An Optimization Approach to State-Delay Identification $ $

doi:10.1109/TAC.2010.2050710

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An Optimization Approach to State-Delay Identification $ $

Journal Article•DOI•

An Optimization Approach to State-Delay Identification $ $

Ryan Loxton¹, Kok Lay Teo¹, Volker Rehbock¹•Institutions (1)

01 Sep 2010-IEEE Transactions on Automatic Control (Institute of Electrical and Electronics Engineers)-Vol. 55, Iss: 9, pp 2113-2119

TL;DR: A dynamic optimization problem in which the state-delays are decision variables and the cost function measures the discrepancy between predicted and observed system output is formulated and the gradient of this problem's cost function can be computed by solving an auxiliary delay-differential system.

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Abstract: We consider a nonlinear delay-differential system with unknown state-delays. Our goal is to identify these state-delays using experimental data. To this end, we formulate a dynamic optimization problem in which the state-delays are decision variables and the cost function measures the discrepancy between predicted and observed system output. We then show that the gradient of this problem's cost function can be computed by solving an auxiliary delay-differential system. By exploiting this result, the state-delay identification problem can be solved efficiently using a gradient-based optimization method.

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Journal Article•DOI•

The control parameterization method for nonlinear optimal control: a survey

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Qun Lin, Ryan Loxton, Kok Lay Teo

01 Oct 2013-Journal of Industrial and Management Optimization

TL;DR: The control parameterization method is a popular numerical technique for solving optimal control problems as mentioned in this paper, which discretizes the control space by approximating the control function by a linear combination of basis functions.

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Abstract: The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.

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226 citations

Cites background from "An Optimization Approach to State-D..."

...Gradient formulae for the cost function with respect to the time-delays are derived in [10, 11, 46]....
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...References [10, 11, 46] consider the problem of choosing the delays to minimize the deviation between predicted...
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Journal Article•DOI•

Theory and applications of optimal control problems with multiple time-delays

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Laurenz Göllmann, Helmut Maurer

01 Oct 2013-Journal of Industrial and Management Optimization

TL;DR: In this article, a discretization method is presented by which the delayed control problem is transformed into a nonlinear programming problem, and the associated Lagrange multipliers provide a consistent numerical approximation for the adjoint variables of the delayed optimal control problem.

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Abstract: In this paper we study optimal control problems with multiple time delays in control and state and mixed type control-state constraints. We derive necessary optimality conditions in the form of a Pontryagin type Minimum Principle. A discretization method is presented by which the delayed control problem is transformed into a nonlinear programming problem. It is shown that the associated Lagrange multipliers provide a consistent numerical approximation for the adjoint variables of the delayed optimal control problem. We illustrate the theory and numerical approach on an analytical example and an optimal control model from immunology.

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90 citations

Additional excerpts

...[4, 5, 6, 18, 19]....
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Journal Article•DOI•

Parameter estimation for nonlinear time-delay systems with noisy output measurements

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Qun Lin¹, Ryan Loxton², Chao Xu², Kok Lay Teo²•Institutions (2)

Curtin University¹, Zhejiang University²

01 Oct 2015-Automatica

TL;DR: An efficient optimization algorithm is proposed for determining optimal estimates for the time-delays and system parameters in a general nonlinear time-delay system and is examined on a dynamic model of a continuously-stirred tank reactor.

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53 citations

Journal Article•DOI•

A unified parameter identification method for nonlinear time-delay systems

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Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang

01 Feb 2013-Journal of Industrial and Management Optimization

TL;DR: The main contribution is to show that the partial derivatives of this cost function can be computed by solving a set of auxiliary time-delay systems and can be solved using existing gradient-based optimization techniques.

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Abstract: This paper deals with the problem of identifying unknown time-delays and model parameters in a general nonlinear time-delay system. We propose a unified computational approach that involves solving a dynamic optimization problem, whose cost function measures the discrepancy between predicted and observed system output, to determine optimal values for the unknown quantities. Our main contribution is to show that the partial derivatives of this cost function can be computed by solving a set of auxiliary time-delay systems. On this basis, the parameter identification problem can be solved using existing gradient-based optimization techniques. We conclude the paper with two numerical simulations.

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48 citations

Cites background or methods from "An Optimization Approach to State-D..."

...This type of parameter identification problem was previously considered in [14] for systems in which each nonlinear component contains at most one unknown delay and no unknown system parameters....
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...The method in [14] is only applicable when each nonlinear term contains a single delay and no unknown parameters....
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...See Appendix B of [14] for more details....
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...Our goal in this paper is to extend the approach pioneered in [14] to these more complicated systems....
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...Note that this problem cannot be solved using the identification method in [14], as the third term on the right-hand side of (45) contains both an unknown parameter and an unknown delay....
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Journal Article•DOI•

A class of optimal state-delay control problems

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Qinqin Chai¹, Qinqin Chai², Ryan Loxton¹, Kok Lay Teo¹, Chunhua Yang² - Show less +1 more•Institutions (2)

Curtin University¹, Central South University²

01 Jun 2013-Nonlinear Analysis-real World Applications

TL;DR: It is shown that this optimal state-delay control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables and an efficient numerical method for determining thecost function’s gradient is developed.

