Relay Control of oscillations amplitudes for systems with dela

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Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001

~

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RELAY CONTROL OF OSCILLATIONS AMPLITUDES FOR SYSTEMS WITH DELAY L. Fridman *,

v.

Strygin **, A. Polyakov **

Division of Postgraduate Study and Investigation, Chihuahua Institute of Technology, Av. Tecnologico 2909, Chihuahua, Chih., 31310, Mexico e-mail:[email protected] ** Department of Applied Mathematics, Voronezh State University, Universitetskaja pl. 1, Voronezh, 394693, RUSSIA

Abstract: Time delay does not allows to realize an ideal sliding mode, but implies oscillations in the space of state variables. The notations of E: stabilization and weak controllability are introduced. The algorithms for local stabilization of oscillation's amplitudes for weak controllable control systems with input and output delay via relay control was suggested. Copyright © 2001 IFAC Keywords: Stabilization, Time delay, Relay control

1. INTRODUCTION

control Fridman et al (1993, 2000) used in for control of amplitudes of motions.

Time delay doesn't allow to design the sliding mode control in the space of state variables (Fridman et al1993, 2000). That is why it may be singled out two main approaches to use relay control for delay systems:

Akian et al (1997) suggested P.!. control algorithms for amplitudes control for one dimensional relay system with delay in the input. Fridman et al (1993) have shown that any solution of equation

• time delay compensation

x(t)

Relay control algorithms for systems with delay in state and control based on delay compensation was suggested by Li and Yu (1999). Roh and Oh (1999) designed the sliding mode control in the space of predictor variables for input delay systems using unit control, but it is necessary to take into account the comments by Nguang (2001).

= kx -

psign[x(t - 1)],

with initial conditions 2 - ek Icp(O)1
is situated in the band

• control of oscillations amplitudes

t

The knowledge of oscillations and periodicity properties of first order systems with relay delay

E

< ln2

Ixl <

p(e k

(1) -

l)/k for all

[0,(0).

Strygin et al (2001) and Fridman et al (2001) was generalized the stabilization condition (1) for controllable systems for multidimensional case.

1 This research was partially supported under the grant of Consejo Nacional de Ciencia y Tecnologia (CONACYT)

The main result of this paper is the generalization of results of Strygin et al (2001) and Fridman

N990704

219

Consider two projectors P and Q, transforming

et al (2001) for the class of so called "weak controllable" system. The notations of E: stabilization and weak controllability are introduced. The algorithm for local stabilization of oscillation's amplitudes for weak controllable systems with input and output delay via relay control is suggested.

2.

E:

For any x E E the following representation x = Px + Qx is true. Then system (2) has the form

(Px) + (Qx) = A(Px + Qx) + Bu + !(x).(3) Denoting Y = Px, z (3) in the form

STABILIZATION

iJ = A+y + B+u + h(Y,z) { i = A-z + B-u + h(Y,z),

Consider the system

dx dt = I(x)

+ Bu,

(2)

Z E E_, A+ = PA, A- = QA, B+ = PB = (b[,bt, ... ,b~), B- = QB, h (Y, z) = P!(y + z), h = Q!(Y + z).

Definition 3. The pair of the matrices {A, B} is said to be weak controllable, if rank(B+) = dim(E+)

In this paper to stabilize the zero solution of system (2) we will try to find relay delayed m dimensional feedback in the form:

4.

u = F(signSl (x(t - 1)), ..., signSk(x(t - 1))), where the pair {S, F} belongs to the class Q

E:

STABILIZATION OF MIMO SYSTEMS

Let us denote by Gn,k the set of (n x n) matrices with the following spectrum's properties:

consisting of the pairs of smooth mappings, transforming

S : Rn -+ Rk, S

CT + = {Ad~=l U {Q'j ± ij3j }j=l' 1 + 2v = k(5)

= (Sl,S2, ... ,Skf.

Ai E (O,L),L = In(2),

F : R k -+ R m .

