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1 Introduction
Hebertt Sira-Ram ÿrez Department
of Electrical Engineering, Mechatronics
Section, CINVESTAV
— IPN Av. IPN
2580 Col. San Pedro Zacatenco, DFCP
07360, Mexico e-mail:
[e-mail]
Instituto Polit ecnico Nacional,
UPIITA,
Av. IPN 2580,
Col. Barrio La Laguna Ticom
an, DFCP 07340, Mexico
e-mail: [e-mail]
Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript
received March 14, 2012; final manuscript received October 15, 2014; published
online December 10, 2014. Editor: Joseph Beaman.
Corresponding author.
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uðaÞ
Þ z ¼ hðx; she is; she is_; …;
The reader is referred to [23,27] for the theoretical basis of the GPI approach.
The endogenous variables {z1,…, zm} are called flat outputs, or linearizing
outputs.
controllers; however, these schemes demand the measurement of time derivatives
of the flat outputs, which may demand additional filtering schemes. On the other
hand, the on-line velocity estima tion strategy by using a model-based integral
reconstructor, avoids the use of traditional observers or noisy derivative estimations.
A flatness based controller using integral reconstructors is the so-called GPI
controller
[23].
(4)
Subsection
2.2
introduces the control scheme associated with the linearization
method.
Since the term, [z * (t)] (b) , plays the role of the nominal input in the Brunovsk
y form, the new input is designed as
ez ¼ zz ðtÞ
(2) There exists a multi-index a ¼ (a1,…, am) such that z can be expressed as
051001-2 / Vol. 137, MAY 2015
x ¼ / ðz; z_;…; z ðcÞ Þ
DEFINITION 3. Given a system
(1),
a nominal input u *, is a feedforward input
such that, for a lack of disturbance inputs, or parameter perturbations, as well as
known initial conditions, the solution of
(1)
is a nominal trajectory x * (t).
(1)
and K (ez) consists in a simple linear control scheme capable of stabilizing the
resulting linearized system (proportional derivative (PD), PID, state feedback
control, etc.)
[30].
The combination of the nominal input u ðtÞ ¼ wðz ðtÞ; z_ ðtÞ;
…; ½z ðtÞ Þ and Eq.
(6)
results in the following feedback control structure:
(2)
(3) All system variables, x, u, y, can be expressed in terms of z and a finite
number of its time derivatives, ie,
Numerical and experimental results are reported in Secs.
5
and 6, including a
comparison with an observer based controller using the differential parameterization
provided by flatness. Finally, in Sec. 7, some concluding remarks are presented.
(5)
2.2 Exact Feedforward Linearization and Control Law Design. Since the flatness
based feedforward linearization works in a region close to the desired trajectory,
the control law to be designed consists of two parts: A feedforward part, u ðtÞ ðbÞ
¼ wðz ðtÞ; z_ ðtÞ;…; ½z ðtÞÞ, and a feedback part, K (ez) given, in fact, by any
stabilizing linear control law processing the flat out
put tracking error. In our design, the GPI control will be used.
þ KðezÞ
x_ ¼ fðx; uÞ
are, respectively, the
DEFINITION 1 (Fliess et al. [17]). System
(1)
is said to be differen tially flat if
there exists a set of m quantities z ¼ (z1,…, zm) having the following properties:
ðbÞ u ¼ wðz ðtÞ; z_ ðtÞ; …; ½z ðtÞ
~
for appropriate multi-indices b and c.
LEMMA 4. The application of the nominal input u * to the system
(1)
with a
consistent set of initial conditions, that is, x (t0) ¼ x * (t0), leads to a trajectory x of
Eq.
(1),
which exists on I; this trajectory corresponds to that of a linear system, in
Brunovsky's canonical form, for which the
input is composed of appropriate time deriva tives of the nominal flat output, ie, x ~
¼ x ðtÞ and hðx ~; u ðtÞ; ðaÞ u_ ðtÞ;…; ½u ðtÞÞ ¼ z ðtÞ.
