Adaptive Control of a Class of Feedforward and Non-feedforward Nonlinear Systems

피드포워드와 비피드포워드 비선형성이 혼재된 비선형 시스템의 적응 제어

Adaptive Control of a Class of Feedforward and Non-feedforward Nonlinear Systems

구 민 성, 최 호 림^*, 임 종 태 (Min-Sung Koo¹, Ho-Lim Choi², and Jong-Tae Lim¹)

1Korea Advanced Institute of Science Technology

2Dong-A University

Abstract: We propose a switching-based adaptive state feedback controller for a class of nonlinear systems that have uncertain nonlinearity. The base of the proposed conditions on the nonlinearity is the feedforward form, then it is extended via a nonlinear function containing all the states and the control input. As a result, more generalized systems containing feedforward and nonfeedforward terms are allowed as long as the ratio condition of the nonlinear function is satisfied. Moreover, the information on the growth rate of nonlinearity is not required a priori in our control scheme.

Keywords: adaptive regulation, feedforward and non-feedforward, switching control

I. INTRODUCTION

There have been various control results on the stabilization/regulation problems of the feedforward systems and many related results in either state or output feedback form still can be found in very recent years [1,4,6-10,13-16]. However, in most of these results, the considered systems and control methods are naturally limited to a class of feedforward systems only. Thus, if the systems contain some additional ‘non-feedforward’ terms, most of the existing results become non-applicable. Notably, in [7], the authors developed a dynamic-gain state feedback controller which allows some non-feedforward and non-triangular terms in the nonlinearity. Thus, the system under consideration is extended into some class of systems that contain both feedforward and some non-feedforward terms. However, the method of [7] is applicable when the growth rates of nonlinear terms are known a priori. The purpose of this paper is to develop a regulating state feedback controller under the newly suggested conditions, which allow feedforward and some non-feedforward terms in the nonlinearity at once. Moreover, in our control scheme, we don’t need to know the growth rates of nonlinear terms. With the new conditions, some nonlinear terms, which cannot be covered by any of aforementioned existing results, will be allowed to consider the regulation problem.

In this paper, we will consider the following class of nonlinear systems

( , , )

x Ax Bu
= + +δ t x u (1)

where x R∈ ⁿ and u R∈ are the state and input of the system, respectively. The system matrices (A, B) is a Brunovsky canonical pair and the nonlinearity is an n × vector such as ( , , )1 δ t x u = [ ( , , ), , ( , , )] .δ1t x u
δn t x u ^T We assume that the origin is the only equilibrium point of the system (1) when u = 0. Define a positive definite matrix Eγ_{( )t} =diag[1, (γ t),
, ( )γ t ⁿ⁻¹], γ( ) 0.t > The mappings δi( , , ) :t x u R R× ⁿ× →R R, i=
1, ,n, are C and ¹ satisfy the following conditions.

Assumption 1: There exist an unknown constant L ≥ and a 0 nonnegative function φ( , , ( ))x u tγ such that

( )

( ) 1

2 1 2

1 2

( , , ) 1 ( , , ( ))

( ) | | ( ) | |

t

n i n

i i

E t x u L x u t

t x t u

γ δ φ γ

γ γ

− − −

= +

≤ +

 

×∑ +  ⁽²⁾

for all x R∈ ⁿandu R∈ .

Assumption 2: There exist functions in the ( , , ( ))φi x u tγ form of

( , ) 1

( , , ( )) ⁿ | |^{ai j}| |ⁱ ( )ⁱ

i j

j

x u t x u^µ t^ν

φ γ γ

=

=∏ ⁽³⁾

where a_{( , )}i j, µ ≥ and i 0, ν is an any real number for i i =
1, , ,

m and j=
such that 1, ,n

1

( , , ( )) ^m i( , , ( ))

x u t i x u t

φ γ φ γ

=

≤∑ ⁽⁴⁾

for all x R∈ ⁿ and u R∈ .

