Closed loop Reference Models for Output Feedback Adaptive Systems

Transcription

1 203 European Control Conference (ECC July 7-9, 203, Zürich, Switzerl Closed loop Reference Models for Output Feedback Adaptive Systems Travis E Gibson, Anuradha M Annaswamy, Eugene Lavretsky Abstract Closed loop reference models have recently been proposed for states accessible adaptive systems They have been shown to have improved transient response over their open loop counter parts The results in the states accessible case are extended to single input single output plants of arbitrary relative degree I INTRODUCTION Recently a class of adaptive controllers with Closed loop Reference Models (CRM for states accessible control has been proposed [] [7] The main feature of this class is the inclusion of a feedback gain in the reference model Without the feedback gain the CRM reduces to the Open loop Reference Model (ORM which is used in classical adaptive control [8], [9] The main advantage of the CRM-adaptive systems is their ability to shape improve the transient response of the adaptive system This was demonstrated in [3] [5] through the use of L 2 norms of the model following error, the derivative of the adaptive parameter the rate of control input, improved further in in [3] [5] with an analytical justification for the reduction in high frequency oscillations which are conspicuously absent in CRM systems References [6], [7] also addressed a peaking phenomenon that occurs in CRM systems In [3] [7], it was shown that the extra design freedom in the adaptive system in the form of the feedback gain in the reference model allowed this improvement Other recent works on states accessible CRM adaptive control can be found in [0], [] This paper addresses the next step in the design of adaptive systems, which is the case when only outputs are available for measurement rather than the entire state It is shown that even with output feedback, the resulting CRM based adaptive systems are first foremost stable, exhibit an improved transient response As in the case when states are accessible, it is shown that this improvement is possible due to the suitable choice of the feedback gain in the reference model Unlike the approach in [2], the classical model reference adaptive control structure is used here, with the focus on single-input single-output systems Similar to the states-accessible case, the CRM-based adaptive systems with output feedback presented here have the advantage of an improved transient response This is made possible by the introduction of an extra degree of freedom in the reference model The introduction of this degree of freedom reduces the burden on the adaptive controller by T E Gibson A M Annaswamy are with the Department of Mechanical Engineering, Massaschusetts Institute of Technology, Cambridge, MA, 0239 USA (tgibson@mitedu E Lavretsky is with The Boeing Company, Huntington Beach, CA USA allowing the reference model to meet the plant half-way Also, this degree of freedom allows the trade-off between speed of adaptation the size of the parametric uncertainty to be relaxed enables reduced oscillations In the specific context of output feedback, the advantage of the CRM over the ORM manifests in the form of judicious selections of the reference model, filters, the feedback gain These allow the underlying transfer function in the error model to not only be strictly positive real, but also to have poles zeros that are arbitrarily fast In addition, the CRM allows the analysis design of a minimal representation of the adaptive system, removes the restriction that ORM-based adaptive systems often possess, which is due to the location of the eigenvalues of the nonobservable states of the underlying adaptive system The CRM-based design overall analysis of stability transient response are presented in this paper Both cases when the relative degree is unity arbitrary are analyzed This paper is organized as follows Section II contains the notation In Section III the control problem is defined Section IV contains the analysis of the ORM with