Adaptive Control 2011 Part 4 pdf

Adaptive Control 68 Fig. 1. Virtual controlled system with a virtual filter ( ) ( ) () () () () ∑ − = − ++−= = − 1i 1j ji ξ i f zi 11 ,tzctztubtξ tztξ j1i (6) where ( ) ∑ ∑ − = −+−−− − − = − − += = +−= −≤≤−= 1i 1j ξ 1jir1iri 2r1 1r 1j ξ 1j0 ξ iri ξ ,cββθ βθ cβac 1ri1,aθc j jr i the system (1) can be transformed into the following virtual system which has 1 f ugiven from a virtual input filter as the control input (Michino et al., 2004) (see Fig.1): () () () () () () () () () () () () ,ttξty tCtξtAt ttubttξαtξ T 1 d 1ηyηy T df zy T 1z1 η 11 wd wcηη wcηc += ++= +++= & & (7) where [] [] T r32 T TT y ξ,,ξ,ξ,, L== ξηξη and [] [ ] T T ξ η T 1 1,0,,0,,0,,0,1 LL ccc == . 1 d c and η d C are a vector and a matrix with appropriate dimensions, respectively. Further, η Ais given by the form of . Q A A η T z u η f ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 c 0 )s(f 1 ( ) sf Controlled s y stem () tu ( ) tu 1 f ( ) tu () ty Virtual controlled s y stem Adaptive output regulation of unknown linear systems with unknown exosystems 69 Since f u Aand η Q are stable matrices, η A is a stable matrix. 3.2 Virtual error system Now, consider a stable filter of the form: ( ) ( ) ( ) () () () ,tuttu tutAt 1 f 1 ffff f c T f f cccc += + = zθ czz & (8) where [] T c 1,0,,0 f L=c and [] .βα,,βα β,,β A 1m0 1m0 f c1mc0 T cc 1m1m c − − −−= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −− = − −×− L L θ I0 1m10 ccc β,,β,β − L are chosen such that f c A is stable. Let's consider transforming the system (7) into a one with u f given in (8) as the input. Define new variables X 1 and 2 X as follows: .ααα ξαξαξαξX y0y1 )1m( y1m )m( y2 1011 )1m( 1 1m )m( 1 1 ηηηηX ++++= ++++= − − − − & L & L (9) Since it follows from the Cayley-Hamilton theorem that ,0IαAαAαA 0m1 1m m1m m m =++++ − − L (10) we have from (2) and (7) that ( ) ( ) ( ) ( ) () () () ,tXtAt tubttXαtX 1η2η2 f z2 T 11z1 cXX Xc += ++= & & (11) where 11 11 f 0 f 1 )1m( f 1m )m( f f uαuαuαuu ++++= − − & L (12) Further we have from (10) that .Xeαeαeαe 101 )1m( 1m )m( =++++ − − & L (13) Adaptive Control 70 Therefore defining [ ] T )1(m e,,ee, − = L & E , the following error system is obtained: ( ) ( ) ( ) () () () () () () () () [] () .t0,,0,1te tXtAt tubttXαtX tXtAt 1η2η f z2 T 11z1 1E E cXX Xc EE 2 L & & & = += ++= += (14) Obviously this error system with the input f u and the output e has a relative degree of m+1 and a stable zero dynamics (because η A is stable). Furthermore, there exists an appropriate variable transformation such that the error system (14) can be represented by the following form (Isidori, 1995): () () () () () () () () () ,tzte tz 1 tt ttu b tAt 1 1eee e ee e e ezzz z T z f z eze = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += 0 ηQη η c 00 zz & & (15) where [ ] T eee 1m1 z,,z + = Lz and 1n z R e − ∈η . Since the error system (14) has stable zero dynamics, e z Q is a stable matrix. Recall the stable filter given in (8). Since we have from (8) that ,uuαuαuαu uβuβuβu ff 0 f 1 )1m( f 1m )m( f f c f c )1m( f c )m( f 11 11 011m =++++= ++++ − − − − & L & L (16) the filter's output signal u f can also be obtained from () () () () [] () t0,,0,1tu tu 1 tAt f fff c f f ccc z 0 zz L & = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += by defining [ ] T )1m( f ff c u,,u,u f − = L & z .Using this virtual filter signal in the variable transformation given in (6), the error system (15) can be transformed into the following form, the same way as the virtual system (7) was derived, with u f as the input. ( ) ( ) ( ) ( ) () () () ,tetQt ttubteαte ηeee e T e f ee bηη ηc += ++= & & (17) Adaptive output regulation of unknown linear systems with unknown exosystems 71 where Fig. 2. Virtual error system with an virtual internal model . Q A Q e e f z T z c e ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 c 0 Since f c Aand e z Q are stable matrices, Q e is a stable matrix. Thus the obtained virtual error system (17) is ASPR from the input u f to the output e. The overall configuration of the virtual error system is shown in Fig.2. 4. Adaptive Controller Design Since the virtual error system (17) is ASPR, there exists an ideal feedback gain ∗ k such that the control objective is achieved with the control input: ( ) ( ) tektu f ∗ −= (Kaufman et al., 1998; Iwai & Mizumoto, 1994). That is, from (8), if the filter signal 1 f u can be obtained by ( ) ( ) ( ) ,ttektu f1 c T f zθ−−= ∗ (18) one can attain the goal. Unfortunately one can not design 1 f u directly by (18), because 1 f uis a filter signal given in (8) and the controlled system is assumed to be unknown. In such cases, the use of the backstepping strategy on the filter (5) can be considered as a countermeasure. However, since the controller structure depends on the relative degree of the system, i.e. the order of the filter (5), it will become very complex in cases where the controlled system has higher order relative degrees. Here we adopt a novel design strategy using a parallel feedforward compensator (PFC) that allows us to design the controller through a backstepping of only one step (Mizumoto et al., 2005; Michino et al., 2004). 4.1 Augmented virtual filter For the virtual input filter (5), consider the following stable and minimum-phase PFC with an appropriate order n f : u )s(f 1 1 f u )s(n )s(d d d )s(d )s(n d d f u 1 f u )s(f u y m y e Controlled s y stem Virtual controlled s y stem Virtual error s y stem Adaptive Control 72 ( ) ( ) ( ) ( ) () () () ,tytAt tubttyaty fffff a f T ffff 21 bηη ηa += ++−= & & (19) Fig. 3. Virtual error system with an augmented filter where Ry f ∈ is the output of the PFC. Since the PFC is minimum-phase A f is a stable matrix. The augmented filter obtained from the filter (5) by introducing the PFC (19) can then be represented by ( ) ( ) ( ) () () () () ,tytuttu tutAt ff u T za zuzu 1fff ffff +== + = zc bzz & (20) where [ ] TT ff T f u ,y, f ηzz = and [] 0,,0,1, ,b, A a 0A A ff f f 21 f f u T z a u z ff T ff u z Lcc 0 b b b0 a0 0 = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −= Here we assume that the PFC (19) is designed so that the augmented filter is ASPR, i.e. minimum-phase and a relative degree of one. In this case, there exists an appropriate variable transformation such that the augmented filter can be transformed into the following form (Isidori, 1995): ( ) ( ) ( ) ( ) () () () ,tu 1 tAt tubttuatu f 2 f 1 f aaaa aa T aaaa ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += ++= 0 ηη ηa & & where A a is a stable matrix because the augmented filter is minimum-phase. )s(f 1 ( ) () sd sn d d Virtual error s y ste m u 1 f u f u e PFC f a u Adaptive output regulation of unknown linear systems with unknown exosystems 73 Using the augmented filter's output f a u , the virtual error system is rewritten as follows (see Fig.3): ( ) ( ) ( ) ( ) ( ) ( ) ( ) () () () .tetQt ttyttubteαte ηeee e T e f c T aee ff bηη ηczθ += +−++= & & (21) 4.2 Controller design by single step backstepping [Pre-step] We first design the virtual input 1 α for the augmented filter output f a u in (21) as follows: () ()() () () () ,tΨtt ˆ tetktα 0c T 1 f +−−= zθ (22) where k(t) is an