Adaptive Control 2011 Part 4 pdf

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:uf t =−k∗

Fig 1 Virtual controlled system with a virtual filter

( ) ( )

=

−+

1 j j i ξ i

f z i

1 1

,tzctztubtξ

tztξ

j 1

1 j

ξ 1 j i r 1 i r i

2 r 1

1 r

1 j ξ 1 j 0 ξ

i r i ξ

,cββ

θ

βθ

cβac

1ri1,aθc

j

j r

tCtξtAt

tt

ubtt

ξαtξ

T 1

d 1 η y η y

T f z y T 1 z 1

η

1 1

w d

w c

η η

w c η

c

+

=

++

=

++

y = ξ ,η , ξ= ξ ,ξ ,L,ξ

ξ η T

1 1,0,L,0 , c c ,0,L,0,1

1 d

c and

η dCare a vector and a matrix with appropriate dimensions, respectively Further, A is given byη the form of

.Q

AA

η

T z u

)s(

Since A and uf Q are stable matrices, η A is a stable matrix η

3.2 Virtual error system

Now, consider a stable filter of the form:

( )t ( )t u ( )t ,u

tutAt

1 f

1 f f f f

f c T f

f c c c c

c z z&

1 m 0

1 m 0

f

c 1 m c 0 T

c c

1 m 1 m c

β L − are chosen such that A is stable cf

Let's consider transforming the system (7) into a one with uf given in (8) as the input Define new variables X1 and X2as follows:

.ααα

ξαξαξ

αξX

y 0 y 1 )

1 m ( y 1 m ) m ( y 2

1 0 1 1 )

1 m ( 1 1 m ) m ( 1 1

η η η

η

++++

,0IαAαA

α

m 1 m m

XαtX

1 η 2 η 2

f z 2 T 1 1 z 1

c X X

X c

+

=

++

=

&

&

(11) where

1 1 1

1 ) m 1 (fm 1) 1 f 0 f m

( f

− L & (12) Further we have from (10) that

.Xeαeαe

α

1 m ) m

− L & (13)

Therefore defining [ (m 1)]T

e,,e

tXtAt

tubtt

XαtX

tXtAt

1 η 2 η

f z 2 T 1 1 z 1

1 E

E

c X X

X c

E E

tz1tt

tt

ubtAt

1

1 e

e e

e e e

e

e

e z

z z

z T z f z e z e

=

0 η Q η

η c 0 0

z z

αu

uβuβu

βu

f f 0 f 1 )

1 m ( f 1 m ) m ( f

f c f c ) 1 m ( f c ) m ( f

1 1 1

1

0 1 1

m

=++++

=

++++

tu1tAt

f

f f f

c f

f c

c c

z

0 z z

=

by defining [ (m 1)]T

f f f

cf = u ,u& ,L,u −

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 uf as the input

( )t Q ( )t e( )t,

tt

ubteαte

η e e e

e T e f e e

b η η

η c

+

=

++

=

&

&

(17)

where

Fig 2 Virtual error system with an virtual internal model

.Q

AQ

e

e f z

T z c 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:uf( )t =−k∗e( )t (Kaufman et al., 1998; Iwai & Mizumoto, 1994) That is, from (8), if the filter signal u can be obtained by f1

( )t k e( )t ( )t ,

f =− ∗ −θ z (18) one can attain the goal Unfortunately one can not design u directly by (18), because f1 u is f1

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

d)

s(d)s(nd

)s

myeControlled

system Virtual controlled system Virtual error system

( ) ( ) ( ) ( ) ( )t A ( )t y( )t,

tubtt

yaty

f f f f f

a f T f f f

b η η

η a

+

=

++

Fig 3 Virtual error system with an augmented filter

where yf∈ is the output of the PFC Since the PFC is minimum-phase AR f is a stable matrix

The augmented filter obtained from the filter (5) by introducing the PFC (19) can then be represented by

( )t ( )t u ( )t y ( )t,u

tutAt

f f u T z a

z u z u

1 f f f

f f f f

b z z&

(20)

f f T

Aa

0AA

f f

f f 2 1 f f

u T z

a

u z f f

T f f

u z

L

c c

0

b b b

0

a 0

1tAt

tubtt

uatu

f

2 f 1 f

a a

a a

a a T a a a a

=

++

=

0 η η

η a

&

&

where Aa is a stable matrix because the augmented filter is minimum-phase

) s (

( ) s d

s n

Using the augmented filter's output u , the virtual error system is rewritten as follows (see afFig.3):

