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1、PART 2Unit 2A stability and the time responseThe stability of a continuous or discrete-time system is determined by its response to input or disturbance. Intuitively, a stable system is one that remains at rest (or in equilibrium) unless excited by an external source and returns to rest if all excit
2、ations are removed. The output will pass through a transient phase and settle down to a steady-state response that will be of the same form as, or bounded by, the input. If we apply the same input to an unstable system, the output will never settle down to a steady-state phase; it will increase in a
3、n unbounded manner, usually exponentially or with oscillation of increasing amplitude. 连续或离散系统的稳定性由其对输入或者干扰的响应决定。直观地说，如果一个系统是稳定的，则其停留在稳态（或者平衡点），除非是受到外部激励，且当外部激励去除后，输出又回到稳态点。输出经过瞬态阶段后将回到与输入有相同形式的稳态或者是在输入的附近。如果我们将同样的输入作用于不稳定的系统，其输出将不会回到稳态，而是以无界的方式增长，通常其幅值是指数增长或者振荡增长。Stability can be precisely defined
4、in terms of the impulse response of a continuous system, Kronrcker delta response of a discrete-time system, as follows: A continuous (discrete-time) system is stable if its impulse response (Kronecker response ) approaches zero as time approaches infinity.系统的稳定性可以用连续系统的脉冲响应或者离散系统的Kronrcker 响应来定义：一个
5、连续（离散）系统是稳定的，如果其脉冲响应（Kronrcker 响应）当时间趋于无穷大时趋于零。An acceptable system must at minimum satisfy the three basic criteria of stability, accuracy, and a satisfactory transient response. These three criteria are implied in the statement that an acceptable system must have a satisfactory time response to sp
6、ecified inputs and disturbances. So although we work in the Laplace and frequency domains for convenience, we must be able to relate these two domains, at least qualitatively, to the time domain.一个可接受的系统必须至少满足：稳定性、精度和满意的瞬态响应这三个指标。在陈述：“一个可接受的系统对指定输入和扰动必须有满意的时域响应”已经包含了这三个指标的含义。因此尽管我们为了方便工作在拉氏域或者频率域，我们
7、必须与时间域（至少是定性的）相联系。With the transfer function in the form of Eq.(2-2A-1), the order of the system in defined as the order of the characteristic function D(s), the highest power of s appearing in D(s) establishes the order of the system.在传递函数所在的方程(2-2A-1)中，系统的阶次定义为特征函数D(s)的阶次，因此D(s)的最高次幂决定了系统的阶次。The f
8、irst term is the forced solution, due to the input, and the second the transient solution, due to the system pole. Fig.2-2A-2 shows this transient as well as c(t). The transient is seen to be a decaying exponential, and the commonly used measured of the speed of decay is the time constant: The time
9、constant is the time in seconds for the decaying exponential transient to be reduced to e-1=0.368 of its initial value. Since when t=T, it is seen that the time constant for a simple lag is T seconds. This is, in fact, the reason a simple lag transfer function is often written in this form. The coef
10、ficient of s then immediately indicates the speed of decay, and it takes 4T seconds for the transient to decay to 1.8% of its initial value.第一项为强迫解，对应于输入；第二项为瞬态解，对应于系统的极点。 在图2-2A-2中，该瞬态解为c(t)。瞬态解看上去为指数衰减的，且通常用于衡量衰减速度的是时间常数：即指数衰减的瞬态解衰减至其初始值的36.8%所需的时间（秒数）。因为，当t=T, ，对于一阶惯性环节，时间常数是T秒。这也是为什么一阶惯性环节要写成这个形
11、式。S的系数立即给出了衰减的速度。而且，当时间为4T时， 瞬态解衰减至初始值的1.8%。B：Steady StateA control system is designed to control the dynamic behavior( the time response) of a plant subject to commands or disturbances. The designer should be fully aware, however, of the role of the steady equations and errors in the overall proces
12、s, as well as their influence on the dynamic behavior of the plant.控制系统设计就是使装置在有指令信号或者干扰时有满意的行为（时域响应）。设计者必须清楚地知道整个过程的稳态方程和误差，以及他们对装置的动态性能的影响。An accuracy of a system is a measure of how well it follows commands. It is an important performance criterion; a guidance system that cannot place spacecraft
13、on a suitable trajectory is obviously useless no matter how well-behaved its transient response.衡量系统的精度之一，就是其如何跟踪给定命令。这是一项重要的性能指标。一个导航系统如果不能将飞行器置于合适的轨迹，那么无论有多好的动态性能，都是没有用。Actual system are also subjected to undesirable inputs, such as noise in command inputs and disturbances arising from changes in
14、the plant parameters or changes in the environment in which the plant is operating. Noise inputs that enter the system with the command input require filtering techniques to remove or suppress them without affecting the command input itself. We shall limit our discussion to disturbance inputs which
15、enter the system at the plant rather than at the controller.实际系统总是容易受到不希望的输入干扰，例如， 命令输入中的噪声以及由于参数改变在被控对象中产生的干扰或者被控对象工作环境变化产生的干扰。随着命令输入进入系统的噪声输入需要滤波器进行驱除或者抑制并不对输入信号产生影响。我们将限于讨论通过被控对象进行系统的噪声而不讨论通过控制器进入系统的噪声。It is often difficult to minimize both components of the error simultaneously. Obviously, it is
16、 necessary to have some knowledge as to the nature of probable disturbance inputs. Both error terms of Eq.(2-2B-7) can be set equal to zero by introducing an integrator into the controller. This additional integrator increases the type of the system ( from 1 to 2, for example) , thus eliminating the velocity error, and by being introduced ahead of the point of entry of the disturbance into the system, eliminates the steady-state err
