# Decentralized Optimal Control of Distributed Interdependent Automata With Priority Structure

**Decentralized Optimal Control of Distributed Interdependent Automata With Priority Structure**

**Data Flowchart**

**Notation**

: **subsystem model**, the **plant *** P ^{i }*, deterministic finite-state automaton.

(1)

(2)

(3)

(4) : **P **^{i }^{ }can be transitioned from state into state if the *input l* is applied.

(5)

It encodes with that the transition is possible with at least one input. (6)

To this one-step reachability, * R^{i}* formalizes the

**possibility**of transferring

*between an arbitrary pair of states. (6) models that state*

**P**^{i }**is reachable from state**

*j***by at least one input sequence in at most**

*h***state**

*n*_{i}**transitions**.

for any transition specified forthrough the state transition function (or the set of state transition matrices ). Possible interpretations of such transition costs are the time, the control effort, and/or the energy required to steerP^{i }by the use of the control input , i.e., can encode state and control costs.P^{i}

.

.

(7)

over all values.

(8)

: The minimal costs for transferring the subsystem from the state .

**Problem 1: Subsystem ***independent* of other subsystems

*independent*of other subsystems

The task is to compute astate-feedbackcontroller,

which realizes for any arbitrary initializationan input sequencethat leads to an admissible run

- The final state is the goal
- The state-feedback control law has the structure: (9)
- The costs of are minimal over all admissible runs to transfer
from to : (10)**P**^{i}

(9): , in order to **trigger** the next state **transition**. The selection of ** K^{i}** according to the solution of (10) produces the , and establishes an

**optimal controller**for

*with*

**P**^{i }**Synthesis Algorithm for Independent Subsystems**

Compute **controller matrix**, and the part of **costs** referring to the **goal state**.

**Control of Distributed Systems with Linear Structure**

.

**Task of Distributed Controller Synthesis**

* Assumption 3 *: Any subsystem is completely controllable: .

** Proposition 2 **: Let be a pair of 2 connected subsystems for which an admissible run is a sequence of state pair according to Definition 2 with and . The structure is completely controllable if

**and**

*P*^{1}**on their own are completely controllable according to Assumption 3.**

*P*^{2}* Proof *: Since the

**transition**of

*are independent of the*

**P**^{2 }**current state**of

**, and since subsystem**

*P*^{1}

**P**^{2}**is completely controllable, a sequence of inputs exists to transfer**

*from an arbitrary initial state .*

**P**^{2 }Thus, * P^{2 }*is able to deliver any arbitrary output sequence (and thus admissible run ) to subsystem

**, i.e., any condition formulated for**

*P*^{1 }**in terms of the dependence matrix is satisfiable by**

*P*^{1 }*. Since*

**P**^{2 }**itself is completely controllable as well, a sequence of input exists which transfers**

*P*^{1 }**into an arbitrary**

*P*^{1 }**goal state**.

**Problem 2: Two subsystems **

For 2 subsystems , let the **goal states** be defined. The control task is to compute 2 local feedback control laws, which generate for any initialization , the input sequences , such that the following hold.

- The admissible runs with and with .
- follow from controllers of the following type:
**(11)**with vectors , and matrix as in problem 1, and - The global path costs are minimal
**(12)**

Thus, the solution is targeted to provide local controllers C^{1} and C^{2} for P^{1} and P^{2}, such that the latter are driven from an arbitrarily chosen initial state into the respective local goal state, while the sum of the transfer costs for both control loops is as small as possible.

**1. **when C^{1} receives the information from P^{1} that state is reached ,

**2. **C^{1} sends the request to C^{2} that P^{2} has to reach as a temporary goal state. This state is encoded in K^{1} in order to realize a cost-optimal path of P^{1} into its goal state .

**3. **Then, C^{2 }realizes a path of P^{2} into . If the path comprises more than one transition, the pair (P^{1},C^{1}) waits in state until P^{2} has reached ( the index **k’** in Fig. 4 is meant to indicate that (P^{2},C^{2}) evolve, while (P^{1},C^{1}) wait in step k).

**4. **When P^{2} attains , C^{2} communicates to C^{1} that the requested state is reached .

**5. **Eventually, the control input supplied by C^{1} together with send by P^{2} triggers the state transition in P^{1} .

**Control of Distributed Systems with Tree Structure**