Analysis of a Supervised Set-Point Control Containing a Compact Continuum of Finite-Dimensional Linear Controllers

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1 Analsis of a Superised Set-Point Control Containing a Compact Continuum of Finite-Dimensional Linear Controllers A. S. Morse Yale Uniersit Ma 13, 2004 Abstract A simple analsis is gien of the behaior of a set-point control sstem consisting of a poorl modelled process, an integrator and a multi-controller superised b an estimator-based algorithm emploing dwell-time switching. For a slowl switched multi-controller implementation of a compact continuum of finite-dimensional linear controllers, bounds are deried for the normed-alue of the process s allowable un-modelled dnamics as well as for the sstem s disturbance-to-tracking error exponentiall-weighted induced L 2 gain. 1 Introduction Much has happened in adaptie control in the last quarter centur. The solution to the classical model reference problem is b now er well understood. Proabl correct algorithms exist which, at least in theor, are capable of dealing with un-modelled dnamics, noise, right-half-plane zeros, and een certain tpes of nonlinearities and a number of excellent texts and monographs hae been written coering man of these adances [1, 2, 3, 4, 5, 6, 7, 8]. Howeer despite these impressie gains, there remain man important, unanswered questions: Wh, for example, is it still so difficult to explain to a noice wh a particular algorithm is able to functions correctl in the face of un-modelled process dnamics and L bounded noise? How much un-modelled dnamics can a gien algorithm tolerate before loop-stabilit is lost? How do we choose an adaptie control algorithm s man design parameters to achiee good disturbance rejection, transient response, etc.? It is our iew that eentuall there will be satisfactor answers to all of these questions, that adaptie control will become much more accessible to non-specialists, that we will be able to much more clearl and concisel quantif un-modelled dnamics norm bounds, disturbance-to-controlled output gains, and so on and that because of this we will see the emergence of a bona fide computeraided adaptie control design methodolog which relies much more on design principals then on trial and error techniques. It is with these ends in mind, that this paper has been written. In the sequel we proide a relatiel uncluttered analsis of the behaior of a set-point control sstem consisting of a poorl modelled process, an integrator and a multi-controller superised b an This research was supported b the National Science Foundation 1

2 estimator-based algorithm emploing dwell-time switching. The sstem has been considered preiousl in [9]. It has been analzed in one form or another in [10, 11, 12, 13] and elsewhere under arious assumptions. It has been shown in [12] that the sstem s superisor can successfull orchestrate the switching of a sequence of candidate set-point controllers into feedback with the sstem s imprecisel modelled siso process so as (i) to cause the output of the process to approach and track a constant reference input despite norm-bounded un-modelled dnamics, and constant disturbances and (ii) to insure that none of the signals within the oerall sstem can grow without bound in response to bounded disturbances, be the constant or not. The objectie of this paper is to re-derie these same results in a much more straight forward manner. In fact this has alread been done in [14] for a superisor control sstem in which the number of candidate controllers is finite, and the switching between candidate controllers is constrained to be slow. These restrictions not onl greatl simplified the analsis in comparison with that gien in [12], but also made it possible to derie reasonabl explicit upper bounds for the process s allowable un-modelled dnamics as well as for the sstem s disturbance-to-tracking error gain. In this paper we also constrain switching to be slow, but allow for infinitel man candidate controllers. Infinitel man controllers becomes essential when the class of candidate nominal process models is so large that it cannot be coered b a finite class of controllers in the sense of [15]. Tpicall this occurs when the nominal process model class contains a large compact continuum. Although the problem considered here differs from that considered in [14], it turns out that the analses in both cases is almost the same. In fact the onl significant difference between the two papers are the was in which each characterize dwell time switching in terms of norm inequalities. See in particular Lemma 1 in this paper and the same numbered lemma in [14]. The oerall superisor control sstem to be considered is described in 2. The main theorem characterizing the sstem s behaior is re-stated in 3. A simple, informal proof of the theorem is carried out in 4. Explicit bounds for the process s allowable unmodelled dnamics as well as for the sstem s disturbance-to-tracking error gain appear (27) and (30) respectiel. 2 The Oerall Sstem The aim of this section is to describe the structure of the superisor control sstem to be considered in this paper. We begin with a description of the process. 2.1 The Process The oerall problem of interest is to construct a control sstem capable of driing to and holding at a prescribed set-point r, the output of a process modelled b a dnamical sstem with large uncertaint. The process is presumed to admit the model of a siso linear sstem Σ P whose transfer function from control input u to measured output is a member of a continuousl parameterized class of admissible transfer functions of the form C P = {ν p δ : δ ɛ p } p P 2