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Abstract: We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.

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33 citations

Cites background or methods from "An Optimization Approach to State-D..."

...30 This computational approach is motivated by our earlier work in [12], which 31...
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...In [12], the method proposed for computing the 172 cost function’s gradient involves solving mn + nr + n differential equations....
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...Furthermore, unlike the method in [12], 251 our new method is applicable to systems with nonlinear terms containing 252 more than one state-delay....
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...196 This problem cannot be solved using the identification method in [12], 197 which is only applicable when each nonlinear term in the system dynamics 198...
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...170 A similar (but less general) parameter identification problem was recently 171 considered in reference [12]....
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References

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Book•

Numerical Optimization

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Jorge Nocedal¹, Stephen J. Wright²•Institutions (2)

Northwestern University¹, University of Wisconsin-Madison²

01 Nov 2008

TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.

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Abstract: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

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17,420 citations

"An Optimization Approach to State-D..." refers methods in this paper

...This is a fundamental result that enables one to solve the identification problem—and thereby obtain accurate estimates for the state-delays—using standard optimization techniques [1], [2]....
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...This algorithm can be readily incorporated into a standard gradientbased optimization method, such as a conjugate gradient method (see [1] and [2])....
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Journal Article•DOI•

Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach

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Yong-Yan Cao¹, Paul M. Frank•Institutions (1)

Zhejiang University¹

01 Apr 2000-IEEE Transactions on Fuzzy Systems

TL;DR: The TS fuzzy models with time delay are presented and the stability conditions are derived using Lyapunov-Krasovskii approach and a stabilization approach for nonlinear time-delay systems through fuzzy state feedback and fuzzy observer-based controller is presented.

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Abstract: Takagi-Sugeno (TS) fuzzy models (1985, 1992) can provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear input/output (I/O) submodels. In this paper, the TS fuzzy model approach is extended to the stability analysis and control design for both continuous and discrete-time nonlinear systems with time delay. The TS fuzzy models with time delay are presented and the stability conditions are derived using Lyapunov-Krasovskii approach. We also present a stabilization approach for nonlinear time-delay systems through fuzzy state feedback and fuzzy observer-based controller. Sufficient conditions for the existence of fuzzy state feedback gain and fuzzy observer gain are derived through the numerical solution of a set of coupled linear matrix inequalities. An illustrative example based on the CSTR model is given to design a fuzzy controller.

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768 citations

"An Optimization Approach to State-D..." refers background or methods in this paper

...A dynamic model for this system is [8]...
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...We demonstrate this in Section V of this note, where we apply our new method to the continuously-stirred tank reactor described in [8]....
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...Yet many delay systems that arise in applications, such as predator-prey systems [6], aerospace systems [7], and continuously-stirred tank reactors [8], are actually nonlinear....
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Book•

A unified computational approach to optimal control problems

[...]

Kok Lay Teo, C. J. Goh, K. Wong

01 Jan 1991

441 citations

Journal Article•DOI•

The elements of integration and Lebesgue measure

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Bartle¹, Robert Gardner•Institutions (1)

State University of New York System¹

01 Dec 1996-The Statistician

TL;DR: In this article, the Lebesgue spaces L p. Modes of Convergence and Integrable Functions are discussed. But they do not consider the nonmeasurable and non-borel sets.

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Abstract: THE ELEMENTS OF INTEGRATION. Measurable Functions. Measures. The Integral. Integrable Functions. The Lebesgue Spaces L p . Modes of Convergence. Decomposition of Measures. Generation of Measures. Product Measures. THE ELEMENTS OF LEBESGUE MEASURE. Volumes of Cells and Intervals. The Outer Measure. Measurable Sets. Examples of Measurable Sets. Approximation of Measurable Sets. Additivity and Nonadditivity. Nonmeasurable and Non--Borel Sets. References. Index.

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396 citations

Journal Article•DOI•

Optimal control drug scheduling of cancer chemotherapy

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R. B. Martin¹•Institutions (1)

University of Western Australia¹

01 Nov 1992-Automatica

TL;DR: This optimal control model of cancer chemotherapy constructs drug schedules that most effectively reduce the size of a tumour after a fixed period of treatment has elapsed using an established numerical solution technique known as control parametrization.

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236 citations

"An Optimization Approach to State-D..." refers methods in this paper

...Although specialized techniques for handling this type of cost function are available (see [10]–[12]), none of them are applicable to Problem P....
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