Linear proportional-integral-derivative (PID) controller stands for the most widespread technique in industrial applications due to its simple structure and easy tuning rules. Recently, considering fractional orders λ and μ, there has been studied the fractional-order PIλDμ (FPID) controller to provide salient advantages in comparison to the conventional integer-order PID, such as, a more flexible structure and a preciser performance. In addition, proportional and derivative (PD) and PID error manifolds have been classically proposed; however, there remains the question on how FPID-like error manifolds perform for the control of nonlinear plants, such as robots. In this paper, this problem is addressed by proposing a PD-IλDμ error manifold for novel vector saturated control. The stability analysis shows convergence into a small vicinity of the origin, wherein, such hybrid combination of integer- and fractional-order error manifolds provides further insights into the closed-loop response of the nonlinear plant. Simulations studies are carried out to illustrate the feasibility of the proposed scheme.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
2.
Rahimian
,
M. A.
, and
Tavazoei
,
M. S.
,
2013
, “
Optimal Tuning for Fractional-Order Controllers: An Integer-Order Approximating Filter Approach
,”
ASME J. Dyn. Syst. Meas. Contr.
,
135
(
2
), p.
021017
.
3.
Liebst
,
B. S.
, and
Torvik
,
P. J.
,
2007
, “
Asymptotic Approximations for Systems Incorporating Fractional Derivative Damping
,”
ASME J. Dyn. Syst. Meas. Contr.
,
118
(
3
), pp.
572
579
.
4.
Barbosa
,
R.
,
Tenreiro-Machado
,
J.
, and
Ferreira
,
I.
,
2004
, “
Tuning of PID Controllers Based on Bode's Ideal Transfer Function
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
305
321
.
5.
Dadras
,
S.
,
Dadras
,
S.
, and
Momeni
,
H. R.
,
2017
, “
Linear Matrix Inequality Based Fractional Integral Sliding-Mode Control of Uncertain Fractional-Order Nonlinear Systems
,”
ASME J. Dyn. Syst. Meas. Contr.
,
139
(
11
), p.
111003
.
6.
Tavazoei
,
M. S.
, and
Haeri
,
M.
,
2010
, “
Stabilization of Unstable Fixed Points of Fractional-Order Systems by Fractional-Order Linear Controllers and Its Applications in Suppression of Chaotic Oscillations
,”
ASME J. Dyn. Syst. Meas. Contr.
,
132
(
2
), p.
021008
.
7.
Muñoz-Vázquez
,
A. J.
,
Parra-Vega
,
V.
, and
Sánchez-Orta
,
A.
,
2014
, “
Free-Model Fractional-Order Absolutely Continuous Sliding Mode Control for Euler-Lagrange Systems
,”
53rd IEEE Conference on Decision and Control
, Los Angeles, CA, Dec. 15–17, pp.
6933
6938
.
8.
Muñoz-Vázquez
,
A. J.
,
Parra-Vega
,
V.
, and
Sánchez-Orta
,
A.
,
2015
, “
Control of Constrained Robot Manipulators Based on Fractional Order Error Manifolds
,”
11th IFAC Symposium on Robot Control
, Salvador, Brazil, Aug. 26–28, pp.
131
126
.
9.
Podlubny
,
I.
,
1999
, “
Fractional-Order Systems and PIλDμ-Controllers
,”
IEEE Trans. Autom. Contr.
,
44
(
1
), pp.
208
214
.
10.
Tavakoli-Kakhki
,
M.
, and
Haeri
,
M.
,
2011
, “
Temperature Control of a Cutting Process Using Fractional Order Proportional-Integral-Derivative Controller
,”
ASME J. Dyn. Syst. Meas. Contr.
,
133
(
5
), p.
051014
.
11.
Badri
,
V.
, and
Tavazoei
,
M. S.
,
2016
, “
Simultaneous Compensation of the Gain, Phase, and Phase-Slope
,”
ASME J. Dyn. Syst. Meas. Contr.
,
138
(
12
), p.
121002
.
12.
Li
,
H.
,
Luo
,
Y.
, and
Chen
,
Y. Q.
,
2000
, “
A Fractional Order Proportional and Derivative (FOPD) Motion Controller: Tuning Rule and Experiments
,”
IEEE Trans. Contr. Syst. Technol.
,
18
(
2
), pp.
516
520
.
13.
Yang
,
H.
,
Jiang
,
Y.
, and
Yin
,
S.
,
2018
, “
Fault-Tolerant Control of Time-Delay Markov Jump Systems With It ô Stochastic Process and Output Disturbance Based on Sliding Mode Observer
,”
IEEE Trans. Ind. Inf.
(in press).
14.
Yin
,
S.
,
Yang
,
H.
, and
Kaynak
,
O.
,
2017
, “
Sliding Mode Observer-Based FTC for Markovian Jump Systems With Actuator and Sensor Faults
,”
IEEE Trans. Autom. Contr.
,
62
(
7
), pp.
3551
3558
.
15.
Samko
,
S.
,
Khilbas
,
A.
, and
Marichev
,
O.
,
1993
,
Fractional Integrals and Derivatives. Theory and Applications
,
Gordon and Breach
,
Yverdon, Switzerland
.
16.
Muñoz-Vázquez
,
A. J.
,
Parra-Vega
,
V.
, and
Sánchez-Orta
,
A.
,
2016
, “
Uniformly Continuous Differintegral Sliding Mode Control of Nonlinear Systems Subject to Hölder Disturbances
,”
Automatica
,
66
, pp.
179
184
.
17.
Parra-Vega
,
V.
,
Arimoto
,
S.
,
Liu
,
Y. H.
,
Hirzinger
,
G.
, and
Akella
,
P.
,
2003
, “
Dynamic Sliding PID Control for Tracking of Robots Manipulators: Theory and Experiments
,”
IEEE Trans. Rob. Autom.
,
19
(
6
), pp.
967
976
.
18.
Matignon
,
D.
,
1996
, “
Stability Results for Fractional Differential Equations With Applications to Control Processing
,”
Multiconference on Computation Engineering in Systems Applications
, pp.
963
968
.
19.
Gutman
,
S.
,
1979
, “
Uncertain Dynamic Systems: A Lyapunov Min-Max Approach
,”
IEEE Trans. Autom. Contr.
,
24
(
3
), pp.
437
449
.
20.
Chen
,
Y. Q.
,
Petráš
,
I.
, and
Xue
,
D.
,
2009
, “
Fractional Order Control—A Tutorial
,”
IEEE
American Control Conference
, St. Louis, MO, June 10–12, pp.
1397
1411
.
21.
Oustaloup
,
A.
,
Mathieu
,
B.
, and
Lanusse
,
P.
,
1995
, “
The CRONE Control of Resonant Plants: Application to a Flexible Transmission
,”
Eur. J. Contr.
,
1
(
2
), pp.
113
121
.
You do not currently have access to this content.