CONTINUOUS SLIDING MODE CONTROL OF A CARTESIAN PNEUMATIC ROBOT

by

YANG XIA

A thesis submitted to the Department of Mechanical Engineering

e of in conformity ivith the requirementsfor ~ h degree Muster of Science (Engineering)

Queen's University Kingston, Ontario Canada Km 3N6

October, 200 1

copjright O Yang Xia, 200L

uisitiorrsand mgraphic Semices 3-

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The author has granted a nonexclusive licence allowing the National Li'brary of Canada to reproduce, 10- distniute or sen copies of this thesis in microform, paper or electronic formats.

The author re* ownership of the copyright in this thesis. Neither the thesis nor substaatial extracts fiom if may be prmted or otherwise reproduced without the author's permission.

L'auteur a accordé une licence non exclusive permettant à la Bhliothéfo + 1 xd 1, the value of

X and f = xd -f(x) - K,sgn(X) have

opposite signs, where sgn is the sign ftnction. The magnitude of the tracking error F thus decays at a certain rate and reaches zero after a finite time interval tmCtr, as illustrated by Figure 2.8. The motion afler t h is called the sliding mode (Utkin et al. 1999). The relay control action demands an infinitely fast switch on the sliding surface (condition when S = O). The switching around the sliding surface generates a point of discontinuity. This can't be implemented in any realistic application, because of imperfections in switching devices due to mechanical delays, dead zones and hysteresis.

These

imperfections may lead to high-fiequency oscillations. The imperfect switching problem must be dealt with for the algorithm to be practical.

2.5.2

SLIDING SURFACES

Sliding mode control is one implementation of VSS and it introduces the concept of S as the sIiding surface or manifold.

Without loss of generality, give a single-input dynamic system as

dm) = f (x) + b(x) u

(2-8)

where the scalar x is the output of interest (e-g., the position of a manipulator), the scalar rr is the control input (e-g., a signal voltage), and x =

1,

f

x'""

Ir

is the state

vector, and n is a number of system order.

In Equation 2.8 the functionflx) is not exactly known generaiiy, but it is bounded by a known continuous function of x. Similady, the control gain b(x) is not exactly known, but is of known sign and is bounded by a known continuous function problem

is

to

get

a=

3,

---

xd-')]' in the presence of mode1 irnprecision on

[Y,

the

state

x

to

track a

S.

The control

specific time-varying state

Air) and b(r).

For

the tracking task using a finite control if, the initial desired state Id(O) must be such that

Define

--.

x=x-x,=[~

--•

x( n - t ) ] ~

as the tracking error vector and a time

varying sliding surface S(t) = O in state space R'"), where

with A, a strictly positive constant, whose value must be properly chosen in surface design. When n = 2 and 3, the sliding surface S(x, t) in Equation 2.10 can be expanded as

and

s =X+2&+t2z

(2.12)

-

The geometric interpretations of Equation 2.1 1 and 2.12 are shown in Figure 2.9 and Figure 2.10. Given the initial condition of Equation 2.9, the perféct tracking x

y is

obtained as long as the system state remains on the surface S afker t>O, which means the traclcing problem is translated into that of keeping the scalar quantity S at zero. Moreover, bounds on S c m be direaly converted into bounds on the tracking error veaor

ii. With the assumption of %(O) = 0 , the positive bounding h c t i o n O / IS(t)l can result in the relation

where

E

= @Xn-'.

(2.14)

Thus the sliding surface S becomes a true index to measure the tracking performance (Slotine and Li, 1991).

Figure 2.9 Graphical interpretation of Equation 2.11 with n = 2 (based on Slotine and Li, 1991).

Figure 2.10 Graphical interpretation of Equation 2.12 with n = 3, (based on Utkin et al, 1999). The States outside of the sliding surface S are guaranteed to converge d e r a finite tirne by choosing the proper control law u. For al1 single input systems a suitable candidate of

Lyapunov functions is V(x)

= %

?2

which is globally positive definite.

As long as

switching feedback gains are properly chosen

where p is a stnctly positive constant, Equation 2.15 can always be established, which indicates that the state trajectory converges to the surface (Figure 2.11) and is restricted to the sufice for al1 subsequent tirne (DECarloet ai, 1988; Slotine, 1991).

Figure 2.1 1 A graphical interpretation of the sliding condition (n = 2), (based on Slotine and Li, 1991).

23.3

CBATTElUNG PHENOMENON

As illustrated in Figure 2.12, chottdng descnbes the phenomenon of finite-eequency,

finite-amplitude oscillations about the sliding sufiace. Due to the discontinuous nature of a control action that switches between two distinctively dinerent system structures, ideal sliding mode control demands infinitely fast switching. The high-fiequency switching can excite unmodeled dynamics such as parasitic dynamics, which are oRen negleaed in the open-loop mode1 design if the associated poles are well damped and outside the desired bandwidth of the control system. At the sarne tirne, irnperfect switching can be the result of a relay with hysteresis or mechanical delay. These two mechanisms lead to the chattenng phenomenon in real SMC applications and the system will oscillate in the vicinity of the switching plane.

Figure 2.12 Chattering phenomenon in an imperfect switching (n (based on Slotine and Li, 1991).

= 2),

To prevent chattering in a real-life system, several solutions have been proposed by many

researchers (Utkin et al, 1999). The most commonly cited approach to reduce the effects of chattenng has been the boundary layer solution (the continuation approach) (Slotine and Sastry, 1983; Slotine, 1984). The basic idea is to replace the discontinuous control

law by a saturation finction sat(S) which approximates the sign term sgn(S) in a boundary layer of the sliding sufiace S = 0.

As show in Figure 2.13 (for case n = 2). the vertical dimension