© The Institution of Engineering and Technology
In this study the authors present a solution to the problem of the quadratic mini–max regulator for polynomial uncertain systems. The main characteristic of this type of problems is that the parameters describing the dynamics of the non-linear plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the solution is given in terms of the worst-case scenario. That is to say, the result of the application of a certain control input (in terms of the cost function value) is associated with the worst or least favourable value of the unknown parameter. Based on the general necessary conditions for mini–max optimality, a closed form for the control is provided, which makes use of a p-linear form tensor representation of the polynomial system. In its turn, this allows one to present the solution in a way similar to the so-called Riccati technique. The final control is shown to be a convex combination (with some weights) of the optimal controls for each fixed parameter of the polynomial system. Several simulation examples are presented to show the effectiveness of our approach including the robust regulation of the well-known Duffing equation, which represents a typical (and challenging to control) non-linear polynomial system.
References
-
-
1)
-
W. Fleming ,
R. Rishel
.
(1975)
Deterministic and stochastic optimal control.
-
2)
-
G. Kallianpur
.
(1980)
Stochastic filtering theory.
-
3)
-
Utkin, V., Shi, J.: `Integral sliding mode in systems operating under uncertainty conditions', Proc. 35th Conf. on Decision and Control, December 1996, Kobe, Japan.
-
4)
-
H. Lam
.
Chaotic synchronisation using output/full state-feedback polynomial controller.
IET Control Theory Appl.
,
2285 -
2292
-
5)
-
T. Yoshida ,
K. Loparo
.
Quadratic regulatory theory for analytic nonlinear systems with additive controls.
Automatica
,
531 -
544
-
6)
-
L.S. Pontryagin ,
V.G. Boltyanskii ,
R.V. Gamkrelidze ,
E.F. Mishchenko
.
(1986)
The mathematical theory of optimal processes.
-
7)
-
R. Bellman
.
(1957)
Dynamic programming.
-
8)
-
H.K. Khalil
.
(1988)
Nonlinear systems.
-
9)
-
X. Li ,
H. Yu ,
M. Yuan ,
J. Wang
.
Design of robust optimal proportional-integral-derivative controller based on new interval polynomial stability criterion and Lyapunov theorem in the multiple parameters’ perturbations circumstance.
IET Control Theory Appl.
,
2427 -
2440
-
10)
-
A. Loria ,
E. Panteley ,
H. Nijmeijer
.
Control of the chaotic duffing equation with uncertainty in all parameters.
IEEE Trans. Circuits Syst.–I Fundam. Theory Appl.
,
1252 -
1255
-
11)
-
Z. Shu ,
L. Lam
.
An augmented system approach to static output-feedback stabilization with H_∞ performance for continuous-time plants.
Int. J. Robust Nonlinear Control
,
768 -
785
-
12)
-
L. Fridman ,
A. Poznyak ,
F. Bejarano
.
Decomposition of the min–max multi-model problem via integral sliding mode.
Int. J. Robust Nonlinear Control
,
559 -
574
-
13)
-
M. Basin ,
J. Perez ,
M. Skliar
.
Optimal filtering for polynomial system states with polynomial multiplicative noise.
Int. J. Robust Nonlinear Control
,
287 -
298
-
14)
-
V. Boltyanski ,
A. Poznyak
.
Linear multi-model time optimization.
Optim. Control Appl. Methods
,
141 -
161
-
15)
-
L. Zhang ,
J. Lam
.
On H_2 model reduction of bilinear systems.
Automatica
,
205 -
216
-
16)
-
A. Truman ,
M. Govan
.
Polynomial LQG synthesis of subrate digital feedback systems.
IET Control Theory Appl.
,
247 -
256
-
17)
-
M. Basin ,
D. Calderon-Alvarez
.
Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances.
Int. J. Control
,
1435 -
1447
-
18)
-
V. Boltyanski ,
A. Poznyak
.
Robust maximum principle in minimax control.
Int. J. Control
,
4 ,
305 -
314
-
19)
-
V. Benes
.
Exact finite-dimensional filters for certain diffusions with nonlinear drift.
Stochastics
,
65 -
92
-
20)
-
X. Yan ,
J. Lam ,
L. Xie
.
Robust observer design for nonlinear interconnected systems using structural characteristics.
Int. J. Control
,
741 -
746
-
21)
-
S. Yau
.
Finite-dimensional filters with nonlinear drift. I: A class of filters including both Kalman-Bucy and Benes filters.
J. Math. Syst. Estim. Control
,
181 -
203
-
22)
-
H. Nijmeijer ,
B. Harry
.
On Lyapunov control of the Duffing equation.
IEEE Trans. Circuits Syst.-I Fund. Theory Appl.
,
473 -
477
-
23)
-
M. Basin ,
D. Calderon-Alvarez ,
M. Skliar
.
Optimal filtering for incompletely measured polynomial states over linear observations.
Int. J. Adapt. Control Signal Process.
,
482 -
494
-
24)
-
T. Kapitaniak
.
(1996)
Controlling chaos.
-
25)
-
W. Wonham
.
Some applications of stochastic differential equations to nonlinear filtering.
SIAM J. Control Optim.
,
347 -
369
-
26)
-
A. Poznyak ,
T. Duncan ,
B. Pasik-Duncan ,
V. Boltyanski
.
Robust maximum principle for multi-model LQ-problem.
Int. J. Control
,
15 ,
1770 -
1777
-
27)
-
M. Jimenez-Lizarraga ,
A. Poznyak
.
Robust Nash equilibrium in multi-model LQ differential games: Analysis and extraproximal numerical procedure.
Optim. Control Appl. Methods
,
2 ,
117 -
141
-
28)
-
R. Kalman ,
R. Bucy
.
New results in linear filtering and prediction theory.
ASME Trans. Part D (J. Basic Eng.)
,
95 -
108
-
29)
-
H. Kushner
.
On differential equations satisfied by conditional probability densities of Markov processes.
SIAM J. Control Optim.
,
106 -
119
-
30)
-
E. Al’brekht
.
On the optimal stabilization of nonlinear systems.
J. Appl. Math. Mech.
,
1254 -
1266
-
31)
-
M. Basin
.
On optimal filtering for polynomial system states.
ASME Trans. J. Dyn. Syst. Meas. Control
,
123 -
125
-
32)
-
A. Poznyak ,
E. Sanchez ,
W. Yu
.
(2001)
Differential Neural networks for robust nonlinear control.
-
33)
-
A.E. Bryson
.
(1999)
Dynamic optimization.
-
34)
-
M. Jimenez-Lizarraga ,
A. Poznyak
.
Quasi-equilibrium in LQ differential games with bounded uncertain disturbances: Robust and adaptive strategies with pre-identification via sliding mode technique.
Int. J. Syst. Sci.
,
7 ,
585 -
599
-
35)
-
H. Kwahernaak ,
R. Sivan
.
(1972)
Linear optimal control systems.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2011.0090
Related content
content/journals/10.1049/iet-cta.2011.0090
pub_keyword,iet_inspecKeyword,pub_concept
6
6