Asian Journal of Control, Vol. 5, No. 4, pp. 467-475, December 2003
467
SLIDING MODES, ∆-MODULATORS, AND GENERALIZED
PROPORTIONAL INTEGRAL CONTROL OF LINEAR SYSTEMS
Hebertt Sira-Ramírez
ABSTRACT
In this article, we attempt a reapproachment between sliding mode control of linear systems and classical control through the possibilities of evading state measurements and circumventing the use of asymptotic state observers in the sliding surface synthesis. This is shown to be possible thanks to
the use of integral state reconstructors combined with iterated integral output
error compensation. The proposed scheme is also robust with respect to unmatched perturbation inputs. A connection between sliding modes and classical analog ∆-modulators, and their natural generalization, is also brought to
attention as a tool for the realization of integral reconstructor based sliding
mode control schemes.
KeyWords: Generalized PI control, sliding mode control, delta-modulators.
I. INTRODUCTION
The many advantages of sliding mode control are
well reported, founded, and illustrated, in the existing
literature which, incidentally, has reached a significantly
advanced state of maturity. Among the shortcomings of
sliding mode control, we simply mention that sliding
mode control technique is, fundamentally, a state-based
discontinuous feedback control technique. The lack of
complete knowledge of the state vector components
forces the designer to use asymptotic state observers, of
the Luenberger, or of the sliding mode type, or to resort
to direct output feedback control schemes. Unfortunately,
the first approach is not robust with respect to unforseen
exogenous perturbation inputs, even if they happen to be
of the “classical type” (by this we mean: steps, ramps,
parabolas, etc.) while the second approach, while being
quite limited in nature, is not applicable in non-minimum
phase systems and it fails in unmatched perturbation
input cases. For general background on sliding mode
control, we refer the reader to the seminal books by Utkin, [8,9] and the recent books by Utkin, Guldner and
Shi [10] and that by Edwards and Spurgeon [1]. Recent
Manuscript received January 9 2002; accepted February 3
2003.
The authors are with Cinvestav-Ipn, Sección de Mecatrónica, Department Ingeniería Eléctrica, Avenida IPN, # 2508,
Col. San Pedro Zacatenco, A.P. 14740 México, D.F., México.
This research was supported by the Centro de Investigación y Estudios Avanzados del IPN (CINVESTAV-IPN), México, D.F., México.
developments, advances and applications, of the sliding
mode control area are found in the book by Perruquetti
and Barbot [6].
In this article, we propose a new approach to the
synthesis of sliding mode feedback control schemes for
linear, time invariant, controllable and observable Single
Input Single Output (SISO) systems. The approach consists in using integral reconstructors of the state vector
for the sliding surface synthesis. Integral reconstructors
were introduced, in the realm of continuous feedback
strategies, addressed as Generalized Proportional Integral (GPI) control schemes, in the work of Fliess [3]. A
complete theoretical account of integral reconstructions
and GPI control has been presented in recent articles by
Fliess et al (see [4,5]). Integral reconstructors use only
inputs, outputs and iterated integrals of such signals
while neglecting the influence of the constant, but unknown, initial conditions and the effect of classical perturbation inputs. Hence, our approach naturally leads to
dynamic input-output feedback schemes for the synthesis
of the sliding surface and, due to its inherent input-output nature, it requires no matching conditions
whatsoever. Due to the unstable nature of the state reconstruction errors associated with the integral reconstructos, a given suitable state-dependent sliding surface
can be synthesized via integral reconstructors provided
its expression is suitably modified with a sufficiently
large linear combination of iterated output error compensation terms. The aim is to synthesize the sliding
surface so as to exhibit an exponentially asymptotically
