A class of optimal state-delay control problems
Summary (1 min read)
2. Problem formulation
- Let T denote the set of all such admissible state-delay vectors.
- Let Z denote the set of all such admissible parameter vectors.
Any vector
- The authors assume that the following conditions are satisfied.
- These time points are called characteristic times in the optimal control literature [2, 17, 18] .
- As the authors will see, cost functions with characteristic times arise in parameter identification problems, where the aim is to minimize the discrepancy between predicted and observed system output at a set of sample times.
- The authors optimal state-delay control problem is defined formally below.
3. Gradient computation
- Problem (P) can be viewed as a nonlinear optimization problem in which the decision vectors τ and ζ influence the cost function J implicitly through the governing dynamic system (1)-( 2).
- Thus, if the gradient of J can be computed for each admissible control pair, then Problem (P) can be solved using existing gradient-based optimization methods, such as sequential quadratic programming (see [20, 21] ).
3.3. Solving Problem (P)
- Evolves forward in time (starting from an initial condition), while the auxiliary system (6)-( 8) evolves backwards in time (starting from a terminal condition).
- Thus, since the state and auxiliary systems evolve in opposite directions, and the auxiliary system depends on the solution of the state system, these two systems cannot be solved simultaneously.
- Instead, the state system is solved first in Step 1, and then the solution of the state system is used to solve the auxiliary system in Step 2.
- In practice, numerical integration methods are used to solve the state and auxiliary systems.
- The integrals in the gradient formulae ( 9) and ( 19) can be evaluated using standard numerical quadrature rules.
4.1. Problem formulation
- Using the algorithm in Section 3.3, only 2n differential equations need to be solved.
- Thus, their new method is ideal for online applications in which efficiency is paramount.
6. Conclusion
- The authors new method is applicable to systems with nonlinear terms containing more than one state-delay.
- The authors have restricted their attention in this paper to systems with time-invariant time-delays.
- Such problems arise in the control of crushing processes [19] and mixing tanks with recycle loops [27] .
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Citations
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Cites background from "A class of optimal state-delay cont..."
...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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90 citations
Additional excerpts
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References
136 citations
"A class of optimal state-delay cont..." refers background in this paper
...Voluntarily introducing delays via delayed feed45 back control can be beneficial for certain types of systems; see, for example, 46 [13, 14, 15]....
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...Nevertheless, it has been shown 216 that introducing delays to an undelayed system can be beneficial, especially 217 for chaotic systems [13, 15, 22]....
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109 citations
"A class of optimal state-delay cont..." refers background in this paper
...These time points 89 are called characteristic times in the optimal control literature [2, 17, 18]....
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104 citations
"A class of optimal state-delay cont..." refers background in this paper
...Assumptions 1 and 2 ensure that system (1)-(2) admits a unique solution 84 corresponding to each admissible control pair (τ , ζ) ∈ T ×Z [16]....
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96 citations
"A class of optimal state-delay cont..." refers methods in this paper
...Well-known tools in5 clude the necessary conditions for optimality [5, 6] and numerical methods 6 based on the control parameterization technique [7, 8]....
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93 citations
"A class of optimal state-delay cont..." refers background in this paper
...evapo2 ration and purification processes [1, 2], aerospace models [3], and human 3 immune response [4]....
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