Sliding Mode in Systems with Parallel Unmodeled High Frequency Oscillations

Copyright © IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995

SLIDING MODE IN SYSTEMS WITH PARALLEL UNMODELED HIGH FREQUENCY OSCILLATIONS K. D. YOUNG* and V. I. UTKIN** "Lawrence Livermore National Laboratory, University of California, Livermore, California 94550 U.S.A. E-mail: kkdyoung@l!nl.gov ""Department of Electrical Engineering and Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210 U.S.A. E-mail: [email protected]

Abstract. In this paper, we examine the effects of parallel unmodeled high frequency parasitic dynamics which may destroy sliding mode. We show that asymptotic observers provide the robustness to the nominal feedback system as in the case of serial parasitic dynamics. Key Words. Sliding Mode ControL Parasitic Dynamics, Asymptotic Observers, Motion Separations.

tion in the boundary layer (Young and K watny, 1982; Asada, 1986; Kwatny and Siu, 1987). This approach replaces the switching function in the discontinuous control by a continuous approximation in the vicinity of the switching manifold. However, although chattering can be removed , the robustness of sliding mode is also compromised. The second approach utilizes a localization of the high frequency phenomenon in the feedback loop (Bondarev, 1985)- it introduces a discontinuous feedback control loop which is closed through an asymptotic observer of the plant. Since the model imperfections of the observer are supposedly smaller than that in the plant, and the control is discontinuous only with respect to the observer variables, chattering is localized inside a high frequency loop which bypasses the plant . However, this approach assumes that an asymptotic observer can be designed such that the observation error converges to zero asymptotically.

1. INTRODUCTION

The utility of feedback control which is discontinuous on the intersection of hypersurfaces of the state space of the dynamic system, or plant , being controlled is primarily due to the superb robustness that the closed loop system possesses in sliding mode - state trajectories in the manifold defined by the surfaces' intersection. The unique insensitivity of the system in sliding mode with respect to system parameter variations and exogenous disturbance is the primary reason why discontinuous feedback control is one of the solutions for the control of uncertain dynamic systems. Despite of its nice properties, discontinuous control, or more commonly known as variable structure control, is mostly restricted to control engineering problems where the control input of the plant is, by the nature of the control actuator, necessarily discontinuous. Such problems include control of electric drives where pulse-width-modulation is not the exception, but the rule of the game. Space vehicle attitude control is another example where reaction jets operated in an on-off mode are commonly used . The third example, which is closely related to the first one, is power converter and inverter feedback control design .

Whereas the robustness issues relating to parasitic actuator and sensor dynamics which are in series with the nominal plant have been resolved using the aforementioned by-pass through the asymptotic observer, there remains a large class of high frequency parasitic dynamics which are in parallel to the plant. This is the case when the plant models flexible mechanical structures in the modal form, and the high frequency modes are generally neglected in the design. In this paper, we consider the robustness of sliding mode control in the presence of these types of parasitics, and show that the effects of high frequency oscillations may be suppressed by using asymptotic observer in the sliding mode controller.

However, in the presence of switching nonidealities, such as switching time delays and small time constants in the actuators, the discontinuity in the feedback control may excite umodeled fast dynamics in the plant, thus producing unacceptable system dynamic behavior which is commonly referred to as chattering. Two different approaches have been proposed to reduce chattering. The first one is the use of continuous approxima-

483

and the sliding mode dynamics on y = 0 are governed by

We first examine the issues with parallel high frequency parasitics using a simple example, and then provide general formulation of the sliding mode control design.

(11) (12)

From the sliding mode condition,

2. ILLUSTRATIVE EXAMPLE

(13)

We consider the following simple example to motivate our problem formulation . The dynamic system

x

This implies that the regulation of x suffers from having a high frequency oscillation component. For a closer look at the quality of closed loop system response, we shall estimate the effects of the parasitic dynamics. Physical reality dictates that high frequency modes have finite energy. We rewrite the parasitic dynamics with respect to the new time variable T = ~ :

(1)

U

(2) (3) has "parallel" parasitic dynamics denoted by the states ql and q2 , and the dominant motion is that of a pure integrator plant whose state is x. Parasitic dynamics have high frequency with relatively slow decaying rate, i.e., A = A(W) ,

lim A =

.

A

hm - =0

00 ,

W ->OO

W --+OO

W

(4)

For large w , the above system is approximately homogeneous, and conservative in the new time scale, i.e., the total energy

For example, A = yfw. Suppose the plant output is consists of two components:

(16)

(5)

is conserved and bounded for any w. This means that ql is O(~) , and q2 is 0(1) . From (9), this implies that an 0(1) feedback control is needed to provide sliding mode.