stable dynamics under, closed loop, ideal sliding condi-
Asian Journal of Control, Vol. 5, No. 4, December 2003
468
u
W
ξ
−W
x
1
_
s
Fig. 1. Classical analog ∆-modulator.
tions. In this article, we show, and advocate, that the
sliding surface design problem is greatly facilitated by
resorting to the system flatness property (For the definitions and theoretical basis of differential flatness, the
interested reader is referred to the work of Fliess and his
coworkers [2]. See also the Appendix).
We bring to the attention of the reader, the close
connection between sliding modes and classical analog
∆-modulators. In particular, we show the relevance of
∆-modulators, and their generalizations, in the dynamic
synthesis of state-dependent sliding surfaces using integral state reconstructors. A complete account of ∆modulators, which never benefited from the theoretical
basis of sliding mode control, is found in the classical
book by Steele [7].
Section 2 presents a review of the simplest analog
∆-modulator and its connection with sliding mode control. In this section we explore the links between
∆-modulators and sliding mode control of simple integration plants naturally involving GPI control schemes.
Section 3 deals with a generalization of ∆-modulators for
the synthesis of sliding surfaces in general linear SISO
controllable and observable systems. Section 4 is devoted to present a design example involving the normalized model of two wheeled, frictionless, cars joined by a
spring. We illustrates the robust and convenient features
of our proposed GPI sliding mode controllers via digital
computer simulations. The appendix presents some generalities regarding the concept of flatness and integral
state reconstructors in observable linear, n-dimensional,
SISO systems.
II. ∆-MODULATORS AND GPI
CONTROLLERS
Consider the basic block diagram of Fig. 1 depicting a classical analog ∆-modulator, traditionally used in
voice encoding systems. The following theorem summarizes the relation of ∆-modulators with sliding mode
control and depicts the basic features of performance of
the basic modulator.
Theorem 2.1. Consider the ∆-modulator of Fig. 1. Given
a bounded C1 signal ξ(t), with bounded first order time
derivative, ξ , there exists a strictly positive gain, W,
such that x(t) → ξ(t) in a finite amount of time th, pro-
vided the following encoding condition is satisfied,
W > sup ξ(t )
(1)
Moreover, from any arbitrary initial value of the tracking,
or local encoding, error e(t0) = x(t0) − ξ(t0), a sliding motion exists on the perfect encoding condition e = 0 for all
t > th, where the quantity th is bounded by th ≤ T, with T
e(t0 )
satisfying, T ≤ t0 +
.
W − sup ξ(t )
Proof. From the figure, the variables in the ∆-modulator
satisfy the following relations:
x = u
u = W sign(ξ − x)
(2)
e = x −ξ
Clearly, e = −W sign (e) − ξ(t ) and since ξ(t ) is assumed to be bounded, choosing W > sup ξ(t ) we have,
for e > 0:
ee = −W e − eξ(t ) = −W e − e ξ sign(e)
≤ −W e + e sup ξ = −(W − sup ξ ) e < 0
(3)
A sliding regime exists on e = 0 for all time t after the
hitting time th (see [8]). Under ideal sliding, or encoding,
conditions, e = 0; e = 0, we have that x = ξ(t) and the
equivalent (average) value of the coded output signal u is
given by ueq = ξ(t ) for all t ≥ th.
The ∆-modulator output u ideally differentiates the
modulator input signal ξ(t) in an equivalent control sense
([8]). The role of the ∆-modulator in sliding mode control schemes avoiding full state measurements of simple
linear plants will be clear from the following examples.
2.1 A second order example
Consider the unperturbed second order plant ÿ = u,
where only the output y and the input u are available for
use on a stabilizing sliding mode feedback scheme. A
traditional sliding mode controller with sliding surface: σ
= y + λy is thus forbidden due to the unavailability of
the signal y . An integral reconstructor for y is obt
tained directly from the expression: yˆ (t ) = ∫0 u ( ρ )d ρ
which, we know, differs from the actual value of y by
the unknown constant initial condition, y 0 , i.e. y = ŷ +
