Leture Remark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the funtion V ( x ) to be positive definite. ost often, our interest will be to show that x( t) as t. For that we will need to establish that the largest invariant set in E is the origin, that is: = { }. his is done by showing that no solution an stay identially in E other than the trivial solution x() t. heorem 4.1 (due to Barbashin and Krasovskii) Let x = be an equilibrium point for (3.1). Let V : D R be a ontinuously n differentiable positive definite funtion on a domain D R ontaining the origin, suh that V x() t S = x D: V x = and suppose that no other solution in D. Let { } an stay in S, other than the trivial solution x( t). hen the origin is loally asymptotially stable. If, in addition, V ( x ) is radially unbounded then the origin is globally asymptotially stable. Note that if V ( x) is negative definite then { } S = and the above theorem oinides with the Lyapunov nd theorem. Also note that the LaSalle s invariant set theorems are appliable to autonomous system only. Example 4. Consider the 1 st order system together with its adaptive ontrol law he dynamis of the adaptive gain ˆk ( t ) is x = ax+ u ˆ u = k t x ˆk = γ x where γ > is alled the rate of adaptation. hen the losed-loop system beomes: ( ˆ x = k() t a) x ˆ k = γ x he line x = represents the system equilibrium set. We want to show that the trajetories approah this equilibrium set, as t, whih means that the adaptive ontroller regulates x() t to zero in the presene of onstant unertainty in a. Consider the Lyapunov funtion andidate: 13
(, ˆ) = 1 + 1 ( ˆ ) V x k x k b γ where b> a. ime derivative of V along the trajetories of the system is given by 1 V ( ˆ) ( ˆ x, k = xx + k b) kˆ= x kˆ a + kˆ b x = x ( b a) γ V x, k ˆ is positive definite and radially unbounded funtion, whose derivative Sine V (, ˆ x k) is semi-negative, the set (, ˆ) : (, ˆ) positively invariant set. hus taking satisfied. he set E is given by ( ˆ) { } Ω = x k R V x k is ompat, Ω =Ω, all the onditions of LaSalle s heorem are {, : } E = x k Ω x=. Beause any point on the line x = is an equilibrium point, E is an invariant set. herefore, in this example = E. From LaSalle s heorem we onlude that every trajetory starting in Ω approahes E, as t, that is () xt as t. oreover, sine (, ˆ) (, ˆ ) Ω an be hosen large enough that ( ˆ) V x k is radially unbounded, the onlusion is global, that is it holds for all initial onditions onstant in the definition of x k beause the x, k Ω. 5. Boundedness and Ultimate Boundedness Consider the nonautonomous system where :, [ ) [, ) D, and f D R (, ) x = f t x (5.1) n is pieewise ontinuous in t, loally Lipshitz in x on D R n is a domain that ontains the origin x =. Note that if the origin is an equilibrium point for (5.1) then by definition: f ( t,) =, t. On the other hand, even if there is no equilibrium at the origin, Lyapunov analysis an still be used to show boundedness of the system trajetories. We begin with a motivating example. Example 5.1 Consider the IVP with nonautonomous salar dynamis x = x+ sin t (5.) xt ( ) = a> > he system has no equilibrium points. he IVP expliit solution an be easily found and shown to be bounded for all t t, uniformly in t, that is with a bound b independent of t. In this ase, the solution is said to be uniformly ultimately bounded (UUB), and b is alled the ultimate bound, (prove it). 14