Copyright© ICROS 2011

* 책임저자(Corresponding Author)

논문접수: 2011. 3. 16., 수정: 2011. 4. 14., 채택확정: 2011. 4. 24.

구민성, 임종태: 한국과학기술원 전기 및 전자공학과 ([email protected]/[email protected])

최호림: 동아대학교 전기공학과([email protected])

※ 이 논문은 2011년도 한국과학기술원 BK21 전자 통신기술사업단 에 의하여 지원되었음. 이 논문은 2009년도 정부(교육과학기술부) 의 재원으로 한국연구재단의 기초연구사업 지원을 받아 수행된 것임(2009-0063942).

Using | |xi ≤γ(t)^{n i−} E xγ_{( )}t , we have, for i=
1, ,m,

( , )

( , ) 1

( 1)

1| | ( ) ( ) .

n i j

i j j

j a

n a

j t

j x γ t^{−= −} E xγ

=

≤ ∑

∏ From Assumption 2 and the

above inequality, the following inequality can be directly derived by using Eγ_{( )}t.

( ) | |

( , , ( ))

( )

i i

i r t

i q

E x u

x u t

t

µ

φ γ γ

≤ γ (5)

where _{( , )}

1 n

i i j

j

r a

=

=∑ ^{and ( , )}

1

( 1)

n

i i i j

j

q ν j a

=

= − +∑ − ^{for all} x R∈ n and u R∈ .

Assumption 3: The following inequality holds

(n−1)r q µ_i− −_{i i}< (6) 1 for all i=
1, , .m

The control problem is to adaptively regulate the system (1) under Assumptions 1 through 3. The proposed conditions are quite extensible since a large choices of φ( , , ( ))x u tγ are possible. Here, we provide some observations on how to apply the proposed conditions and how the non-feedforward terms are included throughout some specific examples for simplicity and clarity.

Example A: (Application of the conditions) We let n = 3,

1( , , )t x u x u2 32 ,

δ = δ2( , ,t x u) 0= , and δ₃( , ,t x u) 0= . By As- sumption 1, we have

2 3 1 2 2 1 2 2

2 2 2

|x u |≤γ( ) | | | | ( ) | | (1t ⁻ x u γ t u≤ +γ( ) | | | | ) ( ) | |t ⁻ x u γ t u

(7)

Thus, we obtain φ( , , ( ))x u tγ =γ( )t ⁻¹| | |x₂ ² u| .² Next, in applying Assumption 2, we give two ways and show that one is satisfactory while the other is not. (i) First, we simply choose

1,

m = i.e., φ( , , ( ))x u tγ =φ1( , , ( ))x u tγ . (ii) Second, we may further divide φ( , , ( ))x u tγ ≤γ( ) | |t ⁻¹ x₂ ⁴+γ( ) | |t ⁻¹ u ⁴ so that

1 4

1( , , ( ))x u t ( ) | |t x2

φ γ =γ ⁻ and φ₂( , , ( ))x u tγ =γ( )t⁻¹| |u⁴. Now, with Assumption 3, it is easy to check that the choice of (i) is satisfied with r =₁ 2, µ = and ₁ 2, q = However, with the ₁ 3.

choice of (ii), the term φ₁( , , ( ))x u tγ =γ( ) | |t⁻¹ x₂ ⁴ does not satisfy Assumption 3 because r =₁ 4, µ = and 1 0, q = Thus, 1 5.

it is important to obtain the proper functions φi( , , ( ))x u tγ because the same system can be viewed completely different depending on the selection.