The control design objective is to regulate x to zero. For the nominal plant where the parasitic dynamics are neglected, the feedback control

u = -uosgn(x)

,U

o> 0

(6)

Since asymptotic observers in sliding mode has been shown to be effective toward improving the closed loop system response for parasitic dynamics which are in series with the plant, we propose to design observer for the plant with parallel parasitics dynamics. We design a standard linear observer

results in sliding mode with x = 0 after a finite time interval since xx < O. However, since the plant output is corrupted with the parasitic mode, the actual control input takes the form

u = -uosgn(y)

(7)

Following from

:t=u+l(y- x )

iJ = u + q2 sliding mode exists on y

(8)

=0

for the nominal plant without the parasitics dynamics where l is the constant feedback gain which is chosen such that the observation error

if

(9)

e=x-x

We note that in contrast to serial parasitic dynamics, the existence of sliding mode is unaffected by parallel parasitic dynamics. Since the parasitic dynamics have high frequency oscillations, the magnitude of velocity q2 may be large, thus it is necessary to have large magnitude of control input to provide sliding mode. Furthermore, the equivalent control u eq J i.e., the solution to iJ = 0, is

u eq = -q2

(17)

(18)

converges to zero asymptotically with a desired decaying rate. The feedback control is discontinuous with respect to an estimated state

u = -uosgn(x) The condition for sliding mode to exist on becomes

(19)

x= 0 (20)

(10)

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with rankB = m, and

The required magnitude U o depends on ql which is of 0 ( Clearly, it is an order of magnitude different from the previous design (9) . Sliding mode dynamics are obtained as follows

t ).

q2 -)..q2 - (w 2 + l)ql - lx

x=0

=}

x

= e,

(25) (26) --+

0 if

W

--+ 00 .

-Aq2 - nql

+ Ru,

q2 E'Rl

The

n are diagonal matrices

+ A l2 X 2 , A 2l XI + A22X2 + B2U ,

AllXI

-Lx - lql

(27)

qi -qi -

(28)

'!/J (C)ql - c2 lx

where

(30)

1

(31)

± J J '!/J (c) + O(c))

The sliding mode dynamics are given by

(40) For a controllable pair (A , B ), there exists Cl such that the eigenvalues of the above system can be arbitrarily assigned (Utkin and Young , 1978) . Let the discontinuous feedback control

(32)

In particular, ql (t ) is a high frequency oscillation time function whose amplitude is O( From (27) , the homogeneous component of x(t ) is decaying with rate l, and the forced component due to ql is of O(~).

t)·

(41 ) where

Uo

is a positive scalar. Then from

(42 )

This simple example illustrates the utility of an asymptotic observer for parallel parasitic dynamics which is primarily in the reduction of the required magnitude of control for enforcing sliding mode, and in the attenuation of the high frequency oscillation component in the regulated variable.

the conditions for the existence of sliding mode Si Si

< 0, i = 1, .. . , m

(44)

We consider the general form of plants with parallel parasitic dynamics:

+ Bu , x

E 'Rn

,u E 'Rm

,

/3.

In the presence of parasitic dynamics, the output of the system contains high frequency components

3. PROBLEM FORMULATION

= Ax

(43)

hold if

for some positive

j;

(38)

(39)

For sufficient small c > 0, the motion of this system is decomposed into slow motion with an eigenvalue -l + 0 (c) , and fast motion with eigenvalues

2c (-1

(37)

where Xl E 'Rn-m , X2 E 'R m , U E 'Rm and B2 is nonsingular. The objective of the control design for the nominal system is to enforce sliding mode on the manifold

(29)

00

(35)

)..;'s satisfy the same condition (4) as in the example, and Wi ' S are of the same order of magnitude, i.e. , there exists Wo such that ~ = 0(1) . The parallel parasitic dynamics are motivated by plants that model mechanical structures with dominant rigid body modes and flexible modes of high resonating frequencies . The dominant dynamics are represented by state vector x while ql and q2 model l uncoupled second order high frequency dynamics . Without loss of generality, the dominant dynamics are represented in the so-called regular form:

vVe introduce scaling to q2 for further analysis of the above system. Let

Note that from (4) that c resulting system is

(34)

ql E 'RI

(36)

and equivalent control

-

=

where A and

(24)

c

q2 ,

q2

(21) (22) (23)

-Lx -lql

with

Ih

(45)

(33)

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the condition for existence of sliding mode is invariant with respect to the parasitic dynamics, the required magnitude of control effort is reduced, and the effects of neglected dynamics on regulation error are further attenuated. We further assume that the number of outputs of the plant is less than the dimension of the nominal plant.