he UUB property of (5.) an be established via Lyapunov analysis and without using x the expliit solution of the state equation. In fat, starting with V ( x ) =, we an alulate the time derivative of V, along the system trajetories. V ( x) = xx = x( x+ sin t) = x + xsin t x + x = x ( x ) It immediately follows that V ( x) <, x > In other words, the time derivative of V is negative outside of the ompat set B = { x }, or equivalently, all the solutions that start outside of B will enter this set within a finite time, and will remain inside the set for all future times. Formally, it an be stated as follows. Choose >. hen all the solutions that start in the set B = { V( x) } B x will remain therein, for all future time. he latter takes plae sine V is negative on the boundary. Hene, the solutions are uniformly bounded. oreover, an ultimate bound of these solutions an also be found. Choose ε suh that < ε < hen V is negative in the annulus set { V ( x) } ( ()) { V( x) ε} ε, whih implies that in this set V x t will derease monotonially in time, until the solution enters the set. From that time on, the trajetory will remain in the set, beause V is x negative on the boundary: V ( x) = ε. Sine V ( x ) =, we an onlude that this solution is UUB with the ultimate bound x ε. Definition 5.1 he solutions of (5.1) are uniformly bounded if there exists a positive onstant, independent of t, and for every a (, ), there is β β ( a) = >, independent of t, suh that x t a x t β, t t (5.3) globally uniformly bounded if (5.3) holds for arbitrarily large a 15
uniformly ultimately bounded with ultimate bound b if there exist positive onstants b and, independent of a,, there is (, ) = a b, independent of t, suh that t, and for every x t a x t b, t t + (5.4) globally uniformly ultimately bounded if (5.4) holds for arbitrarily large a. Figure 5.1: UUB Conept In the definition above, the term uniform indiates that the bound b does not depend on t. he term ultimate indiates that boundedness holds after the lapse of a ertain time. he onstant defines a neighborhood of the origin, independent of t, suh that all trajetories starting in the neighborhood will remain bounded in time. If an be hosen arbitrarily large then the UUB notion beomes global. Basially, the notion of UUB an be onsidered as a milder form of stability in the sense of Lyapunov (SISL). A omparison between SISL and UUB onepts is given below. SISL is defined with respet to an equilibrium, while UUB is not. Asymptoti SISL is a strong property that is very diffiult to ahieve in pratial dynamial systems. SISL requires the ability to keep the state arbitrarily lose to the system equilibrium by starting suffiiently lose to it. his is still too strong a requirement for pratial systems operating in the presene of unknown disturbanes. 16
he main differene between UUB and SISL is that the UUB bound b annot be made arbitrarily small by starting loser to the equilibrium or the origin. In pratial systems, the bound b depends on disturbanes and system unertainties. o demonstrate how Lyapunov analysis an be used to study UUB, onsider a ontinuously differentiable positive definite funtion V ( x ). Choose < ε <. Suppose that the sets Ω ε = { V ( x) ε} and Ω = { V ( x) } are ompat. Let Λ= ε V ( x) =Ω Ω ε { } and suppose that it is known that the time derivative of ( ) V x t along the trajetories of the nonautonomous dynamial system (5.1) is negative definite inside Λ, that is V ( x() t ) W( x( t) ) <, x Λ, t t where W ( x() t ) is a ontinuous positive definite funtion. Sine V is negative in Λ, a trajetory starting in Λ must move in the diretion of dereasing V ( x() t ). In fat, it an be shown that in the set Λ the trajetory behaves as if the origin was uniformly asymptotially stable, (whih it does not have to be in this ase). Consequently, the funtion V ( x() t ) will ontinue dereasing until the trajetory enters the set Ω ε in finite time and stays there for all future time. Hene, the solutions of (5.1) are UUB with the ultimate bound b= max x. A sketh of the sets Λ, Ω, Ω ε is shown in Figure 5.. x Ω ε Figure 5.: UUB by Lyapunov Analysis In many problems, the relation V ( t, x) W( x) is derived and shown to be valid on a domain whih is speified in terms of x. In suh ases, the UUB analysis involves finding the orresponding domains of attration and an ultimate bound. 6. Barbalat s Lemma and Invariane like heorems 17