Example B: (Norm-bound feedforward form) We consider the example in [12] such that n =3, δ1( , , ) st x u = inx3+4 ,x3²

6 2( , , )t x u u ,

δ = and δ₃( , ,t x u) 0= By assumption 1, it is easy . to obtain that

2 6

( ) 1 3 3

5

3 3

( , , ) | sin | 4 | | ( ) | | 4(1 | | | | )(| | ( ) | |)

E t t x u x x t u

x u x t u

γ δ γ

γ

≤ + +

≤ + + + (8)

From (8), we set φ₁( , , ( )) | |x u tγ = x₃ and φ₂( , , ( )) | |x u tγ =u⁵

for Assumption 2. Now, with Assumption 3, we have r = ₁ 1,

1 0,

µ = q =1 2, r =2 0, µ = and 2 5 q = i.e., all conditions 2 0, are satisfied. In fact, this particular example can be easily generalized into the following feedforward form. For

1, , 2 i=
n−

,1 ,2

( , , ) (1 | |^pi | | )(|^pi 2| | | | |)

i t x u Li xn u xi xn u

δ ≤ + + + + +
+ (9)

with δn−₁( , , )t x u ≤Ln−₁(1 | |+ xn ^pn−1,1+| |u ^pn−1,2) | |u and δn( , ,t x ) 0.

u = Here, p p ≥i,1, i,2 1, i=1, ,
n−1. From Assumption 1

and 2, we have ^{,1 ,2}

1

( , , ( )) ⁿ | |n ^pi | |^pi

i

x u t x u

φ γ

=

=∑ + ^{and φ1( , ,x u} ( )) |t xn|^p1,1.

γ = Then, we obtain r₁= p_1,1, µ = and 1 0 q = 1

(n−1) .p1,1 Therefore, it is easy to check that the conditions in Assumption 3 are satisfied. Similarly, for φj( , , ( )),x u tγ j = 2, ,2(
n −1), we can check that this form satisfies Assumptions 1 through 3 via simple algebraic manipulation. Some non- feedforward terms are limitedly allowed because the feedforward form is expressed in terms of norm-bound. For example, let

1( , , )t x u x3sinx1sinx2,

δ = which violates the conditions used in [4,11-14] due to x1 and x2. However, in view of norm-bound, it is obvious that δ₁( , , ) | |t x u ≤ x₃ which belongs to (9). Thus, this norm bound feedforward form itself has some extensions over the existing feedforward conditions.

Example C: (Non-feedforward form) Let n =3,δ1( , , )t x u =

2 15 2 3

3 1 2 sin ,1

x +x x u x δ2( , , )t x u =x u2^{13 4}, and δ₃( , , )t x u =x u₁ ⁴. In this case, it clearly contains both feedforward and non- feedforward terms. By Assumptions 1 and 2, we can obtain that

1( , ,x u t( )) |x3|,

φ γ = φ₂( , , ( ))x u tγ =γ( ) |t⁻¹ x₁|¹⁵| | | | ,x₂ ² u² φ3( ,x

1 3

2 3

, ( )) | || | ,

u tγ = x u and φ₄( , , ( ))x u tγ =γ( ) | || |t x u₁ ³. Then, Assumption 3 is satisfied by taking r =₁ 1, µ =₁ 0, q = ₁ 2,

2 11,

r = 5 µ =2 2, q =2 3, 3 1 ,

r =3 µ =3 3, 3 1 ,

q =3 r =4 1, µ = 4 3, and q = − ₄ 1.

II. ADAPTIVE FEEDBACK CONTROLLER To solve our control problem for any given initial conditions

, (0) ⁿ

x ∈R we introduce an adaptive controller which can tune the gain parameter by using the suitably defined monitoring signals.

On-line tuning feedback controller:

( ( ))

u K= γ t x (10)

where K( ( )) [ / ( ) , , / ( )],γ t = k₁ γ t ⁿ
kn γ t γ( ) 0t > is to be tuned according to the switching logic.

Monitoring signals:

( )

1( ) 1

( )

i i

i i

m t r

i q n

t E x t

µ γ

θ µ

γ

+

= +

=∑ ⁽¹¹⁾

2( ) ( ) ( )

k t

t t Eγ τx d

θ =∫ τ τ ⁽¹²⁾

for t t> where ,_k t_k k =0,1,2 ,
is the moment that switching occurs.

