Overall of the MPC

模型預(yù)測(cè)控制（Model Predictive Control）指一類算法左权，周期性基于當(dāng)幀測(cè)量信息在線求解一個(gè)有限時(shí)間開環(huán)優(yōu)化問題，并將結(jié)果的前部分控制序列作用于被控對(duì)象褐墅。根據(jù)所用模型不同拆檬，分為動(dòng)態(tài)矩陣控制（DMC），模型算法控制（MAC）妥凳、廣義預(yù)測(cè)控制（GPC）竟贯。在智能駕駛方向，重點(diǎn)在于基于狀態(tài)空間模型的模型預(yù)測(cè)控制逝钥。

如下圖屑那，MPC控制器在智能駕駛的應(yīng)用

MPC應(yīng)用

如下為Udacity對(duì)于Model、Cost Function艘款、Constraints的Setup

MPC Setup

如何得到最優(yōu)解持际？
Udacity給出的方案是基于CppAD的ipopt控制器（第三方庫）進(jìn)行二次規(guī)劃求解最優(yōu)解。

Vehicel Models

Kinematic Models

Kinematic models are simplifications of dynamic models that ignore tire forces, gravity, and mass.

Dynamic Models

Vehicle Model

各State的解釋如下

Explanation of x y psi v

關(guān)于CTE（車道偏離）和eψ（橫擺角偏離）的解釋如下：

Cross Track Error

Orientation Error

More Studying：
Dynamic Model Forces
Tire Slip Angle滑移角: Alpha = arctan(wheel lateral velocity / wheel longitudinal velocity)
Tire Slip Ratio滑移率: s = Radius of Wheel * Angular Velocity / Longitudinal Velocity
Tire Models

Additional resources: Kinematic and Dynamic Vehicle Models for Autonomous Driving Control Design presents a comparison between a kinematic and dynamic model.

Cost Function

包括CTE（車道偏離）和eψ（橫擺角偏離）等哗咆。
我認(rèn)為實(shí)際應(yīng)用過程中蜘欲，可以根據(jù)規(guī)劃trajectory符合度和平順性等評(píng)估，包括車速偏離晌柬、碰撞cost姥份、縱向加速度、橫向加速度年碘、向心加速度等澈歉。

Constraints

包括Acceleration加速度和Yaw Rate橫擺角加速度的極限約束。

Length and Duration 預(yù)測(cè)Step數(shù)和步長(zhǎng)

MPC預(yù)測(cè)的control inputs [δ,a]僅限于一定范圍的時(shí)間盛泡，包括N = Number of Timesteps 和 dt = Timestep Duration闷祥。
期望N越大越好，dt越小越好。

Latency系統(tǒng)執(zhí)行延遲

In a real car, an actuation command won't execute instantly - there will be a delay as the command propagates through the system. A realistic delay might be on the order of 100 milliseconds.

This is a problem called "latency", and it's a difficult challenge for some controllers - like a PID controller - to overcome. But a Model Predictive Controller can adapt quite well because we can model this latency in the system.

最優(yōu)解求解器

以上闡述了智駕MPC應(yīng)用的整體思路和架構(gòu)凯砍。但是到算法層面箱硕，如何得到最優(yōu)解是最核心的。當(dāng)然對(duì)于工程應(yīng)用悟衩，可以選擇第三方最優(yōu)化求解器剧罩，當(dāng)前這是一個(gè)非線性規(guī)劃問題

Ipopt

關(guān)于Cppad::ipopt算法核心可見Interior Point OPTimizer 內(nèi)點(diǎn)法
Ipopt is the tool we'll be using to optimize the control inputs [δ1?,a1?,…,δN?1?,aN?1?]. It's able to find locally optimal values (non-linear problem!) while keeping the constraints set directly to the actuators and the constraints defined by the vehicle model. Ipopt requires we give it the jacobians and hessians directly - it does not compute them for us. Hence, we need to either manually compute them or have a library do this for us. Luckily, there is a library called CppAD which does exactly this.

CppAD

CppAD is a library we'll use for automatic differentiation. By using CppAD we don't have to manually compute derivatives, which is tedious and prone to error.
Given a C++ algorithm that computes function values, CppAD generates an algorithm that computes corresponding derivative values (of arbitrary order using either forward or reverse mode).
我把CppAD理解為基于Ipopt的應(yīng)用。

More Learning:
Use Ipopt to Solve a Nonlinear Programming Problem
Nonlinear Programming Using CppAD and Ipopt: Example and Test
IPOPT和CppAD簡(jiǎn)介

參考文獻(xiàn)：
1.Udacity無人駕駛課程
2.https://blog.csdn.net/qq_42258099/article/details/95353986