A Convolution Bound Implies Tolerance to Time-variations and Unmodelled Dynamics

A Con v olution Bound Implies T olerance to Time-v ariations and Un mo delled Dynamics ⋆ Mohamad T. Shahab , Daniel E. Miller ∗ Dept. of Ele ctrica l and Computer Engine ering, University of W aterlo o W aterlo o, O N, Cana da N2L 3G1 Abstract Recent ly it has be en sho wn, in several settings, ho w to carry out adaptiv e con tro l for an L TI plant so that a conv o lution bo und holds on the closed-lo op behavior; this, in turn, has b een leveraged to prov e robustness of the closed-lo op system to time-v a rying parameters and unmodelled dynamics. The go al of this pa per is to show that the same is tr ue for a large class of finite-dimensional, nonlinear plant a nd controller combinations. Keywor ds: A daptive Control, Robustness, Conv o lution B o unds, Time-v aria tions, Unmodelled Dynamics 1. In tro duction In control system design, a common requirement is that the closed-lo op system not only be stable, but also be robust, in the sense that it tolera tes, at the very least, small time-v aria tions in the plant parameter s and a sma ll amount of unmo delled dyna mics . O f course, if the pla n t and controller are b oth linear and time-inv ariant, then such robustness follows from closed-lo op stability—see [ 9 ], [ 1 ]. On the other hand, if either the plant or controller is nonlinear, this is often not the case and/o r it is not easy to prov e. Recent ly it has been prov en, in bo th the po le placement and fir st o rder o ne-step-ahead settings , that if discr ete- time a da ptiv e control is carr ied out in just the r ig h t wa y , then a (stable) co n volution bo und can b e obtained on the closed-lo op b ehavior—see [ 4 ] a nd [ 6 ]; hence, the closed- lo op system acts ‘linear-like’, a nd the conv olution b ound can b e leveraged in a mo dular fashion 1 to prove that tol- erance to small time-v ariatio ns and a small amount of un- mo delled dynamics follows. Of cour se, there the controller is nonlinear a nd the nominal plant is single-input, single- output, a nd L TI. The goa l of this pap er is to gener alize this result to a larger clas s of multi-input m ulti-output pla nts and controllers. T o this end, here we consider a class of finite-dimensiona l, nonlinear plant and co n tro ller combinations; if a conv olu- ⋆ F unding for this researc h was provided by the Natural Sciences and Engineering Researc h Council of Canada (NSER C). ∗ Corresponding author. Email addr esses: m4shahab @uwaterlo o.ca (Mohamad T. Shahab), miller@u waterloo. ca (Daniel E. Mil ler) 1 It is mo dular in the sense that we are able to lev erage the results for the ideal case without reopening its pro of; robustness can be pro v en directly from the con volution b ound. tion bound holds, then w e prove that tolerance to sma ll time-v ariations in the plant par a meters and a small a moun t of unmo delled dynamics follows. An immediate applica- tion of this r esult is to pr o v e robustness of our recently de- signed multi-estimator switching adaptive controllers pre- sented in [ 6 ] a nd [ 8 ]. This result s hould also pr o v e useful in extending our work on the ada ptiv e co n tro l of L TI plants [ 5 ], [ 6 ], [ 8 ], [ 7 ] to that of nonlinear plants, a llo wing us to fo- cus on the ideal pla n t mo del in our analysis, knowing that robustness will come for free. Last of all, this r esult has the po ten tial fo r use in other non-adaptive (but no nlinea r) contexts. W e denote Z , Z + and N as the sets of integers, non- negative integers and natural num b ers, resp ectiv ely . W e will denote the Euclidean-norm o f a vector and the in- duced nor m of a matrix by the subscript-less default no- tation k · k . W e let S ( R p × q ) denote the set o f R p × q -v alued sequences. W e als o let ℓ ∞ ( R p × q ) denote the set of R p × q - v alued b ounde d sequences. If Ω ⊂ R p × q is a bo unded set, we define k Ω k := sup x ∈ Ω k x k . Throughout this pap er, we sa y that a function Γ : R p → R q has a b ounde d gain if there exists a ν > 0 such that for all x ∈ R p , we hav e k Γ( x ) k ≤ ν k x k ; the smallest such ν is the gain, a nd is denoted by k Γ k . F or a clo sed and conv ex set Ω ⊂ R p , the function Pro j Ω {·} : R p → Ω denotes the pro jection onto Ω ; it is well known that the function Pro j Ω is well defined. 