Let f: Rn Rm ! Rn be a smooth mapping. Consider a non linear system of the
form
In this article, a solution is proposed to the problem of trajectory tracking in a
two wheel mobile robot by means of a feedforward linearization associated to a
linear controller of the GPI class. The proposed scheme is free of time derivative
computations on the flat outputs. This alleviates the need for implementation of a
state observer, or signal differentiators, which reduces the complexity of the control
algorithm in imple mentation. Moreover, since it is based on the use of user-defined
trajectory tracking tasks, the proposed controller may avoid a large variety of
singularities, easily arisen in practice. Also, the problem of linearization can be
solved using signals from the feedforward input, which has been a good alternative
for open chain robotic manipulators control (see Ref. [28]), but its appli cation to
mobile robot control has not been reported. In fact, the results for open chain robotic
manipulators control are just based on experimental facts of the application, and
here, the results are theoretically supported by the feedforward linearization
methodology.
DEFINITION 2 (Hagenmeyer and Delaleau [29]). Let I Rþ be a time interval of
the form I ¼ [t0, tf), tf > t0, with tf being either real or þ1. A time function
Transactions of the ASME
One of the most important features of this class of controllers is that only the
outputs of the system are necessary to induce a com plete pole placement strategy
without using observers, which avoids the necessity of time derivatives
measurements
[24,25].
As such, the technique is necessarily restricted to flat
systems. The controller design procedure is analog to that of the PID controller
design, but the control can be written in terms of a classical lead compensator with
better results than traditional PID control
[26].
(3)
2.1 Flatness-Based Exact Feedforward Linearization. The following lemmae
describe the principle of exact feedforward linearization (see Refs.
[22]
and
[30]
for
a detailed analysis).
The structure of the combination of the feedforward and feed back parts of the
controller is given as follows:
where the tracking error ez is defined as
(1) The elements of z are differentially independent.
The control law
(8)
exactly linearizes the system as well as it stabilizes the
tracking error ez to zero. The stability of the control
is a nominal trajectory for the flat output z of Eq.
(1)
if it is suffi ciently differentiable
and such that Eqs. (2) - (4) are defined over I.
This section revisits the concepts of differential flatness, and the exact
feedforward linearization based control scheme, which will be the basis of the
proposed approach for the solution of the main problem. The main result is closely
related to the present section:
and x ¼ [x1,…, xn]
where u ¼ [u1,…, um] input
and state vectors.
(7)
Þ þ KðezÞ
(8)
The remainder of the article is organized as follows: In Sec. 2, some preliminary
concepts are provided. In Sec. 3, the mathematical cal model of the mobile robotic
system is established and the problem formulation is presented. Section
4
presents
the feedfor ward linearization control scheme for the particular mobile robot.
~
LEMMA 5. The application of the nominal input u * (t) to the sys tem
(1)
with a
nonconsistent initial condition, in other words, x (t0) 6¼ x * (t0), which is sufficiently
close to x * ( t0), leads to a trajec tory x of ~
(1),
which exists on an interval of finite
amplitude I1 ¼ [t0, t1) of I. This trajectory remains in the neighborhood of x * (t) on
I1. In other words, the trajectory ~ z of z, corresponding to x and u *
(t), remains in the neighborhood of the nominal trajectory z * (t) on the interval I1.
ðbÞ v ¼ ½z ðtÞ
(6)
ðbÞ
Þ u ¼ wðz; z_; …; Z
ðbÞ
z: I! Rm
2 Definitions and Basic Results
T
T
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Finally, by choosing k3, k2, k1, and k0, such that the polynomial,
p (s), has its roots located on the left half of the complex plane C,
the tracking error ey asymptotically exponentially converges to
zero. Clearly, the decay time constant is determined by the control
gain parameters.
uðtÞ ¼ y € ðtÞ
(21)
(9)
(19)
eyðrÞdr
e_1 ¼
e2 e_2 ¼ k3e2 k2e1 k1q1 q_
^ y_ ¼ y_
For this, we appropriately define state variables q1, q2, whose initial conditions
are conveniently related to the independent term in the right-hand side of Eq.
(15).
Indeed, define
with n being an unknown constant disturbance input. The control problem consists in
obtaining a control law for u, which forces the output, y, to asymptotically track a
given smooth output reference trajectory denoted by y * (t), without using asymptotic
observers.
scheme can be demonstrated when using any type of suitable lin ear controller (see
Ref.
[29]
for a further analysis of the topic). In particular, a detailed analysis of the
stability of this scheme by a simple extended PID control as a feedback control is
given in Ref.
[22].
In our work, the proposed feedback control part consists of a GPI
controller
[27].
Next, the GPI controller design for the particular case of a perturbed
two integrator chain (as it will be the case in the control of the mobile robot) will be
revisited.
The relationship between the structural estimate, ^ y_, of y_, and its actual value,
is given by
Integrating Eq.
