-cients. Restating that, for any real numbers ,

p(x, u) p(x, u) + p(x, 0) + p(0, u)The (optional) output is linear, y = hx.

To begin with, BLS are simpler and better understoodthan most other nonlinear systems. Their study involves aconstant interplay between two profitable viewpoints: look-ing at BLS as time-invariant nonlinear control systems andas time-variant linear systems.

Another answer is that BLS are useful in designing con-trol systems that use a very small control signal to modu-late a large current of electricity of fluid, apply brakes, orchange rates of growth.

A third answer is that the usual linearization of a non-linear control system near an equilibrium point can be im-proved by using a BLS approximation; thus,

x = a(x) + ug(x), with a(xe) = 0; letA = a

x|x=xe, b = q(xe), B =

qx

|x=xeTranslating the origin so that xe = 0, to first order in x

and u separately:

x = Ax + u(Bx + b), y = hx (3)Although some results will be stated for equation 3,

usually we will suppose that b = 0; such BLS are calledhomogeneous bilinear systems, and for later reference,they are given here in both their single-input and k-inputversions

x = Ax + uBx (4)

x = Ax +k

j=1ujBjx (5)

(We will see later how to recover facts about equation 3from the homogeneous case.) If A = 0 in equations 4 or 5,the BLS is called symmetric:

x =k

j=1Bjx (5)

As a control system, equation 4 is time-invariant, and weneed to use controls that can start and stop when we wish.The use of PWC controls not only is appropriate for thatreason but also allows us to consider switched linear sys-tems as BLS, and in that case, only a discrete set of controlvalues such as {1, 1} or {0, 1} is used.

Later we will be concerned with state-dependent feed-back controls, u = u(x), which may have to satisfy condi-tions that guarantee that differential equations like x =Ax + u(x)Bx have well-behaved solutions.

Discrete time bilinear systems (DBLS) are described bydifference equations, rather than ordinary differentialequations. DBLS applications have come from the discrete-time dynamics common in economic and financial model-ing, in which the control is often an interest rate. DBLS alsoare used for digital computer simulation of continuous timesystems like equation 3: Using Eulers point-slope methodwith a time-step , for times k = 0, 1, 2, . . . , the discrete-timesystem is

x(k + 1) = (I + A)x(k) + u(k)(Bx(k) + b) (6)with output y(k) = hx(k). DBLS will be discussed brieflyat appropriate places below. Their solutions are obtainedby recursion from their initial conditions using their dif-ference equations.

Some Application Areas

Reference (2) lists early applications of BLS to nuclearreactors, immunological systems, population growth, andcompartmental models in physiology. For a recently ana-lyzed BLS from a controlled compartmental model, see thework on cancer chemotherapy by Ladzewicz and Schattler(6) and its references.

Using linear feedback u = Kx in a BLS results in aquadratic autonomous system. Recently some scientificallyinteresting quadratic systems, exhibiting chaotic behavior,have been studied by decomposing them into BLS of thistype. Celikovsky and Vanecek (7) have studied the third-order Lorenz system as a BLS with output feedback:

x = Ax + uBx, u(x) = x1, with > 0, > 0, > 0, and

A =( 0

1 00 0

), B =

(0 0 00 0 10 1 0

)

For small all eigenvalues of A are negative, but for > 1one becomes positive and B generates a rotation. This de-scription seems to be characteristic of several such exam-ples of strange attractors and bounded chaos, including theRossler attractor.

Bilinear Systems 3

In electrical engineering, BLS viewpoints can be usedto discuss switched and clocked circuits, in which the con-trol takes on only a discrete set of values like {0, 1} andthe plant is linear in each switched condition. The solu-tions are then easy to compute numerically. Sometimes theduty cycle of a switch is under control, as in motors andDC-to-DC power conversion systems. A simple example isa conventional ignition sparking system for automobiles,in which the primary circuit can be modeled by assigningx1 to voltage across capacitor of value C, x2 to current inthe primary coil of inductance L. The control is a distrib-utor or electronic switch, either open (infinite resistance)or closed (small resistance R) with duty cycle specified bythe crankshaft rotation and timing controls. Then with abattery of emf V,

x1 = 1C

x2 uC

(x1 V ), x2 = 1L

x1, u ={

1/R, closed0, open

Other automotive BLS include mechanical brakes andcontrolled suspension systems, among the many applica-tions discussed in Reference 3.

