基于S函数的simulink仿真
S函数可以用计算机语言来描述动态系统 。在控制系统设计中，S函数可以用来描述控制算法、自适应算法和模型动力学方程。
S函数中使用文本方式输入公式和方程，适合复杂动态系统的数学描述，并且在仿真过程中可以对仿真参数进行更精确的描述、
1.1 S函数简介
S函数是系统函数（system function）的简称。可以用MATLAB代码、C、C++等语言来编写S函数。
1.2 S函数的使用步骤
步骤如下：
- 创建S函数源文件
- 在动态系统的simulink模型框图中添加S-function模块，并且进行正确设置
- 在simulink模型框图中按照定义好的功能连接输入输出端口
1.3 S函数的基本功能及重要参数设定
S函数的基本功能及重要参数设定如下：
- S函数功能模块：各种功能模块完成不同的任务，这些功能模块（函数）称为仿真例程或回调函数（call - back functions），包括初始化（initialization）、导数（mdlDerivative）、输出（mdlOutput）等。
- NumContStates表示S - 函数描述的模块中连续状态的个数。
- NumDiscStates表示离散状态的个数。
- NumOutputs和NumInputs分别表示模块输出和输入的个数。
- 直接馈通（dirFeedthrough）为输入信号是否在输出端出现的标识，取值为0或1。例如，形如 y = k × u y = k×u y=k×u的系统需要输入（即直接反馈），其中， u u u是输入， k k k是增益， y y y是输出，形如等式 y = x , x ˙ = u y = x,\dot{x}=u y=x,x˙=u的系统不需要输入（即不存在直接反馈），其中， x x x是状态， u u u是输入， y y y为输出。
- NumSampleTimes为模块采样周期的个数，S函数支持多采样周期的系统。 除了sys外，还应设置系统的初始状态变量 x 0 x_0 x0、说明变量str和采样周期变量 t s t_s ts。 t s t_s ts变量为双列矩阵，其中每一行对应一个采样周期。对连续系统和单个采样周期的系统来说，该变量为 [ t 1 , t 2 ] [t_1,t_2] [t1,t2]， t 1 t_1 t1为采样周期， t 1 = − 1 t_1 = - 1 t1=−1表示继承输入信号的采样周期， t 2 t_2 t2为偏移量，一般取为0。对连续系统来说， t s t_s ts取为 [ − 1 , 0 ] [-1,0] [−1,0]。
1.4 S函数描述实例
在控制系统设计中，S函数可以用于控制器、自适应律和模型描述。
以模型 J θ ¨ = u + d ( t ) J\ddot{\theta}=u+d(t) Jθ¨=u+d(t)为例，其中， u u u为控制输入， d ( t ) d(t) d(t)为加在控制输入端的扰动，模型输出为 θ 和 θ ˙ \theta和\dot{\theta} θ和θ˙，即转动角度和角速度， J J J为转动惯量，该模型可以描述如下：
x ˙ 1 = x 2 x ˙ 2 = 1 J ( u + d ( t ) ) \begin{align*} \dot{x}_1&=x_2\\ \dot{x}_2&=\frac{1}{J}(u + d(t)) \end{align*} x˙1x˙2=x2=J1(u+d(t))
其中： x 1 = θ , x 2 = θ ˙ x_1=\theta ,x_2=\dot{\theta} x1=θ,x2=θ˙
1 首先，初始化Initialization函数
采用S函数来描述动力学方程，可选取1输人2输出系统，如果角度和角速度的初始值取零，则模型初始化参数写为[0，0]，模型初始化S函数描述如下：（见模板)
2 微分方程描述的mdlDerivative函数
该函数可用于描述微分方程并实现数值求解。在控制系统中，可以采样该函数来描述被控对象和自适应律等，并通过Simulink环境下选择数值分析方法来实现对模型的数值求解
取 J = 2 ， d ( t ) = s i n t J=2，d(t)=sint J=2，d(t)=sint，则采用S函数可以实现模型角度和角速度的求解，描述如下：
matlab
function sys=mdlDerivatives(t,x,u)
J=2;
dt=sin(t);
ut=u(1);
sys(1)=x(2);
sys(2)=1/J*(ut+dt);
sys = [dx1;dx2];
3 用于输出的mdlOutput函数
S函数的mdlOutput函数通常用于描述控制器或模型的输出。采用S函数的mdlOutput模块来描述模型角度和角速度的输出：
matlab
function sys=mdlOutputs(t,x,u)
sys(1) = x(1);
sys(2) = x(2);
最后，给出S函数模板
matlab
function [sys,x0,str,ts,simStateCompliance] = plant(t,x,u,flag,pa)
%SFUNTMPL General MATLAB S-Function Template
% With MATLAB S-functions, you can define you own ordinary differential
% equations (ODEs), discrete system equations, and/or just about
% any type of algorithm to be used within a Simulink block diagram.
