Adaptive stabilisation of extended nonlinear output feedback systems

Z.Ding

Abstract: The paper deals with the adaptive stabilisation for a class of uncertain nonlinear systems whose nonlinear terms are functions of two state variables. The uncertainty is para- metrised by an unknown constant parameter vector affined to smooth nonlinear functions. The model format considered in the paper can be viewed as an extension to the standard output feedback systems with nonlinear terms of the single output which has been extensively studied recently. Based on the principle of adaptive backstepping design, an adaptive control algorithm is proposed to stabilise the system in a feasibility region, using the feedback of the system output which is a vector containing the two state variables.

1 Introduction

Adaptive control of nonlinear systems is an active area of research. Nonlinear adaptive control algorithms are proposed for systems of a number of special formats using state and output feedback. In the early stage of nonlinear adaptive control, the vector field affined to the input is required to match the system vector field so that, under suitable co-ordinates, the nonlinear terms can be cancelled by the input [l]. The condition specified in [l], which is now referred to as the matching condition, is weakened by the extended matching condition in [2]. Further relaxations are reported in [3], where parametric- strict-feedback and less restrictive parametric-pure-feed- back forms are introduced. An important advance from state feedback to output feedback is achieved in [4], where conditions are identified under which nonlinear terms can be transformed into functions of a single system output, in suitable co-ordinates, so that the system is in strict feed- back form. Such conditions do not impose any growth rate restriction on the nonlinearities, such as boundedness, sector or Lipchitz assumptions [ S , 61.

A number of techniques have also been developed along with the algorithms proposed for the suggested model formats, such as adaptive backstepping in [3] and filtered transformation in [4], which are instrumental for designing nonlinear adaptive output feedback control. Further devel- opments from the original algorithms for the parametric- strict(pure)-feedback and the output feedback models have been carried out. Overparametrisation in [3] has been overcome partially in [7] and completely in [8] by the introduction of tuning functions, K-filters have been devel- oped in [9] to replace filtered transforms in [4] for the

0 IEE, 2001 IEE Proceedings online no. 200 104 16 DOZ: 10.1049/ip-cta:20010416 Paper first received 8th August 2000 and in revised form 15th March 2001 The author is with the Department of Mechanical Engineering, Ngee Ann Polytechnic, 535 Clement1 Road, Singapore 599489, Republic of Singapore E-mail: dzt@np.edu.sg

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output feedback model. Recently, efforts have been devoted to making nonlinear adaptive algorithms robust to disturbances [lo, 111 and relaxing the assumption of the sign of the high-frequency gains [12, 131.

However, the nonlinear terms in all the output feedback systems, reviewed so far, for adaptive control, are functions of a single state variable which is defined as the output, and the models are in the strict feedback form. When the nonlinear terms are functions of two state variables, the system violates the strict feedback condition, and therefore the existing output feedback control laws are no longer applicable. In this paper, an adaptive control algorithm is proposed for the stabilisation for a class of uncertain nonlinear systems whose nonlinear terms are functions of two state variables. Except for the extension in the number of state variables contained in the nonlinear terms, other assumptions conceming the system remain the same, such as linear parametrisation of uncertain constant parameters and minimum phase assumptions. Hence, the model format considered in this paper can be viewed as an extension to the standard output feedback systems with nonlinear terms of the single state variable studied in [4, 91. With the two state variables in the nonlinear functions as the output, an output feedback adaptive control design is proposed using adaptive backstepping. Filters similar to those used for the standard output feedback model are designed for state observation and backstepping design. The proposed adap- tive algorithm incorporates a few newly developed techni- ques such as K-filters and tuning functions in the design. The stability analysis guarantees the uniform stability of the adaptive system around the equilibrium, which is similar to the stability result achieved for parametric- pure-feedback systems.

2 Preliminary results

2.1 Extended output feedback system Consider a class of nonlinear systems of the form

y = cx (1)

IEE ProcXontrol Theory Appl.. Vol. 148, No. 3, May 2001

with

ro 1

Then the error E = X - f satisfies

E = Aoe (6) '

The estimate shown in eqn. 5 is not available, due to the fact that e is unknown. However, the expression is useful in the backstepping design. For example, when x3 is needed in the design, it can be expressed, based on eqn. 5 , as

x 3 = i3 + €3

= y 3 + E3 + 5 ( 3 ) 0 + €3

= y 3 + t 3 + =(;)a + E ( 3 ) i j + € 3 (7)

where the subscript (3) denotes the third row of a matrix, 6 denotes a t estimate of 0, and 0 = 0 - 6. The last two terms E ( 3 ) 6 and c3 are unknown and will be tackled in the adaptive control design.

where 0 and b are vectors of constant parameters, p is the relative degree, x E R" is the state, U E R is the input, y E R2 is the output, a and $J~,~, 0 5 i 5 q, 1 5 j 5 n are smooth nonlinear functions with 4i,j(0) = 0.

Remark I : The system in the form of eqn. 1 is different from the standard single-input single-output feedback form considered in [4, 91 which is in the strict feedback form. In this paper, the output is extended from x1 in the standard form to include x2 as well. With this extension, the system eqn. 1 is not in strict feedback form, and therefore the design methods presented in [4, 91 do not apply to it.

Assumption 1: The relative degree p is a known constant.

Assumption 2: a(y) # OVy E R2,

Assumption 3: The parameter vector b is known and the polynomial B(s) = xEo hismpi is Hurwitz.

Remark 2: Assumptions 1 and 2 are among the common assumptions for adaptive control of nonlinear systems [4, 91. Other common assumptions include the sign of the high-frequency gain bo and the minimum phase, i.e. B(s) is Hurwitz. Assumption 3 is introduced here to simplify the notation and the presentation. It is not very difficult to generalise the adaptive algorithm with known b to the systems with unknown b, as long as the sign of the high- frequency gain is known.

2.2 Observer design The basic concept of dynamic swapping in the design of K-filters used for the standard output feedback systems [9] is applied to the system model eqn. 1. A number of filters designed accordingly are summarised in lemma 1 .

Lemma 1: Consider filters defined by

5 = A05 + dJo(Y) + KY (2)

j l = A,? + [O(p-I), 1 7 bTIT4y)u (4)

where A, = A - KC and K E Rn 2, which is chosen such that A,, is Hurwitz; and consider a state estimate given by

i = y + 5 + E 0 (5) IEE Proc.-Control Theory Appl., Vol. 148, No. 3, May 2001

3 Backstepping design

Backstepping design procedures have been introduced in [4, 91 for systems where the nonlinear terms are functions of the single output x l . In the extended output feedback model eqn. 1 considered in this paper, the nonlinear terms are a function of x1 and x,, which violates the strict feedback format. Hence, the control algorithms shown in [4, 91 cannot be applied to the system eqn. 1 . However, the backstepping design principle can still be used to design the adaptive control for the systems, with modifications from the standard backstepping procedures for single- output standard output feedback form. A few notations are defined as follows:

z , =x, (8)

zi = yi - ai-,, i = 3 , . . . , p (10)

aai- ' 3 1 a 6 a. . = -rwj i = 2 , . . . , p , j = 2 , . . .','p (15)

where ai are referred as stabilising functions, I' E Rq 4 is positive definite. Step 1: From eqn. 1 , the dynamics of z1 is described by

21 = x2 + 4 0 , d Y ) + @(l)(Y)0

a1 = -C,Z1 - 40,dY) - @(l)(Y)9.

z, = z, - ClZ] - 0y0

(16)

(17)

Then the choice of the stabilising function

results in -

(1 8) where coefficient c1 is among a set of positive real design parameters ci, i = 1 , . . . , p.

269

Step 2: Considering the dynamic variables in a,, it is clear that a! =al(xl , x2, a), and from eqns. 1 and 7, the following can be obtained:

Based on eqn. 19, the stabilising function a2 is set such that

where coefficient d2 is among a set of positive real design parameters di, i = 2 , . . . , p , and z2 is one of the tuning functions, zi , i = 1, . . . , p , which will be defined later. With the defined stabilising function, the dynamics can be re- arranged as

Step i, i = 3, . . . , p : The remaining steps can be carried out in a similar way to the adaptive backstepping with tuning functions presented in [9]. Omitting the deriving proce- dures, the resultant stabilising functions are as follows:

270

To conclude the design, the tuning functions are defined as i

zi = -rCwjzj, i = I , . . . , p j= 1

The parameter adaptive law and the control input are given by

6 = zp (25)

4 Stability analysis

As presented in Section 3, there are a number of different featuresin the design procedure from the standard single- output feedback systems in [9]. In particular, at step 2, the stabilising function a2 is not assigned directly, which is caused by the fact that the system output is a function of the first as well as the second state variables. The evalua- tion of a2 depends on the factor

(1-3)

From eqn. 17, the following can be obtained:

To ensure the feasibility of the proposed algorithm, the following assumption is made.