(9)
leads to
The main idea of this control approach is the structural recon struction of the
state vector (ie, the reconstruction is carried out modulo initial conditions and modulo
constant, ramp, parabolic errors, ie, modulo the, so called, classical errors). The
approach avoids the use of observers. Instead of observers, some structural state
estimates are produced on the basis of the system inputs, out puts, and their finite
iterated integrals. The accumulation of classi cal errors in the reconstruction prompts
the use of a posteriori added iterated output, or input, integral error compensation.
Thus, one of the advantages of the scheme is the development of linear control laws
without the need for asymptotic observers and a sys tematic approach to the design
of classical compensation networks.
(13)
uðsÞds
PROPOSITION 6. Given the dynamical system described in Eq.
(9),
the dynamical
feedback control law
¼
(14)
y_ ðtÞÞ þ k2ey þ k1 ðt
and
(16)
y_ ðtÞÞ k2ðy y ðtÞÞ
uðsÞds þ y_0 þ nt
2.2.1 GPI Control. In general, the GPI control, or control based on integral
reconstructors
[31],
is a development in the literature on automatic control whose
main line of development rests within the finite dimensional linear systems case, with
some extensions to lin ear delay differential systems and nonlinear systems
[32].
y_0 nt
Let ey ¢ yy * (t) be the tracking error and let u * (t) be the feed forward, nominal,
input, obtained as y € ðtÞ ¼ u ðtÞ. The input error
eyðrÞdr
The integral reconstructor of y_ is defined as
_
q2ðrÞdr
Using the fact that, eu ¼ uu * (t), and applying the Laplace transform, it is obtained
(11)
(12)
^ y_ ¼ ðt
It is not difficult to conclude that the characteristic equation of system
(15)
is given
by,
e € y þ k3ð ^ y_
Consider the following perturbed second-order example:
y_ ¼ ðt
From Eq.
(12)
y_0 þ ðt
Journal of Dynamic Systems, Measurement, and Control
Remark 7. Notice that by resorting to the frequency domain, by means of the
Laplace transform, the controller can be expressed in terms of a classical
compensation network. From Eq.
(13),
we have
(18)
(10)
eyðrÞdrds
(17)
eyðrÞdrds ¼ k3ðy_0 þ ntÞ
e € y þ k3e_y þ k2ey þ k1 ðt
MAY 2015, Vol. 137 / 051001-3
ðyðrÞ y ðrÞÞdrds
Clearly, the unknown initial conditions for q1 and q2 are given by: q1ð0Þ¼ k3 =
k1y_0; q2ð0Þ¼ k3 = k1n. The closed-loop system, with unknown but fixed initial
conditions, is rewritten as
The characteristic polynomial of this last linear time invariant system is just
A ramp type error, given that n is constant, exists between the integral reconstructor
and the actual output derivative. A complementary control action, represented by a
double nested integral of the tracking error must, therefore, be considered. Let us
summa rize the result in the following proposition:
Proof. Applying Eq.
(13)
in Eq.
(9),
the closed-loop error dynamics is governed by
pðsÞ ¼ s
k3ð ^ y_
y € ¼ u þ n
yð0Þ ¼ y0
y_ð0Þ ¼ y_0
is defined as eu ¼ uu ðtÞ ¼ uy € ðtÞ.
(15)
eð4Þ þ k3eð3Þ þ k2e € y þ k1e_y þ k0ey ¼ 0
q_
eyðrÞdrds ¼ 0
(20)
2
0
ðs
0
(where k3; k2; k1; k0 2 R is the set of constant controller gains, and ^ y is defined in Eq.
(11)),
forces the output y to asymptotically exponentially track the given output reference trajectory, y *
(t ).
0
y
2
k1
4
uðtÞ ¼ u ðtÞ k3 ðt k1 ðt
ðyðsÞ y ðsÞÞds k0 ðt
0
ðs
k0 q2 ¼
¼ q2 þ e1
k0 e1 k1
þ k2s
0
k0
q1 ¼ ðt
1
0
0
0
0
0
y
0
0
3
ðuðsÞ u ðsÞÞds k2eyðtÞ
eyðsÞds k0 ðt
k1 ðt 0
k3
n
k1
D
0
þ k1s þ k0
eyðsÞds þ k0 ðt
k1 ðt 0
þ k3s
ðs
0
eyðsÞds þ k0 ðt ðs
0
0
e1 ¼ ey