relative degree Section V contains the analysis of the CRM with relative degree Section VI analysis the arbitrary relative degree case, Section VII closes with our conclusions II NOTATION All norms unless otherwise stated are the Euclidean norm enduced Euclidean norm Let PC [0, denote the set of all bounded piecewiese continuous signal Definition : Let x,y PC [0, The big O notation, y(t = O[x(t] is equivalent to the existence of constants M,M 2 > 0 t 0 R + such that y(t M x(t +M 2 t t 0 Definition 2: Let x,y PC [0, The small o notaion, y(t = o[x(t] is equivalent to the existence of constants β(t PC [0, t 0 R + such that y(t = β(tx(t t t 0 lim t β(t = 0 Definition 3: Let x,y PC [0, If y(t = O[x(t] x(t = O[y(t] Then x y are said to be equivalent denoted as x(t y(t Definition 4: Let x,y PC [0, x y are said to grow at the same rate if sup t τ x(τ sup t τ y(τ Definition 5: The prime notation is an operator that removes the high frequency gain from a transfer function so that W(s k sm +b s m 2 + b m s m +a s m + +a m W (s W(s k / 203 EUCA 365

2 III THE CONTROL PROBLEM Consider the Single Input Single Output (SISO system of equations y(t = W(su(t ( u R is the input, y R is the measurable output, s the differential operator The transfer function of the plant is parameterized as W(s k p Z(s P(s k pw (s (2 k p is a scalar, Z(s P(s are monic polynomials with deg(z(s < deg(p(s The following assumptions will be made throughout Assumption : W(s is minimum phase Assumption 2: The sign of k p is known Assumption 3: The relative degree of W(s is known The goal is to design a control input u so that the output y in ( tracks the output y m of the reference system Z m (s y m (t = W m (sr(t k m r(t (3 P m (s k m is a scalar Z m (s P m (s are monic polynomials with W m (s relative degree n In Sections IV through VI, we present the control designs for n = for an arbitrary n IV n = AND ORM Assumption 4: W m (s is Strictly Positive Real (SPR The structure of the adaptive controller is now presented: ω (t = Λω +b λ u(t (4 ω 2 (t = Λω 2 +b λ y(t (5 ω(t [r(t, ω T (t, y(t, ωt 2 (t]t (6 θ(t [k(t, θ T (t, θ 0 (t, θ T 2 (t] T (7 u = θ T (tω (8 Λ R (n (n is Hurwitzx, b λ R n, ˆk R, ω,ω 2 R n, θ R 2n is adaptive gain vector with k(t R, θ (t R n, θ 2 (t R n θ 0 (t R The update law for the adaptive parameter is then defined as θ(t = γsign(k p e y ω, (9 e y = y y m Before stability is proved, a discussion on parameter matching is needed Let θ c [k c, θ T c, θ 0c, θ T 2c ]T be a constant vector When θ(t = θ c the forward loop feedback loop take the form λ(s λ(s C(θ c ;s D(θ c;s λ(s For simplicity we choose λ(s = Z m (s, but note that this is not necessary the stability of the adaptive system will still hold The closed loop system is now of the form y(t = W cl (θ c ;sr(t with W cl (θ c ;s k c k p Z(sZ m (s (Z m (s C(θ c ;sp(s k p Z(sD(θ c ;s From the Bezout Identity, a θ T [k, θ T, θ 0, θ T 2 ]T exists such that W cl (θ ;s = W m (s Therefore, y(t = k p W m (s(φt (tω(t+k r(t (0 e y (t = k p W m(sφ(tω(t, ( φ(t = θ(t θ (t k = k m /k p A Stability for n = The plant in (2 can be represented by the unknown quadruple, (A p,b p,c p,k p ẋ = A p x+b p u; y = k p c T p x (2 k p c T p (si A pb p = W(s In general one does not need to keep the high frequency gain as a separate variable when writing the transfer function dynamics in state space form In the context of adaptive control however, the sign of k p is important in proving stability is therefore always singled out from the rest of the dynamics Using (2, the dynamics in (0 can be represented as ẋ = A mn x+b mn (φ T (tω +k r; y = k p c T mnx (3 A p +b p θ0 k pc T p b p θ T b p θ T 2 A mn = b λ θ0 k pc T p Λ+b λ θ T b λ θ2 T b λ k p c T p 0 Λ b mn = b p b λ, c mn = c p 0 x x p ω 0 0 with the reference model having an equivalent non minimal representation ω 2 ẋ mn = A mn x mn +b mn k r; y m = k p c T mnx mn with the property that k p c T mn (si A mnb mn = k p W m (s The non minimal error vector is defined as e mn = x x mn satisfies the following dynamics ė mn = A mn e mn +b mn φ T ω; e y = k p c T mn e mn (4 Theorem : Following Assumptions -4, the plant in ( with the reference model in (3, controller in (8 the update law in (9 are globally stable with the model following error asymptotically converging to zero Proof: See [8, 53] 366

3 V n = AND CRM Unlike the ORM, in this case, the reference model is chosen as ẋ m = A m x m +b m k m r+l(y y m, y m = c T m x m (5 (A m,b m,c T m is an m dimensional system in observer canonical form with c T m = [0 0 ] satisfying c T m(si A m b m k m = W m (s The feedback gain l makes the reference model somewhat similar to the Luenberger observer y m (t is now related to the reference commr(t model following error e y (t as y m (t = W m (sr(t+w l (s(y(t y m (t (6 W l (s k l Z l (s P m (s, (7 k l R along with the m order monic polynomial Z l (s are a function of l free to choose Subtracting (6 from (0 results in the following differential relation W e(s e y = k p W e(sφ T ω (8 Z m (s P m (s k l Z l (s (9 Lemma 2: An l can be chosen such that W e(s is SPR for any n = minimum phase transfer function W m (s Proof: The product k l Z l (s a polynomial of order n with n degrees of freedom through l P m (s is a monic polynomial of degree n Therefore, P m (s k l Z l (s is a monic polynomial of order n with n degrees of freedom determined by l Thus for any Z m (s the roots of W e(s can be placed freely in the closed left half plane such that W e (s is SPR Let A e = A mn +Glk p c T mn (20 G transformsx m to the controllable subspace in x mn, which always exist [3] The non minimal error dynamics therefore take the form ė mn (t = A e e mn (t+b mn φ(tω(t (2 Remark : It is worth noting that in the construction of the minimal non minimal systems the location of the gains k p k m switch from being located at the input to the output The non minimal systems is never created thus need not be realized For the case of the minimal reference model in (5 it is critical however that k m appears at the input of the system This is done so that given the canonical form of c m the l in (5 completely determines the zeros high frequency gain of W l (s in (7 Theorem 3: Following Assumptions -3 l chosen as in Lemma 2, the plant in ( with the reference model in (5, controller in (8 the update law in (9 are globally stable with the model following error asymptotically converging to zero Proof: Given that W e(s is SPR, there exists a P e = Pe T > 0 such that A T e P e +P e A e = Q e P e b mn = c mn (22 Q e = Q T e > 0 Thus V = e T mnp e e mn + φt φ γ k p is a Lyapunov function with derivative V = e T mn Q ee mn Barbalat Lemma ensures the asymptotic convergence of e mn to zero A Performance Now that we have proved stability we can return to a minimal representation of the error dynamics in (8 which is ė m = A l e m +b m k p φ T ω, e y = c T m e m; (23 the all the eigen values of A l are the roots to P m (s k l Z l (s, as can be seen from (9 Recall the Anderson version of KY Lemma; A T l P +PA l = gg T 2µP; Pb m = c m (24 µ min i λ i (A l, i = to m (25 The following performance function has a time derivative V p = e T mpe m + φt φ γ k p From (27 it directly follows that e y (t 2 L 2 2µ (26 V p 2µe T mpe m (27 ( λmax (P λ min (P e(0 2 + γ k p φ(0 2 λ min (P (28 Example : The transfer function W e (s must be SPR, therefore, the poles of W e(s are limited by the location of its zeros The order of A m however is free to choose so long as m, thus we can choose m = Therefore making W m (s = k m s+a m b m = k m A m = a m The closed loop reference model transfer function therefore is W e (s = k m (29 s+a m +l l = l, l > 0 From (29, it is clear that there are no zeros limiting the location of the closed loop pole Further more, the Anderson Lemma reduces to the trivial solution of P =, g = 0, µ = a m +l Since there are no zeros to worry about, W e(s is SPR for all l Therefore, µ can can be chosen arbitrarily The bound in (28 for this example simplifies to e y (t L2 2(a m +l ( e(0 2 + φ(0 2 (30 γ k p 367

