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Multisubsystem co-design refers to the simultaneous optimization of physical plant and controller of a system decomposed into multiple interconnected subsystems. In this paper, two decentralized (multilevel and bilevel) approaches are formulated to solve multisubsystem co-design problems, which are based on the direct collocation and decomposition-based optimization methods. In the multilevel approach, the problem is decomposed into two bilevel optimization problems, one for the physical plant and the other for the control part. In the bilevel approach, the problem is decomposed into subsystem optimization subproblems, with each subproblem having the optimization model for physical plant and control parts together. In both cases, the entire time horizon is discretized to convert the continuous optimal control problem into a finite-dimensional nonlinear program. The optimality condition decomposition method is employed to solve the converted problem in a decentralized manner. Using the proposed approaches, it is possible to obtain an optimal solution for more generalized multisubsystem co-design problems than was previously possible, including those with nonlinear dynamic constraints. The proposed approaches are applied to a numerical and engineering example. For both examples, the solutions obtained by the decentralized approaches are compared with a centralized (all-at-once) approach. Finally, a scalable version of the engineering example is solved to demonstrate that using a simulated parallelization with and without communication delays, the computational time of the proposed decentralized approaches can outperform a centralized approach as the size of the problem increases.

In a recent paper [8], the authors of this paper presented a class of multisystem co-design problems with a finite time horizon linear quadratic regulator. The class of problems in Ref. [8] consisted of multiple subsystems coupled through dynamic equations as well as plant design constraints. A decomposition approach based on an indirect method was proposed in Ref. [8], in which control variables were optimized using the equations derived from the necessary optimality conditions. The proposed techniques can be regarded as an extension of the decomposition-based approach of a single-system co-design problem, as in Ref. [3]. In this paper, two decentralized approaches (i.e., multilevel and bilevel) are developed for general multisubsystem co-design problems based on the direct collocation method and optimality condition decomposition, which differs from the analytical target cascading framework [9] for co-design problems. Using a direct method in the proposed approaches, a continuous-time optimal control problem can be transformed into a finite-dimensional optimization problem by the discretization of the time horizon.

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The main contribution of this paper is twofold. First, following the decentralized framework in Ref. [8], a multilevel decentralized approach is proposed to solve multisubsystem co-design problems. It is possible to use decentralized optimal control techniques to solve certain classes of co-design problems. For example, the control and formation optimization of multiple UAVs can be regarded as a co-design problem [22]. Some other methods (e.g., Refs. [23,24]) can be applied to co-design problems with modifications. However, to the best of our knowledge, the papers in the control field do not focus on the physical design of multiple subsystems or the entire system. In this paper, the proposed problem is more general wherein both the physical plant design and control are optimized using decomposition-based methods. In the proposed approach, the direct collocation method is extended and integrated with the optimality condition decomposition technique to solve the control part. Compared with Ref. [8], the new approach has relaxed the model restrictions on the control problem, which makes it possible to obtain an optimal solution for more general co-design problems, including those with nonconvex objective and state constraint functions, as well as nonlinear dynamics in the control part. Second, a bilevel framework is developed for solving multisubsystem co-design problems. For the two cases that the shared plant design variables exist or do not exist, the proposed approaches formulate and solve co-design subproblems for each subsystem. In particular, when there are shared plant design variables, both complicating variables and complicating constraints (as defined in Ref. [25]) exist in the converted optimization problem by the direct collocation method. To the best of our knowledge, no previous literature has addressed a decomposition-based technique for such problems. It is shown that the bilevel approach has a better performance than the multilevel one in terms of both solution quality and computational time. These two decentralized approaches are compared against a centralized (all-at-once) approach. For the tested examples, the bilevel approach obtains exactly the same optimal solution as the centralized one, while the multilevel one obtains a close but slightly worse solution. It is also shown that the decentralized approaches can outperform the centralized approach in terms of computational time as the size of the entire problem increases.

(A1) The coupling is unidirectional [26], i.e., the plant design objective functions and constraints are dependent only on plant design variables, while the control objective functions, dynamics, and constraints are dependent on both plant design and control variables.

(A2) In the plant design part of all subsystems, the objective functions, and inequality constraints are convex and equality constraints are linear functions of local and shared plant design variables. This assumption is made to satisfy the requirement of the dual decomposition method [25].

(A4) The optimal control subproblems of all subsystems are feasible, regardless of the values of plant design variables. It is also possible to explicitly impose the controllability constraints in the proposed optimization approach. However, given the format of the co-design problem as Eq. (1), the controllability constraints are not considered in this paper.

To solve the multisubsystem co-design problem (Eq. (1)) in a decentralized manner, both the plant and control parts are decomposed and solved. A similar multilevel framework as in Ref. [8] is adopted, as shown in Fig. 2. In the plant design part, as in Ref. [8], a dual decomposition approach [25] is used to solve for the plant design variables. In the control part, a decentralized implementation of the direct collocation method presented in Sec. 3 is employed.

In this framework, as shown in Fig. 3, the entire problem is decomposed in terms of subsystems and co-design of each subsystem. The solution steps are described on both cases when shared plant variables ysi,j exist or do not exist.

First, if there is no shared plant design variable, the decentralized implementation in Sec. 3 can be directly used. The bilevel framework is adopted from the optimality condition decomposition. At the bottom level, the co-design problem of each subsystem is solved by employing the direct collocation method. The optimizing variables are coordinated at the top level. The iterations stop when all variables converge.

When there are shared plant variables ysi,j(i) between subsystems, both complicating variables and complicating constraints exist in the converted optimization problem by the direct method. Hence, a bilevel framework incorporating the optimality condition decomposition and the dual decomposition is devised. At the bottom level, the local state, control, and plant design variables as well as a local copy of the shared plant variables are optimized in each subsystem. At the top level, the dual variables associated with the shared plant variables are updated. The iterations continue until the shared plant variable values converge.

The bilevel approach (A2) aims to improve the performance of the multilevel approach. If no shared plant design variable exists, the convergence of the approach can be obtained by the convergence of the optimality condition decomposition method. In the presence of shared physical design variables, a bilevel approach is proposed in this paper to solve problems with both complicating variables and complicating constraints in a decomposed manner.

The numerical example is slightly modified from Example 4.1 in Ref. [8]. The problem has two subsystems, and m1tom5 are the five physical plant design variables. The subsystems SS1 and SS2 are formulated as follows:

In this section, a similar engineering example in Ref. [8] is considered, as shown in Fig. 5; however, here, nonlinearity is introduced into dynamic equations. The plant design part of each subsystem consists of one mass (m), one spring (k), and one damper (c). The control part has a quadratic objective function. The state variable xi(t) = [xi1(t), xi2(t)]T denotes the displacement and velocity of the mass mi, and the control variable ui(t) is the force applied to the mass from 0 to 5 s.

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