adaptive feedback gain and () t ˆ θ is an estimated value of θ , these are adaptively adjusted by () () () () ()() () .0σ,0ΓΓ,t ˆ σtetΓt ˆ 0σ,0γ,tkσteγtk θθ T θθ c θ kkk 2 k f >>=−= >>−= θzθ & & (23) Further, () tΨ 0 is given as follows: ( ) ( ) ( ) ( ) ( ) () ⎪ ⎩ ⎪ ⎨ ⎧ > ≤ = +−= f f 1 y f y f f a0 ff 0 δyif,1 δyif,0 yD tubtΨayDtΨ & (24) where f y δ is any positive constant. Now consider the following positive definite function: ,PΔΓΔ 2 1 kΔ γ2 1 e b2 1 V ee T e 1 θ T2 k 2 e 0 ηηθθ +++= − (25) where () () ,t ˆ Δ,ktkkΔ θθθ −=−= ∗ ∗ k is an ideal feedback gain to be determined later and P e is a positive definite matrix that satisfies the following Lyapunov equation for any positive definite matrix R e . .0RPQQP ee T eee <−=+ Adaptive Control 74 Since Q e is a stable matrix, there exists such P e . The time derivative of V 0 can be evaluated by ( ) [ ] ( ) {} [] () 0 2 3 1 θ min θ 2 2 k k 0 f 1 2 e1emin 2 00 RΔρΓλσ kΔρ γ σ eΨy eωρRλevkV +−− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−−− +−−−−≤ − ∗ θ η & (26) with any positive constant 1 ρ to 3 ρ . Where 1a1 αuω f − = and ( ) [] () . ρ4 Γλσ γρ4 kσ R ρb4 bP2 b α v 2 3 2 1 θ min 2 θ 2 k 2 2 k 0 1 2 e 2 eηee e e 0 2 θ bc − ∗ += + += (27) [Step 1] Consider the error system, 1 ω -system, between f a uand 1 α . The 1 ω -system is given from (21) by . 1 11 1 αubua αuω aa T aaa a 2f f & &&& −++= − = ηa (28) The time derivative of 1 α is obtained as follows: () () () ,ubΨayD ˆ ˆ αα k tk α e α ub e α e α eα e α α a0 ff 1 c c 11 e T e 1 f e 1 c T 1 1 e 1 1 1 f f 1f +−+ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = θ θ z z ηczθ & & & & (29) where T e T 1 b θθ = . Taking (28) and (29) into consideration, the actual control input is designed as follows: [] ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ >−+− ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++− ≤ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++− = f f f f y f 2 0 f a 3 f 2 f 2 ff a 21111 2 a 2 a011 f a 1 y f 21111 2 a 2 a011 a δyif,Ψ yb ε yεyγ b 1 ΨωΨεωuεωc yb ω δyif ,ΨωΨεωuεωc b 1 u η η η (30) Adaptive output regulation of unknown linear systems with unknown exosystems 75 where 0 ε to 3 ε and f γ are any positive constants, and 1 Ψ and 2 Ψ are given by ,ω e α β ˆ ub ˆ e α ˆ e α eα ˆ e α Ψ l α ˆ ˆ α k k α Ψ 1 2 1 1 f e 1 c T 1 1 e 1 2 2 c 2 c 1 2 2 1 2 2 1 1 1f f f ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − ∂ ∂ − ∂ ∂ −= + ∂ ∂ + ∂ ∂ + ∂ ∂ = zθ z z θ θ & & & where l is any positive constant and 11ee β ˆ , ˆ ,b ˆ ,α ˆ θ are estimated values of 11ee β,,b,α θ , respectively, and adaptively adjusted by the following parameter adjusting laws. () () () () () () () () () () () () () () () tβ ˆ σ e α tωγtβ ˆ t ˆ σtω e α tΓt ˆ tb ˆ σtu e α tωγtb ˆ tα ˆ σte e α tωγtα ˆ 1 β 2 1 2 1 β 1 1 θ 1 1 c θ 1 e bf 1 1 b e eα 1 1αe 11 1 f 1 1 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = − ∂ ∂ −= − ∂ ∂ −= − ∂ ∂ −= & & & & θzθ (31) where 111 βθb α βb α σ,σ,σ,σ,γ,γ,γ are any positive constants and 0ΓΓ T θθ 11 >= . 4.3 Boundedness analysis For the designed control system with control input (30), we have the following theorem concerning the boundedness of all the signals in the control system. Theorem 1 Under assumptions 1 to 3 on the controlled system (1), all the signals in the resulting closed loop system with the controller (30) are uniformly bounded. Proof: Consider the following positive and continuous function V 1 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ >+++ +++ ≤+++ +++ = − − ,δyif,y 2 1 βΔ γ2 1 bΔ γ2 1 αΔ γ2 1 ΔΓΔ 2 1 ω 2 1 V δyif,δ 2 1 βΔ γ2 1 bΔ γ2 1 αΔ γ2 1 ΔΓΔ 2 1 ω 2 1 V V