(6)

1 7r Q'j E (0, M), M = o~~t {2 t cos(t + "4)}' (7)

Let us denote as x(t) the solution to the system (2) with initial conditions

= cp(t)

(4)

where Y E E+,

where x E Rn, 1 : Rn -+ Rn is a smooth function, 1(0) = 0, B is a (nxm) matrix, u E R m . Assume that the matrix A = Ix(O) can have the eigenvalues with positive real part.

x(t)

= Qx one can rewrite system

and all the eigenvalues from CT + are simple.

(-I:S t:S 0).

Theorem 1. Assume

Definition 1. The trivial solution to the system (2) is said to be E: stabilizable, if for any E: > o there exist cS > 0, integer k > 0, and the pair {S,F} E Q, such that from the inequality SUPtE[-l,Ojllcp(t)11 < cS it follows that SUPtE[-l,oojllx(t) II < c.

1) Ix(O) E Gn,k; 2) the pair Ux (0), B} is weak controllable. Then the system (2) is E: stabilizable. Let us describe the control algorithm allowing to realize the theorem 1.

Remark 2. It is necessary to remark, that S, F usually depend on E: > O.

= dim E+ = k = 1 + 2v that is why the vectors {bJ} (j = TI,) are linearly independent. This means that the following representation holds: rank B+

3. WEAK CONTROLLABILITY Assume that the spectrum CT(A) of the matrix A = Ix(O) consists of two parts

CT(A)

= CT+ U CT_, hj = Sl/+2j- 1 b[ + S21+2j- 1 bt +

where CT + and CT _ are the sets of matrix A eigenvalues with the positive and negative real parts.

hj

Then the state space E = Rn could be represented in the following form E = E+ $E_, where E+, E_ are the invariant subspaces with respect to A.

= sl/+2j b[ + S21+2j bt +

+ Skl+2j- 1 bt, + Skl+2j bt,

where i = 1, ... , l,j = 1" ... , v. Consider the matrix So consisting of {Sij} coefficients for repre-

220

where B is an inclination angle, k is a friction coefficient, p = 9/ l, where l is a length of pendulum. Linearizing (11) we will have

sentations of matrix A eigenvectors in the basis bt, ... , bt· Let us design the control u in the form: u = ( So 0 0) 0

. (0 1)

(7,

x =

where the function (7 = ((71 (y(t - 1», (7z(y(t 1», ..., (7m(y(t-l)))T we will define bellow. In this case the first equation of system (4) has the form:

+ (~) u(t -1) +

0 0

0 o Al 0 0 0 o al -13 1 0 0 o 13 1 al

0 0 0

x

(~) g(Xl(t»),

(12)

where g(xd = o(xd. Assume that the eigenvalues AI, AZ of the matrix

Denote B+ So = (hI, h z , ... , hI, Ill, In, !Iz, fzz, ... , !Iv,Jzv, 0, ... ,0) = (T 0). Matrix A+ has only simple eigenvalues which, means that T is a nonsingular matrix. Substituting the variables v = T- 1 y into (8), we will have

o o

p -k

A

=

(0 1) p -k

have the different signs, and 0 < Al < L. Substituting (13)

.........

v=

0 0 0

• • • • 0.

0 0

into the system (12) we will have

v

y=

'0'

0 0 0 0 0 0 0 0

a v -13 v 13v a v

-~

+(10)(7 + g(v, z),

u(t - 1)

e+ 0.30 -

(15)

0.04 sin (B)

=u

(16)

u = 0.3a'esign(0(t - 1) - AzB(t - 1)).

where some constants depending from Ai, ai and f( x). Returning to the original system (2) we will have

B(t) = 0.005 cos(2t), O(t)

= -O.01sin(2t) for t E [-1,0).

(17) (18)

1)), (10) Figures 1- 3 shows the behavior of pendulum (16) - (18) for a' = 0.1, e = 0.01.

where B o = (b 1 , bz , ... , bk). Now from Strygin et al (2001) and Fridman et al (2001) one can conclude that the system (10) is e stabilizable.