2. The Setup Here the nomina l plan t is mul ti-input multi-output with finite memory and an a dditiv e disturba nce , such that the uncertain plant para meter enters linearly . T o this end, with an output y ( t ) ∈ R r , an input u ( t ) ∈ R m , a distur- bance w ( t ) ∈ R r , a mo deling parameter o f θ ∗ ∈ S ⊂ R p × r , and a vect or o f input-output data of the for m φ ( t ) =               y ( t ) y ( t − 1) . . . y ( t − n y + 1) u ( t ) u ( t − 1) . . . u ( t − n u + 1)               ∈ R n y · r + n u · m , we consider the pla n t y ( t + 1) = θ ∗ ⊤ f  φ ( t )  + w ( t ) , φ ( t 0 ) = φ 0 ; (1) we assume that f : R n y · r + n u · m → R p has a b ounded gain a nd that S is a b ounded set ; b oth r equiremen ts are r easonable given that w e will r e q uire uniform b ounds in our analysis. W e represent this system by the pair  f , S  . Here we consider a lar ge cla ss of co n tr o llers whic h sub- sumes L TI ones a s well as a large cla ss o f adaptive ones . T o this end, we co nsider a controller with its state partitioned in to tw o parts: • z 1 ( t ) ∈ R l 1 and • z 2 ( t ) ∈ R l 2 , an exogenous signal r ( t ) ∈ R q (t ypically a refer ence signal), together with e quations of the form z 1 ( t + 1) = g 1 ( z 1 ( t ) , z 2 ( t ) , φ ( t ) , y ( t + 1) , r ( t ) , t, t 0 ) , z 1 ( t 0 ) = z 1 0 (2a) z 2 ( t + 1) = g 2 ( z 1 ( t ) , z 2 ( t ) , φ ( t ) , y ( t + 1) , r ( t ) , t, t 0 ) , z 2 ( t 0 ) = z 2 0 (2b) u ( t ) = h ( z 1 ( t ) , z 2 ( t ) , φ ( t ) , r ( t )) . (2c) Here we assume that g 2 : R l 1 × X × R n y · r + n u · m × R r × R q × Z × Z − → X , i.e. if z 2 is initialized in X , then it rema ins in X through- out. Remark 1. This class subsumes finite-dimensional L TI c ontr ol lers: simply set l 2 = 0 so that t he sub-state z 2 dis- app e ars, and make t he functions g 1 and h to b e line ar. Remark 2. This class subsu mes many adaptive c ontr ol lers: simply set l 1 = 0 and let z 2 b e the state of a p ar ameter es- timator c onstr aine d to the set X . W e now pr ovide a definition of the desired linear -lik e closed-lo op prop erty: Definition 1. W e say that ( 2 ) pro vides a con vo- lution b ound for  f , S  with gain c ≥ 1 and de c ay r ate λ ∈ (0 , 1) if, for every θ ∗ ∈ S , t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X ⊂ R l 2 , w ∈ S ( R r ) and r ∈ S ( R q ) , when ( 2 ) is applie d to ( 1 ) , the fol lowing holds:      φ ( t ) z 1 ( t )      ≤ cλ t − τ      φ ( τ ) z 1 ( τ )      + t − 1 X j = τ cλ t − j − 1 ( k r ( j ) k + k w ( j ) k ) + c k r ( t ) k , t ≥ τ ≥ t 0 . (3) Remark 3. The r e ason why we do not fo cus on the exp o- nential stability asp e ct of ( 3 ) is t ha t the cλ t − τ      φ ( τ ) z 1 ( τ )      term c an b e viewe d, in essenc e, as the effe ct of the p ast inputs on the fut u r e, in mu ch t he same way as the ‘zer o- input-r esp onse’ c an b e viewe d in the analysis of L TI sys- tems. Mo r e sp e cific al ly, the cλ t − τ      φ ( τ ) z 1 ( τ )      term c an b e viewe d as having arisen fr om a c onvolution of the p ast in- puts (b efor e time τ ) with cλ t , s o this term c an b e viewe d as a c onvolution sum in its own right. 3. T ol eranc e to Ti m e-V ariation W e now co nsider pla n ts with a p ossibly time-v arying parameter vector θ ∗ ( t ) ins tea d of a static θ ∗ : y ( t + 1) = θ ∗ ( t ) ⊤ f  φ ( t )  + w ( t ) , φ ( t 0 ) = φ 0 . (4) With c 0 ≥ 0 and ǫ > 0 , let s ( S , c 0 , ǫ ) denote the subset of ℓ ∞ ( R p × r ) whose e le ments θ ∗ satisfy: • θ ∗ ( t ) ∈ S fo r every t ∈ Z , • and t 2 − 1 X t = t 1 k θ ∗ ( t +1) − θ ∗ ( t ) k ≤ c 0 + ǫ ( t 2 − t 1 ) , t 2 > t 1 , t 1 ∈ Z . The ab ov e time-v a r iation mo del enco mpasses bo th slow v ariations a nd/or o ccasional jumps; this cla ss is well-kno wn in the a daptiv e control literature, e.g. see [ 2 ]. W e can ex- tend Definition 1 in a natura l wa y to handle time-v a riations. 2 Definition 2. W e say that ( 2 ) provides a con v o - lution b ound for  f , s ( S , c 0 , ǫ )  with gain c ≥ 1 and de c ay r ate λ ∈ (0 , 1 ) if, for every θ ∗ ∈ s ( S , c 0 , ǫ ) , t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X ⊂ R l 2 , w ∈ S ( R r ) and r ∈ S ( R q ) , when ( 2 ) is applie d to ( 4 ) , the fol lowing holds:      φ ( t ) z 1 ( t )      ≤ cλ t − τ      φ ( τ ) z 1 ( τ )      + t − 1 X j = τ cλ t − j − 1 ( k r ( j ) k + k w ( j ) k ) + c k r ( t ) k , t ≥ τ ≥ t 0 . (5) W e now will show that if a controller ( 2 ) provides con- volution b ounds for the pla nt ( 1 ), then the same w ill be true for the time-v arying plant ( 4 ), as long as ǫ is small enough. W e consider tw o ca ses: o ne where there is a de- sired decay rate, and one where there is not. Theorem 1. Su pp ose t hat the c ontro l ler ( 2 ) pr ovides a c onvolution b oun d for ( 1 ) with gain c ≥ 1 and de c ay r ate λ ∈ (0 , 1) . Then for every λ 1 ∈ ( λ, 1) and c 0 > 0 , ther e exist a c 1 ≥ c and ǫ > 0 so that ( 2 ) pr ovides a c onvolution b ound for  f , s ( S , c 0 , ǫ )  with gain c 1 and de c ay r ate λ 1 . Remark 4. This pr o of is b ase d, in p art, on the pr o of of The or em 2 of [ 6 ], which de als with a much simpler setup. Pr o of of The or em 1. Supp ose the controller ( 2 ) provides a conv olution b ound for ( 1 ) with ga in c ≥ 1 and a decay rate of λ . Fix λ 1 ∈ ( λ, 1) and c 0 > 0 ; let t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X , w ∈ S ( R r ) a nd r ∈ S ( R q ) be a rbitrary . Now fix m ∈ N to be a ny num ber satisfying m ≥ ln( c ) + 4 c 0 c k f k λ 1 − λ [ln (1 + 2 