Assumption 4:

11 +%+%a1 ax, ax2 > 0,Vx E Bx ,V6 E Bg

with B, c R” and B, being an open set such that 0 E Bg . Lemma 2: The design procedure in Section 3 is feasible in the set B, x B , with assumption 4.

Remark 3: The condition specified in assumption 4 is similar to the feasibility conditions for extended matching systems in [2] and for parametric-pure-feedback systems in [3]. Actually, by considering x3 as the input, the first and second points of eqn. 1 are in the form of extended matching or parametric-pure-feedback. To evaluate a2 from eqn. 20, the condition

is needed, which is ensured by assumption 4. There may exist some singular points in state space that violate assumption 4, and those points should be excluded from the domain of the proposed control design. Therefore, the feasibility region is not global in general. This, as explained in [2 , 31, is due to the model structure, rather than the adaptive algorithm introduced. If assumption 4 is satisfied for the entire state space, the feasibility region is then global. Indeed, the feasibility region is global for an example shown later. The condition specified in assump- tion 4 is easier to check than the conditions for extended matching or parametric-pure-feedback forms, because there are no state variables involved other than the output. The condition will also be used in the stability analysis.

IEE Proc-Control Theory Appl., Vol. 148, No. 3, May 2001

Theorem I : If a system eqn. 1 satisfies assumptions 1 to 4, then under the adaptive law eqn. 25 and the control input eqn. 26, the origin of the overall adaptive system is uniformly stable, and for every

x(0) E B,.c R", 0 E Bg, limt+, x(t) = 0, limt+, <(t) = 0, limt+, a(t) = 0, limt+, y( t ) = 0.

Proofi It will be shown in this proof that the adaptive algorithm in Section 3, together with the filters in Section 2, stabilise the system eqn. 1.

3

Consider

where

and P is a positive definite matrix, satisfying PA, +ATP = -21,. Similar to the proof of standard back- stepping results in [9], the following can be obtained:

P

v < (29) i= 1

which implies 1 1 z 11 E L , n &_and ( 1 6 11 E L , which indi- cates 11 6 11 EL, as 6 = 8 - 0.

From eqns. 17 and 9, z2 - ClZl = x2 + 40,dY) + @,l,(Y)6 (30)

Based on this equation, assumption 4 guarantees x2 EL, , and it can concluded that xi E L , for i = 1,2 and

11 y 1 1 EL,, which, in turn, ensures 11 < 11 , 11 E 11 EL, . The boundedness of y will now be established. From the

observer design, y = x - < - E8 - E. From this discus- sion, it can be concluded that, for i = 1,2, yi E L , . From eqn. 20, note that a2 is bounded, which further implies that y3 is bounded using 23 = y 3 - a2. Applying this reasoning iteratively, the boundedness of yi can be concluded, for i = 4 , . . . , p - 1, from the boundedness of ai, for i = 3, . . . , p - 1; and then the boundedness of a,, . The boundedness of the remaining elements of y can be established from the filter dynamics eqn. 4. From the dynamics of yP in eqn. 4,

cm = ($I + kp,lYl + k,,lY2 - Yp+l)/bO (31)

Substituting the above into the equation

[Yp+t 9 9 * . , Y J T = P + P i 9 . . . , bmlTyP/bo (32) where p satisfies

From the assumption that ~ ~ o b i ~ f f l - i is Hunvitz, the boundedness of p can be obtained from eqn. 33, which further implies the boundedness of y i , for i = p + 1, . . . , n. Assumption 2 provides the boundedness of U and the

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boundedness of yp+l and u p . Hence, it can be concluded that all the variables remain bounded.