Remark 1: Logic (11) is proposed for preventing the finite time escape phenomenon of controlled system with high-order nonlinearity. Logic (12) is presented for the regulation of the system.

Initialization:

• Set the positive constant 0< < such as c 1 c>(n−1)r_i

i i

q µ

− − for all i=
1, ,m.

• Set positive constants S ≥₀ 1,α > and 1 γ ≥ such as 0 1

0 0

0

(0) ^{i i}

i i

r c

q n

E xγ ^µ

γ µ

γ

+

> + for all i=
1, ,m.

• Set k = at 0 t = 0 0.

Switching logic:

Step 1: Set γ( )t =γk.

Step 2: If θ₁( )t ≥mγk^c or θ₂( )t >kγk² E x tγ_k ( )k

→S_k+₁=αS_k;

→γ_k+₁=S_k+₁γ_k;

→t_k+1= ; t

→k k= + ; (Switching) 1 Go to Step 1;

Else

→γ( )t =γ_k; (No Switching) Go to Step 2;

Here, we address some mathematical notations and setups for Theorem 1 and its proof. Let AK_{( ( ))}γt = +A BK( ( ))γ t . Then, we define K K= (1) and AK =AK₍₁₎. If given that A is _K Hurwitz, from [2], we can obtain a Lyapunov equation of

1 2

( ( )) ( ( )) ( ( )) ( ( )) ( ) ( )

KT t K t K t K t t

A γ P γ +P γ A γ = −γ t E⁻ γ with P_{K( ( ))}γ_t =

( )t K ( )t

E P Eγ γ from A P^T_{K K}+P A_{K K} = − where I denotes an I n n× identity matrix. We note that the proposed controller (10) is equivalently expressed as u=γ(t KE x)⁻ⁿ γ_{( )}t .

Now, we state the main result.

Theorem 1: Under Assumptions 1 through 3, select K such that AK is Hurwitz. Then, the controller (10) with the switching logic asymptotically regulates the closed-loop system (1) for any given initial conditions x(0)∈Rⁿ. Also, γ( )t →γss< ∞ as t → ∞ .

Proof: From (1) and (10), the closed-loop system is

( ( )) ( , , )

K t

x A
= γ x+δ t x u (13)

where γ( )t is nondecreasing and γ( ) 1t ≥ for all t ≥ 0 according to the switching logic. Thus, the system (13) does not suffer from any singularity problem. The vector field of (13) is C¹ with respect to its arguments. So, the system (13) satisfies a local Lipschitz condition in a neighborhood of the initial condition

(0) ⁿ.

x ∈R Moreover, it is certain that γ( )t exists and is finite as long as x remains bounded. Thus, the solution of (13) exists

and is unique on [0, )T for some f T ∈ ∞ [5,9]. Without f (0, ) loss of generality, we assume now on that [0, )T is the f

maximally extended interval of the solution of (13).

The gain function γ( )t evolves as a piecewise constant func tion on t∈[0, )Tf and there are time-intervals ∆ = t_k: { |t t∈( ,t tk k+1] ,} k =0,1,2 ,
such that ( )γ t remains as a constant value γ for each time-interval _k ∆ Obviously, there t_k. are associated switching time denoted as tk , k =0,1,2, ,
i.e., switching occurs at t = tk. As a whole, the closed-loop system can be viewed as a switched system x f x u t
= _k( , , ( ,γ )) k =0,1,2 ,
and each subsystem x f x u γ
= k( , , )k is engaged for each time- interval ∆ where t_k γ( )t is replaced by γ for each _k ∆ tk. Now, for each subsystem x A
= K_{( )}γ_kx+δ( , , ,)t x u we set a Lyapunov function V xk( )=x P^T K_{( )}γ_k x where P_{K( )}γ_k =E P Eγ_{k K} γ_k

and Eγ_k =Eγ_{( )}t |γ_{( )}t=γ_k. Then, we have

2 2

1 E xγ_k V xk( ) 2 E xγ_k

λ ≤ ≤λ (14)

where λ λ₁= _min( )P_K and λ₂=λ_max(P_K), which are independent of γ k.

