StateFeedbackGains

Compute the state feedback gains that place poles at specific locations:

Verify the result:

The state feedback gains for a multiple-input system:

Shift the pole of an integrator to :

Feedback gain for a symbolic first-order system:

The closed-loop system:

Compute the state feedback gains of a symbolic discrete-time system:

The poles of the closed-loop system:

By default, the KNVD method is chosen for inexact inputs:

The solution using Ackermann's formula:

The KNVD method is more robust:

For inexact multiple-input systems, the Ackermann method selects the feedback input that has the controllability matrix with the lowest condition number:

By default the first input is selected:

The gains associated with the second output:

Stabilize a system using state feedback:

A system with two masses, a spring, and a damper:

The choice of poles can be used to tune the response of the system:

The gain gives an overdamped response, while results in a slightly damped system:

Compute the state feedback gains to regulate an inverted pendulum about the upright position:

Displacements and of the cart and the pendulum from an initial cart position of :

The displacements are also regulated when disturbances are present:

The motion of a spacecraft linearized about a circular Kepler orbit:

Determine a set of thruster commands to regain the nominal trajectory after an initial disturbance:

The response plot:

Solve the deadbeat control problem of determining the control input required to bring the output of a discrete-time system to zero in the smallest number of time steps:

With the poles at the origin, the output of a third-order system is zero in 3 time steps:

Use SystemsModelStateFeedbackConnect to obtain the closed-loop system:

A larger displacement between the open and closed loop pole locations requires a larger control effort:

State feedback does not affect the input-blocking properties of a system:

The input Sin is blocked by both systems:

The closed-loop system will be unobservable if the feedback moves any of the poles to the zeros:

The KNVD method does not handle exact systems when the number of feedback inputs is less than the number of states:

Evaluate numerically:

The KNVD method cannot be used when the pole multiplicity exceeds the number of feedback inputs to the system:

Use the Ackermann method instead:

The KNVD method can give different sets of gains on different computer systems:

The closed-loop eigenvalues are the same:

For a MIMO system, the Ackermann method uses only one of the inputs as the feedback input:

With the default KNVD method, the control effort is shared between all the feedback inputs:

Arbitrary pole assignment by state feedback requires a controllable model:

Verify controllability: