Control of nonlinear non minimum phase s

Control of nonlinear non-minimum phase systems using dynamic sliding mode Article in International Journal of Systems Science · February 1999 DOI: 10.1080/002077299292533 · Source: DBLP

Control of nonlinear non-minimum phase

systems using dynamic sliding mode

Article in International Journal of Systems Science · February 1999

DOI: 10.1080/002077299292533 · Source: DBLP

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Control of nonlinear non-minimum phase systems using dynamic

sliding mode

A newly developed dynamic sliding mode control technique for multiple input systems is shown to be useful in the control of nonlinear, non-minimum phase systems where the zero dynamics have no ® nite escape time The system may not be dynamic feedback linearizable To achieve asymptotic performance, unbounded control may be necessary

as determined by the zero dynamics As long as the growth rate of the zero dynamics is

no more than exponential, ultimate bounded performance can be achieved with ® nite control e€ ort Lagrange stability analysis of the closed-loop system resulting from the proposed variable structure scheme is performed Essentially a thin layer is introduced around the sliding surface Outside the layer, the sliding mode controller is used; inside

dynamic compensator It is shown that there is a trade-o€ between control performance and control e€ ort The method is illustrated by the control of the Inverted Double Pendulum which is not dynamic-feedback linearizable and is non-minimum phase and thus constitutes a testing example for the proposed scheme.

The concept of zero dynamics is not essential in linear

control systems design because, as long as the system is

controllable, it can be stabilized with static state

feed-back However, it is generally recognized that zero

dynamics play an important role in nonlinear controller

design when the system is not exactly linearizable

In nonlinear controller design, di€ erent types of zero

dynamics arise from the stability analysis of di€ erent

feed-back is employed are the dynamics of the uncontrolled

states The zero dynamics adopted in Lu and Spurgeon

a generalization of the zero dynamics used by Fliess

closed-loop system resulting from a sliding mode design

is uniformly asymptotically stable However, in many practical situations, a bounded control signal is accept-able There is no need for the control to be asymptoti-cally stable It is thus possible to remove the restriction

of uniform asymptotic stability of the zero dynamics relating to the control and still yield useful results Some e€ ort has been made to control nonlinear non-minimum phase systems The remarkable work in

may be considered as the ® rst step into the control of slightly minimum phase systems Here slightly non-minimum phase means that the linearization has no eigenvalues with strictly positive real parts Isidori and

phase system to approximate the non-minimum phase system These methods fail if the growth rate of the zero

dynamics exceeds certain limits In the paper by Lu et al.

phase system is considered A generalized Lyapunov method is used in considering Lagrange stability of the closed-loop system where the zero dynamics have restricted growth rate However, this method is di cult

to generalize if the number of controls is greater than 2 and the zero dynamics have a high growth rate

0020± 7721/99 $12.00 Ñ 1999 Taylor & Francis Ltd.

Received 23 July 1997 Revised 5 January 1998 Accepted 9 January

1998.

² Control Systems Research, Department of Engineering,

Uni-versity of Leicester, Leicester LE1 7RH, UK e-mail: xyl@sun.engg.le.

ac.uk, tel: + 252 2567; sks@sun.engg.le.ac.uk, tel: +

44-116-252 2531.

From the work in Isidori (1995), it is known that the

zero dynamics essentially depends on the choice of

output A nonlinear system may be linearized with

respect to some outputs but not others Based on this

output to track the reference output while

simul-taneously rendering the zero dynamics acceptable The

output tracking considered is to zero order only which

makes the control task slightly easier The work in

control non-minimum phase linear systems and

lineariz-able nonlinear systems If a system is linearizlineariz-able and

locally weakly controllable, there is no zero dynamics

Thus the system is minimum phase by de® nition The

unicycle control problem in the paper by Sira-Ramirez

system

Nonlinear systems which are linearizable via static or

dynamic feedback and coordinate transformation are

condition for linearizability is very restrictive and thus

the class of nonlinear systems which can be linearized is

very limited It is noted that a nonlinear system which is

is not dynamic feedback linearizable, asymptotic

feed-back linearization appears a very promising alternative

This paper extends the sliding mode control method

mini-mum phase systems to non-minimini-mum phase systems

where the corresponding zero dynamics (i.e the

and a growth rate which is at most exponential It is

not necessary for the system to be feedback linearizable

The inverted double pendulum discussed by Fliess et al.