4 Remark 2: The use of CRMs has two advantages compared to the use of ORMs The first is that the reference model need not be SPR a priori, but only needs to be of appropriate relative degree There are several methods of dealing with non SPR reference models forn =, but these methods require the use of pre filters [4], or augmented error approaches (see [8], Section VI The second advantage is illustrated in Example Using this approach, a reference model can be chosen such that it has no zeros When this is done a CRM is used, the location of the slowest pole of the error model dynamics is free to choose When using ORMs, the location of the slowest eigenvalue of the closed loop error model is not free to choose, as speeding up the reference model eigenvalues without the use of CRMs will require the use of high gain feedback which is equivalent to θ being large if the open loop plant has slow eigenvalues In the next section, we will show that the above advantages help in realizing a bound on the derivatives of the adaptive parameters which leads to a reduction in the oscillations that adaptive systems often exhibit VI ARBITRARY n AND CRM For higher relative degrees it is common to use an augmented error approach, by the original model following error e y is not used to adjust the adaptive parameter, but an augmented error signal which does satisfy the SPR conditions needed for stability The augmented error method used in this result is Error Model 2 as presented in [8, 54], with some changes to the notation For ease of exposition clarity in presentation we present the k p known For the case when k p unknown we refer the reader to [5] A Stability for known high frequency gain We begin by replacing Assumption 2 with: Assumption 2 : k p is known Without loss of generality we choose k m = k p = the control input for the generic relative degree case reduces to Ď ( denotes the vectors, u(t = r(t+ s θ T (tsω(t (3 sω(t [ω T (t, y(t, ω2 T (t] T (32 sθ(t [θ T (t, θ 0 (t, θ2 T (t] T (33 A feedforward time varying adaptive gain k(t is no longer needed thus r(t has been removed from the regressor vector do to the fact that k p = k m = The model following error then, satisfies the following differential relation e y = W e (ss φ T sω (34 the reader is reminded that the prime notation removes the high frequency gain from transfer functions, since k m = k p =, W e(s = W e (s A stable minimally realized filter F(s with no zeros is used to generate the filtered regressor sζ = F(sI sω (35 I is the2n by2n identity matrix,f(s designed with unity high frequency gain, F(s l chosen so that W f(s W e(sf (s (36 is SPR Lemma 4: For any stable F(s an l can be chosen such that W f (s is SPR Proof: The proof follows the same arguments as in Lemma 2 The tuning law for the arbitrary relative degree case uses an augmented error e a, which is generated from the model following error e y an auxiliary error e χ Using the CRM in (5, the augmented auxiliary error are defined as: e a e y +W f(s ( e χ e aζ st ζ s (37 e χ s θ F(s s θ T sω (38 A stable tuning law for the system is then defined as ṡθ = γe a ζ (39 Theorem 5: Following Assumptions, 2 3, with l chosen such that W f (s is SPR, the plant in ( with the reference model in (5, controller in (3 update law in (39 are globally stable with the model following error e y asymptotically converging to zero Proof: The proof proceeds in 4 steps First it is shown that θ(t s e a are bounded that e a, ṡθ L 2 Second, treating θ(t s as a bounded time varying signal, then all signals in the adaptive system can grow at most exponentially Third, if it is assumed that the signals grow in an unbounded fashion, then it can be shown that y, ω ω 2, sω, ζ s u grow at the same rate Finally, from the fact that ṡθ L 2 it is shown that ω 2 sω do not grow at the same rate This results in a contradiction therefore, all signals are bounded furthermore,e y (t asymptotically converges to zero Steps 4 are detailed below Steps -3 follow directly from [8, 55] with little changes Step 4 does involve a modification to the analysis which is addressed in detail next Step : Exping the error dynamics in (37 canceling like terms of W e(s θ st ω we have e a = W e (ss θ T sω +W f (s( s θ ea s ζ Adding subtracting W f (ss θ the equation becomes e a = W f(s ( s φ ea s ζ +δ(t (40 δ(t is an exponentially decaying term do to initial conditions defined as δ(t = W f(s ( s θ (t F(s s θ T sω(t (4 Breaking apart s ζ from its definition in (35 noting that sθ now commutes with F(s we have that δ(t = W f(s ( s θ T (F(s F(sI sω (42 Therefore, if the filter F(s is chosen to have the same initial conditions when constructing s ζ e χ then, δ = 0 368

