Numerical method for solving optimal control problem (0) -- Definition

Basic description

This article gives a complete description of the optimal control problem .

The optimal control problem can be briefly described as ： For a controlled system , Under the constraint conditions , Seek the optimal control quantity to minimize the performance index .

The mathematical description is ： Find control variables $\mathbf{u}(t)\in {\mathbb{R}}^m$, Make the performance index

$$J=\Phi (\mathbf{x}(t_0),t_0,\mathbf{x}(t_f),t_f)+{\int }_{t_0}^{t_f}L(\mathbf{x}(t),\mathbf{u}(t),d)\text{d}t$$

Minimum .

The state variables and control variables satisfy the following constraints ：

$$\begin{array}{cc}& \dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t),t)t\in [t_0,t_f],\\ & \varphi (\mathbf{x}(t_0),t_0,\mathbf{x}(t_f),t_f)=0,\\ & \mathbf{C}(\mathbf{x}(t),\mathbf{u}(t),t)\le 0.\end{array}$$

In style ,$\mathbf{x}(t)\in {\mathbb{R}}^n$ Is the state variable ,$\mathbf{u}(t)\in {\mathbb{R}}^m$ Is the control variable ,$t_0$ Is the initial time ,$t_f$ Is the terminal time .

The boundary condition of the state variable satisfies ：

$$\mathbf{x}\in X\subset {\mathbb{R}}^n,X=\left\{x\in {\mathbb{R}}^n:x_{lower}\le x\le x_{upper}\right\}$$

$x_{lower}$ Is the lower bound of the state variable ,$x_{upper}$ Is the upper bound of the state variable .

The boundary condition of the control variable satisfies ：

$$\mathbf{u}\in U\subset {\mathbb{R}}^m,U=\left\{u\in {\mathbb{R}}^m:u_{lower}\le u\le u_{upper}\right\}$$

$u_{lower}$ Is the lower bound of the control variable ,$u_{upper}$ Is the upper bound of the control variable .

Upper form $\Phi ,L,\mathbf{f},\varphi ,\mathbf{C}$ Defined as ：

Φ :   R n × R × R n × R → R , L :   R n × R m × R → R , f :   R n × R m × R → R n , ϕ :   R n × R × R n × R → R q , f :   R n × R m × R → R c , \begin{aligned} &\mathit{\Phi}: \ \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}, \\ &L: \ \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R} \rightarrow \mathbb{R}, \\ &\boldsymbol{f}: \ \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R} \rightarrow \mathbb{R}^n, \\ &\phi: \ \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^q, \\ &\boldsymbol{f}: \ \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R} \rightarrow \mathbb{R}^c, \\ \end{aligned} ​Φ: Rn×R×Rn×R→R,L: Rn×Rm×R→R,f: Rn×Rm×R→Rn,ϕ: Rn×R×Rn×R→Rq,f: Rn×Rm×R→Rc,​

Optimal control , The control quantity changes in time sequence , The solution result is several curves . After the control curve is determined , The state curve can be determined according to the differential dynamics system .

Part of the

The above optimal control problem generally consists of four parts , Respectively ：

• Performance indicators ;
• Control system differential equation constraints ;
• Boundary constraints ;
• Path Constraint .

Performance indicators

The performance index is the objective function in the optimization problem , However, in the field of optimal control, we call it performance index . Performance index is an important symbol to measure the quality of control system , There are generally three forms , Namely ：

• Mayer Type performance index ;
• Lagrange Type performance index ;
• Bolza Type performance index .

Mayer Type performance index

Also known as constant performance index , Only consider the state variables of the control system at the terminal time point 、 Control variables 、 Indicators of time and its composite relationship , Such as the time it takes for the aircraft to move to the specified position （ Terminal time ） etc. . The mathematical description is ：

$$J=\Phi (\mathbf{x}(t_0),t_0,\mathbf{x}(t_f),t_f).$$

Lagrange Type performance index

Also known as integral performance index , Only emphasize the requirements for the whole control process , This indicator includes the state variables in the whole time domain 、 Integral of control variables and their compound Relations , It can represent the energy consumption of the system , Such as the amount of heat consumption caused by the control process . The mathematical description is ：

$$J={\int }_{t_0}^{t_f}L(\mathbf{x}(t),\mathbf{u}(t),d)\text{d}t.$$

Bolza Type performance index

Also known as composite performance index , yes Mayer The type and Lagrange Type combination , It not only emphasizes the system state at the time of the terminal , It also emphasizes the requirements for the control system process . This form can be transformed into the above two forms under certain conditions , Therefore, when describing the performance index of general optimal control problems Bolza Type performance index . The mathematical description is ：

$$J=\Phi (\mathbf{x}(t_0),t_0,\mathbf{x}(t_f),t_f)+{\int }_{t_0}^{t_f}L(\mathbf{x}(t),\mathbf{u}(t),d)\text{d}t.$$

Control system differential equation constraints

Optimization problems contain many constraints , The optimal control problem is also a special optimization problem , Its special feature is that the constraint conditions have differential equations .

Any control system needs to use differential equations to describe the motion process , For example, the aircraft is under the action of gravity and thrust , Combined with its own quality changes and other characteristics , Establish a dynamic differential equation that can describe its motion law ; The robot manipulator is subjected to torque , Combine your arm length 、 quality 、 Joint and other characteristics , Establish a dynamic differential equation that can describe its motion law . The above equations can be used as differential algebraic equations （Differential Algebraic Equation,DAE） describe , by ：

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t),t)t\in [t_0,t_f].$$

Boundary constraints

It is often necessary to give the initial state or end state in the control system , Such as the height of the rocket when it was just launched 、 Speed, etc （ The initial state ）, You need to specify the height of the rocket at the end 、 Speed, etc （ That is, the end state ）. In the optimal control problem , The above state at a certain time point is called boundary constraint . The mathematical description is ：

$$\varphi (\mathbf{x}(t_0),t_0,\mathbf{x}(t_f),t_f)=0.$$

Path Constraint

The constraints that the control system must meet in the whole time period are called path constraints .

The difference between path constraints and boundary constraints is , Path constraints occur in the entire time period , Boundary constraints occur at a specific point in time .

Common path constraints include ：

• The state variable is on the top of the whole control process 、 Lower limit , Such as aircraft location 、 Speed etc. ;
• The control variable is on the top of the whole control process 、 Lower limit , Such as the output power of the motor 、 Moment, etc ;
• On the function composed of state variables and control variables 、 Lower limit , For example, aircraft or robots need to ensure that they cannot pass through certain specific areas .

The mathematical description of path constraints is ：

$$\mathbf{C}(\mathbf{x}(t),\mathbf{u}(t),t)\le 0.$$

summary

thus , So as to give performance indicators 、 Control system differential equation constraints 、 Mathematical description of boundary constraints and path constraints . The above four parts completely define the optimal control problem .

Solve the optimal control problem , That is to solve the optimization problem that minimizes the performance index under the above three kinds of constraints .