%
% The general form of an MATLAB S-function syntax is:
% [SYS,X0,STR,TS,SIMSTATECOMPLIANCE] = SFUNC(T,X,U,FLAG,P1,...,Pn)
%
% What is returned by SFUNC at a given point in time, T, depends on the
% value of the FLAG, the current state vector, X, and the current
% input vector, U.
%
% FLAG RESULT DESCRIPTION
% ----- ------ --------------------------------------------
% 0 [SIZES,X0,STR,TS] Initialization, return system sizes in SYS,
% initial state in X0, state ordering strings
% in STR, and sample times in TS.
% 1 DX Return continuous state derivatives in SYS.
% 2 DS Update discrete states SYS = X(n+1)
% 3 Y Return outputs in SYS.
% 4 TNEXT Return next time hit for variable step sample
% time in SYS.
% 5 Reserved for future (root finding).
% 9 [] Termination, perform any cleanup SYS=[].
%
%
% The state vectors, X and X0 consists of continuous states followed
% by discrete states.
%
% Optional parameters, P1,...,Pn can be provided to the S-function and
% used during any FLAG operation.
%
% When SFUNC is called with FLAG = 0, the following information
% should be returned:
%
% SYS(1) = Number of continuous states.
% SYS(2) = Number of discrete states.
% SYS(3) = Number of outputs.
% SYS(4) = Number of inputs.
% Any of the first four elements in SYS can be specified
% as -1 indicating that they are dynamically sized. The
% actual length for all other flags will be equal to the
% length of the input, U.
% SYS(5) = Reserved for root finding. Must be zero.
% SYS(6) = Direct feedthrough flag (1=yes, 0=no). The s-function
% has direct feedthrough if U is used during the FLAG=3
% call. Setting this to 0 is akin to making a promise that
% U will not be used during FLAG=3. If you break the promise
% then unpredictable results will occur.
% SYS(7) = Number of sample times. This is the number of rows in TS.
%
%
% X0 = Initial state conditions or [] if no states.
%
% STR = State ordering strings which is generally specified as [].
%
% TS = An m-by-2 matrix containing the sample time
% (period, offset) information. Where m = number of sample
% times. The ordering of the sample times must be:
%
% TS = [0 0, : Continuous sample time.
% 0 1, : Continuous, but fixed in minor step
% sample time.
% PERIOD OFFSET, : Discrete sample time where
% PERIOD > 0 & OFFSET < PERIOD.
% -2 0]; : Variable step discrete sample time
% where FLAG=4 is used to get time of
% next hit.
%
% There can be more than one sample time providing
% they are ordered such that they are monotonically
% increasing. Only the needed sample times should be
% specified in TS. When specifying more than one
% sample time, you must check for sample hits explicitly by
% seeing if
% abs(round((T-OFFSET)/PERIOD) - (T-OFFSET)/PERIOD)
% is within a specified tolerance, generally 1e-8. This
% tolerance is dependent upon your model's sampling times
% and simulation time.
%
% You can also specify that the sample time of the S-function
% is inherited from the driving block. For functions which
% change during minor steps, this is done by
% specifying SYS(7) = 1 and TS = [-1 0]. For functions which
% are held during minor steps, this is done by specifying
% SYS(7) = 1 and TS = [-1 1].
%
% SIMSTATECOMPLIANCE = Specifices how to handle this block when saving and
% restoring the complete simulation state of the
% model. The allowed values are: 'DefaultSimState',
% 'HasNoSimState' or 'DisallowSimState'. If this value
% is not speficified, then the block's compliance with
% simState feature is set to 'UknownSimState'.
% Copyright 1990-2010 The MathWorks, Inc.
%
% The following outlines the general structure of an S-function.
%
switch flag,
%%%%%%%%%%%%%%%%%%
% Initialization %
%%%%%%%%%%%%%%%%%%
case 0,
[sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes;
%%%%%%%%%%%%%%%
% Derivatives %
%%%%%%%%%%%%%%%
case 1,
sys=mdlDerivatives(t,x,u,pa);
%%%%%%%%%%
% Update %
%%%%%%%%%%
case 2,
sys=mdlUpdate(t,x,u);
%%%%%%%%%%%
% Outputs %
%%%%%%%%%%%
case 3,
sys=mdlOutputs(t,x,u);
%%%%%%%%%%%%%%%%%%%%%%%
% GetTimeOfNextVarHit %
%%%%%%%%%%%%%%%%%%%%%%%
case 4,
sys=mdlGetTimeOfNextVarHit(t,x,u);
%%%%%%%%%%%%%
% Terminate %
%%%%%%%%%%%%%
case 9,
sys=mdlTerminate(t,x,u);
%%%%%%%%%%%%%%%%%%%%
% Unexpected flags %
%%%%%%%%%%%%%%%%%%%%
otherwise
DAStudio.error('Simulink:blocks:unhandledFlag', num2str(flag));
end
% end sfuntmpl
%
%=============================================================================
% mdlInitializeSizes
% Return the sizes, initial conditions, and sample times for the S-function.
%=============================================================================
%
function [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes
%
% call simsizes for a sizes structure, fill it in and convert it to a
% sizes array.
%
% Note that in this example, the values are hard coded. This is not a
% recommended practice as the characteristics of the block are typically
% defined by the S-function parameters.
%
sizes = simsizes;
sizes.NumContStates = 2;
sizes.NumDiscStates = 0;
sizes.NumOutputs = 2;
sizes.NumInputs = 1;
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1; % at least one sample time is needed
sys = simsizes(sizes);
%
% initialize the initial conditions
%
x0 = [0,0];
%
% str is always an empty matrix
%
str = [];
%
% initialize the array of sample times
%
ts = [0 0];
% Specify the block simStateCompliance. The allowed values are:
% 'UnknownSimState', < The default setting; warn and assume DefaultSimState
% 'DefaultSimState', < Same sim state as a built-in block
% 'HasNoSimState', < No sim state
% 'DisallowSimState' < Error out when saving or restoring the model sim state
simStateCompliance = 'UnknownSimState';
% end mdlInitializeSizes
%
%=============================================================================
% mdlDerivatives
% Return the derivatives for the continuous states.
%=============================================================================
%
function sys=mdlDerivatives(t,x,u,pa)
k=pa.k;
m=pa.m;
x1=x(1);
x2=x(2);
dx1=x2;
dx2=-k/m*x1^3+u/m;
sys = [dx1;dx2];
% end mdlDerivatives
%
%=============================================================================
% mdlUpdate
% Handle discrete state updates, sample time hits, and major time step
% requirements.
%=============================================================================
%
function sys=mdlUpdate(t,x,u)
sys = [];
% end mdlUpdate
%
%=============================================================================
% mdlOutputs
% Return the block outputs.
%=============================================================================
%
function sys=mdlOutputs(t,x,u)
sys = x;
% end mdlOutputs
%
%=============================================================================
% mdlGetTimeOfNextVarHit
% Return the time of the next hit for this block. Note that the result is
% absolute time. Note that this function is only used when you specify a
% variable discrete-time sample time [-2 0] in the sample time array in
% mdlInitializeSizes.
%=============================================================================
%
function sys=mdlGetTimeOfNextVarHit(t,x,u)
sampleTime = 1; % Example, set the next hit to be one second later.
sys = t + sampleTime;
% end mdlGetTimeOfNextVarHit
%
%=============================================================================
% mdlTerminate
% Perform any end of simulation tasks.
%=============================================================================
%
function sys=mdlTerminate(t,x,u)
sys = [];
% end mdlTerminate