# 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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### 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

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

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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....

[...]

414 citations