With x substituted by y in (39), the switching manifold becomes

We shall analyze the condition for the existence of sliding mode and the resulting behavior on the manifold s = from

°

4. SLIDING MODE CONTROL DESIGN

in which the parasitic dynamics (34) ,(35) are rewritten as

Let the output of the plant be y=Mx+Hql,

(57)

yEnr , r
(48) This measurement equation is motivated by plants which model flexible structure, and only generalized displacements are sensed. We design a standard asymptotic observer for the nominal plant,

The existence of sliding mQde requires the matrix (49)

to be at least nonsingular which may be the case for sufficient small 11 HIli and 11H 211. Even if this matrix is nonsingular, the existence of sliding mode may still be violated with the nominal control

=

Xl

£2

=

+ A12X2 + Ll(y AllXl + A 12 X2 + B 2 u+ AllXl

Mx)

+L 2 (y - Mx)

(59)

and define the switching manifold as

(50)

(60)

because the conditions (43) depend strongly on the matrix multiplying the control input. Thus, parallel parasitic dynamics may destroy sliding mode. However, the nominal sliding mode control (50) may satisfy the conditions (43) if Uo

> allqll + ,8Ilxll

Let the control be of the same form as in (41), however, with s replaced by s: (61)

As follows from

(51)

§ = CAx - uosgn(s)

for some positive a and (3. Thus the required magnitude for control is of 0 (1) since q is of 0 (1) as follows from similar arguments in the example. In sliding mode s = 0 and § = 0, X2

U eq

+ LM(x -

x)+

(62)

+LMHql

sliding mode may be enforced if

-ClXl - ((CIH l + H 2 )q

(52)

(63)

+ Tlql + T2q2

(53)

where e :::::: X - X is the state estimation error. The required magnitude U o depends on ql which is of By eliminating O(...L). Wo

NXl

Xl = (All - A 12 C l )Xl + +A 12 (C l H l + H 2 )q

ql

q2

q2

- Aq2 -

(54) (55)

X2

(56)

el

. .:. .

Since the high frequency dynamics are decoupled, similar arguments used in the example can be applied to arrive at the following conclusions:

using

s=

• ql is of O( ~o

flql

+ RU eq

-e2 - CIXl - Clel , Xl - Xl ,

e2:::::: X2 -

X2 ,

(64)

0, solving for the equivalent control

(65)

)

in § = 0, and substituting these two expressions in (35) and (37), the following equations

has two components: the rate of decay of the homogeneous component is determined by the eigenvalues of All - A12 C l , and the forced component is of O(~J • From (52), X2 depends on q, hence its forced component is of 0(1)

•

(58)

Xl

e

=

(A+LM)e+LHql

Xl = (All - A12C l)Xl + +A12(Clel + e2)

Our goal is to design sliding mode control in the presence of parallel parasitic dynamics such that

(66) (67)

and the parasitic dynamics given by the same set of equations as before, (55, 56), although with a

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Asada H. and J-J . E. Slotine (1986). Robot Analysis and Control, pp.139-157. Kwanty H. G . and T.L. Siu (1987) . "Chattering in Variable Structure Feedback Systems,"

different equivalent control, prescribe the sliding mode dynamics. The following observations can be made:

10th World Congress on Automatic Control Preprints, Munich, FDG pp 385-400. Bondarev, A. G ., S. A. Bondarev, N. E . Kostyleva, and V. 1. Utkin (1985) . " Sliding Modes in Systems with Asymptotic State Observers," Automation and Remote Control, pp. 679-684. Utkin, V. 1. and K. D. Young (1978) . "Methods for constructing discontinuous planes in multidimensional variable-structure systems," Automation and Remote Control, Vol. 39 , No. 10, (P.1), pp. 1466-1470.

• High frequency parasitic dynamics are decoupled from the slow motion of the closed loop system which is consists of the nominal plant dynamics and the state estimation error dynamics. • e has two components: the rate of decay of the homogeneous component is determined by eigenvalues of A + LM, and the forced component is of O( ~ ) • Xl has two compon~nts: the rate of decay of the homogeneous component is determined by eigenvalues of All -A 12 C I , and the forced component is of O( ~) Wo • X2 depends on e, hence its forced component is of O(:-h ) Wo For the case when the output is

y=Mx+Hq ,

(68)

i.e., both generalized displacements and velocities of the parasitic dynamics corrupt the sensor outputs, the above design using asymptotic observer remains to be valid - existence of sliding mode is in variant with respect to the parasitic dynamics. However, the magnitude of control, as it is given by (63) is of 0(1) since U o depends on q which is of 0(1) . The magnitude of the forced component of Xl is also increased to O ( ~) , for X2 , it is of O(~ ) . 0

Wo

5. CONCLUSIONS The presence of high frequency parasitic dynamics which are in parallel to the plant, if ignored in sliding mode control design may violate the condition for existence of sliding mode. As in the case of fast parasitic dynamics which are in series with the plant, asymptotic observer provides the means for enforcing sliding mode in the observer space, thus preserving the properties of the nominal feedback system. High frequency oscillating components which are neglected in the design are filtered by the observer, hence their effects on the nominal system are attenuated. Although it is not analyzed in this paper, the utility of the observer for plants with both parallel and serial parasitic dynamics is expected to be the same.

6. REFERENCES Young K.D . and H. G. Kwatny (1982) . "Variable Structure Servomechanism Design and its Application to Overspeed Protection Control," Automatica, VoLl8, No. 4, pp 385-400.

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