# Neural-network-based decentralized control of continuous-time nonlinear interconnected systems with unknown dynamics

**Neural-network-based decentralized control of continuous-time nonlinear interconnected systems with unknown dynamics**

**– Math and Optimal Control**

**Problem formulation**

Consider a continuous-time nonlinear large-scale system *∑ *composed of N interconnected subsystems described by

(1)

where

*x _{i}*(

*t*) ∈

*R*: state.

^{n}^{i }The overall state of the large-scale system *∑ *is denoted by

*u _{i }*[

*x*(

_{i}*t*) ] ∈

*R*: control input vector of the ith subsystem.

^{m}^{i }*f _{i}* : continuous nonlinear internal dynamics function.

*f*(0)=0.

_{i}*g*_{i}[ x_{i}(t) ] : input gain function

*Z _{i}* [

*x*(

*t*) ] : interconnected term for the ith subsystem.

The ith isolated subsystem

(2)

**Decentralized control law**

**Optimal control**

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**Reinforcement Learning and Optimal Control Methods for Uncertain Nonlinear Systems**

*Page 27-29 2.3 Infinite Horizon Optimal Control Problem* is the same as Definition 1.

Notation:

: state.

: control input.

(2-5)

Cost function for the system Eq. 2-5:

(2-6)

where t : initial time.

*r*(x,u) ∈* R *: immediate or local cost for the state and control.

(2-7)

where Q(x) ∈* R *continuously differentiable and positive definite.

*R* ∈* R ^{m x m}* : positive-definite symmetric matrix.

Optimal value function:

(2-8)

where

: set of admissible controls.

Bellman’s principle of optimally can be used to derive the following optimality condition

(**2-9**)

which is a nonlinear partial differential equation (PDE), also called the **HJB equation**.

Optimal control: (using convex local cost in Eqs. 2-7 and 2-9.)

(2-10)

For the control-affine dynamics of the form

(2-11)

Eq. 2-10 -> in terms of the system state

(2-12)

The **HJB** in **Eq. 2-9** can be rewritten in terms of the optimal value function by substituting for the local cost in Eq. 2-7, the system in Eq. 2-11 and the optimal control in Eq. 2-12, as

(**2-13**)

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