c k f kkS k ) + ln (2) − ln( λ + λ 1 )] ln(2 λ 1 ) − ln( λ + λ 1 ) , (the ratio na le for this c hoice will b e more clear sho rtly), and set ǫ = c 0 m 2 ; let θ ∗ ∈ s ( S , c 0 , ǫ ) b e arbitrary and a pply the controller ( 2 ) to the time-v ar ying plant ( 4 ). T o pro ceed, we analyze the clo sed-loo p s ystem b eha vior on int erv als of leng th m , which we further analyze in g r oups of m 2 . T o pr oceed, let ¯ t ≥ t 0 be a rbitrary . Define a sequence { ¯ t i } by ¯ t i = ¯ t + i m, i ∈ Z + . W e can rewrite the time-v a rying plant as y ( t + 1) = θ ( ¯ t i ) ⊤ f ( φ ( t )) + w ( t ) + [ θ ( t ) − θ ( ¯ t i )] ⊤ f ( φ ( t )) | {z } =: ˜ n i ( t ) , t ∈ [ ¯ t i , ¯ t i +1 ) . On the interv al [ ¯ t i , ¯ t i +1 ] , we can reg ard the plant as time- in v aria n t, but with an extra disturbance; so by hypothesis,      φ ( t ) z 1 ( t )      ≤ cλ t − ¯ t i      φ ( ¯ t i ) z 1 ( ¯ t i )      + t − 1 X j = ¯ t i cλ t − j − 1 ( k r ( j ) k + k w ( j ) k + k ˜ n i ( j ) k ) + c k r ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ] , i ∈ Z + . (6) T o ana lyze this difference inequality , we firs t construct an asso ciated difference equation: ψ ( t +1) = λψ ( t )+ k r ( t ) k + k w ( t ) k + k ˜ n i ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ) , with a n initial condition o f ψ ( ¯ t i ) =      φ ( ¯ t i ) z 1 ( ¯ t i )      . Using the fact that c ≥ 1 , it is straightforw ard to prov e that      φ ( t ) z 1 ( t )      ≤ cψ ( t ) + c k r ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ] . (7) Now w e a nalyze this equation for i = 0 , 1 , . . . , m − 1 . Case 1 : k ˜ n i ( t ) k ≤ λ 1 − λ 2 c k φ ( t ) k for all t ∈ [ ¯ t i , ¯ t i +1 ) . Using the ab o ve b ound ( 7 ) and the fact that λ 1 − λ ∈ (0 , 1) , we obtain ψ ( t + 1) ≤ λψ ( t ) + k r ( t ) k + k w ( t ) k + k ˜ n i ( t ) k ≤ λψ ( t ) + k r ( t ) k + k w ( t ) k + λ 1 − λ 2 c k φ ( t ) k ≤ λψ ( t ) + k r ( t ) k + k w ( t ) k + λ 1 − λ 2 ( ψ ( t ) + k r ( t ) k ) ≤ λ 1 + λ 2 ψ ( t ) + 2 k r ( t ) k + k w ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ) , (8) which means that | ψ ( t ) | ≤  λ 1 + λ 2  t − ¯ t i | ψ ( ¯ t i ) | + t − 1 X j = ¯ t i  λ 1 + λ 2  t − j − 1 (2 k r ( j ) k + k w ( j ) k ) , t = ¯ t i , ¯ t i + 1 , . . . , ¯ t i +1 . (9) This, in turn, implies that there exists c 2 ≥ 2 c so that      φ ( ¯ t i +1 ) z 1 ( ¯ t i +1 )      ≤ c  λ 1 + λ 2  m      φ ( ¯ t i ) z 1 ( ¯ t i )      + ¯ t i +1 − 1 X j = ¯ t i c 2  λ 1 + λ 2  ¯ t i +1 − j − 1 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( ¯ t i +1 ) k . (10) Case 2 : k ˜ n i ( t ) k > λ 1 − λ 2 c k φ ( t ) k for some t ∈ [ ¯ t i , ¯ t i +1 ) . 3 Since θ ∗ ( t ) ∈ S fo r t ≥ t 0 , we see that k ˜ n i ( t ) k ≤ 2 kS k k f ( φ ( t )) k ≤ 2 k f kkS k× k φ ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ) . This means that ψ ( t + 1) ≤ λψ ( t ) + k r ( t ) k + k w ( t ) k + k ˜ n i ( t ) k ≤ λψ ( t ) + k r ( t ) k + k w ( t ) k + 2 k f kkS k k φ ( t ) k ≤ (1 + 2 c k f kkS k ) | {z } =: γ 3 ψ ( t ) + (1 + 2 c k f kkS k ) k r ( t ) k + k w ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ) , (11) which means that | ψ ( t ) | ≤ γ t − ¯ t i 3 | ψ ( ¯ t i ) | + t − 1 X j = ¯ t i γ t − j − 1 3 ( γ 3 k r ( j ) k + k w ( j ) k ) , t = ¯ t i , ¯ t i + 1 , . . . , ¯ t i +1 . (12) Setting t = ¯ t i +1 and using ( 7 ) yields      φ ( ¯ t i +1 ) z 1 ( ¯ t i +1 )      ≤ cγ m 3      φ ( ¯ t i ) z 1 ( ¯ t i )      + ¯ t i +1 − 1 X j = ¯ t i cγ ¯ t i +1 − j − 1 3 ( γ 3 k r ( j ) k + k w ( j ) k ) + c k r ( ¯ t i +1 ) k ≤ cγ m 3      φ ( ¯ t i ) z 1 ( ¯ t i )      + cγ 3  2 γ 3 λ 1 + λ  m × ¯ t i +1 − 1 X j = ¯ t i  λ 1 + λ 2  ¯ t i +1 − j − 1 ( k r ( j ) k + k w ( j ) k ) + c k r ( ¯ t i +1 ) k . (13) This completes C a se 2. A t this p oin t we combine Ca se 1 and 2. W e would like to a na lyze m in terv als o f length m . On the interv al [ ¯ t, ¯ t + m 2 ] , there are m subin terv als of length m ; furthermor e, bec a use of the choice of ǫ we hav e that ¯ t + m 2 − 1 X j = ¯ t k θ ( j + 1) − θ ( j ) k ≤ c 0 + ǫm 2 ≤ 2 c 0 . It is easy to s e e that there are at most N 1 := 4 c 0 c k f k λ 1 − λ subin terv als which fall into the category of Case 2 , with the remainder falling int o the category of Case 1; it is clear from the for m ula for m that m > N 1 . If we use ( 10 ) and ( 13 ) to ana lyze the b ehavior of the closed-lo op s ystem on the in terv al [ ¯ t, ¯ t + m 2 ] , we end up with a crude bo und of      φ ( ¯ t + m 2 ) z 1 ( ¯ t + m 2 )      ≤ c m γ N 1 m 3  λ 1 + λ 2  m ( m − N 1 )      φ ( ¯ t ) z 1 ( ¯ t )      + 2 m  2 γ 3 λ 1 + λ  m ( c 2 γ m +1 3 ) m  2 λ 1 + λ  ( m +1) m × ¯ t + m 2 − 1 X j = ¯ t  λ 1 + λ 2  ¯ t + m 2 − j − 1 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( ¯ t + m 2 ) k . (14) F rom the c hoice of m a b ov e, it is easy to show that m 2 ln  2 λ 1 λ 1 + λ  ≥ m ln( c ) + N 1 m ln( γ 3 ) + N 1 m ln  2 λ + λ 1  ; this immediately implies that c m γ N 1 m 3  2 