For autonomous systems, LaSalle s invariane set theorems allow asymptoti stability onlusions to be drawn even when V is only negative semi-definite in a domain Ω. In that ase, the system trajetory approahes the largest invariant set E, whih is a subset of all points x Ω where V ( x ) =. However the invariant set theorems are not appliable to nonautonomous systems. In the ase of the latter, it may not even be lear how to define a set E, sine V may expliitly depend on both t and x. Even when V = V ( x) does not expliitly depend on t the nonautonomous nature of the system dynamis preludes the use of the LaSalle s invariant set theorems. Example 6.1 he losed-loop error dynamis of an adaptive ontrol system for 1 st order plant with one unknown parameter is e = e+ θ w( t) θ = ew t () where e represents the traking error and wt is a bounded funtion of time t. Due to the presene of wt (), the system dynamis is nonautonomous. Consider the Lyapunov funtion andidate V ( e, θ ) = e + θ Its time derivative along the system trajetories is V e, θ = ee + θθ = e e+ θ w t + θ ew t = e ( ) ( ) his inequality implies that V is a dereasing funtion of time, and therefore, both et and θ () t are bounded signals of time. But due to the nonautonomous nature of the system dynamis, the LaSalle s invariane set theorems annot be used to onlude the onvergene of et () to the origin., then we may expet that the trajetory of the system In general, if V ( t x) W( x) approahes the set { W( x ) = }, as t. Before we formulate main results, we state a lemma that is interesting in its own sake. he lemma is an important result about asymptoti properties of funtions and their derivatives and it is known as the Barbalat s lemma. Definition 6.1 (uniform ontinuity) A funtion f (): t R R is said to be uniformly ontinuous if ε > = ( ε) > t t1 f ( t) f ( t1) ε (6.1) Note that t 1 and t play a symmetri role in the definition of the uniform ontinuity. Lemma 6.1 (due to Barbalat) 18
Let f (): t R R be differentiable and has a finite limit as t. If f () t is uniformly ontinuous then f () t as t. Lemma 6. If f () t is bounded then f () t is uniformly ontinuous. An immediate and pratial orollary of Barbalat s lemma an now be stated. Corollary 6.1 If f (): t R R is twie differentiable, has a finite limit, and its nd derivative is bounded then f () t as t. In general, the fat that derivative tends to zero does not imply that the funtion has a limit. Also, the onverse is not true. f t C f t Example 6. As t, f () t sin ( ln t) os ln t = does not have a limit, while f () t =. t t t f t e sin e f t = e t sin e t + e t sin e t. As t, () =, while Example 6.3 Consider an LI system x = Ax+ Bu with a Hurwitz matrix A and a uniformly bounded in time input u( t ). hese two fats imply that the state x() t is bounded. hus, the state derivative x ( t) is bounded. Let y = Cx represent the system output. hen y = Cx is bounded and, onsequently the system output y () t is a uniformly ontinuous funtion of time. oreover, if the input u() t = u is onstant then the output y ( t ) tends to a limit, as t. he latter ombined with the fat that y is uniformly ontinuous implies, (by Barbalat s Lemma), that the output time derivative y asymptotially approahes zero. o apply Barbalat s lemma to the analysis of nonautonomous dynami systems we state the following immediate orollary. Corollary 6. (Lyapunov-like Lemma) V = V t, x is suh that If a salar funtion V ( t, x ) is lower bounded V ( t, x) x = f ( t, x ) 19
V ( t, x) then V ( t x) is uniformly ontinuous in time,, as t. Notie that the first two assumptions imply that V ( t, x ) tends to a limit. he latter oupled with the 3 rd assumption proves (using Barbalat s lemma) the orollary. Example 6.4 Consider again the losed-loop error dynamis of an adaptive ontrol system from Example 6.1. Choosing V ( e, θ ) = e + θ, it was shown that along the system trajetories: V ( e, θ ) = e. he nd time derivative of V is V ( e, θ) = 4ee = 4e( e+ θ w( t) ) Sine wt () is bounded by hypothesis, and etand θ ( t) were shown to be bounded, it is lear that V is bounded.. Hene, V is uniformly ontinuous and by the Barbalat s lemma (or the Lyapunov-like lemma), V whih in turn indiates that the traking error et tends to zero, as t. Often, the Barbalat Lemma is formulated with respet to an integral of a funtion, rather than to the funtion itself. Lemma 6.3 (due to Barbalat) If f : R + R is uniformly ontinuous for all t, and if the limit of the integral exists and finite, then t ( τ ) lim f dτ (6.) t lim f t = (6.3) t Definition 6. he L norm of a funtion f : R + R is defined as: p for all p = [ 1, ), while denotes the L norm of f. 1 p p τ (6.4) f () t = f ( τ ) d p max f ( t) f t = (6.5) t
