Linearize a textual RLC circuit model:

Linearize an RLC circuit block diagram model:

Linearize an acausal RLC circuit:

Linearize a DAE model:

Linearize a DAE model symbolically:

Linearize a model without inputs:

Linearize a model without states:

Linearize a model without outputs:

Linearize around an equilibrium:

Linearize around initial values:

Linearize around equilibrium with given state and input constraints:

Linearize around equilibrium with state constraints:

Linearize around given partial states and inputs, using initial values for remaining values:

Linearize around an equilibrium:

Linearize around initial values:

Linearize around equilibrium with given state and input constraints:

Linearize around initial values from a simulation:

Linearize around final values from a simulation:

Linearize around final values from a simulation run to steady state:

The "SymbolicDerivative" method uses a fully specified operating point from a simulation:

The "NumericDerivative" method uses an operating point specified by state and input values:

Linearizing symbolically allows keeping some parameters symbolic:

Use "ReduceIndex" to turn off index reduction when linearizing symbolically:

Turning off index reduction results in a descriptor StateSpaceModel:

Control progress reporting with ProgressReporting:

Compare responses from a model and its linearization at an equilibrium point:

Linearize around the equilibrium point:

Compare the stationary output response with a nonlinear model:

Compare the y1 output:

Test the stability of a linearized system from eigenvalues of the system matrix:

Since there is an eigenvalue with a positive real part, the system is unstable:

Plotting the output response also indicates an unstable system:

Test the stability of a linearized system from poles of the transfer function:

Since there is a pole with a positive real part, the system is unstable:

Do a frequency analysis using a linear model:

By plotting for the linearized transfer function :

Verify the result using Fourier on simulated data:

Compute from :

Alternatively, the imaginary parts of the eigenvalues give the resonance peaks:

Linearization takes place at time 0:

Linearize with the switch connecting at time 0:

If the switch is not connected at time 0, the result is different:

Design a PID controller using a linearized model:

Define a PID controller and closed-loop transfer function:

Select PID parameters for appropriate step response:

Design a lead-based controller for a DC motor based on its linearization:

Define a PI-lead controller transfer function:

Open-loop transfer function:

Select controller parameters:

Use selected parameters and close the loop with the PI-lead controller:

Design a controller using pole placement:

Place the closed-loop poles:

Compute the closed-loop state-space model:

Show the step response:

Design an LQ controller:

Define state and input weight matrices:

Define LQ controller gain:

Closed-loop state-space model:

Closed-loop step response:

Design a state estimator:

Compute estimator gains and the estimator state-space model:

The state and output response to a unit step on the inputs:

Observer state response:

Compare each state and its estimate: