线性系统

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In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.

在系统论中,线性系统是基于线性算子的系统的数学模型。

Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.

Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.

线性系统通常具有比非线性系统简单得多的特征和性质。

As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be

As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be

线性系统作为一种数学抽象或理想化的概念,在自动控制理论、信号处理和通信领域有着重要的应用。例如,无线通信系统的传播介质通常是

modeled by linear systems.

modeled by linear systems.

由线性系统模拟。


Definition

A general deterministic system can be described by an operator, [math]\displaystyle{ H }[/math], that maps an input, [math]\displaystyle{ x(t) }[/math], as a function of [math]\displaystyle{ t }[/math] to an output, [math]\displaystyle{ y(t) }[/math], a type of black box description. Linear systems satisfy the property of superposition. Given two valid inputs

A general deterministic system can be described by an operator, [math]\displaystyle{ H }[/math], that maps an input, [math]\displaystyle{ x(t) }[/math], as a function of [math]\displaystyle{ t }[/math] to an output, [math]\displaystyle{ y(t) }[/math], a type of black box description. Linear systems satisfy the property of superposition. Given two valid inputs

一个一般的确定性模型可以由一个操作符来描述,它把一个输入映射成一个输出,< math > x (t) </math > ,< math > y (t) </math > ,一种黑盒子的描述。线性系统满足叠加性质。给定两个有效输入

[math]\displaystyle{ x_1(t) \, }[/math]

[math]\displaystyle{ x_1(t) \, }[/math]

X _ 1(t) ,</math >

[math]\displaystyle{ x_2(t) \, }[/math]

[math]\displaystyle{ x_2(t) \, }[/math]

[ math > x _ 2(t) ,[ math ]

as well as their respective outputs

as well as their respective outputs

以及他们各自的产出

[math]\displaystyle{ y_1(t) = H \left \{ x_1(t) \right \} }[/math]

[math]\displaystyle{ y_1(t) = H \left \{ x_1(t) \right \} }[/math]

< math > y _ 1(t) = h left { x _ 1(t) right } </math >

[math]\displaystyle{ y_2(t) = H \left \{ x_2(t) \right \} }[/math]

[math]\displaystyle{ y_2(t) = H \left \{ x_2(t) \right \} }[/math]

[ math > y _ 2(t) = h left { x _ 2(t) right }

then a linear system must satisfy

then a linear system must satisfy

那么线性系统必须满足

[math]\displaystyle{ \alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} }[/math]

[math]\displaystyle{ \alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} }[/math]

< math > alpha y _ 1(t) + beta y _ 2(t) = h left { alpha x _ 1(t) + beta x _ 2(t) right } </math >

for any scalar values [math]\displaystyle{ \alpha \, }[/math] and [math]\displaystyle{ \beta \, }[/math].

for any scalar values [math]\displaystyle{ \alpha \, }[/math] and [math]\displaystyle{ \beta \, }[/math].

对于任何标量值 < math > alpha,</math > 和 < math > beta,</math > 。


< ! -- 插入描述叠加和缩放属性的图片 -- >


The system is then defined by the equation [math]\displaystyle{ H(x(t)) = y(t) }[/math], where [math]\displaystyle{ y(t) }[/math] is some arbitrary function of time, and [math]\displaystyle{ x(t) }[/math] is the system state. Given [math]\displaystyle{ y(t) }[/math] and [math]\displaystyle{ H }[/math], the system can be solved for [math]\displaystyle{ x(t) }[/math]. For example, a simple harmonic oscillator obeys the differential equation:

The system is then defined by the equation [math]\displaystyle{ H(x(t)) = y(t) }[/math], where [math]\displaystyle{ y(t) }[/math] is some arbitrary function of time, and [math]\displaystyle{ x(t) }[/math] is the system state. Given [math]\displaystyle{ y(t) }[/math] and [math]\displaystyle{ H }[/math], the system can be solved for [math]\displaystyle{ x(t) }[/math]. For example, a simple harmonic oscillator obeys the differential equation:

系统由方程式 < math > h (x (t)) = y (t) </math > 定义,其中 < math > y (t) </math > 是时间的任意函数,< math > x (t) </math > 是系统状态。如果给定“数学”和“数学” ,系统可以解出“数学”和“数学”。例如,一个简单的谐振子服从微分方程:

[math]\displaystyle{ m \frac{d^2(x)}{dt^2} = -kx }[/math].

[math]\displaystyle{ m \frac{d^2(x)}{dt^2} = -kx }[/math].

[ math > m frac { d ^ 2(x)}{ dt ^ 2} =-kx.


If

If

如果

[math]\displaystyle{ H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t) }[/math],

[math]\displaystyle{ H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t) }[/math],

< math > h (x (t)) = m frac { d ^ 2(x (t))}{ dt ^ 2} + kx (t) </math > ,

then [math]\displaystyle{ H }[/math] is a linear operator. Letting [math]\displaystyle{ y(t) = 0 }[/math], we can rewrite the differential equation as [math]\displaystyle{ H(x(t)) = y(t) }[/math], which shows that a simple harmonic oscillator is a linear system.

then [math]\displaystyle{ H }[/math] is a linear operator. Letting [math]\displaystyle{ y(t) = 0 }[/math], we can rewrite the differential equation as [math]\displaystyle{ H(x(t)) = y(t) }[/math], which shows that a simple harmonic oscillator is a linear system.

那么,h </math > 是一个线性算子。让 y (t) = 0 </math > ,我们可以将微分方程重写为 < math > h (x (t)) = y (t) </math > ,这表明一个简单的谐振子是一个线性系统。


The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation.

The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation.

受复杂输入影响的结果系统的行为可以描述为对较简单输入的响应之和。在非线性系统中,不存在这种关系。

This mathematical property makes the solution of modelling equations simpler than many nonlinear systems.

This mathematical property makes the solution of modelling equations simpler than many nonlinear systems.

这种数学性质使得建模方程的求解比许多非线性系统简单。

For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function [math]\displaystyle{ x(t) }[/math] in terms of unit impulses or frequency components.

For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function [math]\displaystyle{ x(t) }[/math] in terms of unit impulses or frequency components.

对于时不变系统,这是脉冲响应或频率响应方法的基础(见线性时不变系统理论) ,这些方法用单位脉冲或频率分量来描述一个一般的输入函数。


Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

线性时不变系统的典型微分方程很好地适用于在连续情况下使用拉普拉斯变换变换进行分析,在离散情况下使用 z 变换进行分析(特别是在计算机实现中)。


Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

另一种观点认为,线性系统的解决方案包括一系列函数,这些函数在几何意义上起着向量的作用。


A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

线性模型的一个常见用途是通过线性化来描述非线性。这通常是为了数学上的便利。


Time-varying impulse response

The time-varying impulse response h(t2,t1) of a linear system is defined as the response of the system at time t = t2 to a single impulse applied at time t = t1. In other words, if the input x(t) to a linear system is

The time-varying impulse response h(t2,t1) of a linear system is defined as the response of the system at time t = t2 to a single impulse applied at time t = t1. In other words, if the input x(t) to a linear system is

线性系统的时变脉冲响应 h (t < sub > 2 ,t < sub > 1 )定义为系统在 t = t < sub > 2 时对 t = t < sub > 1 时单脉冲的响应。换句话说,如果线性系统的输入 x (t)是


[math]\displaystyle{ x(t) = \delta(t-t_1) \, }[/math]

[math]\displaystyle{ x(t) = \delta(t-t_1) \, }[/math]

X (t) = delta (t-t _ 1) ,</math >


where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

其中 δ (t)表示狄拉克δ函数,系统的相应响应 y (t)是


[math]\displaystyle{ y(t) |_{t=t_2} = h(t_2,t_1) \, }[/math]

[math]\displaystyle{ y(t) |_{t=t_2} = h(t_2,t_1) \, }[/math]

[ math > y (t) | { t = t2} = h (t2,t1) ,</math >


then the function h(t2,t1) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

then the function h(t2,t1) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

那么函数 h (t < sub > 2 ,t < sub > 1 )就是系统的时变脉冲响应。由于系统在输入之前不能作出反应,因此必须满足以下因果关系条件:


[math]\displaystyle{ h(t_2,t_1)=0, t_2\lt t_1 }[/math]

[math]\displaystyle{ h(t_2,t_1)=0, t_2\lt t_1 }[/math]

(t _ 2,t _ 1) = 0,t _ 2 < t _ 1 </math >


The convolution integral

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:

任何一般连续时间线性系统的输出都与一个积分的输入有关,由于存在因果条件,这个积分可以写在一个双无穷大的范围内:


[math]\displaystyle{ y(t) = \int_{-\infty}^{t} h(t,t') x(t')dt' = \int_{-\infty}^{\infty} h(t,t') x(t') dt' }[/math]

[math]\displaystyle{ y(t) = \int_{-\infty}^{t} h(t,t') x(t')dt' = \int_{-\infty}^{\infty} h(t,t') x(t') dt' }[/math]

< math > y (t) = int _ {-infty } ^ { h (t,t’) x (t’) dt’ = int _ {-infty } ^ { infty } h (t,t’) x (t’) dt’ </math >


If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h() is a function only of the time difference τ = t-t' which is zero for τ<0 (namely t<t'). By redefinition of h() it is then possible to write the input-output relation equivalently in any of the ways,

If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h() is a function only of the time difference τ = t-t' which is zero for τ<0 (namely t<t'). By redefinition of h() it is then possible to write the input-output relation equivalently in any of the ways,

如果系统的性质不依赖于它运行的时间,那么可以说它是时不变的,h ()仅仅是 τ < 0(即 t < t’)时间差 t = t-t’的函数。通过重新定义 h () ,就可以用任何一种方法等价地写出输入输出关系,


[math]\displaystyle{ y(t) = \int_{-\infty}^{t} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d \tau = \int_{0}^{\infty} h(\tau) x(t-\tau) d \tau }[/math]

[math]\displaystyle{ y(t) = \int_{-\infty}^{t} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d \tau = \int_{0}^{\infty} h(\tau) x(t-\tau) d \tau }[/math]

< math > y (t) = int _ {-infty } h (t-t’) x (t’) dt’ = int _ {-infty } h (t-t’) x (t’) dt’ = int _ {-infty } h (t’) x (t’) dt’ = int _ {-infty } h (t) x (t-tau) d tau = _ {0}{ infty } h (tau) x (t-tau) d tau </math >


Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

线性时不变系统最常用的拥有属性是脉冲响应函数的拉普拉斯变换,即传递函数,它是:


[math]\displaystyle{ H(s) =\int_0^\infty h(t) e^{-st}\, dt. }[/math]

[math]\displaystyle{ H(s) =\int_0^\infty h(t) e^{-st}\, dt. }[/math]

[math]\displaystyle{ H(s) =\int_0^\infty h(t) e^{-st}\, dt. }[/math]


In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function:

In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function:

在应用中,这通常是 s 的有理代数函数因为 h (t)对于负 t 是零,积分可以等价地写在双无穷范围上,s = iω 遵循频率响应函数的公式:


[math]\displaystyle{ H(i\omega) = \int_{-\infty}^{\infty} h(t) e^{-i\omega t} dt }[/math]

[math]\displaystyle{ H(i\omega) = \int_{-\infty}^{\infty} h(t) e^{-i\omega t} dt }[/math]

< math > h (i omega) = int _ {-infty } ^ { infty } h (t) e ^ {-i omega t } dt </math >


Discrete time systems

The output of any discrete time linear system is related to the input by the time-varying convolution sum:

The output of any discrete time linear system is related to the input by the time-varying convolution sum:

任何离散时间线性系统的输出都与时变卷积和的输入有关:


[math]\displaystyle{ y[n] = \sum_{m =-\infty}^{n} { h[n,m] x[m] } = \sum_{m =-\infty}^{\infty} { h[n,m] x[m] } }[/math]

[math]\displaystyle{ y[n] = \sum_{m =-\infty}^{n} { h[n,m] x[m] } = \sum_{m =-\infty}^{\infty} { h[n,m] x[m] } }[/math]

< math > y [ n ] = sum _ { m =-infty } ^ { n }{ h [ n,m ] x [ m ]} = sum _ { m =-infty } ^ { infty }{ h [ n,m ] x [ m ]} </math >


or equivalently for a time-invariant system on redefining h(),

or equivalently for a time-invariant system on redefining h(),

或者等价于重新定义 h ()的时不变系统,


[math]\displaystyle{ y[n] = \sum_{k =0}^{\infty} { h[k] x[n-k] } = \sum_{k =-\infty}^{\infty} { h[k] x[n-k] } }[/math]

[math]\displaystyle{ y[n] = \sum_{k =0}^{\infty} { h[k] x[n-k] } = \sum_{k =-\infty}^{\infty} { h[k] x[n-k] } }[/math]

< math > y [ n ] = sum _ { k = 0} ^ { infty }{ h [ k ] x [ n-k ]} = sum _ { k =-infty } ^ { infty }{ h [ k ] x [ n-k ]}} </math >


where

where

在哪里


[math]\displaystyle{ k = n-m \, }[/math]

[math]\displaystyle{ k = n-m \, }[/math]

[ math ] k = n-m,[ math ]


represents the lag time between the stimulus at time m and the response at time n.

represents the lag time between the stimulus at time m and the response at time n.

表示时间 m 的刺激和时间 n 的反应之间的滞后时间。


See also

Category:Systems theory

范畴: 系统论

Category:Dynamical systems

类别: 动力系统

Category:Mathematical modeling

类别: 数学建模

Category:Concepts in physics

分类: 物理概念


Notes


References

This page was moved from wikipedia:en:Linear system. Its edit history can be viewed at 线性系统/edithistory