λ + λ 1  N 1 m ≤  2 λ 1 λ 1 + λ  m 2 ⇔ c m γ N 1 m 1  λ 1 + λ 2  m ( m − N 1 ) ≤ λ m 2 1 . Since λ 1 + λ 2 < λ 1 , it follows from ( 14 ) that there exists a constant γ 4 so that      φ ( ¯ t + m 2 ) z 1 ( ¯ t + m 2 )      ≤ λ m 2 1      φ ( ¯ t ) z 1 ( ¯ t )      + γ 4 ¯ t + m 2 − 1 X j = ¯ t λ ¯ t + m 2 − j − 1 1 ( k r ( j ) k + k w ( j ) k ) + γ 4 k r ( ¯ t + m 2 ) k . (15) Now let τ ≥ t 0 be a rbitrary . By setting ¯ t = τ , τ + m 2 , τ + 2 m 2 , . . . , in succe s sion, it follows fr o m ( 15 ) that      φ ( τ + q m 2 ) z 1 ( τ + q m 2 )      ≤ λ qm 2 1      φ ( τ ) z 1 ( τ )      + γ 4 τ + q m 2 − 1 X j = τ λ τ + q m 2 − j − 1 1 ( k r ( j ) k + k w ( j ) k ) + γ 4 k r ( τ + q m 2 ) k , q ∈ Z + . (16) So  φ ( t ) z 1 ( t )  is well-behaved at t = τ , τ + m 2 , τ + 2 m 2 , etc; we ca n use ( 9 ) of Case 1 , ( 12 ) of Case 2 and ( 7 ) to prov e that nothing un tow a rd happ ens be tw een these times. W e conclude that there exists a constant γ 5 so that      φ ( t ) z 1 ( t )      ≤ γ 5 λ t − τ 1      φ ( τ ) z 1 ( τ )      + γ 5 t − 1 X j = τ λ t − j − 1 1 ( k r ( j ) k + k w ( j ) k ) + γ 5 k r ( t ) k , t ≥ τ . (17 ) Since τ ≥ t 0 is a r bitrary , the desired bound is prov en.  A ca reful examination of the a bov e pro of reveals that ǫ → 0 as c 0 → 0 and as c 0 → ∞ . If we do not care ab out the decay rate, then we can remov e this drawback. 4 Theorem 2. Su pp ose t hat the c ontro l ler ( 2 ) pr ovides a c onvolution b oun d for ( 1 ) with gain c ≥ 1 and de c ay r ate λ ∈ (0 , 1) . Then ther e exists an ǫ > 0 such that for every c 0 ≥ 0 , t her e exist λ ∗ ∈ (0 , 1) and γ > 0 so that ( 2 ) pr ovides a c onvolution b ound for  f , s ( S , c 0 , ǫ )  with gain γ and de c ay r ate λ ∗ . Pr o of of The or em 2 . Suppo se the co n tro ller ( 2 ) provides a conv o lution bo und for ( 1 ) with ga in c ≥ 1 and a decay rate of λ . Fix λ 1 ∈ ( λ, 1 ) ; let t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X , w ∈ S ( R r ) a nd r ∈ S ( R q ) b e arbitrary . The goal is to prov e that for a sma ll- enough ǫ , the co n tr o ller ( 2 ) provides a convolution b ound for  f , s ( S , c 0 , ǫ )  for every c 0 ≥ 0 . So at this p oint we will analyze the closed-lo op system for an a rbitrary ǫ > 0 , c 0 ≥ 0 , and θ ∗ ∈ s ( S , c 0 , ǫ ) . T o pro ceed, let ¯ t ≥ t 0 be arbitrary . F o r m ∈ N , w e will first analyze closed- loop b eha vior on interv a ls of leng th m ; define a se q uence { ¯ t i } by ¯ t i = ¯ t + i m, i ∈ Z + . W e can rewrite the time-v a rying plant as y ( t + 1) = θ ( ¯ t i ) ⊤ f ( φ ( t )) + w ( t ) + [ θ ( t ) − θ ( ¯ t i )] ⊤ f ( φ ( t )) | {z } =: ˜ n i ( t ) , t ∈ [ ¯ t i , ¯ t i +1 ) . On the in terv al [ ¯ t i , ¯ t i +1 ] , we rega rd the plant as time- in v aria n t, but with a n extra disturbance: so we obtain      φ ( t ) z 1 ( t )      ≤ cλ t − ¯ t i      φ ( ¯ t i ) z 1 ( ¯ t i )      + t − 1 X j = ¯ t i cλ t − j − 1 ( k r ( j ) k + k w ( j ) k + k ˜ n i ( j ) k ) + c k r ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ] , i ∈ Z + . (18) Using the s ame idea as in the pro of of Theo r em 1, we define the differe nce equation ψ ( t +1) = λψ ( t )+ k r ( t ) k + k w ( t ) k + k ˜ n i ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ) with ψ ( ¯ t i ) =      φ ( ¯ t i ) z 1 ( ¯ t i )      ; it follows that      φ ( t ) z 1 ( t )      ≤ cψ ( t ) + c k r ( t ) k , t ∈ [ ¯ t i , ¯ t i +1 ] . (19) Case 1 : k ˜ n i ( t ) k ≤ λ 1 − λ 2 c k φ ( t ) k for all t ∈ [ ¯ t i , ¯ t i +1 ) . Arguing in an identical ma nner to the pr oof of Theo rem 1, we obtain the following tw o b ounds: | ψ ( t ) | ≤  λ 1 + λ 2  t − ¯ t i | ψ ( ¯ t i ) | + t − 1 X j = ¯ t i  λ 1 + λ 2  t − j − 1 (2 k r ( j ) k + k w ( j ) k ) , t = ¯ t i , ¯ t i + 1 , . . . , ¯ t i +1 ; (20) this, in tur n, implies that there exists c 2 > c so tha t      φ ( ¯ t i +1 ) z 1 ( ¯ t i +1 )      ≤ c  λ 1 + λ 2  m      φ ( ¯ t i ) z 1 ( ¯ t i )      + ¯ t i +1 − 1 X j = ¯ t i c 2  λ 1 + λ 2  ¯ t i +1 − j − 1 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( ¯ t i +1 ) k . (21) Case 2 : k ˜ n i ( t ) k > λ 1 − λ 2 c k φ ( t ) k for some t ∈ [ ¯ t i , ¯ t i +1 ) . Arguing in an iden tical manner to the pro of of Theorem 1, we obtain the following t w o b ounds: there exists γ 3 > 0 so that | ψ ( t ) | ≤ γ t − ¯ t i 3 | ψ ( ¯ t i ) | + t − 1 X j = ¯ t i γ t − j − 1 3 ( γ 3 k r ( j ) k + k w ( j ) k ) , t = ¯ t i , ¯ t i + 1 , . . . , ¯ t i +1 , (22)      φ ( ¯ t i +1 ) z 1 ( ¯ t i +1 )      ≤ cγ m 3      φ ( ¯ t i ) z 1 ( ¯ t i )      + ¯ t i +1 − 1 X j = ¯ t i cγ ¯ t i +1 − j − 1 3 ( γ 3 k r ( j ) k + k w ( j ) k ) + c k r ( ¯ t i +1 ) k ≤ cγ m 3      φ ( ¯ t i ) z 1 ( ¯ t i )      + c 2 γ 3  2 γ 3 λ 1 + λ  m × ¯ t i +1 − 1 X j = ¯ t i  λ 1 + λ 2  ¯ t i +1 − j − 1 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( ¯ t i +1 ) k . (23) This completes C a se 2. A t this p oin t we combine Case 1 and 2. W e would lik e to analyze ¯ N ∈ N interv als o f length