Obtain the linearized system:

Design a controller using the linear system:

The closed-loop system:

Simulate the response to a unit step input:

Obtain the residual system:

Compute the eigenvalues of the residual system:

Explicitly specify the new state and feedback variables:

The results in terms of the specified variables:

Obtain a property directly:

Obtain multiple properties directly:

The transformed system has two subsystems:

The entire transformed system:

It can also be assembled from lsys and rsys:

Find the nonlinear state feedback controller :

The controller designed using the exactly linearized system:

Obtain the controller for the original system:

Find a nonlinear state estimator:

Compute the estimator gains for the exactly linearized system:

Obtain the estimator for the original system:

Plot the estimated state trajectories:

Compare the actual and estimated state trajectories:

Find the controlled closed-loop system:

The feedback gains computed using the linearized system:

Obtain the closed-loop system:

Simulate the closed-loop system:

The closed-loop system using a dynamic controller:

The linear system:

The linear controller and estimator design:

The controller for the original system:

The closed-loop system:

The full controller for the original system:

The feedback gains computed using the linearized system:

The full controller for the original system:

The inputs to the full controller are the reference inputs and the state feedback:

Plot the control input to the system:

The feedback compensator:

The compensated system consists of independent single-input, single-output loops:

The first output is influenced only by the first input:

And the second output is influenced only by the second input:

Other variants of the feedback compensator:

The systems model of the inverse feedback compensator:

The inverse feedback compensator as a list of transformation rules:

The decoupling matrix:

The matrix must be invertible for any control design to be valid:

Obtain the decoupling matrix from the inverse feedback transformation:

If possible, a precompensator is computed to make the decoupling matrix nonsingular:

The precompensator in series with the system results in a well-defined vector of relative orders:

The decoupling matrix of just the original system is singular:

If possible, for scalar systems a postcompensator is computed:

With the postcompensator, the system is exactly feedback linearizable:

The post compensator is a combination of the outputs of the system:

Different outputs result in different linear systems:

The zero dynamics system:

It can also be obtained from the residual system by deleting all its inputs:

The zero dynamics manifold:

Compute it from the inverse state transformation:

A system with no residual dynamics rsys:

Since there are no residual dynamics, linear control design can be used:

Simulate a step response for the closed-loop system:

A system with stable residual dynamics:

The residual dynamics are stable, so feedback-design-based linear methods are valid:

Use linear feedback design, and find the feedback law:

Simulate the system:

A system with unstable residual dynamics rsys:

The eigenvalues have a positive real part, so the rsys is unstable:

For such nonminimum phase systems, the designs based on linear techniques are not valid:

The designed feedback is unable to stabilize the intrinsically ill-behaved system:

For linear systems, an identity transformation and feedback are used by default:

The Burnovsky form moves the hidden modes to the residual system:

For nonlinear systems, the result is in Burnovsky form by default:

It may be possible to decouple the input from the residual dynamics:

Design a controller that stabilizes a magnetic levitation system using exact linearization, and compare with a design based on approximate linearization:

The affine model can be obtained directly from the governing equations:

The system without feedback is unstable, here with initial value {0.2,0.,0.1}:

It is completely feedback linearizable, since it has no residual dynamics:

The controller design based on the linear system:

The closed-loop system using original state variables:

Simulate the system with initial value {0.3,0.,0.31305}:

Design based on the approximately linearized system:

The closed-loop system with the controller based on linearization:

Simulation of the system with controller based on approximate linearization:

The design based on feedback linearization has a better response:

Find a stabilizing controller for a two-wheeled inverted pendulum (e.g. Segway) using voltages to the two DC wheel motors as inputs:

The AffineStateSpaceModel of the system with states {θ,θ',ψ,ψ',ϕ,ϕ'}:

Feedback linearize the system:

The zero dynamics have purely oscillatory behavior:

Design a controller using the linearized subsystem:

Compute the state response of the closed-loop system:

The plots show that the oscillations are in the pendulum's pitch:

The two wheels are at rest in steady state:

And so is the yaw motion of the pendulum:

Design a vibration controller for a flexible structure, and compute the control effort expended. A two-mode model of a flexible structure with no damping: »

The two modes:

The model is completely feedback linearizable:

Design a controller using the linear system to suppress the vibrations:

The modes of the closed-loop system:

The control effort:

Design a controller to suppress the oscillations in an axial flow compressor. A model of the compressor with throttle as input: »

Simulations show the presence of surge-like oscillations:

Feedback linearize the system:

Design a controller to suppress the oscillations:

The closed-loop system:

Simulations show that the oscillations have been suppressed:

Design a controller to improve an isothermal continuous stirred-tank reactor process: »

The affine system with states and inputs :

The system is completely feedback linearizable:

Design a controller based on the linearized system:

The closed-loop system:

The response of the closed-loop system from a non-equilibrium initial condition:

The response of the uncontrolled system is far worse:

Assume that the inputs to the system are random and normally distributed:

The response of the controlled system:

The response of the open-loop system:

Again, the response of the controlled system is better:

Design a controller for a room cooling system: »

The air and water flow are the inputs and are the states:

Feedback linearize the system:

The zero dynamics are stable:

Compute a controller using the linearized system:

The closed-loop system:

The temperature in the room from various initial conditions as a function of time in hours:

The outside temperature is not present in the closed-loop system, as it is offset by the feedback:

The control inputs to the system can be determined using the full controller:

The outside air temperature on a warm day:

The temperature plot:

The control inputs, which are the air and water flows:

Plot the control inputs:

Design a controller for better speed response in an induction motor subject to varying loads: »

The model of the motor with inputs :

Simulations show the effect of torque on the angular velocity:

The model with an integrator in the q axis:

Feedback linearize the system:

Design a state feedback controller for the linear system:

The closed-loop system:

Simulate the closed-loop system for various torque values:

Design a controller that regulates quantities in a wind energy conversion system: »

Its feedback linearization:

The zero dynamics are stable:

The linear system consists of a double integrator and three single integrators:

A controller that gives desired closed-loop transfer functions in each linearized and decoupled loop:

The closed-loop system also has the same linear characteristics plus the zero dynamics:

The response of the system to a unit step applied to each input in turn:

The first input affects only the first output:

And the closed-loop behaves as the designed linear system:

This is true of the three other decoupled SISO loops as well: