Output-sliding control for a class of nonlinear systems

ISA Transactions 40 (2001) 123±131

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Output-sliding control for a class of nonlinear systems Y.J. Huang *, H.K. Way Institute of Electrical Engineering, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, Taiwan 32026, ROC

Abstract In this paper, a systematic output-sliding control design methodology for nonlinear multivariable systems is presented. The control law consists of a continuous nominal control and a discontinuous switching control. The former is the equivalent control for the system with all uncertainties at zero and the latter is designed for nonzero uncertainties. Simulation results indicate that the proposed switching control law drives the system state trajectories onto the chosen switching surface in ®nite time and the output tracking is achieved. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Variable structure system; Robust control; Nonlinear system; Sliding mode

1. Introduction In recent years, the variable structure control (VSC) approach has received increasing attention for robust controller design for nonlinear systems in the presence of both parametric uncertainties and external disturbances [1±6]. The underlying principle behind VSC is to alter the system dynamics along the switching surfaces in the state space in which the desired dynamic behavior is assumed so that the system states are attracted to these surfaces and maintained thereafter. Several advantages such as simplicity, adaptation to various perturbations from modeling and disturbance can then be achieved. Using output variables to construct the switching surfaces, [2] presented a new VSC approach for nonlinear systems but the result was not complete. Lately, Huang and Yeung [7,8] developed

* Corresponding author. Tel.: +886-3-463-8800, ext 410; fax: +886-3-463-0336. E-mail address: [email protected] (Y.J. Huang).

the so-called output-sliding control (OSC) with practical solutions. However, this method is limited to linear systems. In this paper, the OSC approach for nonlinear multivariable systems is presented. It is believed that a straightforward extension of most of the design procedures illustrated so far is possible, provided the restriction of our consideration to a particular class of multivariable nonlinear systems as speci®ed in the following section. Special emphasis is on the differential geometric approach [9], which is the mathematical fundamental of this strategy. The switching surfaces are de®ned individually for each output channel. The control law consists of a continuous nominal control and a discontinuous switching control. The nominal portion is determined using the nominal values of the system parameters, while the switching portion deals with the parameter variations and disturbances. The regular OSC design methodology is presented in the following section. In Section 3, a numerical tracking problem example with parameter variations is carried out. Finally, Section 4 gives the conclusion.

0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0019-0578(00)00041-0

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Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

2. Main result

In fact, one can generate a linear combination of the previous derivatives of yi without introducing any information about the controls, i.e.

2.1. I/O linearization Consider a multivariable nonlinear system which can be described as X : x ˆ f…x† ‡ gk …x†uk …t†; m

kˆ1

yi …t† ˆ hi …x†; i ˆ 1; 2;


; m:

…1†

in which f…x† and g…x† ˆ ‰g1 …x†; g2 …x†;


; gm …x†Š are smooth vector ®elds, u ˆ ‰u1 ; u2 ;


; um ŠT and h1 …x†; h2 …x†;


; hm …x† are smooth functions de®ned on an open set of Rn . Utilizing the Lie derivative notation, the derivative of each output channel can be written as y…1† i ˆ

@hi …x† …f…x† ‡ gk …x†uk †; i ˆ 1; 2;


; m; @x kˆ1 m X

…5†

One can generate a linear I/O relation between the output derivatives and the new input for the channel vi by adding the ri derivatives of yi as vi ˆ

ri X ik y…k† i kˆ0

ˆ

ri m X X ik Lkf …hi …x†† ‡ iri Lgk …Lfri 1 …hi …x†††uk …x†y…k† i …x† kˆ0

kˆ1

ˆ li …x† ‡ ii …x†u:

…6†

Lgk hi …x†uk :

For shorthand notations, de®ne …2†

kˆ1 i …x† †
gk …x† elements is di€erent If any of the …@h@x from zero, one stops di€erentiating yi ; if not, one continues to di€erentiate until stage ri which is the minimum order of the derivative of yi where the coecient of at least one uk is not zero. The integer ri is called the relative degree of the nonlinear system (1) for any input uk that does not appear in the expression of the derivative of yi until the ri th di€erentiation of the output channel yi . One can express this derivative in the Lie derivative notation as:

m X dri yi y 1 ri ˆ L … h … x † † ‡ < dLf i …hi …x††; gk …x† > uk ; f r i dt kˆ1

…3†

or, ˆ

1

m X

ˆ Lf hi …x† ‡

yi…ri †

1

yi yi X d k yi X ik k ˆ ik Lkf …hi …x††: dt kˆ0 kˆ0

Lrf i …h…x††

m X ‡ …Lgk Lrf i 1 hi …x††uk ;

…4†

kˆ1

where ri is the relative degree of the nonlinear system (1).

yi…k† …x† ˆ li …x† ˆ

dk yi ; dtk

ri X ik Lkf …hi …x††; kˆ0

h i ii …x† ˆ iri Lg1 …Lrf i 1 …hi …x†††


Lgm …Lrf i 1 …hi …x††† : …7† Augmenting the system to include all output channels, one has v…x† ˆ l…x† ‡ J…x†u;

…8†

where  T J…x† ˆ iT1 …x†;


; iTm …x† lT …x† ˆ ‰l1 …x†;


; lm …x†ŠT v…x† ˆ ‰v1 ;


; vm ŠT ;

…9†

Assuming that J(x) is invertible, the control law that linearizes the system is given by u ˆ …J…x†† 1 ‰v…x†

l…x†Š:

…10†

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

It is obvious that to linearize the system we must ®nd an invertible m  m Jacobian matrix J as in [2] 2 < dLrf 1 1 …h1 …x††; g1 …x†>


< dLrf 1 6 6< dLr2 1 …h …x††; g…x† > 2 6 f J…x†ˆ 6 6 .. 6 . 4 < dLfrm 1 …hm …x††; g…x†> 2 3 Lg1 Lrf 1 1 h1 …x†


Lgm Lrf 1 1 h1 …x† 6 7 6 L Lr2 1 h …x†


L Lr2 1 h …x† 7 2 gm f 2 6 g1 f 7 7: ˆ6 6 7 . . 6 7 . 4 5 Lg1 Lfrm 1 hm …x†


Lgm Lrf m 1 hm …x†

1

…h1 …x††; gm …x†>

3 7 7 7 7 7 7 5

…11†

a. J is nonsingular; b. the zero-dynamics are globally exponentially stable. 2.2. OSC design Using the consequence developed in the previous statement, one can de®ne the sliding variable i as a function of the derivatives of yi up to the ri th order, where ri is the equivalent linearizability index for the output yi . De®ne  as a set of switching surfaces i , which also represent the desired dynamics of the errors in the sliding mode, as functions of the output space vector, i.e.  ˆ ‰1 ; 2 ;


; m ŠT ;

…12†

i ˆ

…13†

i …p†ei ;

where ˆ pri ‡ i;1 pri 1 ‡


‡ …t ei ˆ …yi ydi †dt:

i …p†

0

i;ri 1 p

chosen by the designer to specify the output error dynamics in the sliding mode according to (13). The above system is said to be in the ``sliding motion'' when   0. This can be achieved in ®nite time t0 if : i  i 4

"i ji j; "i > 0;

‡

i;ri ;

…14†

d is the di€erential operator, yd ˆ The symbol p ˆ dt ‰yd1 ; yd2 ;


; ydm ŠT denotes the desired output. The coecients are constants i;1 ;


; i;ri 1 ; i;ri

…15†

holds for all subsequent time t > 0. The larger "i is, the shorter the time t0 will be to reach the sliding motion. In the sliding mode, one has i …p†…yi

Since VSC algorithms are now widely used due to their robust properties and potential to decouple the high dimensional design problem into a set of independent subproblems of lower dimension [5,9,10]. The following usual assumptions are made:

125

ydi †  0;

i ˆ 1; 2;


; m:

…16†

In order to meet the sliding condition (15), let the control law be formulated as u ˆ uo ‡ us ;

…17†

where uo is the continuous nominal control and us is the discontinuous switching control. The former is the equivalent control for the nominal system and the latter deals with the parametric variations and disturbances. Let the symbol
represent the varying part, i.e. fi …x† ˆ fio …x† ‡
fi …x†, hi …x† ˆ hio …x†, gk …x† ˆ gko …x†. Note that (1),(12) and (13) give rise to :  ˆ a…x† ‡ Ju

…p†yd ;

…18†

where 2

3 ri j L h … x † j l f1 6 jˆ0 7 6 7 6 r2 7 6P 7 r2 j 7 L h … x † a…x† ˆ 6 j 2 f 2 6 jˆ0 7 6 7 6P 7 r m 4 5 rm j j Lfm hm …x† 2

rl P

jˆ0

< dLrf i

1

…h1 …x††; gl …x† >


< dLfri

6 6 < dLr2 1 …h2 …x††; g…x† > f Jˆ6 6 4 < dLfrm 1 …hm …x††; g…x† >  …p† ˆ diag 1 …p†; 2 …p†;


; 2 3 1 …p†yd1 6 …p†y 7 d2 7 6 2 7 …p†yd ˆ 6 .. 6 7 4 5 . m …p†ydm

m …p†

1

…hl …x††; gm …x† >

3 7 7 7; 7 5

;

…19†

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Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

Since one has assumed that J is nonsingular and refers to this case as the regular case, let Jo 1 ‰ao …x†

uo ˆ

…p†yd Š;

…20†

hand side of (22). This can be achieved by formulating us as us ˆ ‰u1s ; u2s ;


; ums ŠT uis m X K…t† …Jo 1 †i; j sgnj …i ˆ 1; 2;


; m†;

ˆ

where

…25†

jˆ1

Jo …x† ˆ 2 < dLrfo1 1 …h1 …x††; g1 …x† >


< dLrfo1 6 6 < dLr2 1 …h …x††; g…x† > 2 6 fo 6 6. 6 .. 4 < dLrfom 1 …hm …x††; g…x† > 2 r1 3 P r1 j L h …x† 1; j 1 f 1o 6 jˆ0 7 6 7 6P 7 6 r2 7 r2 j 6 2; j Lf2o h2 …x† 7 6 jˆ0 7 ao …x† ˆ 6 7: 6 7 .. 6 7 6 7 . 6 rm 7 4P 5 rm j m; j Lfmo hm …x†

where

1

…h1 …x††; gm …x† >

jˆ0

3 7 7 7 7 7 7 5


J ˆ J

(21)

…22†

…26†

The reason for the selection of the above control law can be validated as follows: Substitution of (25) into (22) yields :  i ˆ
ai …x† ‡
Ji uo K…t† m X …
J
Jo 1 †i; j sgnj sgni †sgni ; …1 ‡

…27†

where Jio and
Ji represent the ith row of Jo and
J, respectively, and …
†i; j denotes the …i; j†th element of …
†. It can be seen from (24) and (26) that the following inequality is valid

…23†

The value of
J depends on the system uncertainties. Assuming that the sum of the absolute values of all of the elements of each row of the product
J
Jo 1 is less than 1, i.e. m X …
J
J 1 † 1 < 1; for i ˆ 1; 2;


; m: o

‡1;  > 0 1;  < 0;

jˆ1

ao …x†; Jo :

jˆ1

k0 > 0…k0 is a constant†:

where
a…x† ˆ a…x†

 sgn ˆ

Assume that Jo does not degenerate into a singular matrix and thus the inverse matrix Jo 1 exists. Substituting (20) and (21) into (18) yields :  ˆ
a…x† ‡ Jus ‡
Juo ;

supf
ai …x† ‡
Ji uo g ‡ k0 ; K…t†5  m X 4inff1 ‡ …
J
Jo 1 †i; j sgnj sgni g;