ex-amples It is demonstrated that for such systems to

achieve asymptotic performance, unbounded control

using a variable structure control, Lagrange stability

can be achieved for the closed-loop system (Lu and

control by using variable structure control around the

sliding surfaces This method may be considered as a

type of approximate sliding mode method Outside a

thin layer, the dynamic sliding mode controller

used; within the layer, the controller is designed to

expo-nentially stabilize the near zero dynamics which are the

dynamics of the compensator when the outputs and

their derivatives are su ciently small The design

method is demonstrated using the two input Inverted Double Pendulum which is both non-minimum phase and not dynamic feedback linearizable

The paper is structured as follows Section 2 provides some background for dynamic sliding mode control and

a bound estimate for a type of unstable system; section 3 describes a dynamic sliding mode control design method; section 4 is for partial stability analysis which

is related to the performance of the closed-loop system; section 5 gives Lagrange stability analysis of the closed-loop system resulting from a variable structure control design; control of the Inverted Double Pendulum is used

as an illustrative example in section 6

The following notation will be used throughout

This section introduces the class of systems to be

con-sidered The two design stages in the dynamic sliding mode approach developed for nonlinear minimum

choice of sliding mode surface and the choice of sliding reachability condition, are brie¯ y reviewed A bound estimate lemma for a type of nonlinear unstable system is given for later Lagrange stability analysis in

section 5 A concept of partial asymptotic stability which

will help in the understanding of non-minimum phase systems with zero dynamics whose growth rate is at most exponential Because of the discontinuous sliding reachability condition, di€ erential equations with dis-continuous right hand side will be involved Necessary work in this area will be reviewed (Filippov 1964, Paden

2.1 Some concepts in dynamic sliding mode

The control of nonlinear non-minimum phase systems where the highest order derivatives of the control appear linearly are considered in this paper However, ® rst several related concepts are de® ned for more general nonlinear di€ erential I± O systems It was shown by

observ-ability and regularity conditions, any nonlinear system representation

x f x,u,t,

y h x,u,t,

di€ erential I± O system representation

y n1

1 u 1 y,u^,t

y n p

where

u u1, ,u b 1 1

1 , ,u m, ,u b m 1

m

T ,

1 , ,u b m

m

T , u uT,u b T T

and

y y1, ,y n1 1

1 , ,y p, ,y n p 1

De® nition 2.1: The di€ erential I± O system (1)is called

proper if

(2) Allu i ,., , i 1, ,m,areC1-functions;

(3) Regularity Condition

det ¶ u 1, ,u m

¶ u b 1

1 , ,u b m

is de® ned as

u 1 0,u^,t 0

u p 0,u^,t 0

3

,u m, ,u b m 1

viewed as the level of control required to maintain the

states near zero For non-minimum phase systems, it is

necessary to extend this concept to the case when

dynamics An advantage of these dynamics will be that

the unstable behaviour of the uncontrolled states is

transferred to the control It is this advantage which

makes it possible to design a variable structure

con-troller to curb the control variable whilst ensuring

ulti-mately bounded regulation

y y1, ,y n1

1 , ,y m, ,y n m

par-ameters The following concept arises

De® nition 2.3: The near zero dynamics of (4)is de® ned as

u 1 " t ,u^,t " n1 1

u m " t ,u^,t " n m

or equivalently by the Implicit Function Theorem

u1b r 1 " t ,u,t

u b

4

M, c, T > 0 there holds

c is called the bound of exponential growth rate.