5.2 e stabilization

01 weak

controllable system

Consider the system Xl) _ ( sin(0.7xl +O.lxz) +0.3 x z ) ( XZ -O.lsin(xl) + In(l- Xl - 1.7xz)

5. EXAMPLES

5.1 Invertible pendulum

+ ( !1 )

Consider the problem of an invertible pendulum stabilization with the help of relay delayed control. The model of the pendulum has the form:

e+ kO - psin(B) = u(t - 1).

1».

Consider the case when

1»,

= vI+zj-dt-1) sin13j +vI+zj(t-1) cos 13j , (7r(v(t -1)) = O,r = k + I,m, i = 1,2, ... , l, j = 1,2, ... ,11. Here a~ are + BS(7([PBoS ot 1 Px(t -

= ka'esign(Yl(t -

e+ kO - psin(B) = ka'esign(O(t -1) - AzB(t -1»).

ii;(t-1)

I(x)

(14)

Returning to the system (11), we will have

(71+Zj-l(V(t -1» = -a;+jesign('lj;(t -1», 'lj;(t-1) = VI+Zj-l (t -1) cos 13 j -vI+Zj (t-1) sin 13 j ,

x=

(i) g(Yl(t) + yz(t).

Let us design u(t - 1) in form

(7i(V(t - 1» = -a~esign(vi(t - 1»,

= -af+jesign(ii;(t -

1)

(9)

where g(v,z) = T- 1 !I(Tv,z). Let us design (7(v(t - 1» in form

(71+Zj(v(t - 1))

(~1 ~Z) Y - ~ (i) u(t -

u(t - 1)

It is obvious that 1(0) = 0 and

Ix(O)

(11)

221

= (0.7 -1.1

0.4) -1.7

(19)

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0,0045 0,004

-- -\\-- ,-------

_c

; __ -- .. ---.. --- -- -.---.-- __ c_ .... --: .. -- -- _:--- --.;..

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0,0025

--,-- --

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.. _, .. __ .. _· ........,· ..

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:

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---- ------- ,--. ---- -, .. -- ....I

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0,002

*

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"",0035 ·0,004 -O,0045

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t--· -- -- ,"--- -- .. ---- -- .. --------.. ----

-- .. --" ---- ---- .... ---:- .. ---- .. ------,

10

.0

30

20

so

60

90

IlO

'0

-OJJj

-

--

-

--

.. -

-

100

5

ID

1S

to

25

])

35

40

I

Fig. 1. Inclination angle

I

-

.(;,01· ..s

50

SS

60

65

10

15

80

as

90

95

100

I

e

Fig. 4.

Xl

coordinate

0,002F~::-:c-:-:-:-:-~~-:-:-:~:-:-:-:-~:::-::~~~:=~cc:-~~~::-:! 0,0018 O,0016

t·----·........ :.. -

0.035··

0,0014

0,0012t

.. --

----- :-.--

--

--

0,001

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O,OOO8t -- - ..... -- --- ... : -- .. -- -.-- --- -- .... --

---'--.--"-- .. -- .. -----,

-

0,02

0,0006

o,tl1S·

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~O.OOO2

D.~

..

£.0.0002° :l

x

•0,0004

~ .0.005

"".0006 -0,0008 -0,001

-.

0,03 '"

,--

-O.~

t-· .. ------ .. ---:-· .. ------ ..·

.tI,015 .- •.•..•..•...•......•..•...•..••

-0,0012 -0,0014

.(102

-0,0016

-OJ]25

·0,0018

.c,OO

-o,002'--

----------------' 30

20

'0

'0

so

60

90

IlO

'0

'00

I

•••••••••••••••• -•••••-_ ••

-O,D3S ID

Fig. 2. Angular speed iJ 0.0009

'"

50

60

I

"

..