5 for all time For this reason we ignore the affect of choosing different filter initial conditions The interested reader can see how one can prove stability in augmented error approaches δ(0 0 [8, pg 23], with the addition of an extra term in the Lyapunov function A non minimal representation of e a is given as ė an = A e e an +b an ( sφ ea s ζ, ea = c T an e an (43 c T an (si A e b an W f (s (44 Given that W f (s is SPR, there exists a P a = P T a > 0 such that A T e P a +P a A e = Q a P a b an = c an (45 Q a = Q T a > 0 Consider the Lyapunov cidate V = e T anp a e an + φ T φ γ Differentiating along the system dynamics in (43 substitution of the tuning law from (39 results in V e T anq a e an 2e 2 s aζ T ζ s Therefore, e an, θ s L e an, ṡθ L 2 Step 2: The plant dynamics can be expressed as ẋ = A mn x+b mn ( s φ T (tω +r; y = c T mnx (46 with an appropriate choice of a C can be expressed as ẋ = ( A mn +b mn s φ T (tc x+b mn r (47 From Step it is known that φ s is bounded, therefore x grows at most exponentially Futhermore, for r piecewise continuous, x ζ s are both piecewise continuous as well Step 3: If it is assumed that all signals grow in an unbounded fashion then it can be shown that sup y(τ sup ω (τ sup ω 2 (τ sup sω sup ζ s sup u(τ [8, 55] Step 4: Rewrinting (38 in terms of sω we have that (48 e χ s θ T F(sI sω F(s s θ T sω (49 given that ṡθ L 2 F(s is stable the following holds [ ] e χ (t = o sup sω(τ (50 The above bound follows from the Swapping Lemma [8, Lemma 2] From (39 the fact that ṡθ L 2 we have that e aζ s L2 Given that W f (s is asymptotically stable, [8, Lemma 29] can be applied it follows that ] W f (s( (e aζ s T ζ s = o [sup ζ(τ s (5 The plant output can be written in terms of the reference model model following error as y(t =y m (t+e y (t =W m (sr(t+(+w l (se y(t Using (37, e y (t = e a W f (s( e χ e a s ζ the above equation exps as y(t =W m (sr(t+(+w l (se a (+W l(sw f(s ( e χ e a s ζ Using (50 (5 noting that +W l (s is asymptotically stable [8, Lemma 29] can be applied again y(t =W m (sr(t+(+w l (se a [ ] [ ] +o sup ζ(τ s +o sup sω(τ Given that r e a are piecewise continuous bounded we finally have that [ ] y(t = o sup sω(τ (52 This contradicts (48 therefore all signals are bounded Furthermore, from (43 it now follows that ė an is bounded given that e an L 2, from Step, it follows that e an asymptotically converges to zero therefore lim t e a (t = 0 From (50 it follows that e χ asymptotically converges to zero Therefore, lim t e y (t = 0 The above analysis differs from the analysis for the ORM output feedback adaptive control do to the fact that one can not a priori assume that y m (t is bounded, do to the feedback of e y into the reference model B Performance when k p known Just as in the n = case, with stability proved a Lyapunov performance function can be studied that uses a minimal representation of the dynamics That being said, consider the minimal representation of the dynamics in (40 ė am = A l e am +b am ( sφ ea s ζ, ey = c T am e am (53 in observer canonical form so that c T am = [0 0 ] c T am(si A l b am W f(s Recall the Anderson version of KY Lemma; A T l P p +P p A l = gg T 2µP p ; P p b am = c am (54 µ is defined in (25 The following performance function V p = e T am P φ pe am + st φ s γ has a time derivative V p 2µe T am P pe am 2e 2 s aζ T ζ s From (VI-B it directly follows that e a (t 2 L 2 ( λmax (P p 2µ λ min (P p e(0 2 + φ(0 s 2 (55 γ λ min (P p ṡθ(t 2 L 2 2 ( γ 2 λ max (P p e(0 2 +γ s φ(0 2 (56 Ultimately we would like to compute the L 2 norm of e χ e y Given that these norms will depend explicitly on the specific values of the filter reference model, we perform that analysis in the following example 369