As all the variables are bounded, 11 z 11 EL, which, together with 1 1 z 11 E L2 n L, , implies lim,,,z(t) = 0 and in particular, limt+,xl(t) = 0, from Barbalat's lemma. From eqn. 30 and assumption 4, it can be concluded that lim,,,x2(t) = 0 and, therefore, lim,,,y(t) = 0, which further implies limt+,<(t) = 0, limt+,E(t) = 0. Following the same procedure as when establishing the boundedness of y, it can be concluded that y converges to zero, which further implies that x and ai, i = 1, . . . , p converge to zero.

5 An example

Consider a nonlinear system

4 1 = t 2 - t : + t ? O + ~

43 = t T 8 - 2 ~ 4, = c3 - 2r:e (34)

If c1 is taken as the output, then the system isin the strict feedback form. However, the adaptive control algorithms introduced in [9, 41 cannot be applied here, because the system is nonminimum phase with respect to the output c1 . The proposed design in this paper can be applied, when the system is transformed to the extended output feedback format by the state transform

-1 1

0 -1 x = [-: -1 ;] 5 (3 5)

After the state transformation, the system is described by

i 1 = x2 + (x2 -x,)3 x2 = x3 + ( ~ 2 - ~ 1 ) ~ 8 + U (36) x3 = -(x2 - ~ , ) ~ 8 + 2u

The filters are designed as

i = A,< + [(x2 - x , ) ~ , 0, oiT + KY k = A,= + [0, (x2 - x ~ ) ~ , -(x2 - x , ) ~ ] ~

= A,? + [O, 1, 2ITu

Following the design procedure in Section 3, the parameter adaptive law and the control input are obtained as

b = r ( i + 3(X2 - x1)2)((x2 - x ~ ) ~ + qZ2 (37)

1 1 f 3(X2 - Xi)'

U =

[-zl - c2z2 - (c1 - 3 ( X 2 - x~)~)(x~ + (x2 - x , )~ ) I

- d2(1 + 3(x2 - ~ 1 ) ~ ) 2 2 - y3 - (3 - ( ( ~ 2 - + 3,)g (3 8)

For the system eqn. 36,

= ( 1 + 3 ( ~ 2 - ~ 1 ) ~ ) > 0 V Y E R ~

which is independent of 8. Therefore the domain of attraction is global.

In the simulation study, the following parameters were set: c1 = c2 = d2 = r = 1 and K = [k, k2] with k, = [9, 26, 24IT and k2=0, for 8=1 and 5(0)=[1, -1, 01. The simulation results for the system under the proposed control are shown in Figs. 2 and 3. For comparison, the open-loop response is shown in Fig. 1.

27 1

6 Conclusions 25

20

15

-

-

- I

-5

-6

-7

,

-

-

-

I

5

0

I I I I I I I I , I

0 1 2 3 4 5 6 7 8 9 1 0 time, s

Fig. 1 Open-loop response __ I , , -.- 1 2 , - - - I 3

1.5r

I \ I \

‘.5r I

-1.5 1‘) ’ I 0 1 2 3 4 5 6 7 8 9 10

Fig. 2 system control input U time, s

-8 I I , I I I I I 1 ,

0 1 2 3 4 5 6 7 8 9 1 0 time, s

Fig. 3 - t,, -.- 52, --- I 3

System response under the proposed control

This paper has considered adaptive stabilisation for a class of uncertain nonlinear systems whose nonlinear terms are functions of two state variables. The control algorithm uses the two state variables as the output, and the other state variables are not needed. The adaptive backstepping has been successfully applied to design the control input, together with K-filters and tuning functions. There are a number of modifications in the proposed algorithm to accommodate the extension in the number of the state variables in the nonlinear terms, compared with the stan- dard algorithm for the output feedback systems in [9]. As the system is not in the strict feedback form, the stability region is not global in general. In fact, the feasible region depends on the range of the unknown parameters and the nonlinear terms in the system. The stability result achieved by the proposed algorithm is comparable to the results for parametric-pure-feedback and extended matching systems. In the example, the system considered is in the standard output feedback form. However, the standard output feed- back adaptive control algorithm [9] cannot be applied, due to the difficulty that the system is of nonminimum phase with respect to the state variable in the nonlinear functions. After a state transform, the system is in the extended output feedback form and the proposed control design can be applied.

7 Acknowledgments

The author would like to thank the reviewers for their useful comments to the earlier version of the paper and the encouragement for writing the revised version.

8 References

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