( )t Q ( )t e( )t

tt

ytt

ubteαte

η e e e

e T e f c T a e

b η η

η c z

4.2 Controller design by single step backstepping

[Pre-step] We first design the virtual input α for the augmented filter output 1

f a

u in (21) as follows:

0σ,0γ,tkσteγtk

θ θ T θ θ

c θ

k k k

2 k

y f

y f f

a 0 f f 0

δyif,1

δyif,0yD

tubtΨayDtΨ&

(24)

where δ is any positive constant yf

Now consider the following positive definite function:

,PΔΓΔ2

1kΔγ

1eb

1

k

2 e

0= + + θ − θ+η η (25)

where

( )t k , Δ ˆ( )t ,k

e e

Since Qe is a stable matrix, there exists such Pe

The time derivative of V0 can be evaluated by

k 0 f

1

2 e 1 e min 2 0 0

RΔρΓλσ

kΔργ

σeΨy

eωρ

RλevkV

4

Γλσγρ4kσR

ρb

bP2b

αv

2 3

2 1 θ min 2 θ 2 k 2

2 k 0

1 2 e

2 e η e e e

e 0

2

θ

b c

−

∗+

=

++

a

α u ω

a a

T a a a a

2 f

ˆˆαα

ktk

αube

αe

αeαe

αα

a 0 f f

1 c c 1 1

e T e 1 f e 1 c T 1 1 e 1 1

1

f f

1 f

+

−+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂

=

θ θ

z z

η c z

f f

y f 2

f a

3 f 2 f 2 f f a

2 1 1 1 1 2 a 2 a 0 1 1 f a 1

y f

2 1 1 1 1 2 a 2 a 0 1 1 a

δyif,Ψyb

εyεyγb1

ΨωΨεωu

εωcybω

δyif

,ΨωΨεωu

εωcb1

u

η η η

(30)

where ε0 to ε3 and γf are any positive constants, and Ψ1and Ψ2are given by

,ωe

αβˆubˆe

αˆ

e

αeαˆe

αΨ

lα

ˆˆ

αkk

αΨ

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

1 f

f f

∂

∂+

∂

∂

=

z θ

z z

tˆσtωe

αtΓtˆ

tbˆσtue

αtωγtbˆ

tαˆσtee

αtωγtαˆ

1 β

2 1 2 1 β 1

1 θ 1 1 c θ 1

e b f 1 1 b e

e α 1 1 α e

1 1

1 f

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 V1

+

++

+

≤+

++

++

1βΔγ

1bΔγ1

αΔγ

1ΔΓΔ2

1ω2

1V

δyif,δ2

1βΔγ

1bΔγ1

αΔγ

1ΔΓΔ2

1ω2

1V

V

f 1

1

f f

1 1

y f 2

f 2 1 β

2 e b

2 α 1 1 T 1 2 0

y f 2

y 2 1 β

2 e b

2 α 1 1 T 1 2 0

1

θ θ

θ θ

(32)

where

( )t , Δβ βˆ ( )t β ,ˆ

Δ

btbˆbΔ,αtαˆαΔ

1 1 1 1 1 1

e e e e e e

and δ is any positive constant yf

From (26) and (32), the time derivative of V1 for y ≤f δyfcan be evaluated by

2 e 3 b b 2 e 2 α α

2 1 1 θ min θ 2 1

2 3 1 θ min θ 2 2 k k

2 e 0 1 e min 2 0

1

ReyΨyβΔμγσ

bΔμγ

σαΔμγσ

ΔμΓλσωc

ΔρΓλσkΔργσ

μρRλeεl4

1vkV

1 1

1 1

βσγμ4bσγμ4ασμ

4

Γλσε4

3RR

β 4

2 2 β b 3

2 2

α 2

2 2 α 2 1 1

2 1 min 2

1 0

2 0 f 2

5

0 f 5 0

μ4Ψyμ

2ΨyeμeΨ

V& ≤− + (35) fory ≤f δyf, where

[ ] [ ]

[ ]

[ ] [ ]

[ ]

