A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

1.
Stengel
,
R.
, 1994,
Optimal Control and Estimation
,
Dover
,
New York
.
2.
Bertsekas
,
D.
, 1995,
Dynamic Programing and Optimal Control
,
Athena Scientific
,
Belmont, MA
.
3.
Zhou
,
K.
,
Doyle
,
J.
, and
Glover
,
K.
, 1996,
Robust and Optimal Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
4.
Kirk
,
D.
, 2004,
Optimal Control Theory: An Introduction
,
Dover
,
New York
.
5.
Hargraves
,
C.
, and
Paris
,
S.
, 1987, “
Direct Trajectory Optimization Using Nonlinear Programming and Collocation
,”
J. Guid. Control Dyn.
0731-5090,
10
, pp.
338
342
.
6.
von Stryk
,
O.
, 1991, “
Numerical Solution of Optimal Control Problems by Using Direct Collocation
,”
Proceedings of the Conference in Optimal Control and Variational Calculus
, Oberwolfach.
7.
Bellman
,
R.
, 1957,
Dynamic Programming
,
Princeton University Press
,
Princeton, NJ
.
8.
Miele
,
A.
, 1980,
Gradient Algorithms for the Optimization of Dynamic Systems, Control and Dynamic Systems
,
,
New York
, pp.
1
52
.
9.
Chernousko
,
F.
, and
Lyubushin
,
A.
, 1982, “
Method of Successive Approximation for Solution of Optimal Control Problems
,”
Opt. Control Appl. Methods
0143-2087,
3
, pp.
101
114
.
10.
Rosen
,
O.
, and
Luus
,
R.
, 1992, “
Global Optimization Approach to Nonlinear Optimal Control
,”
J. Optim. Theory Appl.
0022-3239,
73
(
3
), pp.
547
562
.
11.
Esposito
,
W.
, and
Floudas
,
C.
, 2000, “
Deterministic Global Optimization in Nonlinear Optimal Control Problems
,”
J. Global Optim.
0925-5001,
17
, pp.
97
126
.
12.
Bardi
,
M.
, and
Capuzzo-Dolcetta
,
I.
, 1997,
Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations
,
Birkhäuser
,
Boston, MA
.
13.
von Stryk
,
O.
, and
Bulirsch
,
R.
, 1992, “
Direct and Indirect Methods for Trajectory Optimization
,”
Ann. Operat. Res.
0254-5330,
37
(
1
), pp.
357
373
.
14.
Vlassenbroeck
,
J.
, and
van Dooren
,
R.
, 1988, “
A Chebyshev Technique for Solving Nonlinear Optimal Control Problems
,”
IEEE Trans. Autom. Control
0018-9286,
33
(
4
), pp.
333
340
.
15.
Shih
,
D.
, and
Kung
,
F.
, 1986, “
Optimal Control of Deterministic Systems Via Shifted Legendre Polynomials
,”
IEEE Trans. Autom. Control
0018-9286,
31
(
5
), pp.
451
454
.
16.
Bosarge
,
W. E.
, Jr.
,
Johnson
,
O. G.
,
McKnight
,
R. S.
, and
Timlake
,
W. P.
, 1973, “
The Ritz-Galerkin Procedure for Nonlinear Control Problems
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429,
10
(
1
), pp.
94
110
.
17.
Beard
,
R. W.
, 1998, “
Successive Galerkin Approximation Algorithms for Nonlinear Optimal and Robust Control
,”
Int. J. Control
0020-7179,
71
(
5
), pp.
717
743
.
18.
Polak
,
E.
, 1973, “
An Historical Survey of Computational Methods in Optimal Control
,”
SIAM Rev.
0036-1445,
15
, pp.
553
584
.
19.
Betts
,
J.
, 1998, “
Survey of Numerical Methods for Trajectory Optimization
,”
J. Guid. Control Dyn.
0731-5090,
21
(
2
), pp.
193
207
.
20.
Pukhov
,
G. E.
, 1980,
Differential Transforms of Functions and Equations
,
Naukova Dumka
,
Kiev
, in Russian.
21.
Zhou
,
J.
, 1986,
Differential Transformation and Its Application for Electrical Circuits
,
Huazhong University Press
,
Wuhan
, in Chinese.
22.
Chen
,
C.
, and
Liu
,
Y.
, 1998, “
Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method
,”
J. Optim. Theory Appl.
0022-3239,
99
(
1
), pp.
23
35
.
23.
Jang
,
M.
,
Chen
,
C.
, and
Liu
,
Y.
, 2001, “
Two-Dimensional Differential Transform for Partial Differential Equations
,”
Appl. Math. Comput.
0096-3003,
121
, pp.
261
270
.
24.
Chen
,
C.
, and
Ho
,
Y.
, 1999, “
Transverse Vibration of a Rotation Twisted Timoshenko Beam Under Axial Loading Using Differential Transformation
,”
Int. J. Mech. Sci.
0020-7403,
41
, pp.
1339
1365
.
25.
Bert
,
C.
, and
Zeng
,
H.
, 2004, “
Analysis of Axial Vibration of Compound Bars by Differential Transformation Method
,”
J. Sound Vib.
0022-460X,
275
, pp.
641
647
.
26.
Mei
,
C.
, 2006, “
Differential Transformation Approach for Free Vibration Analysis of a Centrifugally Stiffened Timoshenko Beam
,”
ASME J. Vibr. Acoust.
0739-3717,
128
, pp.
170
175
.
27.
Hwang
,
I.
,
Li
,
J.
, and
Du
,
D.
, 2008, “
A Numerical Algorithm for Optimal Control of a Class of Hybrid Systems: Differential Transformation Based Approach
,”
Int. J. Control
0020-7179,
81
(
2
), pp.
277
293
.
28.
Rudin
,
W.
, 1976,
Principles of Mathematical Analysis
,
McGraw-Hill
,
New York
.
29.
Athans
,
M.
, and
Falb
,
P.
, 1966,
Optimal Control: An Introduction to the Theory and Its Applications
,
McGraw-Hill
,
New York
.
30.
Kress
,
R.
, 1998,
Numerical Analysis
,
Springer
,
New York
.
31.
Boyd
,
J.
, 1989,
Chebyshev and Fourier Spectra Methods
,
Lecture Notes in Engineering
,
Springer-Verlag
,
Berlin
.
32.
Beard
,
R.
, 1995, “
Improving the Closed-Loop Performance of Nonlinear Systems
,” Ph.D. thesis, Rensselaer Polytechnic, Troy, NY.
33.
Hull
,
D.
, 2003,
Optimal Control Theory for Applications
,
Springer
,
New York
.
34.
Xu
,
X.
, and
Antsaklis
,
P.
, 2004, “
Optimal Control of Switched Systems Based on Parameterization of the Switched Instants
,”
IEEE Trans. Autom. Control
0018-9286,
49
, pp.
2
16
.
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