y 0 . A modified sliding surface, inspired on the GPI control technique (see [5]), can now be proposed as:
t
t
σˆ = yˆ + λ1 y + λ0 ∫0 y ( ρ ) d ρ = ∫0 u ( ρ )d ρ + λ1 y
t
+λ0 ∫0 y ( ρ )d ρ
(4)
S.R. Hebertt: Sliding Modes, ∆-Modulators, and Generalized Proportional Integral Control
∆-Modulator
∆-Modulator
GPI Compensator
PI Compensator
0
λ1+
0
Plant
− σ^
u
W
λ0
__
s
−W
469
1
_
s2
− σ^
(a+λ
) ___
λ
_____ (b+λ2) + 1 + 0 2
s s
y
ξ
u
W
−W
Plant
1
__
s2
y
1
_
s
y
1
_
s
Fig. 3. GPI ∆-modulator sliding mode controller scheme for a perturbed plant.
Fig. 2. GPI sliding mode controller exhibiting a ∆-modulator.
2.2 A perturbed second order plant
The proposed sliding mode controller is clearly
synthesized via an output PI compensating signal feeding a ∆-modulator as depicted in Fig. 2. According to the
results of Theorem 2.1, under ideal sliding conditions,
the average value of the output signal, ueq, of the
∆-modulator is given by ueq = −λ1 y − λ0 y , i.e. the second order plant is, on the average, being regulated by a
PD controller.
Note that the integral output error compensation
term in equation (4) compensates for the constant error,
incurred in the estimation of the state y through the
integral reconstructor, when the sliding condition σ̂ = 0
is enforced. The modified sliding surface, σ̂ , is seen to
be equivalent to
t
σˆ = y + λ1 y + λ0 ∫0 y ( ρ )d ρ − y 0
which, under ideal sliding conditions: σˆ = 0 ; σˆ = 0 ,
yields the following equivalent control and the following
asymptotically stable closed loop dynamics, provided
appropriate choice of the design parameters λ0 and λ1 is
made.
ueq = −λ1 y − λ0 y, y + λ1 y + λ0 y = 0
t
Indeed, defining ζ = ∫0 y ( ρ )d ρ − y 0 / λ0 , the ideal sliding condition σ̂ = 0 yields the following linear homogeneous system with one unknown initial condition,
y = −λ1 y − λ0ζ , y (0) = y0
y
ζ = y, ζ (0) = 0
λ0
clearly, y + λ1 y + λ0 y = 0.
A sliding mode controller for the stabilization of
the system is then of the form:
u = −W sign(σˆ )
t
t
σˆ = ∫0 u ( ρ )d ρ + λ1 y + λ0 ∫0 y ( ρ )d ρ
= λ1 y + ∫ [u ( ρ ) + λ0 y ( ρ )]d ρ
The above developments allow one, for instance, to
readily propose a sliding mode feedback controller for
the following perturbed second order plant ÿ = ay + by +
u + ξ1(t − τ), with ξ being a constant, unknown,
non-zero perturbation input appearing at the unknown
instant τ > 0, and a and b being known parameters. We
know that a PID controller of the form: u = −(b + λ2) y −
t
(a + λ1)y − λ0 ∫0 y ( ρ )d ρ , exponentially asymptotically
stabilizes y to zero, irrespectively of ξ and τ, for an appropriate choice of the set of parameters, λ0, λ1 and λ2.
The closed loop dynamics is given by (s3 + λ2s2 + λ1s +
λ0)y = 0. If we regard the proposed PID controller as an
average (equivalent) controller synthesis, then the results
of Theorem 2.1 allow us to propose the following GPI
compensation plus ∆-modulation scheme which recovers,
on the average, the previous PID controller features.
u = −W sign σˆ
t
t
σˆ = ∫0 u ( ρ )d ρ + (b + λ2 ) y + (a + λ1 ) ∫0 y ( ρ )d ρ
t
ρ
+λ0 ∫0 ∫0 y (ζ ) dζ d ρ
Indeed, according to the results of the theorem, the
equivalent control ueq, obtained from the condition σ̂ =
0 is given by
t
ueq = −(b + λ2 ) y − (a + λ1 ) y − λ0 ∫0 y ( ρ )d ρ
(5)
t
Note that ∫0 [u ( ρ ) + ay ( ρ )]d ρ + by qualifies as an integral reconstructor, ŷ , of the state, y . Such reconstructor is related to the actual value y of the unmeasured state by the expression y = yˆ + y 0 − by0 + ξ(t –
τ)1(t – τ). The sliding surface σ̂ is then equivalent to
t
t
ρ
σˆ = y + λ2 y + λ1 ∫0 y ( ρ )d ρ + λ0 ∫0 ∫0 y (ζ )dζ d ρ
− y 0 + by0 − ξ (t − τ )1(t − τ )
(6)
Under the sliding condition: σ̂ = 0, the closed loop
dynamics is given by the following linear system with
Asian Journal of Control, Vol. 5, No. 4, December 2003
470
some unknown initial states and subject to an exogenous
sudden step input perturbation, υ, at time τ.
y = −λ2 y − λ1ζ 1 , y (0) = y0
− σ^
0
−1
−1
SP(s )+r(s )
λ
y − by0
ξ
, υ = 1(t − τ )
ζ1 = y + 0 ζ 2 + υ , ζ 1 (0) = − 0
λ1
λ1
λ1
ζ = y, ζ (0) = 0
2
Generalized
∆-Modulator
W
−W
"Classical
perturbation"
input
ξ
Plant
y
u
c(sI − A)−1b
SQ(s−1)
y
2