3 where P is a compact subset of a finite dimensional space, ν p = α p β p is a prespecified, strictl proper, nominal transfer function, ɛ p is a real non-negatie number, δ is a proper stable transfer function whose poles all hae real parts less than the negatie of a prespecified stabilit margin λ > 0, and is the shifted infinit norm δ = sup δ(jω λ) ω IR It is assumed that the coefficients of α p and β p depend continuousl on p and for each p P, that β p is monic and that α p and β p are coprime. All transfer functions in C P are thus proper, but not necessaril stable rational functions. Prompted b the requirements of set-point control, it is further assumed that the numerator of each transfer function in C P is nonzero at s = 0. The specific model of the process to be controlled is shown in Figure 1. u d ν p* δ Figure 1: Process Model Here is the process s measured output and d is a disturbance. 2.2 The Sstem to Be Superised Presumed gien is an continuousl parameterized famil of off-the-shelf loop controller transfer functions K = {κ p : p P} with at least the following propert: Stabilit Margin Propert: For each p P, λ is greater than the real parts of all of the closedloop poles 1 of the feedback interconnection κ p ν p 1 s Figure 2: Feedback Interconnection Also presumed gien is an integer n C 0 and a continuousl parameterized famil of n C -dimensional realizations {A p, b p, f p, g p }, one for each κ p K. These realizations are required to be chosen so that for each p P, (c p, λi A p ) is detectable and (λi A p, b p ) is stabilizable. As noted in [9], there are a great man different was to construct such realizations, once one has in hand an upper bound n κ on the McMillan Degrees of the κ p. Gien such a famil of realizations, the sub-sstem to be superised 3

4 r e T Σ (σ) C 1 s u Σ P Figure 3: Superised Sub-Sstem is thus of the form shown in Figure 3 where Σ C (σ) is the n C -dimensional state-shared dnamical sstem ẋ C = A σ x C b σ e T = f σ x C g σ e T, (1) called a multi-controller, is the input to the integrator e T is the tracking error and σ is a piecewise constant switching signal taking alues in P. u =, (2) e T = r, (3) 2.3 The Superisor The problem of interest is to construct a proabl correct superisor which is capable of generating σ so as to achiee 1. global boundedness {of all sstem signals} in the face of an arbitrar but bounded disturbance inputs and 2. set-point regulation {i.e., e T 0} in the eent that the disturbance signal is constant. To understand the idea behind the superisor we ultimatel intend to consider, assume temporaril that P is a finite set with m elements and consider the sstem shown in Figure 4. Here each p is a suitabl defined estimate of which estimator p 1 estimator p 2 p1 p2 e p1 e p2 π p1 π p2 estimator p i pi e pi π pi Σ S σ estimator p m pm e pm π pm Figure 4: Estimator-Based Superisor would be asmptoticall correct if ν p were the process model s transfer function and there were no noise or disturbances. For each p P, e p = p denotes the pth output estimation error and π p is a norm - squared alue of e p or a performance signal which is used b the superisor to assess the potential performance of controller p. Σ S is a switching logic whose function is to determine σ on the basis of the current alues of the π p. The underling decision making strateg used b such a superisor is basicall this: From time to time select for σ, that candidate control index q whose corresponding performance signal π q is the smallest among the π p, p P. Motiation for this idea is obious: the nominal process model whose associated performance 1 B the closed-loop poles are meant the zeros of the polnomial sρ pβ p γ pα p, where αp β p transfer functions ν p and κ p respectiel. and γp ρ p are the reduced 4