We say that f Lp if () p f t exists and finite. In partiular, f L dτ <. if f ( τ) he next statement diretly follows from Lemma 6.3, if one hooses f () t g() t p Corollary 6. If gg, L and g() t Lp, for some p = [ 1, ), then g( t) lim =. t =. 1
Part II: System ID ethods 1. Introdution Read [Ioannou], Chapters 1. he goal of any parameter estimation method is to infer the values of unertain system parameters from the measurements of input and output signals of the system. Parameter estimation an be done either on-line or off-line. he latter may be preferable if the parameters are onstant and there is a suffiient time for estimation to be performed. However, for parameters that vary during operation, on-line parameter estimation is neessary to keep trak of the parameter values. Sine problems in robust and adaptive ontrol ontext usually involve time-varying parameters, on-line estimation methods are thus more relevant. In what follows, several basi methods of on-line estimation will be presented. Note that, although the main purpose of the on-line estimators may be to provide parameter estimates for adaptive / self-tuning ontrol, they an also be used for other purposes, suh as system health monitoring and failure detetion.. Stati Linear in Parameters odel An estimation model that relates the available data to the unknown parameters forms the basis of any parameter estimation method. A quite general model for parameter estimation appliations is in the linear parameterization form: y t =Θ Φ x t + ε t (.1) where () m n y t R, x() t R are the system output and input signals, the ( N m) N m Θ R ontains the unknown parameters to be estimated, x( t) R N matrix Φ (alled the regressor ) represents the N dimensional vetor of hosen basis funtions, and ε t R denotes the non-parametri unertainties in the system (suh as noise, m () modeling errors, et). Note that in (.1) both y ( t ) and x( t) Φ are known quantities. his implies that (.1) is simply is a system of linear equations in terms of the unknown Θ. For every time instant t there is suh a system. So if ontinuous measurements of y () t and Φ ( x() t ) are available on-line then, in general, there will be more equations than the unknowns. However, this will failitate parameter estimation in the presene of inevitable noise and modeling error. Φ for a period of time, and solves the equations one and for all. In on-line estimation, equations are In off-line estimation, one ollets the data points of y ( t ) and x( t)
solved reursively, implying that the estimated value of ˆ ˆ ( t) set of data y () t and Φ x() t beomes available. Θ=Θ is updated one a new Stati linear in parameters model (.1), although simple, is atually quite general, as seen from the following example. Example.1: Linear parameterization of 1 st order dynamis Consider the 1 st order system x = x+ ax+ bu+ξ ( t) (.) x( ) = where x, u are the salar state and input respetively, a and b are the unknown onstants to be identified on-line using the available measurements of x and u. Also in (.), ξ () t represent bounded unknown disturbane, (suh as proess noise). he very first step in the design of an on-line parameter identifier is to lump the unknown parameters a and b in a vetor, that is one needs to rewrite (.) in the form of (.1). oward that end, one an easily integrate (.) and get: () ( ) t t t t τ t τ t τ x t = x + a e x τ dτ + b e u τ dτ + e ξ τ dτ () () ε () xf t uf t t (.3) or, equivalently: x() t = axf ( t) + buf ( t) + ε ( t) (.4) Finally, the system an be written in the desired form of (.1) : Θ ( t) () t xf x = ( a b) + ε () t u (.5) f Example.: Linear parameterization of airraft pith dynamis Consider the following pith dynamis of a rigid-body, fixed-wing airraft. IY q = ( α) + q q+ + pr( Iz Ix) + ( r p ) Ixz I is the pith inertia, ( q, q) Φ (, ) aero IC p r (.6) where Y are the pith aeleration and angular rate. Also in (.6), aero denotes the airraft total aerodynami pithing moment whih is omprised of the baseline omponent ( α ), pith damping q q, and pith inrement due to elevator defletion. Finally, (, ) IC pr represents the sum of ross-oupling terms 3
that are due to the angular roll and yaw rates, ( p, r ), saled by the airraft inertia omponents ( x, z, xz) I I I. Suppose that the aerodynami pithing moment omponents α, q, q, are unknown and need to be estimated on-line using available input-output measurements ( α, p, q, r, ) and q, respetively. It is further assumed that the airraft inertia terms are onstant and known. Consequently, the inertial oupling term IC ( pr, ) in the right hand side of (.6) is known, as well. Rewriting (.6) using aerodynami oeffiients yields: I Y IC q q C C C q QS = α + + IY V (.7) In (.7), Q denotes the dynamis pressure, S is the airraft wing referene area, and is the mean geometri hord. For simpliity, the unknown pith ontrol effetiveness C and the pith rate damping oeffiient C are assumed to be onstant. On the other q hand, the unertain aerodynami pithing moment omponent C ( α ) depends on the airraft angle of attak α. Suppose that the pithing moment an be written as: C ( α ) = θ Φ α + ε α (.8) N where θ R is the N dimensional vetor of unknown onstant parameters, and N ( α ) R basis funtions, suh as splines, polynomials, RBF-s, et. Also in (.8), ε Φ is the N dimensional regressor vetor that onsists of appropriately hosen α is the non-parametri approximation error that an be made small by hoosing a suffiiently large number of basis funtions N. Substituting (.8) into (.7) gives the desired linear in parameters model. I Y IC q q C C q QS = θ Φ α + + + ε α IY V (.9) or equivalently: Φ ( α ) I Y IC q q ( θ C C = ) q + ε (.1) QS IY V Θ yt () Ψ ( α, qv,, ) Note that the error term ε may ontain other non-parametri unertainties in the system, suh as measurement noise, numerial inauraies et. As a result, the airraft pith dynamis relation (.6) is represented using the linear-in-parameters model: y t =Θ Ψ t + ε t (.11) 4
Note that the stati model (.11) requires knowledge of the pith aeleration signal q. ost often in real-life appliations, vehile rotational aelerations are not available as on board measurements. In this ase, an alternative modeling approah an be utilized. We rewrite (.6) in the form, aero ( α, q, ) + IC ( p, r) q = af q+ af q+ (.1) IY where a f >, and integrate both sides, (assuming zero initial onditions). t t t aero ( α( τ), q( τ), ( τ) ) IC ( p( τ), r t t t ( τ τ τ τ )) q() t = a e q( τ ) dτ + e dτ + e dτ (.13) hen, f I Y I Y qf () t aero f () t IC f () t () IC f f f aero f yt () q t t a q t = t (.14) where y () t is the known output, while aero f ( t ) represents the vehile filtered pith aerodynamis. his term an be modeled as a linear in unknown parameters funtion. Φ ( α ) q QS aero f = ( θ C C ) q + ε f (.15) V IY Θ Ψ ( α, qv,, ) hus, again we arrive at a stati model in the form of (.11). Remark 1: In some pratial appliations, using a filtered version of the derivative, suh f s as q = q, in plae of the derivative q works well beause the signal q that is τ q s + 1 being filtered is ontinuous. herefore, the signal energy is at low frequenies, where the speified filter is a good approximation to the derivative, ( 1 defines the bandwidth of τ q the derivative filter). It is evident that the derivative approximation error an be folded into the bounded non-parametri unertainty vetor ε. 5