The unperturbed ideal sliding motions exhibits a characteristic polynomial q(s) given by: q(s) = s3 + λ2s2 + λ1s +
λ0, which can be made Hurwitz, and the equilibrium
point of the perturbed ideal sliding dynamics is seen to
ξ
be given by: y = 0, ζ1 = 0, ζ2 = − .
λ0
Fig. 4. GPI Generalized ∆-modulator sliding mode controller scheme
for a classically perturbed linear plant.
K
F
M
III. GPI SLIDING MODE CONTROL AND
GENERALIZED ∆-MODULATORS
The previous developments point to the fact that
state dependent sliding surfaces can be effectively designed on the basis of state integral reconstructors, which
are linear combinations of iterated integrals of inputs and
outputs. Since the integral reconstructors neglect the
initial state conditions and the external perturbation inputs, the integrally reconstructed sliding surface must be
sufficiently compensated in order to counteract their
de-stabilizing effects, when the ideal sliding conditions
are valid.
It is not difficult to show (See the Appendix) that
in an observable linear system of the form: x = Ax + bu;
y = cx, the state vector x can always be reconstructed,
modulo the effect of neglected initial conditions and
classical external perturbation inputs, in terms of vectors
of integral polynomials: P(s−1) and Q(s−1) as follows:
xˆ = P ( s −1 ) y + Q ( s −1 )u
(7)
A sliding surface, σ, which may have been synthesized
from a viewpoint such as atness, of the form: σ = Sx with
S being a constant row vector, can then be modi_ed by
considering,
σˆ = Sxˆ + r ( s −1 ) y = [ SP( s −1 ) + r ( s −1 )] y + SQ( s −1 )u
(8)
where r(s−1) is an integral polynomial scalar function
which compensates for the neglected state initial conditions and the influence of classical external perturbation
inputs. The complete control scheme is shown in Fig. 4
where the ∆-modulator of the previous examples is now
addressed as a “Generalized ∆-modulator”.
The sliding surface synthesis problem may then be
stated as follows: Given a controllable and observable
linear plant y = c(sI − A)−1bu = [n(s)/d(s)]u whose state
vector x defining the realization: x = Ax + bu; y = cx,
M
x1
x2
Fig. 5. A two mass spring system.
can always be integrally parameterized, modulo initial
conditions, by the integral reconstructor x̂ = P(s−1)y +
Q(s−1)u, find a row vector S and an integral polynomial
r(s−1) such that the closed loop dynamics, corresponding
to the ideal sliding conditions σ̂ = 0, σ̂ = 0, is exponentially asymptotically stable.
The involved relations are:
σˆ = [ SP( s −1 ) + r ( s −1 )] y + SQ( s −1 )u
u = −W sign σˆ
(9)
d ( s) y = n( s)u + ξ
The choice of the row vector S and the iterated integral compensation term r(s−1) is greatly facilitated by
resorting to the system flatness property, (see [2]) which
due to the controllability assumption is, therefore, guaranteed (see the Appendix).
IV. A MASS-SPRING SYSTEM
Consider the following unperturbed normalized
model of two wheeled carts joined by a spring shown in
Fig. 5
x1 = u + ( x2 − x1 )
x2 = −( x2 − x1 )
(10)
y = x2
A state dependent exponentially stabilizing sliding
surface can be shown to be given by
σ ( x1 − x2 ) + γ 2 ( x1 − x2 ) + γ 1 x2 + γ 0 x2
(11)
S.R. Hebertt: Sliding Modes, ∆-Modulators, and Generalized Proportional Integral Control
with the coefficients γ2, γ1 and γ0 chosen so that the characteristic polynomial s3 + γ2s2 + γ1s + γ0 is Hurwitz. Note
that the state expression for σ is motivated by the
desirable closed loop dynamics
y (3) + γ 2 y + γ 1 y + γ 0 y = 0
The system is found to be observable since y = x2 qualifies as the flat output. The complete differential parameterization of the states and the input in terms of the at
output is given by x2 = y, x2 = y , x1 = ÿ + y, x1 = y(3)
+ y , u = y(4) + ÿ. The last flatness relations, the system
equations and the observability relations: y = x2, y = x2 ,
ÿ = −(x2 − x1) and y(3) = − ( x2 − x1 ) , help us in obtaining
the integral reconstructors of the state. These are found
to be1:
(2)
(3)
1 = ( ∫ u ) + 2( ∫ u ) − ( ∫ u )
xl1 = ( ∫ u ) − y, xl
(12)
(3)
2 = ( ∫ u ) − 2( ∫ y )
xl2 = y, xl
or, in complex variable, integral polynomial vector notation:
 xl1 
 s −2 
   −1 
 −1 −3 