'00

Fig. 5. X2 coordinate

~~~~~~~-.-:::-:-~~~~~~~~-:-:-:--:-~

0,0008 0,0007

0,0006

0,025 '

0,0005 0,0004

.. ----

------ ---- .. --. -- .. -- .. ----·-- ..·---:·---- .. -· .. -- ....1

._._-_._~--

0,02

0,0003

S

=

0,0002

0,015

-:-

0,0001

~_

, ............•................. - ..

_-, ..

--

~

10 ;

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U -O,()(x)2

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f--- - .. --

"".0004~

o b

-_

--

:

-- ..,

D·

<

--

~4,005

-0,0005

...

"",0006 -O,OCXJ7t .. -----,---

------

--

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--- .. --

-------, .QP1S'

·O,OlXIEl

"",0009

.. ---- - .•.•.•.• _-

L::.:.:.:......:.:..:.:.:::.:.:.:......:.:..:.:.:~i=:..:.:.......:....:.:.:.:..:.:.::..:.:....:.:.:....:.:.:.::.:.:.:...:.:.:.::.:.:.:;i '0

20

JO

so

60

'0

IlO

90

'00

I

ID

Fig. 3. Relay control law u The pair Ux(O),b} is not controllable, but this pair is weak controllable. It allows us to design the control law in the following form:

u

= -a'csign[l1xI(t -

1)

30

.

60

10

. ..

Fig. 6. Relay control law u

+ 2X2(t -1)].

CONCLUSION

Figures 4 - 6 are shown the behavior of control system (19) for c = 0.02, a' = 0.3 and initial functions x~(t) = O.Olet,xg(t) = 0.2sin(t) (-1:S t:S 0).

The sufficient conditions are found for stabilization of oscillation's amplitudes for weak controllable systems with input or output delay via relay control.

222

100

REFERENCES Akian, M., P.-A. Bliman and M. Sorine (1997). P.I. control of periodic oscillations of relay systems. Proceedings of Conference on Control and Chaos, St-Petersburg, Russia. Bartolini, G., W. Caputo, M. Cecchi, A. Ferrara and L. Fridman (1997). Vibration damping in elastic robotic structure via sliding modes. Int. Journ. of Robotic Systems, 14, , 675-696. Choi, S.-B. and J. K. Hedrick (1996). Robust Throttle Control of Automotive Engines. ASME J. of Dynamic Systems, Measurement and Control1l8, 92-98. Drakunov, S.V. and V.I. Utkin (1993). Sliding mode control in dynamic systems, Int. Journ. of Control, 55, 1029-1037. Fridman, L., E. Fridman and E. Shustin (1993). Steady modes in an autonomous system with break and delay. Differential Equations, 29, 1161-1166. Fridman, E., L. Fridman and E. Shustin (2000). Steady modes in the relay control systems with delay and periodic disturbances. ASME Journal of Dynamical Systems, Control and Measurement, 122,4, 732-737. Fridman, L.,V. Strygin and A. Polyakov (2001). Stabilization of Oscillations Amplitudes via Relay Delay Control. Proc. of 40 th Conference on Decision in Control, Orlando, FL. Gouaisbalt, F., W. Perruquetti, Y. Orlov and J.-P. Richard (1999). Sliding Mode Controller Design for Linear Time-Delay Systems . Proceedings of European Control Conference ECC'1999, Carlsruhe, Germany, 1999. Li, X. and S. Yurkovitch (1999). Sliding Mode Control of Systems with Delayed States and Controls, in: Variable Structure Systems, Sliding Mode and Nonlinear Control, K.D. Young, U. Ozguner (Eds.), Lecture Notes in Control and Information Sciences, 247, Springer, Berlin, 93-108. Nguang, S. K. (2001). Comments on "Robust stabilization of uncertain input delay systems by sliding mode control with delay compensation". Automatica, 37, 1677. Roh, Y.- H. and J.-H. Oh (1999). Robust stabilization of uncertain input delay systems by sliding mode control with delay compensation. Automatica, 35, 1861-1865. Strygin, V., L.Fridman and A. Polyakov (2001). Local stabilization of relay systems with delay. Doklady Mathematics, 64, 106 - 108.

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