5 signal is the smallest, best approximates what the process is and thus the candidate controller designed on the basis of that model ought to be able to do the best job of controlling the process. The actual superisor to be considered has a different realization than that shown in Figure 4, one which can be implemented een when P contains a continuum of points. Internall the superisor we want to discuss consists of three subsstems: a multi-estimator Σ E, a weight generator Σ W, and a dwell-time switching logic Σ D. Σ E is a n E -dimensional linear dnamical sstem of the form Σ E x E Σ W W Σ D σ Figure 5: Estimator-Based Superisor [ AE 0 ẋ E = 0 A E ] [ ] [ ] be 0 x E (4) 0 b E where n E = 2(nν 1) and (A E, b E ) is a parameter-independent, n ν 1-dimensional siso, controllable pair with λi A E stable. Here n ν is an upper bound on the McMillan Degrees of the ν p, p P. In [9] it is explained how to construct a continuous function p c p so that for each p P, {[ ] [ ] [ ] } AE 0 be 0 c 0 A E 0 p,, c b p E is a stabilizable realization of 1 s ν p whose uncontrollable eigenalues hae real parts less than λ. The c p are used in the definition of Σ W which will be gien in a moment. The c p also enable us to define output estimation errors e p = cp x E, p P (5) While these error signals are not actuall generated b the superisor, the pla an important role in explaining how the superisor functions. The superisor s second subsstem, Σ W, is a causal dnamical sstem whose inputs are x E and and whose state and output W is a weighting matrix which takes alues in a linear space W. W together with a suitabl defined performance function Π : W P IR determine a scalar-alued performance signal of the form π p = Π(W, p) (6) which is iewed b the superisor as a measure of the expected performance of controller p. Σ W and Π are defined b [ ] [ ] xe xe Ẇ = 2λW (7) and Π(W, p) = [ c p 1 ] W [ c p 1 ] (8) respectiel. The definitions of Σ W and Π are prompted b the obseration that if π p are gien b (6), then because of (5), (7) and (8). π p = 2λπ p e 2 p, p P (9) The superisor s third subsstem, called a dwell-time switching logic Σ D, is a hbrid dnamical sstem whose input and output are W and σ respectiel, and whose state is the ordered triple {X, τ, σ}. Here X is a discrete-time matrix which takes on sampled alues of W, and τ is a continuous-time ariable called a timing signal. τ takes alues in the closed interal [0, τ D ], where τ D is a pre-specified positie number called a dwell 5

6 time. Also assumed pre-specified is a computation time τ C τ D which bounds from aboe for an X W, the time it would take a superisor to compute a alue p = p X P which minimizes Π(X, p). Between eent times, τ is generated b a reset integrator according to the rule τ = 1. Eent times occur when the alue of τ reaches either τ D τ C or τ D ; at such times τ is reset to either 0 or τ D τ C depending on the alue of Σ D s state. Σ D s internal logic is defined b the computer diagram shown in Figure 6 where p X denotes a alue of p P which minimizes Π(X, p). Initialize σ τ = 0 τ = τ D τ C n σ = p X X = W τ = τ D τ C τ = τ D n n Π(X, p ) < Π(X, σ ) X Figure 6: Computer Diagram of Σ D Let us note that implementation of the superisor just described can be accomplished een when P contains infinitel man points. Howeer for the required minimization of Π(X, p) to be tractable, it will tpicall be necessar to make assumptions about both c p and P. For example, if c p is an affine linear function and P is a finite union of conex sets, the minimization of Π(X, p) will be a conex programming problem. In the sequel we call a piecewise-constant signal σ : [0, ) P admissible if it either switches alues at most once, or if it switches more than once and the set of time differences between each two successie switching times is bounded below b τ D. We write S for the set of all admissible switching signals. Because of the definition of Σ D, it is clear its output σ will be admissible. This means that switching cannot occur infinitel fast and thus that existence and uniqueness of solutions to the differential equations inoled is not an issue. 3 Discussion The oerall sstem just described, admits a block diagram description of the form Σ W W x E Σ E Σ D d r e T σ Σ (σ) C 1 s u Σ P Figure 7: Superisor Control Sstem 6