Along the trajectory of each subsy stem, we have

1 2

( ) 1 2

1

( ) 2 ( , , )

2 ( , ,

)

k k

k

k k

T

k k K

k K

V x E x x P t x u

E x

P E x E t x u

γ γ

γ

γ γ

γ δ

γ

δ

−

−

= − +

≤ − +

(15)

The inequality of (15) is obtained by using A_K^T_{( )}γ_kP_{K( )}γ_k +

1 2

( )_k ( )_{k k}.

K K k

P γ A γ = −γ⁻Eγ Regarding the last term Eγ_kδ( , , )t x u₁ of (15), by Assumption 1, we have (recall that u=γk⁻ⁿKE xγ_k )

2

( , , )1 (1 ) (1 ( , , )

k k k k

Eγδ t x u ≤L n + K γ⁻ +φ x uγ E xγ (16) Using (15)-(16), for t∈ ∆ we have t_k,

( )

( ) ²

( ) 2 1 ( , , ) _k

k k k k

V x
≤ −γ⁻ γ −σ +φ x uγ E xγ (17)

where σ =2L n(1+ K P) K .

From Assumption 2, using u=γk⁻ⁿKE xγ_k , for t∈ ∆ we tk, obtain

1 1

( , , )

i i i i

k k

i i i

r r

m m

k q i q n

i k i k

E x u E x

x u

µ µ

γ γ

φ γ ρ µ

γ γ

+

= = +

≤∑ ≤∑ ⁽¹⁸⁾

where ρ =i K ^µi, i=
1, ,m.

Note that, for each time interval,

1

i i

k

i i

m r

kc i kq n

E x m

µ γ

µ γ

γ

+

= + ≤

∑ ^holds

by the proposed switching rule. That is, the condition (2) becomes the linear growth condition for each time interval. Thus, there is no finite escape time for each time interval, which means overall

finite escape phenomenon does not occur with our switching rule under Assumptions 1 through 3. The remaining part is to show that (i) only a finite number of switching occurs, (ii) the system regulation is followed afterward.

(I) Finite switching: We show that switching occurs only finitely. It is clear that γ and _k S are nondecreasing and _k c < 1 from the switching logic and the initialization. Let k^* be the smallest number of switchings such that the following inequalities are satisfied

* (1 ^c*) 1

k m k

γ −σ +ρ γ ≥ (19)

* 2

1 c cr

Sk λ λ

 −

≥  

  (20)

12

* 2

1

k λ

λ

≥   

  (21)

where

[1, ]

max{ },i

i m

ρ ρ

= ∈

[1, ]

1 max{ }

2i m i i ,

r r µ

= ∈ + and

[1, ]

ma (x{

i m

c n

= ∈ −

1)r q µi− −i i}.

By the switching logic, for t∈ ∆t_k*, we obtain

*

*

*

1( ) 1

i i

k

i i

r

m c

q n k

i k

t E x m

µ γ

θ µ γ

γ

+

= +

=∑ ≤ ⁽²²⁾

Using (17)-(19), (22) and the continuity of x t for ( ), t∈ ∆t_k^*, we obtain

12 2

* *

* * *

12 2

* *

* *

( )

2 1

( )

2 1

) (

( )

k k

k k

k k

k

t t k

t t k

E x E x t e

E x t e

γ

γ γ

γ γ

λ λ λ λ

− +

−

− −

+

− −

≤   

 

=   

 

(23)

For t∈ ∆t_k*₊1, similarly to (23), we have

12 2

*1 *1

*1 *1 *

( )

2 1 1

( ) ^{k k}

k k

t t

Eγ x λ Eγ x tk e ^γ λ

−

+ +

+ +

− −

+

≤   

  (24)

From (24), t∈ ∆t_k*₊1, we have

* 1

*

* 2 *1

*1 *1

*

1 1 1

( )

2 1

2

1 1 1

( )