6 Example 2: In this example we consider a relative degree 2 plant The reference model is chosen as W m (s = the filter is chosen as s 2 +b s+b 2 (57 F(s = s+f (58 The reference model gain is exped as l = [ ] T l l 2 Then W e (s = s 2 (59 +(b +l s+(b 2 +l 2 W f (s = s+f s 2 +(b +l s+(b 2 +l 2 (60 Since, k p = k m =, then W m (s = W m (s, W e (s = W e (s W f(s = W f (s For stability to hold W f (s must be SPR from (60 it is clear that the SPR condition can be satisfied by choosing l f appropriately More importantly though, we see that the slowest eignvalue of W f (s can be arbitrarily placed thus the µ in (25 can be arbitrarily increased ( ( e 2 e χ (t 2 χ (0 e 2 χ (0 L f 4f 2 + sω(t 2 θ(t 2L2 f 3 (6 A detailed proof of this expression is given in [5, Appendix A] Furthermore, we have the following bound for the model following error e y (t 2 L 2 2 e a (t 2 L 2 +2 e ζ (t 2 L 2 (62 e ζ (t W f (se χ (t can be bounded as ( e 2 ( e ζ 2 L 2 3m 2 ζ (0 2µ + eχ (0 2 + sω(t 2 θ(t 2L2 4µf µf 2 (63 The bound in (63 is given in [5, Appendix B] Remark 3: Now we compare the norms in (6 (63 for an ORM CRM system note that increasing both f µ decreases the two norms For the ORM system l = 0, therefore µ is solely a function of b b 2 in (60 The coefficients b b 2 can not be arbitrarily changed without affecting the matching parameter vector θ In the presence of persistence of excitation, θ(t θ large θ will directly imply a large control input Furthermore, one can not arbitrarily change the reference model poles, as the reference model is a target behavior for the plant, in which case the control engineer may not want to track a reference system with arbitrarily fast poles Therefore, given that b b 2 are not completely free to choose this also limits the value of f as W f (s must always be SPR In the CRM case b b 2 can be held fixed l, l 2 f can be adjusted so that the poles of W f (s are arbitrarily fast W f (s is still SPRTherefore, the added degree of freedom through l in the CRM adaptive systems allows more flexibility in decreasing the L 2 norm of e y Remark 4: In the above, we have derived bounds on the L 2 norm of the tracking error That the same error has finite L bounds is easily shown using Lyapunov function arguments the fact that projection algorithms ensure exponential convergence of the error to a compact set, similar to the analysis in [3] [6] VII CONCLUSION This work shows that with the introduction of CRMs the adaptive system can have improved transient performance in terms of reduction of the L 2 norm of the model following error the derivative of adaptive perimeter The analysis techniques used followed from the states accessible CRM adaptive systems in [3] [5] Not reported in this work are techniques used to analytically support the reduction in oscillations when CRMs are used, as performed in [6], [7] Extending the results from [6], [7] to the output feedback case are the subject of ongoing investigation ACKNOWLEDGMENT This work was supported by the Boeing REFERENCES [] T-G Lee U-Y Huh, An error feedback model based adaptive controller for nonlinear systems, in Proceedings of the IEEE International Symposium on Industrial Electronics, 997 [2] E Lavretsky, R Gadient, I M Gregory, Predictor based model reference adaptive control, AIAA JGCD, 200 [3] T E Gibson, A M Annaswamy, E Lavretsky, Closed loop reference model adaptive control: Stability, performance robustness, arxiv:204897v6, 202 [4], Closed loop Reference Model Adaptive Control, Part I: Transient Performance, in American Control Conference, 203 [5], Closed loop Reference Model Adaptive Control: Composite control Observer Feedback, in The th IFAC International Workshop on Adaptation Learning in Control Signal Processing, 203 [6] A M Annaswamy, T E Gibson, H S Hussain, E Lavretsky, Practical adaptive control, in The 6th Yale Workshop on Adaptive Learning Systems Yale Workshop on Adaptive Learning Systems, 203 [7] T E Gibson, A M Annaswamy, E Lavretsky, On adaptive 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