7 The following theorem is proed in [12]: Theorem 1 Let τ C 0 be fixed. Let τ D be an positie number no smaller than τ C. There are positie numbers ɛ p, p P, for which the following statements are true proided Σ P has a transfer function in C P. 1. Global Boundedness: For each constant set-point alue r, each bounded piecewise-continuous disturbance input d, and each sstem initialization, u, x C, x E, W, and X are bounded responses. 2. Tracking and Disturbance Rejection: For each constant set-point alue r, each constant disturbance d, and each sstem initialization, tends to r and u, x C, x E, W, and X tend to finite limits, all as fast as e λt. The theorem implies that the oerall superisor control sstem shown in Figure 7 has the basic properties one would expect of a non-adaptie linear set-point control sstem. It will soon become clear if it is not alread that the induced L 2 gain from d to e T is finite as is the induced L gain from d to an state ariable of the sstem. 4 Analsis The aim of this section is to re-derie Theorem 1 in a much more straight forward manner than in [12]. This will be done for a superisor control sstem in which the switching between candidate controllers is constrained to be slow in a sense to be made precise in 4.1. This restrictions not onl greatl simplifies the analsis, but also make it possible to derie reasonabl explicit bounds for the process s allowable un-modelled dnamics as well as for the sstem s disturbance-to-tracking-error gain. In the sequel we will inariabl ignore initial condition dependent terms which deca to zero as fast as e λt, as this will make things much easier to follow. A more thorough analsis which would take these terms into account can carried out in essentiall the same manner. 4.1 Slow Switching Assume that r is a constant and let x denote the composite state [ ] xe x = x C (10) where x E is the shifted state [ A 1 x E = x E E b ] E r (11) 0 It is then possible to show in a straightforward manner, that for an q P and an gien piecewise constant switching signal σ : [0, ) P, whether generated b Σ D or not, the relationships between e q, e σ,, and e T determined b (1)-(5) are gien b a sstem of equations of the form e σ = c σq x e q ẋ = A σσ x h σ e σ = f σσ x g σ e σ e T = e σ c σ x where d = column{b E, 0, 0, b C }, b = column{0, b E, b C, 0}, and for all p, l P, f pl = [ gp c l f p ] h p = bgp d c pl = [ cp c l 0 ] c l = [ cl 0 ] (13) (12) 7

8 and A pl = block diagonal {AE, A E, A C, A C } d [ c l 0 ] bf pl One readil erifiable and important propert of the matrices defined aboe is that for each p, l P, (c pl, λia pl ) is a detectable matrix pair [9]. This is a consequence of certaint equialence, the Stabilit Margin Propert, the requirements that λi A E be a stabilit matrix and that for p P, {λi A p, b p, f p, g p } be a stabilizable and detectable sstem. Since c pp = 0, p P, this means that for each such p, λi A pp must be a stabilit matrix [9]. In the sequel we will assume the following. Slow Switching Assumption:The dwell time τ D is large enough so that for each admissible switching signal σ : [0, ) P, λi A σσ is an exponentiall stable matrix. It is possible to compute an explicit lower bound for τ D for which this assumption holds [9]. 4.2 Block Diagram I Using the diagram of Σ P in Figure 1 together with (1)-(5) and (12), it is not difficult to erif that, up to initial condition dependent terms decaing to zero as fast at e λt, the relationships between d, e σ,, and e T are as shown in the block diagram in Figure 8 where ω E is the characteristic polnomial of the estimator matrix A E [9]. In deeloping this diagram we e represented the sstem defined b (12) as two separate subsstems, namel ẋ 1 = A σσ x 1 h σ e σ ẋ 2 = A σσ x 2 h σ e σ = f σσ x 1 g σ e σ e T = c σ x 2 e σ where x 1 = x 2 = x. Note that the signal in the block diagram labelled z, will tend to zero if d is constant because of the zero at s = 0 in the numerator of the transfer function in the block drien b d. A σσ h σ sα p d z e p e σ e ω E T c σ 1 β p ω E δ A σσ f σσ h σ g σ Figure 8: Block Diagram I 4.3 Norms It is especiall useful to introduce the following. For an piecewise-continuous function z : [0, ) IR n, and an times t 2 > t 1 0, let us write z {t1,t 2} for the exponentiall weighted 2-norm z {t1,t 2} = t2 e 2λt z(t) 2 dt Note that e 2λT π p (T ) = e p 2 {0,T }, T 0, p P, because of (9) and the assumption that W (0) = 0. t 1 8