( )

i i

k

i i

i i

i i i i

k k k

i i

r m i kq n

r r r t t

m k

i kq n

E x

t

E x t

e

µ γ

µ

µ µ µ

γ γ

µ

θ γ

λ

λ γ

+

+ + +

+

= ++

+ + +

− −

+

= ++

=

 

≤   ×

 

∑

∑

(25)

By the properties of γk*₊1=Sk*₊1γk* and S_k*₊1≥ 1,

* * *

* 1

* * *

* * *

2( 1) 2

1 1 1

1

2( 1) 2( 1) 2

1 1

1 ( 1)

1 1

( ) ( )

( )

( )

k

k

n i

k k i k

i

n n i

k k i k

i n

k k

E x t x t

S x t

S E x t

γ

γ

γ

γ

+

−

+ + +

=

− −

+ +

=

−

+ +

=

≤

≤

∑

∑ ⁽²⁶⁾

From (26), the term

* 1 *

* 1 1

( ) ^{i i}

k

i i

r k q n k

E x tγ ^µ

γ µ +

+ + + +

for i=
1, ,m, in the right term of the inequality (25), is bounded as

( )

* *

*1 *

*

* * *

* *

*

*

1 1 ( 1)( )

1

1 1

1 ( 1)

1

( ) ( )

( )

i i i i

k k i i

i i i i i i

i i

k i i i

i i

r r

k k n r

q n q n q n k

k k k

r

k n r q

q n k

k

E x t E x t

S S

E x t

S

µ µ

γ γ µ

µ µ µ

µ

γ µ

µ

γ γ

γ

+

+ +

+ + − +

+ + + +

+ +

+

+ − − −

+ +

≤

≤

(27)

By substituting (27) into (25), for t∈ ∆t_k*₊1, we obtain

* *

*

*

* *

*

*

2 (( 1) ) 1

1 2 1

1 1

2 1

1 1 1

( )

( )

( )

i i

i i

i i i k

i i

i i

k

i i

r r

m n r q k

k q n

i k

r c m k r

k q n

i k

E x t

t S

E x t S

µ µ

µ γ

µ

µ γ

µ

θ λ

λ γ

λ

λ γ

+ +

− − − + + +

=

+ + + +

=

≤   

 

 

≤  

 

∑

∑

(28)

From (22) and (28), for t∈ ∆t_k*₊1, we obtain

* *

1 2 1

1

( )

r

c c

k k

t λ S m

θ γ

λ ⁺

 

≤  

  (29)

Since c c− > and 0 ^{2 *1}₁

1

1

r c c

Sk

λ λ

− −

+

  <

   from (20), we have

( )

* *

* * * *

2 1 1

( )

2

1 1 1

1

r

c c

k k

r c c c c

k k k k

S

S S

λ γ

λ

λ γ γ

λ

+

− −

+ + +

  

 

≤   <

 

(30)

Therefore, from (29) and (30), we obtain, for t∈ ∆tk*₊1

1( )t m _k^c* 1

θ < γ ₊ (31)

Also, from (21) and (24), we have, for t∈ ∆tk*₊1

* 1 * 1 12

* *

* 1

* *

* 1 2

2 2

1 1

1

* 2

1 1

( ) ( )

( )

( 1) ( )

k k

k

k t

t

k k

k k

t E x d

E x t

k E x t

γ

γ

γ

θ τ τ

λ γ

λ γ

+ +

+

+

+ +

+ +

≤

 

≤  

 

≤ +

∫

(32)

From (31)-(32) with our switching logic, it is clear that no further switching occurs for t t> _k*₊1.

(II) System regulation: We will show that if the switching is finite, then the closed-loop system (13) is asymptotically regulated.

Let t be the final switching time where kf k is the finite number f

of switchings for t∈[0,Tf .) From the finite number of switchings, it is clear that γ( )t converges to a finite value γ _kf for t∈[0, ).Tf Also, by the switching logic, for t∈(t Tk_f, f),