−
1
 xl
1 
2s 
s − s  u

y
=
+
 
 0 
l


1
 x2 


 l   −2 s −1 
s −3 


x
 2 
(13)
(2)
σ=m
y (3) + k5 yˆ + k4 yˆ + k3 y + k2 ( ∫ u ) + k1 ( ∫ y )
+ k0 ( ∫
(3)
(14)
y)
which is equivalent to the following expression, for
some unknown constants ρ1, ρ2 and ρ3
σˆ = y (3) + k5 y + k4 y + k3 y + k2 ( ∫ y ) + k1 ( ∫
+ k0 ( ∫
(3)
y ) + ρ1 + ρ 2 t + ρ3t
(2)
y)
(i)
We use the following notation: ( ∫ φ ) =
(1)
t
with ( ∫ φ ) = ( ∫ φ ) = ∫0 φ (σ )d σ .
ζ 1 = (∫ y) +
k
k1
ρ
ρ
(∫ ζ 2 ) + 1 , ζ 2 = (∫ y) + 0 (∫ ζ 3 ) + 2 ,
k2
k2
k1
k1
ζ 3 = (∫ y) +
ρ3
k0
(16)
we have
y (3) = −k5 y − k4 y − k3 y − k2ζ 1 ,
y (0) = y0 , y (0) = y 0 , y (0) = y0
ρ
k
ζ1 = y + 1 ζ 2 , ζ 1 (0) = 1
k2
k2
(15)
(17)
k
ρ
ζ2 = y + 0 ζ 3 , ζ 2 (0) = 2
k1
k1
ρ
ζ3 = y, ζ 3 (0) = 3
k0
The characteristic polynomial of this linear homogeneous system coincides with the desired one. Suitable
choice of the design parameters renders the characteristic
polynomial Hurwitz. The proposed choice of σ̂ results
then in:
(3)
(2)
σˆ = (k4 − 2)( ∫ u ) + k5 ( ∫ u ) + ( ∫ u ) + (k3 − 2k5 ) y
(2)
y ) + k0 ( ∫
(3)
(18)
y)
or, in complex Laplace transform notation
4 − 2k4 + k2 k1 k0 

σˆ = (k3 − 2k5 ) +
+ 2 + 3y
s
s
s 

k − 2 k 1
+  4 3 + 52 +  u
s
s
 s
(19)
4.1 Simulation results
Figure 6 depicts the controlled responses of the
normalized system to a given set of adverse initial conditions. The chosen closed loop characteristic polynomial, corresponding to σ̂ = 0 was set to be, p(s) = (s2 +
2ζωns + ω n2 )3, with ξ = 0.8, ωn = 1.2. The gain W was
set to 12.
4.2 Unmatched perturbations
2
Under ideal sliding motions on σ̂ = 0, the closed loop
system is a linear homogeneous system in y with charac
1
teristic polynomial, s6 + k5s5 + k4s4 + k3s3 + k2s2 + k1s +
k0. Indeed, defining,
+(4 − 2k4 + k2 )( ∫ y ) + k1 ( ∫
The integral reconstructors of the state differ from the
actual values of the states by at most a second order time
polynomial of the form: α + β t + γ t2. Thus, the modified sliding surface requires three iterated integrals of the
output stabilization error. We propose then a sliding surface achieving, in closed loop, an exponentially asymptotically stable dynamics for the at output y. Such a surface is given by
471
t
σ1
0
0
∫∫
σ
"∫0 i −1 φ (σ i )d σ i … d σ 1
To see that the proposed synthesis technique requires no matching condition of the exogenous perturbation inputs, consider the case in which a constant, unknown, perturbation input suddenly affects the underactuated part of the system. Assume such a perturbation
appears at the unknown time, τ, as follows,
Asian Journal of Control, Vol. 5, No. 4, December 2003
472
0.5
0.5
x1(t)
x2(t)
0
−0.5
0
2
0
−2
−4
0
ξ
0
2
4
6
8
10
12
−0.5
0
2
5
x1(t)
10
15
20
25
30
10
15
20
25
30
10
15
20
25
30
0
σ(t)
σ(t)
−2
2
20
4
6
8
10
12
−4
0
5
20
u(t)
u(t)
0
0
−20
0
x2(t)
2
4
6
8
10
12
Fig. 6. GPI sliding mode controlled responses of two car system.
−20
0
5
Fig. 7. GPI sliding mode controlled responses of perturbed two car
system.
x1 = u + ( x2 − x1 )
x2 = −( x2 − x1 ) + ξ 1(t − τ )
(20)
y = x2
Then it can be seen that the integral reconstructors of the
state differ from their actual values, at most, by a third
order time polynomial of the form a + β t + γ t2 + κt3.
This implies that the sliding surface expression must be
of the form:
(2)
σˆ = m
y (3) + k6 yˆ + k5 yˆ + k4 y + k3 ( ∫ y ) + k2 ( ∫ y )
+ k1 ( ∫
(3)
y ) + k0 ( ∫
(4)
(21)
y)
The integral input-output parameterization of such a
sliding surface is now given by:
4 − 2k5 + k3 k2 k1 k0 