9 For an time-aring SISO linear sstem Σ of the form = c(t)x d(t)u, ẋ = A(t)x b(t)u we write A b c d for the induced norm sup{ u {0, } : u U} where u is Σ s zero initial state, output response to u and U is the space of all piecewise continuous signals u such that u {0, } = 1. The induced norm of Σ is finite wheneer λi A(t) is {uniforml} exponentiall stable. We note the following easil deried facts. If e λt u {0,t} is bounded on [0, ) {in the L sense}, then so is u proided d = 0 and λi A(t) is exponentiall stable. If u is bounded on [0, ) in the L sense, then so is e λt u {0,t}. If u 0 as t, then so does e λt u {0,t}. Let us note that for an gien admissible switching signal σ, each of the four blocks in Figure 8 represents an exponentiall stable linear sstem. It is conenient at this point to introduce certain worst case sstem gains associated with two of these blocks. In particular, let us define for p P sα p a p = 2 c = A σσ h σ 2 sup ω E σ S c σ 1 where, as defined earlier, is the shifted infinit norm and S is the set of all admissible switching signals. In the light of Figure 8, it is eas to see that e T {0, t} c 2 e σ {0, t}, t 0, (14) and z {0,t} a p 2 d {0,t}, t 0 (15) The inequalit in (14) bounds the norm of e T in terms of the norm of e σ whereas (15) bounds the norm of z in terms of the norm d. To deelop a bound for the norm of e T in terms of the norm of d, it would therefore be enough to establish a bound for the norm of e σ in terms of the norm of z. As a first step toward this end, we shall make use of the following result which is a direct consequence of dwell time switching. 4.4 Dwell-Time Switching Lemma 1 Suppose that P is a compact subset of a finite dimensional space, that p c p is a continuous function taking alues in IR 1 ne, that c pq is gien b (13), that W is generated b (7), that the π p, p P, are defined b (6) and (8), that W (0) = 0, and that σ is the response of Σ D to W. For each q P, each real number µ > 0 and each fixed time T > 0, there exists piecewise-constant signals f : [0, ) IR 1 (nenc) and ψ : [0, ) {0, 1} such that f(t) µ, t 0 (16) and (1 ψ)(e σ fx) ψe q {0,T } 0 ψ(t)dt n E (τ D τ C ) (17) { ( ) ne } 1 supp P c pq 1 2n E e q {0,T } (18) µ This lemma is proed in the appendix. Lemma 1 captures the essential properties of dwell time switching needed to analze the sstem under consideration. The ke inequalit is (18). The terms inoling f and ψ in this inequalit present some minor difficulties which we will deal with next. 9

10 4.5 Block Diagrams II and III Let us note that for an piece-wise continuous matrix-alued signal f : [0, ) IR 1 (n En C), it is possible to re-write the equations in (12) with q = p as e σ = fx ē ẋ = (A σσ h σ f)x hσ ē (19) = (f σσ g σ f)x gσ ē e T = ( f c σ )x ē where ē = (c σp f)x e p (20) Note that the matrix λi A σσ h σ f will be exponentiall stable for an σ S if f µ, t 0, where µ is a sufficientl small positie number. It is alwas possible to find such a number because A pp and h p are bounded on P and because λi A σσ is exponentiall stable for all admissible switching signals. In the sequel we will assume that µ is such a number and that F is the set of all piece-wise continuous signals f satisfing f µ, t 0. To account for f in (18), we will use in place of the diagram in Figure 8, the diagram shown in Figure 9. In deeloping this diagram we e represented the sstem defined b (19) as two separate subsstems, namel ẋ 1 = (A σσ h σ f)x1 h σ ē ẋ 2 = (A σσ h σ f)x2 h σ ē = (f σσ g σ f)x1 g σ ē e σ = fx 2 ē where x 1 = x 2 = x. d sα p ω E z A σσ h σ f e p ē e σ hσ f 1 β p ω E δ A σσ h σ f f σσ g σ f hσ gσ Figure 9: Block Diagram II Let us note that for an gien f F and an admissible switching signal σ, each of the four blocks in Figure 9 represents an exponentiall stable linear sstem. It is conenient at this point to introduce additional worst case sstem gains associated with two of these blocks. b p = 2 β p ω E sup A σσ h σ f sup f F σ S f σσ g σ f In the light of Figure 9, it is eas to see that hσ gσ d = A σσ h σ f hσ sup sup f F σ S f 1 e σ {0, t} d ē {0, t}, t 0, (21) and b p e p {0, t} ɛ p ē {0, t} z {0, t}, t 0, (22) 2 where ɛ p is the norm bound on δ. The inequalit in (21) bounds the norm of e σ in terms of the norm of ē whereas the inequalit in (22) bounds the norm of e p in terms of the norms of ē and d. To deelop a bound 10