…24†

K…t†…1 ‡

m X jˆ1

…
J
Jo 1 †i; j sgnj sgni †

5
ai …x† ‡
Ji uo > 0: Therefore, (27) and (28) imply that :  i4

k0 …1 ‡

m X …
J
Jo 1 †i; j sgnj sgni †; for i > 0; jˆ1

jˆ1

This assumption is practically true for a robot arm. The purpose here to select a suitable switching control us such that each row of Jus dictates the sign of the corresponding row on the right

…28†

X : …
J
Jo 1 †i; j sgnj sgni †; for i < 0:  i 5k0 …1 ‡ m

jˆ1

…29†

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

"

In other words, (15) holds with : i  i 4

m X k0 …1 ‡ …
J
Jo 1 †i; j sgnj sgni †:

Jˆ …30†

jˆ1

Thus, the control law given by (20), (25) and (26) guarantees that the sliding motion is reached and sustained. The calculation of the switching controller gain in every sample is practical since it possesses no varying uncertainty. In the case of
J
Jo 1 ˆ 0, the control law (25) can be modi®ed to be us ˆ ‰u1s ; u2s ;


; ums ŠT ; m k…t† X …J 1 † sgnj ; uis ˆ ki jˆ1 o i; j

…31†

where k…t†5supf
ai …x† ‡
Ji uo g ‡ k0 ; m X …
J
Jo 1 †i; j sgnj sgni g: ki 5inff1 ‡

…32†

Note that the constants ki , i ˆ 1; 2;


; m, are positive.

x2

1 ˆ 0

0

2

 :

The matrix J is nonsingular. The switching surfaces are de®ned as : 1 ˆ e1 ‡ 5e1 ; : 2 ˆ e2 ‡ 10e2 ‡ 25e2 ; where ei ˆ obtains

„t

0 …yi

ydi †dt. Using (20) and (21), one

uo ˆ " x51 x2 ‡ 5x1 0:5… x21 ‡ 2x2 cosx3 ‡ 4sinx3 cosx3 †  :  …yd1 ‡ 5yd1 † ‡ : : 0:5…y d1 ‡ 10yd1 ‡ 25yd1 †

8 < 1; sat ˆ ; : 1;

Consider the following system …1 ‡ : 6 x ˆ 4 …1 ‡



# 5… x2

2sinx3 †

12:5x3

14 1 <  < 1: 41

The switching control us is determined from (31) and (32) as

3. Illustrative example


a†x51
b†x21

#

The sign  function in (31) is replaced by the function sat , where

jˆ1

2

Lg1 L0f h1 …x† Lg2 L0f h1 …x† Lg1 L1f h2 …x† Lg2 L1f h2 …x†

127

3

2

1 7 6 5 ‡ 40 0

x2 2sinx3 2 3 0:5  1 6 7 x0 ˆ 4 3:0 5; y ˆ 0 0:5

 3

0 7 2 5u; 0

 0 0 x: 0 1

In this example, the uncertainties are supposed to be
a ˆ
b ˆ 1: A calculation by means of the Lie di€erential approach shows that the relative degrees r1 of the output y1 is 1 and r2 of the channel y2 is 2. Thus

us ˆ

2:0 0

2 3  … x5 ‡ 5:0†sat 1 0 0:25 5: 4 1 2 2:5 … x21 ‡ 5:0†sat 0:25

A simulation was carried out for both the nominal and parameter varying cases. In the nominal tracking case, the desired output paths are yd1 ˆ sin…0:5t† and yd2 ˆ cos…0:5t†. The initial condition at t ˆ 0 is x…0† ˆ ‰0:5 30:5ŠT . High frequency components of the chattering are undesirable because they may excite unmodeled high-frequency plant dynamics, which could result in unforeseen instabilities. The chattering of the control inputs is to be reduced because of the boundary layer technique. Here the boundary

128

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

layer technique is used with i ˆ 0:25. To drive the state into the desired switching surfaces, the switching gains are chosen as k0 ˆ 5:0. The simulation results are shown in Figs. 1±7. The details for the output y1 tracking the desired path yd1 are presented in Figure 6. One can see that output robustness is obvious and the tracking performance is excellent. Using the same initial condition and desired output paths in the case above, assume that those parameter variations
a and
b range from 1 to 1. In other words, the parameters vary by 100%. The switching gains are set to be k1 ˆ 0:5, k2 ˆ 0:2 and k0 ˆ 5:0. The simulation results are shown in Figs. 8±14. Again, output robustness is observed,

Fig. 3. Sliding variable sv1 in the nominal case.