+ 2 + 3 + 4y
σˆ = (k4 − 2k6 ) +
s
s
s
s 

 k − 2 k 1
+  5 3 + 62 +  u
s
s
 s
4.4 The case of unavailable at output
Consider now the following perturbed system with
a different output signal,
x1 = u + ( x2 − x1 ) + ξ 1(t − τ )
x2 = −( x2 − x1 )
y = x1
The system is observable from the given output and
hence an integral reconstructor of the state vector is possible to obtain. The integral reconstructors of the unperturbed state components are given by:
(2)
(3)
2 = −( ∫ u ) + 2 y
xl2 = ( ∫ u ) − y, xl
(3)
1 = ( ∫ u ) + ( ∫ u ) − 2( ∫ y )
xl1 = y, xl
(22)
Note that under such a constant perturbation, the perturbed differential parameterization of the state is: x1 = ÿ
+ y − ξ1(t − τ). This means that in steady state x1 will
converge towards the unknown constant value −ξ. The
output y = x2 will, nevertheless, converge to zero as desired.
4.3 Simulation results
Figure 7 depicts the performance of the proposed
sliding mode feedback controller with the desired characteristic polynomial, corresponding to σ̂ = 0, set to be,
p(s) = (s2 + 2ζωns + ω n2 )3(s + r), with ζ = 0.85, ωn = 1, r
= 1.5 and W = 10. The perturbation amplitude was prescribed as ξ = 0.15. The perturbation appears at time τ =
12 time units.
(23)
(24)
It is easy to see that the discrepancy between the
integrally reconstructed states and their actual values,
due to the effect of the initial conditions and of the constant unmatched perturbation input, is, at most, of the
form: α + β t + γ t2. This means that the expression of a
state dependent sliding surface must be compensated by
a linear combination involving at most three iterated
integrals of the output stabilization error. Contrary to our
previous design example we assume the objective is now
to stabilize the variable y = x1 to zero.
Note that the perturbed differential parameterization of the output y = x1, in terms of the at output x2, is
given by: y = x2 + x2.
Consider then, as a sliding surface coordinate function, the expression:
(2)
(3)
1 + λ3 y + λ2 ( ∫ y ) + λ1 ( ∫ y ) + λ0 ( ∫ y )
σˆ = xl
(25)
In terms of the actual value of the at output, the sliding
S.R. Hebertt: Sliding Modes, ∆-Modulators, and Generalized Proportional Integral Control
surface coordinate function σ̂ is equivalent to
σˆ = x1 + λ3 y + λ2 ( ∫ y ) + λ1 ( ∫
(2)
y ) + λ0 ( ∫
(3)
y)
(26)
+ r1 + r2 t + r3t 2
On σ = 0, the resulting closed loop dynamics is given by
y (4) + λ3 y (3) + λ2 y + λ1 y + λ0 y = 0
(27)
which can be guaranteed to be an exponentially asymptotically stable dynamics upon an appropriate choice of
the design coefficients.
V. CONCLUSIONS
A new approach to sliding mode control of linear
time-invariant SISO controllable and observable systems
has been presented. The proposed technique, which
evades the need for asymptotic state observers, exploits
an input-output viewpoint, much as in classical linear
control theory, granted by the help of integral state reconstructors involving only iterated integrals of input
and output signals aided by output error iterated integral
compensation. A suitable generalization of analog
∆-modulators, widely used in the early days of voice
encoding systems, allows one to obtain a dynamic input-output based synthesis of the sliding surface and
demonstrates a natural connection with the recently introduced GPI control design technique. The proposed
design approach suffers from none of the limitations of
direct output based sliding mode control and it is robust
even with respect to unmatched classical exogenous
perturbation inputs due to its underlying input-output
character. We presented several simple design examples
illustrating the advantages and potential of the approach.
The proposed GPI-sliding mode control schemes,
through generalized ∆-modulators, are most adequately
implemented when the flatness of the system is suitably
exploited in the prescription of a desired homogeneous
(closed loop) linear dynamics for the at output.
The results here explained are easily extendable to
multivariable continuous-time linear systems and also to
linear discrete time systems. These will be the subject of
forthcoming publications.
APPENDIX
A.1 Flatness in linear systems
Flatness has been introduced in [2] from a general
nonlinear multivariable systems viewpoint. A smooth
single input system, in state space representation, x =
f(x, u), x ∈ Rn, u ∈ R, is said to be flat if there exists a
scalar output y, which is a function of the state x, such
473
that there exist an n-vector function φ and a scalar function ψ such that: x = φ(y, y , …, y(n−1)) and u = ψ(y, y , …,
y(n)). We say that y completely differentially parameterizes all system variables (states, input and, eventually,
the system output) and usually refer to φ and ψ as a differential parametrization. Evidently, in the adopted context, a flat single input system is (locally) exactly linearizable by means of static state feedback and state and
input coordinates transformations, i.e., it is (locally)
equivalent to the linear Brunovsky system, y(n) = υ, with
υ being the transformed input. Flat outputs are thus devoid of any zero dynamics.
In the context of single input linear systems of the
form: x = Ax + bu, the system is at if and only if it is
controllable in the sense of Kalman, i.e. the controllability matrix C = [b, Ab, … An−1b] is invertible. Furthermore,
a flat output may be determined as the linear combination of the states obtained from the last row of the inverse of the controllability matrix, i.e.
y = [0 " 0 1]C −1 x
Evidently, any constant multiple of y also qualifies as a
flat output and, also, the equivalence result is global.
A.2 Integral reconstructors
Here we use the developments in [3] and [5] to
demonstrate that for any observable linear Single Input
Single Output (SISO) system the state vector can always
be expressed in terms of iterated integrals of the inputs
and the outputs, modulo the influence of initial conditions. Such estimators are known as “integral reconstructors” of the state or, also, as an “integral input-output parameterization” of the state vector. Due to the initial conditions, integral reconstructors differ from the
actual value of the state vector by constant errors and
iterated integrals of such constant errors. This classical,
unstable, reconstruction errors (constants, ramps, parabolas, etc.) can always be stably compensated in feedback control schemes using such reconstructors by the
suitable addition of a sufficiently large number of iterated integrals of the output stabilization or tracking error.
Consider the observable linear, time-invariant,
SISO system
x = Ax + bu, x(0) = x0 , y = cx
(A.1)
Integrating the system, in the sense of Mikusiński (see
[5]) which neglects the effect of initial conditions, we
obtain
x
u
x = A +b
s
s
(A.2)
Iterating on this functional relation once more we
obtain:
Asian Journal of Control, Vol. 5, No. 4, December 2003
474
x(t ) = A2
x
u
u
+ Ab 2 + b
2
s
s
s
Iterating a total of n − 1 times, we obtain the following
implicit representation of the state vector:
u ( s)
 x( s)  n −1
x( s) = An −1  n −1  + ∑ Ai −1b i
s
 s  i =1
(A.3)
On the other hand, consider the output signal y and
its successive time derivatives
" 0  1 
 

" 0  s 
u ( s ) (A.4)
⋅
% 0  # 
 

" cb   s ( n − 2) 
Integrating n − 1 times, the obtained expression (A.4),
we have:
 1

 s n −1



 1
x( s)

 sn−2
=
+
(
)
y
s
O
M


s n −1

 #

 1



 s




 u(s)





(A.7)

 1

 s n −1



 1
 y(s) − M  s n − 2



 #

 1



 s








 u (s) 










(A.8)
where P and Q are vectors of integral polynomials. We
address the expression (A.8), which does not take into
account the influence of the initial states, the integral
reconstructor of the state vector and denote it by x̂ .
Such an “open loop” estimate of the state is based only
on iterated integrals of inputs and outputs.
The integral state reconstructor may be used, in
principle, on an input-output synthesis of a sliding surface σ = Sx, as long as the expression for σ involving the
state estimates is complemented with additional compensation terms which counteract the effect of the neglected initial states values and, possibly, of the unaccounted classical exogenous perturbation inputs. Such
compensation terms only require further iterated integrations of the output stabilization error (or of the output
tracking error in output tracking problems).
REFERENCES
(A.5)
Thanks to the observability of the system we can
uniquely solve for the quantity x(s)/sn−1 from (A.5) to
obtain:
 1
 s n −1

 1
x( s)
−1  n − 2
=
O
s

s n −1
 #
 1

 s
i =1








 u ( s)  






 



u ( s)
si
x = P ( s −1 ) y + Q ( s −1 )u
 1 


s 
u(s)
= Ox ( s ) + M 
 # 
 ( n − 2) 
I
s
 1
 s n −1

 1
 sn−2

 #
 1

n −1
+ ∑ Ai −1b
 1

 s n −1



 1

 sn−2
(
)
y
s
M
−



 #

 1



 s
In other words, we have that the state vector x can always be expressed as:
 1 
 c 




s

 y ( s ) =  cA  x( s )
 # 
 # 
 ( n −1) 
 n −1 
s

 cA 
0
 0

cb
0
+
 #
#
 n−2
 cA b "