f 1 1 ff 1 1 y f 2 f 2 1 β 2 e b 2 e α 1 1 θ T 1 2 10 y f 2 y 2 1 β 2 e b 2 e α 1 1 θ T 1 2 10 1 θθ θθ (32) Adaptive Control 76 where () () () () ,βtβ ˆ βΔ,t ˆ Δ btb ˆ bΔ,αtα ˆ αΔ 111111 eeeeee −=−= −=−= θθθ and f y δ is any positive constant. From (26) and (32), the time derivative of V 1 for f y f δy ≤ can be evaluated by [] () [] () [] () ()() 1 f 0 f 2 14 β β 2 e3 b b 2 e2 α α 2 11 1 θ min θ 2 11 2 3 1 θ min θ 2 2 k k 2 e01emin 2 01 ReyΨyβΔμ γ σ bΔμ γ σ αΔμ γ σ ΔμΓλσωc ΔρΓλσkΔρ γ σ μρRλe εl4 1 vkV 1 1 11 +−− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− −−− −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− −−− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−−≤ − − ∗ θ θ η & (33) with any positive constants 0 μ to 4 μ . Where [ ] ( ) . γμ4 βσ γμ4 bσ γμ4 ασ μ4 Γλσ ε4 3 RR β 4 22 β b 3 2 e 2 b α2 2 e 2 α 2 1 1 2 1 θ min 2 θ 1 01 11 +++++= − θ Here we have () () () 2 5 5 2 0 f 2 5 0 f 50 f eμ μ4 Ψy μ2 Ψy eμeΨy + − + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + −−=−− (34) with any positive constant 5 μ . Furthermore, for f y f δy ≤ , since ( ) 0tΨ 0 = & is held, there exists a positive constant M Ψ such that ( ) ( ) M0 f ΨtΨty ≤− . Therefore the time derivative of V 1 can be evaluated by 11a1 RVαV +−≤ & (35) for f y f δy ≤ , where Adaptive output regulation of unknown linear systems with unknown exosystems 77 [] [] [] () [] [] () [] .δ μ4 Ψ RR μ γ σ γ2,μ γ σ γ2 ,μ γ σ γ2, Γλ μΓλσ 2,c2 , Γλ ρΓλσ 2,ρ γ σ γ2, Pλ μρRλ mins 2,s,μ εl4 1 vkb2minα 2 y 5 2 M 11 4 β β β 3 b b b 2 α α α 1 θ max 1 1 θ min θ 1 1 θ mac 3 1 θ min θ 2 k k k emax 01emin a a50ea f 1 1 1 11 ++= ⎥ ⎥ ⎦ ⎤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎢ ⎢ ⎢ ⎣ ⎡ −− = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−−= − − − − ∗ For f y f δy > , the time derivative of V 1 is evaluated as [] () [] () [] () ,eyeΨ Ψεyεyγyya RβΔμ γ σ bΔμ γ σ αΔμ γ σ ΔμΓλσ ωcΔρΓλσkΔρ γ σ μρRλe εl4 1 vkV f 0 2 03 2 f 2 f 2 2 fffff 2 ff 1 2 14 β β 2 e3 b b 2 e2 α α 2 11 1 θ min θ 2 11 2 3 1 θ min θ 2 2 k k 2 e01emin 2 01 21 1 1 11 −+ −−−+− + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−−− −−− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− −−− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−−≤ − − ∗ ηηa θ θ η & (36) and thus we have for f y f δy > that ,RVαV 21 b 1 +−≤ & (37) where . 4ε RR γ2,s, ε4 1 a 1 εl4 1 vkb2minα 2 2 f 12 f a 3 f 0e b 2 1 a += ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−−−= ∗ (38) [...]... Regulation With Adaptive Internal Model IEEE Trans on Automatic Control, Vol .46 , No.8, pp 1178—11 94, 0018-9286 G, Feg & M, Palaniswami (1991) Unified treatment of internal model principle based adaptive control algorithms Int J Control, Vol. 54, No .4, pp 883—901, 0020-7179 H, Kaufman.; I, Bar-Kana & K, Sobel (1998) Direct Adaptive Control Algorithms-2nd ed., Springer-Verlag, 0-387- 948 84- 8, New York I,... CDC, pp 745 — 749 , 0-7803-7061-9, USA, December, Orlando, Florida R, Michino.; I, Mizumoto.; M, Kumon & Z, Iwai (20 04) One-Step Backstepping Design of Adaptive Output Feedback Controller for Linear Systems, Proc of ALCOSP 04, pp 705-710, Yokohama, Japan, August S, Sastry & M, Bodson (1989) Adaptive Control Stability, Convergence, and Robustness, Prentice Hall, 0-13-0 043 26-5 V, O, Nikiforov (1996) Adaptive. .. ˆ b ˆ β ˆ θ ˆ θ1 Fig 6 Adaptively adjusted parameters 83 Adaptive Control 84 A very good control result was obtained and we can see that a good control performance is maintained even as the frequencies of the disturbances were changed at 50 [sec] Figures 7 and 8 show the simulation results in which