Remark 2 2: Clearly if the trajectories of (4)are

To control non-minimum phase behaviour with direct

dif-ferential I± O systems need to be restricted to the fol-lowing form

u b y,u,t , 5 where the highest order derivative of the control appears

1 , ,y n m

u a y,u,t : Rn Rb R Rm m

u b y,u,t : Rn Rb R Rm

De® nition 2 1’: The system (5)is proper if

functions;

(c) Regularity Condition

u N d u u0

now on

u b

u 1

a y,u,t u b y,u,t

z b 1 q 1 " t ,z,t

z b q m " t ,z,t

9

u a " t ,z,t M a z M a0

u b " t ,z,t M b z M b0

u 1

10

exists T > 0 such that

From

d

2 2 z dtd z 2zTÇz

it is deduced that

d

and

t

t0

From the Gronwall± Bellman Inequality

t

t0

c z t0

c0

c0

The proof for the second assertion is trivial and thus

z 1

2

z 1

n1

z 1

n1 u 1 z ,u^,t

z m1

2

z m1

n m1

z m1

n m1 u m1 z ,u^,t

z m

2

z m

n m u m z ,u^,t

11

1 , , z i

n i y i, ,y n i 1

i ,i 1, ,m

andz z 1, , z m T

2.2 Sliding surface and sliding reachability condition

Based on the above GCCF, the sliding surface should

be chosen to yield the required performance of the

reduced order system when the ideal sliding mode is

reached In this paper, the following direct sliding

sur-face is used

s i

n i

j 1

a j iz i

to the MIMO case in the work of Lu and Spurgeon

The choice of sliding reachability condition should

guarantee that the sliding surface is reached in ® nite

time or asymptotically and that an expression for the

control may be easily recovered For this purpose, the

modi® ed as follows

De® nition 2.5: A strong sliding reachability condition is

de® ned by

· 1 m 1 sgn s1

: :

It is clear that

i 1

2.3 Partial asymptotic stability

Consider the pair of di€ erential equations

x f t,x,z,

The following results relating to partial asymptotic

x t0,w0,t ² t > T

Then

respect to x.

2.4 Di€ erential equations with discontinuous right hand side

Due to the presence of uncertainty which may be discontinuous and the use of a discontinuous control action in sliding mode design, di€ erential equations with discontinuous right hand side must be considered Such equations are extensively studied by Filippov

Consider the system

con-dition:

there exists a Lebesgue integrable function B T t such

Under Condition B, for any initial condition

Typical examples of such functions are sign function

con-sidered in this paper automatically satisfy Condition B

3.1 Design method

Step 2 Choose a strong sliding reachability condition,

Step 3 Choose the sliding gain matrix K such that

j 1 a j i

Lyapunov equation

to

n1 1

j 1 a j1z 1

j 1

n i 1

j 1 a j iz i

j 1

j 1

u a y,u,t u b

u b y,u,t 18

Now set

n1 1

j 1 a j1z 1

j 1

n i 1

j 1 a j iz i

j 1

j 1

u a y,u,t u b

1 , ,u b m

u 1

g s u b y,u,t

n1 1

j 1 a j1z 1

j 1

n i 1

j 1 a j iz i

j 1

j 1

,

20

if the Regularity Condition is satis® ed Note that

in canonical form by introducing the pseudo-state vari-ables as

z11 z21

z b11 1 z b11

z b11 p1 z ,z,t

z b m m 1 z b m m

z b m m p m z ,z,t ,

21

where

z i z1i, ,z b i i u i,uÇ i, ,u b i 1

i 1, ,m

z z1, ,z m T

22

written as:

z F z ,z,t

z Pz ,z,t

F z ,z,t z 1

2 , , z 1

n1,u 1; ; z m

2 , , z m

Pz ,z,t z21, ,z b11,p1; ;z2m, ,z b m m,p mT

23

4 S tability of direct sliding mode

This section formally analyses the partial asymptotic

stability of the loop system First, the

closed-loop system is transformed into a proper form for

sta-bility analysis Then a fundamental lemma concerning

the stability of a particular nonlinear system is proved

Finally, the partial stability of the closed-loop system

can then be proved

z 1

1 , , z 1

n1 1, , z m

1 , , z m

coordi-nate transformation, the typical ith non-degenerate

block given by

z i

2

z i

n i

z i

n i u i z ,u^,t

is transformed as

z i

2

z i

n i 1

n i 1

j 1

a j iz i

z j

1 u j z ,u^,t

is transformed as

can be written equivalently as

z Az Ds

z Pz ,s,z,t

Pz ,s,z,t z21, ,z b11,p1; ;z2m, ,z b m m,p mT,

24

where A and D are as de® ned in Step 3 of the algorithm.

4.1 Some stability results

Some stability results for a particular nonlinear

system are presented here These will enable the stability

Lemma 4 1: Let s s1, ,s m and consider the fol-lowing time invariant nonlinear system

(1) g s g 1 s , ,g m s T is a strong sliding reach-ability condition which satis® es

the following Lyapunov equation:

Consider the following Lyapunov function candidate

which is positive de® nite and radially unbounded

2G x,s s, g

2G x,s s,Ks

xT,sTQ x

2GT

v1

v2

" vT1Cv1 2vT1HTv2 vT2Kv2

This proves that V is negative de® nite along the trajec-Ç

Because V is essentially time invariant, the discussion

above implies that

globally exponentially stable by the Lyapunov Theorem

(29),

Thus

The following result veri® es that for a linear stable

system, arbitrarily given decay rate can be achieved via

the selection of one parameter Consider

x t0

W w1, ,w n

may be expressed as

1

Thus

and

1 i n Re ¸ i L ,

1 i n ¸ i L >|

will lead to

4.2 Partial exponential stability of closed-loop system

Partial asymptotic stability implies that to achieve asymptotic performance, unbounded, but no ® nite escape time, control e€ ort might be necessary In the following theorem, the exponential boundedness for

z t will be relaxed to no ® nite escape time, i.e.

Assumption 1 is not necessarily satis® ed

j 1 a j i ¸ j 1 with a n i i 1 for

i 1, ,m D0 diag d1, ,d m1, d i 0, ,0,1 ,

Proof: First note that (24)is equivalent in stability with

Lemma 4.1

regulation may be achieved possibly with unbounded

but no ® nite escape time control e€ ort The next stage

will be to introduce the variable structure control to

e€ ort This will lead to Lagrange stability of the

closed-loop system

dynamic feedback design from section 4 is

dynamic compensator with an attenuation

rate greater than the growth rate of the near

theor-etically because of the choice of sliding surface

and sliding reachability condition In practice,

for implementation This is equivalent to

surface Note that in the paper by Slotine and

surface is used to reduce chattering The

con-troller within the layer is also variable structure

in nature and is designed as follows

If Assumption 1 is satis® ed, z t , i.e the trajectories of

the dynamic compensator, are exponentially bounded

with growth rate c as in Lemma 2.1 The control

follows:

z1i z2i

z2i z3i

z b i i 1 z b i i

b i

j z j i,

32

distinct roots, i 1, ,m µsatis® es

µ ¸min> c,

i b i,1 i m ¸ k i L i ,

Regularity Condition

rewritten as follows:

z A0z

z u a z ,z,t P0 z ,z,t u b z ,z,t

z Pz ,z,t

33

n1, , z m

form

A0i

(2)If E < ²0,

z A0z ,

z u a z ,z,t Çz b u b z ,z,t ,

34

z b

µ

b 1

j z j1

µ

b i

j z j i

µ

b m

The Lagrange stability analysis is now carried out for