One advantage of piecewise constant control in BLSis that the solutions, being piecewise linear, are readilycomputed. For that reason, in aerospace and process en-gineering, control designs with gain scheduling (see GAINSCHEDULING) are an area where BLS methods should beuseful; such control schemes change the F matrix and equi-librium point to permit locally linear control. Recent workon hybrid systems (finite-state machines interacting withcontinuous plants) also falls into the category of switchedlinear systems.

STABILIZATION I: CONSTANT CONTROLS

This section will introduce an important engineering de-sign goal, stability, and the beginning of a running dis-cussion of stabilization. Stabilization is an active area ofresearch, in an effort to find good design principles.

A matrix F is called a Hurwitz matrix if all n ofthe eigenvalues of F lie in the left half of the com-plex plane; i.e., R(i) < < 0. Then x = Fx is said to beexponentially stable (ES); as time increases, all solutionsare bounded and ||x(t)|| < ||x(0)||et. If even one eigenvaluelies in the right half plane, almost all solutions will growunboundedly and the system is called unstable. Multi-ple imaginary eigenvalues = j can give tmcos(t) (reso-nance) terms that also are unstable. Warning: Even if thetime-varying eigenvalues of a time-variant linear differ-ential equation all lie in the left half plane, that does notguarantee stability!

If A is a Hurwitz matrix, equation 4 is ES when u = 0.Suppose that A is not Hurwitz or that is too small; findinga feedback control u such that equation 4 is ES with adesirable is called stabilization.

The problem of stabilization of BLS and other non-linear control systems is still an active area of engi-neering research. In this section, we consider only theuse of constant controls u = in equation 4; the resultof applying this feedback is a linear dynamical systemx = (A + B)x. To find a range of values for that will

stabilize, the BLS is somewhat difficult, but for smalln one can find PA+B() and test possible values of by the RouthHurwitz stability test for polynomials (seeSTABILITY THEORY, ASYMPTOTIC). For n = 2

PA+B() = 2 (tr(A) + tr(B)) + det(A + B), sotr(A) + tr(B) < 0 and det(A + B) > 0 (7)

guarantee stability. Graphing these two expressionsagainst is an appropriate method for finding good val-ues of . A complete set of conditions A and B that arenecessary and sufficient for stabilizability of second-orderBLS with constant feedback were given by Chabour et al.(8).

Other criteria for stabilization by constant control havebeen found, such as this one from Luesink and Nijmeijer(9). Suppose the eigenvalues of A are i, i = 1,. . . ,n, and theeigenvalues of B are i. If there is some nonsingular matrixP, real or complex, for which P1AP and P1BP are simul-taneously upper triangular, then the eigenvalues of A + Bare i + i, i = 1, . . . , n. If some real satisfies the n linearinequalities R(i + i) < 0, it will be the desired constantcontrol. For more about such triangularizable BLS, see thesection below on The Lie Algebra of a BLS.

SOLUTIONS OF BILINEAR SYSTEMS

From one viewpoint a BLS with a specific nonconstantcontrol history {u(t), t 0} should be thought of as a time-variant linear differential equation. We will use the single-input case of equation 4 as an example, with A + u(t)B astime-variant matrix. The solution depends on the initialtime t0 at which the state is x(t0), and is of the form x(t) =(t, t0)x(t0), where (t, t0) is called a transition matrix.For more about the general theory of these matrices, seeMULTIVARIABLE SYSTEMS, Chap. 9 of Kailath 10, or vol. 1,Chap. 4 of Reference 3. Once having written that expres-sion for x(t), it can be observed that must satisfy thematrix differential equation

= (A + u(t)B), (t0, t0) = IIt has the composition property, also called the semigroupproperty (t, t1)(t1, t0) = (t, t0).

However, as a control system, equation 4 is time-invariant, by definition. Then the most convenient fami-lies of admissible controls for BLS are the PWC and otherpiecewise-defined controls; such a control can be specifiedby its values on an interval of definition of duration , forinstance {u(t), t (t0, t0 + )}. From the time-invariance ofthe system, a basic interval of definition, (0,), can be usedwithout any loss. Given a particular control signal u on(0,), its time shift by can be denoted u(t) = u(t ), on(, + ), as is usual in system theory.