 1

 s n −1


−1   1
n −1 
x=A
O  n−2

 s

 #



 1


(A.6)
We can now combine the expression (A.6) with (A.3) to
obtain:
1. Edwards, C. and S. K. Spurgeon, Sliding Mode Control, Taylor and Francis, London (1998).
2. Fliess, M., J. Lévine, P. Martin, and P. Rouchon,
“Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples,” Int. J. Contr., Vol.
61, pp. 1327-1361 (1995).
3. Fliess, M., R. Marquez, and E. Delaleau, “State
Feedbacks without Asymptotic Observers and Generalized PID regulators,” Nonlinear Control in the
Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, W.
Respondek (Eds.), Lecture Notes in Control and Information Sciences, Springer, London (2000).
4. Fliess, M., and R. Márquez, “Continuous Time Linear Predictive Control and Flatness: A Module-Theoretic Setting with Examples,” Int. J. Contr.,
Vol. 73, pp. 606-623 (2000).
5. Fliess, M., R. Marquez, E. Delaleau, H. Sira- Ramírez,
“Correcteurs Proportionneles Intégraux Généralisés,”
ESAIM, Contr. Optim. Calculu. Variat., Vol 7, No. 2, pp.
S.R. Hebertt: Sliding Modes, ∆-Modulators, and Generalized Proportional Integral Control
23-41 (2002).
6. Perruquetti, W. and J. P. Barbot, Sliding mode control in Engineering, Marcel Dekker Inc., New York
(2002).
7. Steele, R., Delta Modulation Systems, Pentech Press,
London (1975).
8. Utkin, V.I., Sliding Modes and Their Applications in
Variable Structure Systems, Mir Publishers, Moscow
(1978).
9. Utkin, V.I., Sliding Modes in Control and Optimization, Springer-Verlag, Berlin (1992).
10. Utkin, V.I., J. Guldner, and J. Shi, Sliding Mode
Control in Electromechanical Systems, Talylor and
Francis, London (1999).
H. Sira-Ramírez was born in San
Cristóbal (Venezuela). He obtained
the Electrical Engineer’s degree
from the Universidad de Los Andes
in Mérida (Venezuela) in 1970. He
later obtained the M.Sc. in EE and
the Electrical Engineer degree, in
1974, and the Ph.D. degree, also in
EE, in 1977, all from the Massachusetts Institute of
Technology (Cambridge, USA). He has been a Visiting
Professor at the University of Illinois at Urbana-Champaign (USA), Purdue University (USA), The Universities of Sheffeld and Leicester (UK), the Laboratory
Heudyasic of the Universitè de Compiegne (CNRS,
Compiegne-France), the Laboratoire des Signaux et
Systèmes (CNRS, Plateau de Moulon-France), The Institute Nationale des Sciences Appliquées (Toulousse-France), The Universidad Nacional del Sur (Argentina), The Universidad de La Plata (Argentina), The
Ecole Central de Lille (Lille-France), The Centre Automatique et Systèmes of the Ecole Nationale Superieure
des Mines de Paris (Fontainebleau, France), The University of Delaware (Newark-DEL, USA), The Laboratoire
d’Auotmatique de Grenoble (CNRS, Grenoble-France)
475
and the Laboratoire STIX of the Ecole Polytechnique
(Palaiseau-France).
Dr. Sira-Ramírez worked for 28 years at the Universidad de Los Andes from which he was, Head of the
Control Systems Department, Head of the Graduate
Studies in Control Engineering and Vicepresident of the
University. Since 1998, he became a Professor Emeritus
of said University. Currently, he is a Titular Researcher
in the Centro de Investigación y Estudios Avanzados del
Instituto Politécnico Nacional (CINVESTAV-IPN) in
México City (México).
Dr. Sira-Ramírez is a Senior Member of the Institute of Electrical and Electronics Engineers (IEEE), a
Distinguished Lecturer from the same Institute and a
Member of the IEEE International Committee. He is also
a member of the Society for Industrial and Applied
Mathematics (SIAM), of the International Federation of
Automatic Control (IFAC) and of the American Mathematical Society (AMS). He has published more than 250
technical articles; over a 100 of them in credited journals
and the rest in international conferences. Dr.
Sira-Ramírez has authored chapters in eighteen contributed books and he is a coauthor of the books, Passivity
Based Control of Euler-Lagrange Systems published by
Springer-Verlag, in 1998, Algebraic Methods in Flatness,
Signal Processing and State Estimation, Lagares 2003
and Differentially Flat Systems, Marcel Dekker 2004. He
is the recipient of several regional and national scientific
awards from his native country of Venezuela. He has
been a member of the Academia de Mérida (Venezuela),
where he served as Second and First Vice-president for
several years. He is a level III member of the Sistema
Nacional de Investigadores (Conacyt, México) and a
level IV member of the Programa de Promoción del Investigador (Fonacyt, Venezuela).
Dr. Sira-Ramírez is interested in the theoretical and
practical aspects of feedback regulation of nonlinear
dynamic systems with special emphasis in Variable
Structure feedback control techniques.