# AffineStateSpaceModel

AffineStateSpaceModel[{a,b,c,d},x]

represents the affine state-space model , .

gives an affine state-space model corresponding to the system model sys.

AffineStateSpaceModel[eqns,{{x1,x10},}{{u1,u10},},{g1,},t]

gives the affine state-space model obtained by Taylor input linearization about the dependent variable xi at xi0 and input uj at uj0 of the differential equations eqns with outputs gi and independent variable t.

# Details and Options

• AffineStateSpaceModel is also known as an input linear model.
• AffineStateSpaceModel can represent any system where the control input occurs affinely, but still allows for advanced analysis and control design.
• The following short input forms can be used:
•  AffineStateSpaceModel[{a,b,c},x] output given by AffineStateSpaceModel[{a,b},x] output given by
• AffineStateSpaceModel[{a,b,},x,u,y,t] explicitly specifies the input variables u, output variables y, and independent variable t.
• AffineStateSpaceModel allows for operating values for the states x and inputs u.
• AffineStateSpaceModel[,{{x1,x10},},{{u1,u10},},] is used to indicate the operating values for the system. »
• In the following systems can be converted:
•  NonlinearStateSpaceModel approximate Taylor conversion StateSpaceModel exact conversion TransferFunctionModel exact conversion
• A system of ODEs with state equations and output equations is linearized at .
• The input-linearized system has state , input , and output , with state equations and output equation . The coefficient functions are given by , , , and , all evaluated at .
• A system of DAEs with state equations and output equations is linearized at .
• The input-linearized system has state , input , and output , with state equations and output equation . The coefficient functions are given by , , , , and , all evaluated at and .
• Differential equations that include higher-order derivatives of states and inputs are reduced to the cases above by introducing additional states.
• In computations where the operating points xi0 and uj0 are Automatic, it is assumed to be zero in functions such as OutputResponse and conversion to StateSpaceModel, or generic in functions such as ControllableModelQ.
• The following option can be given:
• AffineStateSpaceModel can be used in functions such as OutputResponse and SystemsModelSeriesConnect.

# Examples

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## Basic Examples(1)

The affine system with output :

 In[1]:=
 Out[1]=

Using equation-form input:

 In[2]:=
 Out[2]=

Its response to a unit-step input:

 In[3]:=
 Out[3]=