the adaptively adjusted parameters in the controller were kept constant after 40 [sec] After the disturbances... ηy + cd 1 w , β1 β1 (44 ) where z c f 1 = [1,0 ,L ,0]z c f and we set z( k ) (0 ) = z( k ) (0 ), k = 0 ,L , m We have from (40 ) and (41 ) β1 ξ1 that ( ) & ξ 1 − α z ξ 1 = z( m + 1 ) + β c m − 1 − α z z( m ) ξ1 ξ1 ( ) + β c m − 2 − α zβ c m − 1 z( m − 1) + L ξ1 ( ) & + β c 0 − α zβ c 1 z ξ 1 − α z z ξ 1 T T = bz u f1 + cξ ηy + cd 1 w Further, we have from (43 ), (44 ) and (8) that (45 ) Adaptive output regulation... Adaptive Output Feedback Control of Uncertain Nonlinear systems, Proc of 16th IFAC World Congress, DVD, Prague, July R, Marino & P, Tomei (2000) Robust Adaptive Regulation of Linear Time-Varying Systems IEEE Trans on Automatic Control, Vol .45 , No.7, pp 1301—1311, 0018-9286 86 Adaptive Control R, Marino & P, Tomei (2001) Output Regulation of Linear Systems with Adaptive Internal Model, Proc of the 40 th... sin (2t ) ⎤ ⎡ 2 sin (4 t ) ⎤ ⎢ 2 cos(2 t ) ⎥ ⎢ ⎥ ⎥ ⇒ w = ⎢ 4 cos (4 t ) ⎥ w=⎢ ⎢ 0.5 sin (5t )⎥ ⎢ 0.5 sin (20 t )⎥ ⎢ ⎥ ⎢ ⎥ ⎢2.5 cos(5t )⎥ ⎢2.5 cos(20 t )⎥ ⎣ ⎦ ⎣ ⎦ Figure 5 is the tracking error and Fig.6 shows the adaptively adjusted parameters in the controller output Fig 4 Simulation results with the proposed controller Fig 5 Tracking error with the proposed controller input Adaptive output regulation... Exosystems, Proc of the 40 th IEEE CDC, pp 65—70, 0-7803-7061-9, USA, December, Orlando, Florida Z, Iwai & I, Mizumoto (19 94) Realization of Simple Adaptive Control by Using Parallel Feedforward Compensator Int J Control, Vol.59, No.6, pp 1 543 —1565, 0020-7179 4 Output Feedback Direct Adaptive Control for a Two-Link Flexible Robot Subject to Parameter Changes Selahattin Ozcelik and Elroy Miranda Texas A&M University-Kingsville,... perturbation theory A decomposition based robust controller is proposed with respect to the slow subsystem, and H ∞ control is applied to the fast subsystem The proposed control method has been implemented on a two-link flexible manipulator for precise end-tip tracking control In this work a direct adaptive controller is designed and the effectiveness of this adaptive control algorithm is shown by considering... (1997a) Adaptive servomechanism controller with implicit reference model Int J Control, Vol.68, No.2, pp 277—286, 0020-7179 V, O, Nikiforov (1997b) Adaptive controller rejecting uncertain deterministic disturbances in SISO systems, Proc of European Control Conference, Brussels, Belgium Z, Ding (2001) Global Output Regulation of A Class of Nonlinear Systems with Unknown Exosystems, Proc of the 40 th IEEE... changed, the control performance deteriorated output input Fig 7 Simulation results without adaptation after 40 [sec] Adaptive output regulation of unknown linear systems with unknown exosystems 85 Fig 8 Tracking error without adaptation 6 Conclusions In this paper, the adaptive regulation problem for unknown controlled systems with unknown exosystems was considered An adaptive output feedback controller . 4 μ . Where [ ] ( ) . γ 4 βσ γ 4 bσ γ 4 ασ 4 Γλσ 4 3 RR β 4 22 β b 3 2 e 2 b α2 2 e 2 α 2 1 1 2 1 θ min 2 θ 1 01 11 +++++= − θ Here we have () () () 2 5 5 2 0 f 2 5 0 f 50 f eμ 4 Ψy μ2 Ψy eμeΨy. Fig. 6. Adaptively adjusted parameters feedback gain k(t) α ˆ b ˆ β ˆ θ ˆ 1 ˆ θ Adaptive Control 84 A very good control result was obtained and we can see that a good control. Automatic Control, Vol .46 , No.8, pp. 1178—11 94, 0018-9286 G, Feg. & M, Palaniswami. (1991). Unified treatment of internal model principle based adaptive control algorithms. Int. J. Control,