11 for the norm of e σ in terms of the norm of z, it would therefore be enough to establish a bound for the norm of ē in terms of the norm of z. In iew of (22), this can be accomplished deeloping a bound for the norm of ē in terms of a of the norm of e p, proided ɛ p is sufficientl small. As a first step toward this end, we shall make use of Lemma 1. Let us fix T > 0. In iew of (18) there is a piecewise constant signals f F and ψ : [0, ) {0, 1} satisfing (17), such that (1 ψ)ē ψe p {0,T } γ e p {0,T } (23) where ( ) ne 1 supp P c pp γ = 1 2n E µ and, as in (20), ē = e σ fx. What we need is a bound for the norm of ē. What we hae in (23) is a bound for the norm of (1 ψ)ē ψe p. To get what we need, we first note from (19) and (20) that ē e p = (c σp f)x ẋ = (A σσ h σ f)x hσ ē (24) Consider the block diagram in Figure 10 which depicts this sub-sstem together with an additional block and summing junctions representing the formulas ē = ψē (1 ψ)ē and ē = e p ē e p (1 ψ)ē e p ē ψ ē A σσ h σ f hσ ē e p c σp f 0 Figure 10: Block Diagram III Let us define for q P q = sup sup f F σ S sup t 0 t 0 w q (t, τ)e λ(tτ) 2 dτ where w q (t, τ) = c σ(t)q Φ(t, τ)h σ(τ) and Φ(t, τ) is the state transition matrix of A σσ. Note that each q is finite because of the Slow Switching Assumption and the fact that f F. Using Cauch-Schwartz it can easil be shown with q so defined that ψ(w q ē) {0, t} q t where w q ē is the zero initial state output response of (24). 0 ψ 2 ē 2 {0, µ} dµ, t 0 (25) From Figure 10 it is clear that ē = ψ(e p w p ē) (1 ψ)ē 11

12 Rearranging terms and taking norms we thus obtain ē {0, t} (1 ψ)ē ψe p {0, t} ψ(w p ē) {0, t}, t 0 Moreoer (1 ψ)ē e p {0,t} (1 ψ)ē e p {0,T }, t [0, T ]. Using (23) we thus get ē {0, t} γ e p {0,T } ψ(w p ē {0, t}, 0 t T Taking squares ē 2 {0, t} 2 γ2 e p 2 {0, T } 2 ψ(w p ē) 2 {0, t}, 0 t T Using (25) with q = p t ē 2 {0, t} 2 γ2 e p 2 {0, T } 2 p ψ 2 ē) 2 {0, µ} dµ, 0 t T Hence b the Bellman-Gronwall Lemma 0 ē 2 {0, T } 2 γ2 e p 2 {0, T } e2 p T 0 ψ2 dt From this, (17), and the fact that ψ 2 = ψ, we arrie at an expression for the norm of ē in terms of e p, namel ē {0, T } 2 γe p ne(τdτc) e p {0, T } (26) 4.6 Stabilit Margin Examination of (26) and (22) reeals that if ɛ p satisfies the small gain condition ɛ p < e p n E(τ Dτ C) b p γ (27) then (26) and (22) can be combined to gie This and (21) gie finall ē {0, T } e σ {0,T } 2 e p n E (τ D τ C ) γ d 2 e p n E (τ D τ C ) γ ɛ p b p ɛ p b p z {0, T }, (28) z {0, T }, (29) Note that this inequalit holds for all T 0. The inequalit in (27) proides an explicit bound for the allowable process dnamics. 4.7 Global Boundedness The global boundedness condition of Theorem 1 can now easil be justified as follows. Suppose d is bounded on [0, ). Then so must be e λt z {0, t}. Hence b (29), e λt e σ {0, t} must be bounded on [0, ) as well. This, the differential equation for x in (12), and the exponential stabilit of λi A σσ then impl that x is also bounded on [0, ). In iew of (10) and (11), x E and x C must also be bounded. Next recall that the zeros of ω E {i.e., the eigenalues of A E } hae negatie real parts less than λ, and that the transfer function β p ω E δ in Figure 8 is strictl proper. From these obserations and the block diagram in Figure 8 one readil concludes that e p is bounded on [0, ). Hence from the formulas in (12) for e σ, and e T one concludes that these signals are also bounded. In iew of (3), must be bounded. Thus W must be bounded because of (7). Finall note that u must be bounded because of the boundedness of and and because of the obserabilit of the cascade interconnection of (2) with an minimal realization of Σ P. This, in essence, proes Claim 1 of Theorem 1. 12

13 4.8 Conergence Now suppose that d is a constant. As alread noted, the signal z shown in Figure 8 must tend to zero as fast as e λt because of the zero at s = 0 in the numerator of the transfer function from d to z. This implies that z {0, } <. Therefore e σ {0, } < because of (29). Hence e σ must tend to zero as fast as e λt. So therefore must x because of the differential equation for x in (12). In iew of (10) and (11) x E and x C must tend to zero as well. From Block Diagram I in Figure 8 it now can be seen that e p tends to zero. Hence from the formulas in (12) for e σ, and e T one concludes that these signals must tend to zero as well. In iew of (3), must tent to r. Thus W must approach a finite limit because of (7). Finall note that u tend to a finite limit because and do and because of the obserabilit of the cascade interconnection of (2) with an minimal realization of Σ P. This, in essence, proes Claim 2 of Theorem A Bound on the Disturbance - to - Tracking - Error Gain B combining the inequalities in (14), (15) and (29) we obtain an inequalit of the form e T {0, T } g p d {0, T }, T 0 where g p = ca p e p n E (τ D τ C ) γ ɛ p b p (30) Thus g p bounds from aboe the oerall sstem s disturbance - to - tracking - error gain. 5 Concluding Remarks The formula for g p in (30) and the stabilit margin bound in (27) are probabl the most explicit discoered so far for an estimator-based adaptie control sstem with the properties outlined in Theorem 1. We beliee that een simpler expressions than these can be found for the sstem under consideration. Results such as these suggest that a bona fide, input-output performance theor for adaptie control ma be within our reach. 6 Appendix In the sequel, σ is a fixed switching signal, t 0 = 0, ti denotes the ith time at which σ switches and p i is the alue of σ on [t i1, t i ); if σ switches at most n < times then t n1 = and pn1 denotes σ s alue on [t n, ). An time X takes on the current alue of W is called a sample time. We use the notation t to denote the sample time just preceding time t, if t > τ D τ C, and the number zero otherwise. Thus, for example, t 0 = 0 and t i = t i τ C, i > 0. To proe Lemma 1 will need the following result which can be easil deduced from the discussion about strong bases in section VIII part B. of [12]. Lemma 2 Let ɛ be a positie number and suppose that X = {x 1, x 2,... x m } is an finite set of ectors in a real n-dimensional space such that x m > ɛ. There exists a subset of m n positie integers N = {i 1, i 2,..., i m }, each no larger than m, and a set of real numbers a ij, i M = {1, 2,... m}, j N such that x i a ij x j j N ɛ, i M 13

14 where a ij = 0, i M, j N, i > j (1 sup X )n a ij, i M, j N ɛ Proof of Lemma 1: Let k be that integer for which T [t k1, t k ) and define K = {1, 2,..., k}. There are two cases to consider: Case I: Suppose that c piq µ for i K. In this case set ψ(t) = 0, t 0, f(t) = cσ(t)q for t [0, t k ), and f(t) = 0 for t > t k. Then (17) and (16) hold and e σ = fxe q for t [0, T ). Therefore e σ fx {0,T } = e q {0,T } and (18) follows. Case II: Suppose that there is a largest integer m K such that c pmq > µ. We claim that there is a nonnegatie integer m n E, a set of m positie integers N = {i 1, i 2,..., i m }, each no greater than k, and a set of piecewise constant signals γ j : [0, ) IR, j N, such that c σ(t)q γ j (t)c pjq µ, 0 t T (31) j N where for all j N γ j (t) = 0, t (t j, ) (32) ( ) ne 1 supp P c pq γ j (t), t [0, t j ) (33) µ To establish this claim, we first note that {c p1q, c p2q,..., c pmq} {c pq : p P} and that {c pq : p P} is a bounded subset of an n E dimensional space. B Lemma 2 we thus know that there must exist a subset of m n E integers N = {i 1, i 2,..., i m }, each no greater than m, and a set of real numbers g ij, i M = {1, 2,..., m}, j N such that where c p iq j N g ij c pjq µ, i M (34) g ij = 0, i M, j N, i > j, (35) g ij (1 sup p P c pq ) ne, i M, j N (36) µ Thus if for each j N, we define γ j (t) = g ij, t [t i1, t i ), i M, and γ j (t) = 0, t > t m then (31) - (33) will all hold. To proceed, define f(t) for t [0, t m ) so that f(t) = c piq j N g ij c pjq, t [t i1, t i ), i M and for t > t m so that c piq t [t i1, t i ) i {m 1,..., k} f(t) = 0 t > t k 14