m ; for now we let ¯ N be fr e e . W e see that ¯ t + m ¯ N − 1 X j = ¯ t k θ ( j + 1) − θ ( j ) k ≤ c 0 + ǫm ¯ N . Let N 1 denote the num ber of interv als of the for m [ ¯ t i , ¯ t i +1 ) which lie in [ ¯ t, ¯ t + m ¯ N ] which fall into Cas e 2; it is ea sy to see that N 1 satisfies N 1 × λ 1 − λ 2 c ≤  c 0 + ǫm ¯ N  k f k ⇒ N 1 ≤  2 c k f k λ 1 − λ  × c 0 +  2 c k f k λ 1 − λ  × ǫ × m ¯ N ; (24) 5 observe tha t N 1 depends on bo th c 0 and ǫ . Using ( 21 ) and ( 23 ) we obtain      φ ( ¯ t + m ¯ N ) z 1 ( ¯ t + m ¯ N )      ≤ c ¯ N  λ 1 + λ 2  m ( ¯ N − N 1 ) γ mN 1 3      φ ( ¯ t ) z 1 ( ¯ t )      + 2 ¯ N  2 γ 3 λ 1 + λ  ¯ N ( c 2 γ m +1 3 ) ¯ N  2 λ 1 + λ  ( m +1) ¯ N × ¯ t + m ¯ N − 1 X j = ¯ t  λ 1 + λ 2  ¯ t + m ¯ N − j − 1 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( ¯ t ( q +1) m ) k . (25) A t this p oint , we will c ho ose quantities m, ǫ a nd ¯ N , in that order, so that the key g ain c ¯ N  λ 1 + λ 2  m ( ¯ N − N 1 ) γ mN 1 3 < 1 . First of all, we a pply the b ound on N 1 given in ( 24 ) to this key gain: c ¯ N  λ 1 + λ 2  m ( ¯ N − N 1 ) γ mN 1 3 = h c  λ 1 + λ 2  m i ¯ N   2 γ 3 λ 1 + λ  N 1  m ≤ h c  λ 1 + λ 2  m i ¯ N "  2 γ 3 λ 1 + λ  h 2 c k f k λ 1 − λ  c 0 +  2 c k f k λ 1 − λ  ǫm ¯ N i # m . (26) No w c ho ose m s o that c  λ 1 + λ 2  m =: λ 2 < 1 , i.e. a n y m > ln( c ) ln(2) − ln( λ 1 + λ ) . So rewriting ( 26 ), we now obta in c ¯ N  λ 1 + λ 2  m ( ¯ N − N 1 ) γ mN 1 3 ≤ "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × c 0 m # "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × ǫ × m 2 # × λ 2 ! ¯ N . Now observe that lim ǫ → 0 "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × ǫ × m 2 # = 1 , so now c ho ose ǫ > 0 so that "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × ǫ × m 2 # × λ 2 | {z } =: λ 3 < 1; notice that ǫ is indep enden t o f c 0 . With this choice w e now hav e c ¯ N  λ 1 + λ 2  m ( ¯ N − N 1 ) γ mN 1 3 ≤ "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × c 0 m # × λ ¯ N 3 . Last of all, now choo se ¯ N so that "  2 γ 3 λ 1 + λ   2 c k f k λ 1 − λ  × c 0 m # × λ ¯ N 3 | {z } =: λ 4 < 1; any ¯ N > 2 cc 0 m k f k [ln(2 γ 3 ) − ln( λ 1 + λ )] ( λ − λ 1 ) ln( λ 3 ) will do. Observe that ¯ N dep ends on c 0 . So incorp orating all of the ab ov e, there exists γ 4 > 0 (whic h clearly depends on c 0 via ¯ N ) so that w e can rewr ite ( 25 ) as      φ ( ¯ t + m ¯ N ) z 1 ( ¯ t + m ¯ N )      ≤ λ 4      φ ( ¯ t ) z 1 ( ¯ t )      + γ 4 ¯ t + m ¯ N − 1 X j = ¯ t  λ 1 + λ 2  ¯ t + m ¯ N − j − 1 ( k r ( j ) k + k w ( j ) k ) + γ 4 k r ( ¯ t + m ¯ N ) k . (27) Now let τ ≥ t 0 be arbitrar y . By s e tting ¯ t = τ , τ + m ¯ N , τ + 2 m ¯ N , . . . , in success ion, with λ 5 := max  λ 1 m ¯ N 4 , λ 1 + λ 2  (whic h clearly dep ends o n c 0 via ¯ N ) it follows from ( 27 ) that      φ ( ¯ t + q ¯ N m ) z 1 ( ¯ t + q ¯ N m )      ≤ λ q ¯ N m 5      φ ( ¯ t ) z 1 ( ¯ t )      + γ 4 ¯ t + q ¯ N m − 1 X j = ¯ t λ ¯ t + q ¯ N m − j − 1 5 ( k r ( j ) k + k w ( j ) k ) + γ 4 k r ( ¯ t + q ¯ N m ) k , q ∈ Z + . (28) So  φ ( t ) z 1 ( t )  is well-behaved at t = τ , τ + m 2 , τ + 2 m 2 , etc; we ca n use ( 20 ) of Cas e 1, ( 22 ) o f Ca se 2 and ( 19 ) to pr o v e that nothing un tow a rd happ ens be tw een these times. W e conclude that there exists a constant γ 5 so that      φ ( t ) z 1 ( t )      ≤ γ 5 λ t − τ 5      φ ( τ ) z 1 ( τ )      + γ 5 t − 1 X j = τ λ t − j − 1 5 ( k r ( j ) k + k w ( j ) k ) + γ 5 k r ( t ) k , t ≥ τ . (29 ) Since τ ≥ t 0 is a r bitrary , the desired bound is prov en.  4. T ol eranc e to Unmo del led Dynamics W e now consider the time-v a rying plant ( 4 ) with the term d ∆ ( t ) ∈ R r added to represent unmo delled dynamics: y ( t + 1) = θ ∗ ( t ) ⊤ f  φ ( t )  + w ( t ) + d ∆ ( t ) , φ ( t 0 ) = φ 0 . (30) Here we consider (a generalized version of ) a class of un- mo delled dynamics which is common in the adaptive c o n- trol literatur e —s ee [ 3 ] and [ 6 ]. With g : R n y · r + n u · m → R a map with a b ounded g ain, β ∈ (0 , 1) and µ > 0 , we consider m ( t + 1 ) = β m ( t ) + β | g ( φ ( t )) | , m ( t 0 ) = m 0 (31a) k d ∆ ( t ) k ≤ µm ( t ) + µ | g ( φ ( t )) | , t ≥ t 0 . (31b) 6 It turns out that this model subs umes a lar ge cla ss of clas- sical additive uncer ta in ty , mult iplicative uncer tain ty , and uncertaint y in a coprime fac to rization, with a strict ca usal- it y co nstrain t; s ee [ 6 ] for a more detailed explanation. W e will now show that if the controller ( 2 ) provides a co n vo- lution b ound for  f , s ( S , c 0 , ǫ )  , then a degree o f tolerance to unmo delled dynamics can be pr o ven. Theorem 3. Su pp ose t hat the c ontro l ler ( 2 ) pr ovides a c onvolution b ound for  f , s ( S , c 0 , ǫ )  with a gain c 1 and de c ay r ate λ 1 ∈ (0 , 1) . Then for every β ∈ (0 , 1) and λ 2 ∈ (max { λ 1 , β } , 1) , t her e exist ¯ µ > 0 and c 2 > 0 so that for every θ ∗ ∈ s ( S , c 0 , ǫ ) , µ ∈ (0 , ¯ µ ) , t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X ⊂ R l 2 , r ∈ S ( R q ) and w ∈ S ( R r ) , when the c ontr ol ler ( 2 ) is applie d to the plant ( 30 ) with d ∆ satisfying ( 31 ) , the fol lowi ng holds:         φ ( t ) z 1 ( t ) m ( t )         ≤ c 2 λ t − t 0 2         φ 0 z 1 0 m 0         + t − 1 X j = t 0 c 2 λ t − j − 1 2 ( k r ( j ) k + k w ( j ) k ) + c 2 k r ( t ) k , t ≥ t 0 . (32) Remark 5. This pr o of is b ase d, in p art, on the pr o of of The or em 3 of [ 6 ], which de als with a much simpler setup. Pr o of of The or em 3 . Fix β ∈ (0 , 1) a nd λ 2 ∈ (max { λ 1 , β } , 1) and let θ ∗ ∈ s ( S , c 0 , ǫ ) , t 0 ∈ Z , φ 0 ∈ R n y · r + n u · m , z 1 0 ∈ R l 1 , z 2 0 ∈ X , w ∈ S ( R r ) and r ∈ S ( R q ) be arbitrar y . So by h yp othesis:      φ ( t ) z 1 ( t )      ≤ c 1 λ t − τ 1      φ ( τ ) z 1 ( τ )      + t − 1 X j = τ c 1 λ t − j − 1 1 ( k r ( j ) k + k w ( j ) k + k d ∆ ( j ) k ) + c 1 k r ( t ) k , t ≥ τ ≥ t 0 . (33) T o conv ert this inequa lit y to an equa lit y , we consider the asso ciated difference equations ˜ φ ( t +1) = λ 1 ˜ φ ( t )+ c 1 k r ( t ) k + c 1 k w ( t ) k + c 1 µ ˜ m ( t )+ c 1 µ k g k ˜ φ ( t ) , ˜ φ ( t 0 ) = c 1      φ 0 z 1 0      , together with the difference e q uation based on ( 31a ): ˜ m ( t + 1 ) = β ˜ m ( t ) + β k g k ˜ φ ( t ) , ˜ m ( t 0 ) = | m 0 | . Using induction toge ther with ( 33 ), ( 3 1a ), and ( 31b ), we can prov e that      φ ( t ) z 1 ( t )      ≤ ˜ φ ( t ) + c 1 k r ( t ) k , (34a) | m ( t ) | ≤ ˜ m ( t ) , t ≥ t 0 . (34b) If we combine the difference equations for ˜ φ ( t ) and ˜ m ( t ) , we obtain  ˜ φ ( t + 1) ˜ m ( t + 1 )  =  λ 1 + c 1 k g k µ c 1 µ β k g k β  | {z } =: A cl ( µ )  ˜ φ ( t ) ˜ m ( t )  +  c 1 0  ( k r ( t ) k + k w ( t ) k ) , t ≥ t 0 . (35) Now w e s ee that A cl ( µ ) →  λ 1 0 β k g k β  as µ → 0 , and this matrix has eigenv a lues of { λ 1 , β } which are b oth less that λ 2 < 1 . Using a standa r d Ly apunov argument, it is easy to prove that there exist ¯ µ > 0 a nd γ 1 > 0 such that fo r all µ ∈ (0 , ¯ µ ] , we hav e   A cl ( µ ) k   ≤ γ 1 λ k 2 , k ≥ 0 ; if we use this in ( 35 ) and then apply the b ound in ( 3 4 ), it follows that         φ ( t ) z 1 ( t ) m ( t )         ≤ c 1 γ 1 λ t − t 0 2         φ 0 z 1 0 m 0         + t − 1 X j = t 0 c 1 γ 1 λ t − j − 1 2 ( k r ( j ) k + k w ( j ) k ) + c 1 k r ( t ) k , t ≥ t 0 (36) as desired.  5. Applications In this section, we will apply Theorems 1–3 to v ario us adaptive con trol problems. In these examples, it turns out that w e do not need z 1 as part o f the controller. 5.1. First-O r der One-S t ep-A he ad A daptive Contr ol Here we cons ider the 1st-order linea r time-inv aria n t plant y ( t + 1) = ay ( t ) + bu ( t ) + w ( t ) , =  a b  |{z} =: θ ∗ ⊤ ⊤  y ( t ) u ( t )  | {z } =: φ ( t ) + w ( t ) , y ( t 0 ) = y 0 . (37) W e ha ve y ( t ) ∈ R as the output, u ( t ) ∈ R as the input, and w ( t ) ∈ R as the noise o r disturbance. Here, θ ∗ is unknown but lies in close d and b ounded set S ⊂ R 2 ; to ensure controllabilit y we require that  a 0  / ∈ S for any a ∈ R . The control ob jective is to track a reference signa l 7 y ∗ ( t ) a symptotically; we assume that w e k now it one step ahead. In [ 4 ] the case o f S being co n vex is considered. An adaptive controller is designed bas ed on the idea l pro jec- tion algorithm, a nd it is pr o ven that a conv olution b ound is pr o vided. In that pa per this is leveraged to prove a degree of tolerance to time-v a riation a nd unmo delled dy- namics, though the results there are not q uite as stro ng a s those provided b y Theo rems 1–3 . Now w e turn to the more g eneral case o f S not co n vex. This w as consider ed in [ 8 ] a nd a con volution b ound w as prov en 2 , but nothing w as prov en ab out r obustness to time- v ariation and to unmo delled dynamics. Here we will show that the co n tr oller prop osed there fits into the framework of this pap er, so that Theore ms 1–3 can b e applied. In this cas e , it is pr o ven in [ 8 ] that S c an b e covered by t wo conv ex and compact sets S 1 and S 2 so that, for e very  a b  ∈ S 1 ∪ S 2 we have that b 6 = 0 . T o pro ceed, we use t wo par ameter estimator s—one for S 1 and one for S 2 —and then use a s witc hing ada ptive controller to switch betw een the estimates as nece ssary . F or each i ∈ { 1 , 2 } and given an estimate ˆ θ i ( t ) at time t > t 0 , we hav e a prediction err or of e i ( t + 1 ) := y ( t + 1) − ˆ θ i ( t ) ⊤ φ ( t ); estimator upda tes ar e computed by ˇ θ i ( t + 1) = ( ˆ θ i ( t ) if φ ( t ) = 0 ˆ θ i ( t ) + φ ( t ) k φ ( t ) k 2 e i ( t + 1) otherwise (38) ˆ θ i ( t + 1) = Pro j S i  ˇ θ i ( t + 1)  . (39) W e partition ˆ θ i ( t ) in a natural w ay by ˆ θ i ( t ) =:  ˆ a i ( t ) ˆ b i ( t )  . W e define a switching sig na l σ : Z → { 1 , 2 } to choose which parameter es timates to use in the cont rol law at any point in time. Namely , with σ ( t 0 ) ∈ { 1 , 2 } , the choice is σ ( t + 1) = arg min i ∈{ 1 , 2 } | e i ( t + 1 ) | , t ≥ t 0 , (40) i.e. it is the index corr esponding to the smallest prediction error. Next we apply the Certaint y Equiv alence Principle to yield u ( t ) = − ˆ a σ ( t ) ( t ) ˆ b σ ( t ) ( t ) y ( t ) + 1 ˆ b σ ( t ) ( t ) y ∗ ( t + 1 ) . (41) W e obser v e here that the co n tr o ller ( 38 )–( 41 ) fits int o the paradigm of Section 2 : w e set X = S 1 × S 2 × { 1 , 2 } , z 1 ( t ) = ∅ , 2 T echnically sp eaking, the b ound ( 3 ) was only pro ven for t ≥ τ = t 0 . How ev er, since the contro ller is time-i n v arian t, the extension to t ≥ τ ≥ t 0 follows imm ediat ely . z 2 ( t ) =   ˆ θ 1 ( t ) ˆ θ 2 ( t ) σ ( t )   , r ( t ) = y ∗ ( t + 1 ) . In [ 8 ] it is pr o ven that ( 38 )–( 41 ) provides a conv olution bo und for ( 37 ); by Theo rems 1–3 w e immediately see tha t the same is true in the prese nce o f time-v ariatio n and/or unmo delled dyna mics. 5.2. Pole-Plac ement A daptive Contr ol In this section, we consider the Pole-Placemen t Adap- tiv e Control pro blem. W e consider the n th -order linear time-in v aria n t pla n t y ( t + 1) = n − 1 X j =0 a j +1 y ( t − j ) + n − 1 X j =0 b j +1 u ( t − j ) + w ( t ) =           y ( t ) . . . y ( t − n + 1) u ( t ) . . . u ( t − n + 1)           ⊤ | {z } =: φ ( t ) ⊤           a 1 . . . a n b 1 . . . b n           | {z } =: θ ∗ + w ( t ) , t ≥ t 0 (42) with φ ( t 0 ) = φ 0 . W e have y ( t ) ∈ R as the output, u ( t ) ∈ R as the input, and w ( t ) ∈ R as the noise or disturbance. Here, θ ∗ is unknown but lies in a known s e t S ⊂ R 2 n . Asso ciated with this plant model a re the p olynomials A ( z − 1 ) = 1 − a 1 z − 1 − a 2 z − 2 · · · − a n z − n , and B ( z − 1 ) = b 1 z − 1 + b 2 z − 2 · · · + b n z − n ; W e impo se the following assumption: Assumption 1. S is c omp act, and for e ach θ ∗ ∈ S , the c orr esp onding p olynomials A ( z − 1 ) and B ( z − 1 ) ar e c o- prime. The o b jectiv e here is to obtain so me for m of stability with a secondary ob jective that of asymptotic tracking of a r eference signal y ∗ ( t ) ; the plant may b e non-minimum phase, which limits the tra c k ing goal. In [ 6 ] the case of S conv ex is cons ider ed. An adaptive controller is designed based on a mo dified version of the ideal pro jection algorithm, and it is pr o v en that a convo- lution b ound is provided; this is leveraged there to pr o v e a degree of tolerance to time-v a riation and unmo delled dy- namics, muc h like that pr o vided by Theorems 1 a nd 3. Now we turn to the more general cas e of S not conv ex. This was also c o nsidered in [ 6 ] s ub ject to Assumption 2. S ⊂ S 1 ∪ S 2 with S 1 and S 2 c omp act and c onvex, and for e ach θ ∗ ∈ S 1 ∪ S 2 , the c orr esp onding p olynomials A ( z − 1 ) and B ( z − 1 ) ar e c oprime. 8 A conv olution b o und was proven, but nothing was prov en ab out ro bustness to time-v aria tio n and to unmo delled dy- namics. Her e we will show that the co n tr oller prop osed there fits in to the framework of this pap er, so that Theo- rems 1– 3 can b e applied. T o pro ceed, we use tw o parame- ter es timato rs—one for S 1 and o ne for S 2 , and then use a switchin g adaptive controller to switch be tw een these e sti- mates as necessa ry; to prove that the approach works, all closed-lo op p oles are placed a t the o rigin. The para meter estimation is pro jection-alg orithm-based and similar to that of the previo us sub-section. F or i ∈ { 1 , 2 } and given an es timate ˆ θ i ( t ) at time t > t 0 , we have a prediction error of e i ( t + 1 ) := y ( t + 1) − ˆ θ i ( t ) ⊤ φ ( t ); estimator upda tes ar e computed by ˇ θ i ( t + 1 ) = ( ˆ θ i ( t ) + φ ( t ) k φ ( t ) k 2 e i ( t + 1) if k φ ( t ) k 6 = 0 ˆ θ i ( t ) otherwise (43) ˆ θ i ( t + 1) = Pro j S i  ˇ θ i ( t + 1)  . (44) W e partition ˆ θ i ( t ) a s ˆ θ i ( t ) =:  ˆ a i, 1 ( t ) · · · ˆ a i,n ( t ) ˆ b i, 1 ( t ) · · · ˆ b i,n ( t )  ⊤ ; asso ciated with ˆ θ i ( t ) are the p olynomials ˆ A i ( t, z − 1 ) = 1 − ˆ a i, 1 ( t ) z − 1 − ˆ a i, 2 ( t ) z − 2 · · ·− ˆ a i,n ( t ) z − n , and ˆ B i ( t, z − 1 ) = ˆ b i, 1 ( t ) z − 1 + ˆ b i, 2 ( t ) z − 2 · · · + ˆ b i,n ( t ) z − n . W e design a strictly pro p er controller by choosing its de- nominator and numerator p olynomials, resp ectiv ely , b y ˆ L i ( t, z − 1 ) = 1 + ˆ l i, 1 ( t ) z − 1 + ˆ l i, 2 ( t ) z − 2 · · · l i,n ( t ) z − n , and ˆ P i ( t, z − 1 ) = p i, 1 ( t ) z − 1 + p i, 2 ( t ) z − 2 · · · + p i,n ( t ) z − n satisfying ˆ A i ( t, z − 1 ) ˆ L i ( t, z − 1 ) + ˆ B i ( t, z − 1 ) ˆ P i ( t, z − 1 ) = 1 , (45) i.e. we pla c e the closed-lo op p oles at zero . A switching signal σ : Z → { 1 , 2 } is used to choose which parameter estimates to us e in the co n tro l law at any point in time. W e up date σ ( t ) only every N ≥ 2 n steps; to this end, we define a sequence of switching times as follows: we initialize ˆ t 0 := t 0 and then define ˆ t ℓ := t 0 + ℓN , ℓ ∈ N . The