.δμ4

ΨRR

μγ

σγ,μγ

σγ

,μγ

σγ,Γλ

μΓλσ2,c2

,Γλ

ρΓλσ2,ργ

σγ,Pλ

μρRλmins

2,s,μεl4

1vkbminα

2 5

2 M 1 1

4 β

β β 3 b

b b

2 α

α α 1

max

1 1 θ min θ 1

1 θ mac

3 1 min θ 2 k

k k e

max

0 1 e min a

a 5 0

e a

f

1 1 1

1 1

++

Ψεyεyγyy

a

RβΔμγ

σbΔμγσ

αΔμγ

σΔμΓλσ

ωcΔρΓλσkΔργσ

μρRλeεl4

1vkV

f 0

2 3 2 f

2 f 2 2 f f f f f 2 f f

1 2 4 β

β 2 3 b b

2 e 2 α α 2 1 1 1 θ min θ

2 1 2 3 1 min θ 2 2 k k

2 e 0 1 e min 2 0

1

2 1

1 1

1 1

−+

−

−

−+

V&1≤− b 1+ 2 (37) where

.4εRR

γ,s,ε4

1a

1εl4

1vkbminα

2

2 f 1 2

f a 3 f 0

e b

Finally, for an ideal feedback gain k∗ which satisfies

,ε4

1a

1,μmaxv,vεl4

1vk

3 f 5 1

1 0

V&1≤− 1+ (39) where α=min[αa,αb]>0, R=max[R1,R2] Consequently it follows that V1 is uniformly bounded and thus the signals e( ) ( ) ( ) ( ) ( )t,ω1 t,ηet,yf t,ηf t and adjusted parameters k( ) ( )t,θˆ t,

( ) ( ) ( ) ( )t,ˆ t,bˆ t ,βˆ t

αˆe θ1 e 1 are also uniformly bounded

Next, we show that the filter signal

f c

z and the control input u are uniformly bounded

Define new variable z as follows: ξ1

1 ξ c )

1 m ( ξ c ) m (

1 1 m

− L (40)

w c η

f z y T ξ 1 z

ξ& = + + + (41)

,Cξ

Aη y η 1 dη

η& = + + (42) where ξ1 and η have been given in (7) Further define y z by β1

1 f 1

z& = + + (43)

,η

βη

β

y T ξ β c ) 1 m ( β c ) m (

β 1 + m 1 1 − + + 0 1 =c η +c 1w

− L (44) where zcf1 =[1,0,L,0]zcfand we set z(βk)( )0 z(ξk)( )0,k 0, ,m

b

zαzβαβ

zβαβ

zαβz

ξαξ

T d y T ξ f z

ξ z ξ c z c

) 1 m ( ξ c z c

) m ( ξ z c ) 1 m ( ξ 1 z 1

1 1

1 1 1 0

1 1 m 2 m

1 1 m 1

w c η

+

−+

−+

=

−

− +

( ) ( )

.u

b

zαzβαβ

zβαβzαβz

T d y T ξ f z

β z β c z c

) 1 m ( β c z c ) m ( β z c ) 1 m ( β

1 1

1 1 1 0

1 1 m 2 m 1 1

m 1

w c η

−+

ξ ξ ξ

ξ = z1,z& 1,L,z1−

β β β

l1 1 2ξ

β= η 1,η& 1,L,η 1−

1tA

d y T ξ β

=

& (48) From (8) and (48), we obtain

( ) ( ) ( ) ( )

1tu1bttbA

ttb

T d y T ξ f

z β c z c

β c z

1 1

f f

f

w c η c 0 0

η z

η z

ttbAttb

1

f f f

d y ξ

1 z f 1 z

β c z c β

c z

w c η c

η z η

z

++

+

−+

+

≤+ &

t

z

T β c z z

T f

Since it follows from (19) and (24) that

( )t Ψ( )t a (y( )t Ψ( )t) ( )t

f 0 f f 0

( )t αz β bz c β( )t

z& = + + (54)

from (44) with bzz +cf ηβ( )t as the input and z as the output, since this system is β

minimum-phase and the inequality (53) is held, we have from the Output/Input Lp Stability Lemma (Sastry & Bodson, 1989) that the input bzz +cf ηβ( )t in the system (54) can be evaluated by

with appropriate positive constants lz1 and lz2 From the boundedness of zβ( )t and ηβ( )t ,

we can conclude that zcf( )t is uniformly bounded and then the control input u(t) is also uniformly bounded Thus all the signals in the resulting closed loop system with the controller (30) are uniformly bounded

1

1.01.01.011.01.0

1.01.01.0

1.0

1u10015.02.5

5.05.21.5

5.05.01

w z

( ) ( ) ( ) ( )⎥⎥⎥

w (57)

Before designing a controller, we first introduce the following pre-filter:

as

b+ (58)

in order to reduce the chattering phenomenon to be expected by switching the controller given in (30) Therefore, the considered controlled system has a relative degree of 4

Since the relative degree of the controlled system is 4, we consider a 3rd order input virtual filter in (5) Further we consider a stable internal model filter (8) of the order of 4

For the input virtual filter, in this simulation, we consider a first order PFC:

ubya

y&f =− f1 f+ a

in order to make an ASPR augmented filter

The design parameters for the pre-filter (58), the input virtual filter (5) and the internal model filter (8) are set as follows:

625β,500β,150β,20β

125β,75β,15β

1000ba

3 2

1

c

2 1 0

Further design parameters in the controller given in (23), (24), (30) and (31) are designed by

.100γε,01.0εεε,1000c

100γγγ,I5000Γ

Γ

1.0σσσσ,05.0σ,5.0l

10δ,01.0σ,500γ

f 3 2

1 0 1

β b a 4 θ

θ

β b a θ θ

y k

k

1

1 1

Figure 4 shows the simulation results with the proposed controller In this simulation, the

disturbance w is changed at 50 [sec]:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

.t20cos5.2

t20sin5.0t4cos4t4sin2

t5cos5.2t5sin5.0t2cos22tsin

Fig 4 Simulation results with the proposed controller

Fig 5 Tracking error with the proposed controller

Fig 6 Adaptively adjusted parameters

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 were changed, the control performance deteriorated

Fig 7 Simulation results without adaptation after 40 [sec]

input output

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 with an adaptive internal model was proposed for single input/single output linear minimum phase systems In the proposed method, a controller with an adaptive internal model was designed through an expanded backstepping strategy of only one step with a parallel feedforward compensator (PFC)

7 References

A, Isidori (1995) Nonlinear Control Systems-3rd ed., Springer-Verlag, 3-540-19916-0, London

A, Serrani.; A, Isidori & L, Marconi (2001) Semiglobal Nonlinear Output Regulation With

Adaptive Internal Model IEEE Trans on Automatic Control, Vol.46, No.8, pp

1178—1194, 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, 0-387-94884-8, New York

I, Mizumoto.; R, Michino.; M, Kumon & Z, Iwai (2005) One-Step Backstepping Design for

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

R, Marino & P, Tomei (2001) Output Regulation of Linear Systems with Adaptive Internal

Model, Proc of the 40th IEEE CDC, pp 745—749, 0-7803-7061-9, USA, December,

Orlando, Florida

R, Michino.; I, Mizumoto.; M, Kumon & Z, Iwai (2004) 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, 0-13-004326-5

V, O, Nikiforov (1996) Adaptive servocompensation of input disturbances, Proc of the 13th

IFAC World Congress, Vol.K, pp 175—180, San-Francisco

V, O, Nikiforov (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 40th IEEE CDC, pp 65—70, 0-7803-7061-9, USA, December,

Orlando, Florida

Z, Iwai & I, Mizumoto (1994) Realization of Simple Adaptive Control by Using Parallel

Feedforward Compensator Int J Control, Vol.59, No.6, pp 1543—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, Texas

USA

1 Introduction

Robots today have an ever growing niche Many of today’s robots are required to perform tasks which demand high level of accuracy in end effector positioning The links of the robot connecting the joints are large, rigid, and heavy These manipulators are designed with links, which are sufficiently stiff for structural deflection to be negligible during normal operation Also, heavy links utilize much of the joint motor’s power moving the link and holding them against gravity Moreover the payloads have to be kept small compared to the mass of the robot itself, since large payloads induce sagging and vibration in the links, eventually bringing about uncertainty in the end effector position In an attempt to solve these problems lightweight and flexible robots have been developed These lightweight mechanical structures are expected to improve performance of the robot manipulators with typically low payload to arm weight ratio The ultimate goal of such robotic designs is to accurate tip position control in spite of the flexibility in a reasonable amount of time Unlike industrial robots, these robot links will be utilized for specific purposes like in a space shuttle arm These flexible robots have an increased payload capacity, lesser energy consumption, cheaper construction, faster movements, and longer reach However, link flexibility causes significant technical problems The weight reduction leads the manipulator

to become more flexible and more difficult to control accurately The manipulator being a distributed parameter system, it is highly non-linear in nature Control algorithms will be required to compensate for both the vibrations and static deflections that result from the flexibility This provides a challenge to design control techniques that:

a) gives precise control of desired parameters of the system in desired time,

b) cope up with sudden changes in the bounded system parameters,

c) gives control on unmodeled dynamics in the form of perturbations, and

d) robust performance

Conventional control system design is generally a trial and error process which is often not capable of controlling a process, which varies significantly during operation Thus, the quest for robust and precise control led researchers to derive various control theories Adaptive control is one of these research fields that is emerging as timely and important class of controller design Area much argued about adaptive control is its simplicity and ease of