15 Then (16) holds because of (31) and the assumption that c piq µ for i {m 1,..., k}. The definition of c T implies that c σ(t)q f(t) = γ j (t)c pjq, t [0, T ) j N and thus that e σ(t) (t) e q (t) f(t)x(t) = j N γ j (t)(e pj (t) e q (t)), t [0, T ) (37) For each j N define t j = { t j T if j < k if j = k The definition of dwell-time switching then implies that for j N, π pj ( t j1 ) π q ( t j1 ), q P, π pj ( t j τ C ) π q ( t j τ C ), q P if t j t j1 > τ D Using the fact that e 2λt π p (t) = e p 2 {0,t}, p P, t 0, we obtain e pj 2 {0, t j1 } e q 2 {0, t j1 }, q P e pj 2 {0, t j τ C} e q 2 {0, t j τ C}, q P, if t j t j > τ D (38) For each j N, let φ j : [0, ) {0, 1} be that piecewise-constant signal which is zero eerwhere except on the interal [ t j, t j ), if t j t j1 τ D or [ t j τ C, t j ), if t j t j1 > τ D In either case φ j has support no greater than τ D τ C and is idempotent {i.e., φ 2 j = φ j}. It follows that if ψ = 1 (1 φ j ), (39) j N then ψ T is also idempotent and has support no greater than m(τ D τ C ). In iew of the latter propert and the fact that m n E, (17) must be true. The definitions of the φ j impl that for j N and l P From this and (38) we obtain for all q P (1 φ j )e l 2 {0, t j} = { (1 φj )e l 2 {0, t j1 } if t j t j1 τ D (1 φ j )e l 2 {0, t j τ C} if t j t j1 > τ D if t j t j1 τ D and (1 φ j )e pj 2 {0, t j} = e p j 2 {0, t j1 } e q 2 {0, t j1 } e q 2 {0, t j} (1 φ j )e pj 2 {0, t j} = e p j 2 {0, t jτ C } e q 2 {0, t jτ C } e q 2 {0, t j} if t j t j1 > τ D. From this and the fact that e q 2 {0, t j} e q 2 {0,T }, q P, j N 15

16 there follows (1 φ j )e pj {0, t j} e q {0,T }, j N, q P From this and the triangle inequalit (1 φ j )(e pj e q ) {0, t j} 2 e q {0,T }, j N, q P (40) But From (37) (1 ψ)(e σ e q fx) {0,T } = (1 ψ)γ j (e pj e q ) (1 ψ)γ j (e pj e q ) {0,T } (41) j N {0,T } because of (32). In iew of (33) where j N (1 ψ)γ j (e pj e q ) {0,T } = (1 ψ)γ j (e pj e q ) {0,tj} (1 ψ)γ j (e pj e q ) {0,T } γ (1 ψ)(e pj e q ) {0,tj} (42) γ = (1 sup p P c pq ) ne µ Now (1 ψ)(e pj e q ) {0,tj} (1 φ j )(e pj e q ) {0,tj} because of (39). From this, (40) and (42) it follows that (1 ψ)γ j (e pj e q ) {0,T } 2 γ e q {0,T } In iew of (41) and the fact that m n E, it follows that But so b the triangle inequalit (1 ψ)(e σ e q fx) {0,T } 2n E γ e q {0,T } (1 ψ)(e σ fx)) ψe q = (1 ψ)(e σ e q fx) e q (1 ψ)(e σ fx)) ψe q {0,T } (1 ψ)(e σ e q fx) {0,T } e q {0,T } Therefore and (18) is true. (1 ψ)(e σ fx) ψe q {0,T } (1 2n E γ) e q {0,T } References [1] G. C. Goodwin and K. S. Sin. Adaptie Filtering Prediction and Control. Prentice Hall, [2] K. J. Ȧström and B. Wittenmark. Adaptie Control. Addison-Wesle, [3] Shankar Sastr and Mark Bodson. Adaptie Control: Stabilit, Conergence and Robustness. Prentice Hall, Englewood Cliffs, New Jerse, [4] K. S. Narendra and A. M. Annaswam. Stable Adaptie Sstems. Prentice Hall, [5] J. W. Polderman and I. Mareels. Adaptie Sstems - An Introduction. Birkhäuser, [6] P. A. Ioannou and J. Sun. Robust Adaptie Control. Prentice Hall,

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