application of the control variables and description of the process by the method of SDRE and PD controls will be detailed in the next section.2.2. SDRE Control Design. The technique of SDRE control by nonlinear feedback has been applied in many nonlinear problems, estimating the states satisfying the aiming functions[25–28]. The SDRE controller uses the LQR (Linear Quadratic Regulator) method to find the suboptimal gain in function of State-Dependent Riccati Equations, recalculating them along with the application on nonlinear systems of (32)[11].

Therefore, the equation of motion of the crank-connecting-rod of (31) can be rewritten in matrix form as given by whereThe quadratic performance measure for the feedback control (Ufe) problem is given by   is the desired orbit, 𝑥∗

2 is the desired

velocity, and Q(x) and R(x) are positive definite matrices.The minimization of functional (34) implies the minimization of the system deviation (32) of the desired state and of the applied feedback control (Ufe). Assuming full-state feedback (Ufe), the control law is given by [29]

where P(X) is the solution of the Riccati equation:

Another important factor to consider is that the matrix A(X) cannot violate the controllability of the system. Crank connecting-rod system (32) is controllable if the rank of the matrix M is 2:

The SDRE technique to obtain a suboptimal solution for dynamic control problem has the following procedure [29]:

(1) Define the state-space model with the state-dependent coefficients A(X) and B(X).

(2) Define the initial condition x(0) = x0 so that the rank of M is 𝑛 and choose the coefficients of weight matrices Q and R, where the matrices Q and R determine the relative importance of parameter error and energy expenditure.

(3) Solve P(X) for the Riccati equation which will be described in the sequence of this method for the states as a function of time.

(4) Define e𝑆 the objective functions for the calculation of the parameter errors.

(5) Calculate the input signal Ufe.

(6) Integrate the equation obtained in step (1) and update the state of the system as a function of time with the results.

这是一份开放存取的文章，在知识共享署名许可下发布，允许在任何媒介中无限制地使用、分发和复制，只要原始的作品被正确引用。汽车的可再生能源一直是世界各国研究的动力。化石燃料的减少和城市污染的增加，都需要更高效、更清洁的能源。通过这种方式，这项工作将展示应用于自行车的压缩空气发动机。发动机由两个气缸组成，通过曲柄连杆机构连接到自行车车轮。为了控制自行车的速度，实现了由两种控制组成的控制策略:反馈和前馈控制。对于反馈控制，考虑了状态依赖的Riccati方程(SDRE)控制和比例微分(PD)控制，考虑了速度自行车变化的3例:10 km/h, 20 km/h, 30 km/h。通过拉格朗日能量法得到了系统的运动方程。为了分析系统的动力学特性，进行了数值模拟控制器的效率。

1. 介绍

连杆

在往复式活塞发动机中，连杆或连杆连接活塞或曲轴。这种连杆是在1174-1200年间发明的，当时一位名叫al-Jazari的穆斯林发明家、工程师和工匠建造了5台机器。土耳其的阿图克王朝的国王泵水-其中一个包括连接杆。通过将曲轴连接到连杆上，将旋转运动传递到往复运动中，这在《精巧机械装置的知识手册》中描述过。双作用往复活塞泵是第一个提供自动运动的机器，但是它的机械装置和其他装置，也将有助于发动工业。

复合连杆

多缸多缸发动机，例如v - 12型布置，在曲轴的有限长度上，在许多连杆轴颈上具有很少的空间，这是难以解决的问题，而这个问题的后果就是这个发动机会被视为无效。在大多数道路汽车发动机中，最简单的解决方案，，是使用简单的连杆，这就要求杆轴承要变窄，增加轴承负荷和在高性能发动机中失败的风险。这也意味着相反的气缸不是完全一致的。在某些类型的发动机中，主杆带有一个或多个螺栓。在其他气缸上的从动连杆的小的末端，径向引擎通常在同一气缸内具有用于一个气缸和所有其它气缸的副杆的主杆。某些V型发动机的设计使用一个主/从杆对每一对相反的气缸，缺点是辅助杆的行程比主杆稍短，这增加了V型发动机的振动。高性能航空发动机的通常解决方案是“叉”连杆，一根连杆在大一端分裂成两根，另一根则变薄。据《华尔街日报》报道，罗尔斯•罗伊斯•梅林使用了这种风格。