switchin g signal is given by σ ( t ) = σ ( ˆ t ℓ ) , t ∈ [ ˆ t ℓ , ˆ t ℓ +1 ) , ℓ ∈ Z + . (46) Now define the control ga ins ˆ K i ( i ) ∈ R 2 n that ar e also only upda ted every N ≥ 2 n steps: ˆ K i ( t ) := [ − ˆ p i, 1 ( ˆ t ℓ ) · · · − ˆ p i,n ( ˆ t ℓ ) − ˆ l i, 1 ( ˆ t ℓ ) · · · − ˆ l i,n ( ˆ t ℓ )] , t ∈ [ ˆ t ℓ , ˆ t ℓ +1 ) , ℓ ∈ Z + ; (47) also define the filtered refer ence s ignal r 2 ( t ) := n X j =1 ˆ p σ ( ˆ t ℓ ) ,j ( ˆ t ℓ ) y ∗ ( t − j + 1) , t ∈ [ ˆ t ℓ , ˆ t ℓ +1 ) , ℓ ∈ Z + . F or each i , define a p erformance sig nal J i ( ˆ t ℓ ) := ( 0 if φ ( j ) = 0 for all j ∈ [ ˆ t ℓ , ˆ t ℓ +1 ) max j ∈ [ ˆ t ℓ , ˆ t ℓ +1 ) ,φ ( j ) 6 =0 | e i ( j +1 | k φ ( j ) k otherwise . (48) With σ ( ˆ t 0 ) ∈ { 1 , 2 } , we set σ ( ˆ t ℓ +1 ) = arg min i ∈{ 1 , 2 } J i ( ˆ t ℓ ) , ℓ ∈ Z + , (49) and define the control law by u ( t ) = ˆ K σ ( t − 1) ( t − 1) φ ( t − 1) + r 2 ( t − 1) . (50) W e observe here that the controller ( 43 ),( 44 ), ( 45 ), ( 48 ), ( 49 ) and ( 50 ) fits int o the para digm of Section 2 ; we can rewr ite the controller in the form of ( 2 ) as follows. First we set X = R N × R N × S 1 × S 2 × { 1 , 2 } . F or t ≥ t 0 , we then set z 1 ( t + 1) = ˆ K σ ( t ) ( t ) φ ( t ) + r ( t ) , z 2 ( t ) =       z 21 ( t ) z 22 ( t ) ˆ θ 1 ( t ) ˆ θ 2 ( t ) σ ( t )       , u ( t ) = z 1 ( t ) , with r ( t ) = r 2 ( t ); fo r t ≥ t 0 and i = 1 , 2 , we then set z 2 i ( t + 1) =     0 1 1 . . . 1 0     z 2 i ( t ) +     0 0 . . . 0 1     × ( | e i ( t +1) | k φ ( t ) k φ ( t ) 6 = 0 0 otherwise , and for t > t 0 , we set 3 σ ( t ) =        σ ( t − 1 ) t − t 0 N / ∈ N arg min i ∈{ 1 , 2 } k z 2 i ( t ) k ∞ | {z } = J i ( t − N ) t − t 0 N ∈ N . 3 Here w e use k z 2 i ( t ) k ∞ to denote the ∞ - nor m of the v ector z 2 i ( t ) . 9 In [ 6 ] it is proven 4 that this adaptive controller provides a conv o lution bo und for ( 42 ); by Theorems 1–3 we see that the sa me is true in the presence o f time-v a r iation and/or unmo delled dyna mics. 6. Summary and Conclus ion In this pap er w e hav e shown that for a class of non- linear plant and controller combinations, if a conv o lution bo und on the closed-lo op behavior can b e prov en, then tolerance to small time-v ar iations in the plant pa r ameters and a sma ll amount of unmo delled dynamics fo llows im- mediately . W e applied the result to prov e robustness of our rec e ntly designed mult i-estimator switc hing adaptive controllers presented in [ 6 ] and [ 8 ]. W e exp ect this to b e applicable to other a daptiv e control paradigms, such as the ada ptiv e control of nonlinear pla n ts; this will allow one to fo cus on the ideal plant in the ana lysis knowing that r obustness will come for free. This r esult also has the po ten tial to b e applied in more g eneral nonlinear contexts. References [1] Deso er, C. , 1970. Slowly v arying discrete system x i +1 = A i x i . Electronics Letters 6, 339–340 . [2] Kreisselmeier, G., 1986 . Ada ptiv e con trol of a class of slowly time-v arying plan ts. Systems & Con trol Letters 8, 97–103. [3] Kreisselmeier, G., Anderson, B., 1986. Robust mo del reference adaptiv e contr ol. IEEE T ransactions on Automatic Control 31, 127–133. [4] Mil l er, D.E., 2017a. A parameter adaptiv e cont roller which pro- vides exponential stability: The first order case. Systems & Con- trol Letters 103, 23–31. [5] Mil l er, D.E., 2017b. Classical discrete-time adaptiv e cont rol re- visited: Exp onen tial stabilization, in: 2017 IEEE Conference on Con trol T echnology and Applications (CCT A), IEEE. pp. 1975– 1980. [6] Mil l er, D. E., Shahab, M .T., 2018. Classical p ole placemen t adap- tiv e cont rol revisited: linear-like conv olution b ound s and exp o- nen tial stability . Mathematics of Contro l, Signals, and Systems 30, 19. [7] Mil l er, D.E., Shahab, M.T., 2019. Classi cal d-Step-Ahead Adap - tiv e Control Revisited: Linear-Like Con volut ion Bounds and Exponential Stability , in: 2019 American Control Conference, IEEE, Philadelphia. [8] Shahab, M.T., Miller, D. E., 2018. Multi- Estimator Based Adap- tiv e Cont rol which Prov ides Exp onen tial Stability: The Fi rst- Order Case, in: 2018 IEEE Conferenc e on Decision and Contr ol, IEEE. pp. 2223–2228. [9] Zames, G., 1966. On the input-output stabilit y of time-v arying nonlinear feedbac k systems Part one: Conditions deriv ed usi ng concept s of loop gain, conicity , and p ositivit y. IEEE T ransactions on Automatic Control 11, 228–238. 4 T echnically sp eaking, the b ound ( 3 ) is only prov en for t ≥ τ = t 0 . Ho wev er, s i nce the cont roller is p eriodic of perio d N ≥ 2 n , it follows immediately that the s ame bound ( 3 ) holds for t ≥ τ ≥ t 0 for all τ ∈ { t 0 + N , t 0 + 2 N , t 0 + 3 N , . . . } . Since the contro ller has a b ounded gain, nothing un to w ard can happen f or other τ ’s; it i s easy to pro v e that ( 3 ) will s till hold for a suitably larger choice of c (but with the same λ ). 10