Fig. 1. Input u1 in the nominal case.

Fig. 4. Sliding variable sv2 in the nominal case.

Fig. 2. Input u2 in the nominal case.

Fig. 5. Tracking plots of y1 and yd1 in the nominal case.

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

Fig. 6. Detailed tracking plots of y1 and yd1 in the nominal case.

Fig. 7. Tracking plots of y2 and yd2 in the nominal case.

Fig. 8. Input u1 in the case of parameter variation.

129

Fig. 9. Input u2 in the case of parameter variation.

Fig. 10. Sliding variable sv1 in the case of parameter variation.

Fig. 11. Sliding variable sv2 in the case of parameter variation.

130

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131

Fig. 12. Tracking plots of y1 and yd1 in the case of parameter variation.

Fig. 14. Tracking plots of y2 and yd2 in the case of parameter variation.

4. Conclusion

Fig. 13. Detailed tracking plots of y1 and yd1 in the case of parameter variation.

and the overall controlled system shows the output track and the desired output trajectories in an excellent fashion. From the simulation results it is obvious that the controlled system is insensitive to parameter variations. The e€ectiveness of the sliding mode controller in maintaining the motion along the sliding surfaces under parameter perturbations approximates those with no parameter variations. In addition, it is also demonstrated that the a€ect of the parameter variations is eliminated quickly due to the design strategy of the proposed OSC.

In this study, an output-sliding control design strategy for a class of nonlinear systems was developed. The control law consists of a continuous nominal control and a discontinuous switching control. The former is the equivalent control determined by only the nominal values of the system parameters and the latter deals with the parameter variations and disturbances. The multivariable I/O linearization technique was utilized. For the sake of simplicity and without loss of generality, only the non-interacting multivariable control example was presented to show the e€ectiveness, credibility and robustness of the OSC approach. Simulation results illustrate that the proposed OSC is indeed an excellent candidate for the robust control of uncertain systems. References [1] J.-J.E. Slotine, S.S. Sastry, Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators, International Journal of Control 38 (1983) 465±492. [2] B. FernaÂndez, J.K. Hedrick, Control of multivariable nonlinear systems by the sliding mode method, International Journal of Control 46 (1987) 1019±1040. [3] R.A. Decarlo, S.H. Zak, G.P. Matthews, Variable structure control of nonlinear multivariable system: a tutorial, Proc, IEEE 76 (1988) 212±232.

Y.J. Huang, H.K. Way / ISA Transactions 40 (2001) 123±131 [4] H. Sira-Ramerez, Nonlinear variable structure systems in sliding mode: the general case, Transactions on Automatic Control, IEEE 34 (1989) 1186±1188. [5] Y.C. Chen, S. Chang, Output tracking design of ane nonlinear plant via variable structure system, Transactions on Automatic Control, IEEE 37 (1992) 1823±1828. [6] S.V. Drakunov, V.I. Utkin, Sliding mode control in dynamic system, International Journal of Control 55 (1992) 1029±1037. [7] Y.J. Huang, K.S. Yeung, A robust control scheme against

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all parameter variations and disturbances, International Journal of System Science 25 (1994) 1621±1629. [8] Y.J. Huang, K.S. Yeung, Output-sliding control design of a class of multivariable systems, International Journal of System Science 25 (1994) 1373±1389. [9] A. Isidori, Nonlinear Control Systems: An Introduction, Springer-Verlag, New York, 1985. [10] Y.C. Chen, P.L. Lin, S. Chang, Design of output tracking via variable structure system: for plants with redundant inputs, IEE Proceedings-D 139 (1992) 421±428.