# A class of optimal state-delay control problems

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.

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.

## 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

189 citations

### 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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80 citations

### Additional excerpts

...[4, 5, 6, 18, 19]....

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### Cites background from "A class of optimal state-delay cont..."

...This algorithm was extended in [27, 28] to cater for more general nonlinear systems with unknown system parameters in addition to unknown delays....

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##### References

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### "A class of optimal state-delay cont..." refers background in this paper

...Such problems arise in the control of crushing processes 257 [19] and mixing tanks with recycle loops [27]....

[...]

...Although the optimal control of time-delay systems has been the subject 97 of numerous theoretical and practical investigations [2, 8, 19, 5], most re98 search has focussed on the simple case when the delays are fixed and known....

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392 citations