The concatenation of two controls u and v with respec-tive durations 1 and 2 is written u o v and is anotheradmissible control with duration 1 + 2:

(u v)(t) ={

u(t), t [0, 1)v1 (t), t [1 < t 2]

For the general multi-input BLS of equation 5, the controlis a k-component vector u = [u1, . . . , uk], and the transitionmatrix satisfies = (A +k

j=1 ujBj). Concatenation is

4 Bilinear Systems

defined in the same way as for scalar controls: u o v is ufollowed by the translate of v.

The time-invariance of the BLS leads to useful proper-ties of the transition matrices. The transition matrix de-pends on the control u and its starting time, so the matrix should be labeled accordingly as (u; t, t0), and the statetrajectory corresponding to u is

x(t) = (u; t, t0)x(0)Given two controls and their basic intervals

{u, 0 < t < } and {v, 0 < t < }, the composition prop-erty for BLS transition matrices can be written in a niceform that illustrates concatenation (u followed by thetranslate of v)

(v ; , )(u; , 0) = (u v; , 0) (8a)A transition matrix always has an inverse, but it is notalways a transition matrix for the BLS. However, if the BLSis symmetric (5) and the admissible controls U are sign-symmetric (i.e., if u U, then u U), the transition matrix(u; , 0) resulting from control history {u(t), 0 t } hasan inverse that is again a transition matrix, obtained byusing the control that reverses what has been done before,u (t) = u( t), t 2;

(u ; 2, )(u; , 0) = I (8b)An asymmetric BLS, x = (A + u1B + + ukBk)x, is like asymmetric one in which one of the matrices B0 = A has aconstant control u0 1, whose sign cannot be changed.The controllability problem for asymmetric BLS involvesfinding ways around this obstacle by getting to I some otherway.

From a mathematical viewpoint, the set of transitionmatrices for equation 4 is a matrix semigroup with identityelement I. See the last section of this article.

MORE ABOUT TRANSITION MATRICES

Transition matrices for BLS have some additional proper-ties worth mentioning. For instance, in the rare situationthat A, B1,. . . ,Bk all commute, the transition matrix has acomforting formula

(u; , 0) = eAt+ t

0

ki=1 ui(s)Bids

Warning: If the matrices do not commute, this formula isinvalid!

The solution of a single-input inhomogeneous BLS likeequation 3, x = Ax + u(Bx + b), is much like the solutionof a linear system. If (u; t, 0) is the solution of the homo-geneous matrix system

= A + uB, (0) = Ithen for equation 3 with initial condition x(0),

x(t) = (u; t, 0)x(0) + t

0

(u; t, s)u(s)bds

One advantage of using piecewise constant controls isthat they not only approximate other signals, but suggesta construction of the transition matrix. For a PWC control

u given by m constant pieces {u(t) = u(k1), k1 t < k}on intervals that partition {0 t < m = T }, the transitionmatrix for X = (A + uB)X is clearly

(u; T, 0) =m

k=1e(A+u(k1)B)k (9)

This idea can be carried much further with more anal-ysis: More general (measurable) inputs can be approxi-mated by PWC controls, and in analogy to the definitionof an integral as a limit of sums, the solution to equation 4for measurable inputs can be written as (in an appropri-ate sense) the limit of products like equation 9, called aproduct-integral.

The representation given by equation 9 generalizes tothe multi-input BLS equation 5 in the obvious way. Withequation 9, one can also easily verify the composition and(for A = 0) inverse properties. To emphasize that exponen-tial formulas for noncommuting matrices have surprisingbehavior, here is a standard example in which you shouldnotice that A2 = 0 and B2 = 0.

A =(

0 10 0

)and B =

(0 01 0

); AB BA =

(1 00 1

);

eAteBt =(

1 t0 1

)(1 0t 1

)=(

1 + t2 1t 1

), but

e(A+B)t = exp(

0 tt 0

)=(

cosh(t) sinh(t)sinh(t) cosh(t)

)

OBSERVABILITY AND OBSERVERS

This section is concerned with BLS that have an outputmeasurement y with m < n components, so that x(t) isnot directly available. For many purposes in control (sta-bilization, prediction of future outputs, and optimization),it is necessary 1) to ensure that different states can be dis-tinguished and 2) to obtain estimates of x(t) from avail-able information. An important question is whether aninputoutput history

HT = {u(t), y(t)|0 t T }will uniquely determine the initial or final state.

Let C be an m n matrix (think of it as m row vectors);its null-space {x|Cx = 0} is denoted by C . The BLS is them-output system given by

x = (A + uB)x, y = Cx. (10)The initial state is not known, only the history HT .

Suppose that u(t), t 0, is given. Call two states x, x Rnu-indistinguishable on the interval (0, T) if the two cor-responding outputs are equal on that interval, i.e., ifC(u; t, 0)x = C(u; t, 0)x, 0 t T . This relation, writtenx ux, is transitive, reflexive, and symmetric (an equiva-lence relation) so it partitions the state space into disjointsets (see Chap. 5 of Reference 4); it is also linear in thestate. Therefore we need only be concerned with the set ofstates u-indistinguishable from the origin, namely,

Iudef={x|x u0} = {x|C(u; t, 0)x = 0, 0 t T }

which is a linear subspace called the u-unobservable sub-space of the BLS; the u-observable subspace is the quotient

Bilinear Systems 5

space Ou = Rn/Iu. That can be rephrased as Rn = IuOu.If Iu = 0, we say that the given system is u-observable. InGrasselli and Isidori (11) u-observability is given the nameobservability under single experiment.

If two states x, z are u-indistinguishable for all admis-sible u, we say they are indistinguishable and write x z.The set of unobservable states is the linear subspace

I = {x|C(u; t, 0)x = 0 for all u U}Its quotient subspace (complement) O = Rn/I is called theobservable subspace, and the system is called observable ifI = 0. The unobservable subspace is invariant for the BLS;trajectories that begin in I remain there. If the largest in-variant linear subspace of C is 0, the BLS is observable;this is also called observability under multiple experi-ments, because to test it one would have to have dupli-cate systems, each initialized at x(0) but using its owncontrol u.

Theorem 1 of Reference 11 states that for PWC controlsand piecewise continuous controls, the BLS is observableif and only if a u exists for which it is u-observable. Theproof constructs a universal input u that distinguishes allstates from 0 by concatenating at most n + 1 inputs: u =u0o o un. At the kth stage in the construction, the set ofstates indistinguishable from 0 is reduced in dimension bya well-chosen uk . The test for observability that comes outof this analysis is that the rank of the matrix (C; A, B) isn, where

(C; A, B) = col[C, CA, CB, CA2, CAB, CBA, CB2, . . .]That is, (C; A, B) contains C and all matrices obtained byrepeated multiplications on the right by A and B. This isthe first theorem on the existence of universal inputs, andthe idea has been extended to other nonlinear systems bySontag and Sussmann.

The simplest situation in which to look for observabil-ity criteria is for a system with input zero, an autonomoustime-invariant linear system x = Ax, y = Cx. It is no sur-prise that the Kalman rank criterion for observability isappropriate for such systems. The (time-invariant) observ-ability Gramian is W = col[C, CA, . . . , C(A)n1]; we say {C,A} is an observable pair if rank(W) = n, and that is bothnecessary and sufficient for linear system observability.

How can we extend this to the case where the inputis unknown? To derive the answer, from williamson (12),choose our admissible controls to be polynomials in t ofdegree n on any fixed time interval. Assume x(0) = 0. Itis still necessary that rank(W) = n, to preserve observ-ability when u = 0. Repeatedly differentiate y = Cx att = 0 (not that one would do this in practice) to obtainY

def= col{y0, y0, . . . , y(n1)0 }. If y0 = CAx(0) + u(0)CBx(0) = 0for some u(0), the information from y0 would be lost; thisgives a necessary condition that CB = 0; continuing thisway, necessarily CAB = 0, and so on. All the necessary con-ditions for observability can be summarized as

rank(W) = n and cAkB = 0, 0 k n 2 (11)To show the sufficiency of these conditions for observability,just note that no matter what control u is used, it and itsderivatives do not appear in any of the output derivatives,so Y = Wx(0) and x(0) = W1Y from the rank condition.

State Observers

Given a u-observable system with A, B known, it is pos-sible to estimate the initial state (or current state) fromthe history HT . The theory of time-variant linear systems(see Reference 10 or vol. 1 of Reference 3) shows that Ou isthe range of the time-variant version of the observabilityGramian,

WTdef= T

0

(u; t, 0)CC(u; t, 0)

If rank(WT ) = n, the initial state can be recovered; in ournotation

x(0) = W1T (T ) T

0

(u; t, 0)Cy(t)dt

The current state can be obtained from (u; t, 0)x(0) or bymore efficient means. Even though observability may failfor any constant control, it still may be possible, using somepiecewise constant control u, to achieve u-observability.Frelek and Elliott (13) pointed out that the Gramian couldbe optimized in various ways (e.g., by minimizing its con-dition number) with PWC controls, using a finite sequenceof u values, to permit accurate recovery of the entire stateor certain preferred state variables. The larger the linearspan of the trajectory, the more information is acquiredabout x(0).

One recursive estimator of the current state is anasymptotic state observer. There are many variations onthis idea; see KALMAN FILTERS AND OBSERVERS. A state ob-server can be regarded as a simplification of the Kalmanfilter in which no assumptions about noise statistics aremade, nor is a Riccati equation used, and so can be ex-tended to nonlinear systems for which no Kalman filtercan be found.

The asymptotic state observer to be described is suffi-ciently general for BLS. For a given control system, it isa model of the plant to be observed, with state vector de-noted by z and an input proportional to the output error.(Grasselli and Isidori (14) showed that there is nothingto be gained by more general ways of introducing an er-ror term.) To show how this works, we generalize slightlyto allow an inhomoge-neous BLS. Here are the plant, ob-server, and error equations; K is an n m gain matrix atour disposal.

x = Ax + u(Bx + b), y = Cx (12a)

z = Az + u(Bz + b) + uK(y Cz); let e = z x (12b)

e = (A + uB uKC)e (12c)

Observer design for linear systems is concerned with find-ing K for which the observer is convergent, meaning thate(t) 0, under some assumptions about u. (Observabil-ity is more than what is needed for the error to die out; aweaker concept, detectability, will do. Roughly put, a sys-tem is detectable if what cannot be observed is asymptot-ically stable.) At least three design problems can be posedfor equation 12.

6 Bilinear Systems

1. Design an observer that will converge for all choicesof (unbounded) u in U. This requires only the condi-tions of equation 11, replacing B with B KC, butthen the convergence depends only on the eigenval-ues of A; using the input has gained us no informa-tion.

2. Assume that u is known and fixed; in which case, themethods of finding K for observers of time-variantlinear systems are employed, such as Riccati-typeequations. There is no advantage to the BLS formin this problem.

3. Design a control and simultaneously choose K to getbest convergence. Currently, this is a difficult nonlin-ear programming problem, although Sen (15) showsthat a random choice of the values of PWC u shouldsuffice; from the invariance of the problem under di-lation (zooming in toward the origin), it seems likelythat a periodic control would be a good choice. Thisobserver design problem is much like an identifica-tion problem for a linear system, but identificationalgorithms are typically not bilinear.

Using digital computer control, one is likely to have nota continuous history H but the history of a PWC input udand a sampled output

Hd = {(ud(t1), y(t1)), (ud(t2), y(t2)), . . .}and the BLS can be dealt with as a discrete-time BLS. Us-ing only a, finite sequence of N > n values, the initial statez [or current state x(tN )] can be estimated by the least-squares method, which at best projects z onto the closest zin the observable subspace:

z = arg minz

Nk=1

y(tk) C(ud ; tk, 0)z2

CONSEQUENCES OF NONCOMMUTIVITY

The noncommutativity of the coefficient matrices A, B ofa BLS is crucial to its controllability, raising questionsthat suggest for their answers some interesting mathemat-ical tools: Lie algebras and Lie groups, named for Norwe-gian mathematician Sophus Lie, pronounced lee. See theReading List at the end of this article for books about them.

Lie Brackets

If A and B do not commute, then solving equation 4 ismore interesting and difficult. In addition to A, B, we willneed AB BA, which is written [A, B] and is called theLie bracket of A and B. To see how this matrix might comeinto the picture, and obtain a geometric interpretation ofthe Lie bracket, consider the two-input BLS with piecewiseconstant controls

x = uAx + vBx; u, v {1, 0, 1} (13a)Starting at any x(0), use a control with four control seg-ments (u, v), each of small duration > 0:

{1, 0}, {0, 1}, {1, 0}, {0, 1} on intervals (13b)

{0, }, {, 2}, {2, 3}, {3, 4}, respectively (13c)

The final state is (using the Taylor series of Eq. (7) for theexponential and keeping only terms up to degree 2)

x(4) = 7eBeAeBeAx(0)= (I + 2[A, B] + 3(higher order terms) + )x(0)

(13d)