Compute the optimum feedback gain matrix for a continuous-time system:

Calculate the optimal control gains for an unstable system:

Compare the open- and closed-loop poles:

Compute the optimal state-feedback gain matrix for a discrete-time system:

Calculate the feedback gains for controlling a two-input system using the first input:

A set of feedback gains for a stabilizable but uncontrollable system:

Compute the feedback gains for a continuous-time state-space model:

The feedback gains for a discrete-time system:

Compute the optimal gain matrix when the cost function contains state-control coupling:

The LQR gains for a system in which only the fourth and fifth inputs are feedback inputs:

The gains for a system with the first two inputs as feedback inputs and a cost function with cross-coupling:

Find the optimal gains for a descriptor state-space model:

Compute the gains for a NonlinearStateSpaceModel:

The closed-loop system:

The state response is stable:

Compute a set of state feedback gains that stabilizes an unstable system:

The response of the closed-loop system to a step input:

The open-loop system is unstable:

Compute the Hessian of the cost function's integrand to determine the weighting matrices:

The natural response of the closed-loop system:

Without feedback, the system is highly oscillatory:

Design a regulator for the discrete-time model of a mixing tank system:

The response of the system to an impulse at the second input:

Find the optimal feedback gains for a continuous-time system:

The same solution can be obtained from the solution to the continuous-time Riccati equation:

Find the optimal feedback gains for a discrete-time system:

The same solution can be obtained from the solution to the discrete-time Riccati equation:

Find the loop gain transfer function for an LQR design:

Its Nyquist plot lies outside the unit circle centered at :

Consequently, the gain margin and the phase margin satisfies and :

The gains associated with a state increase as its weight is increased:

The higher penalty of the second state reduces its overshoot:

Find the optimal gains for a stable system as input cost is varied:

The gain converges to zero as approaches zero, or approaches infinity:

As the gain decreases, the closed-loop poles move closer to the open-loop ones:

Find the optimal gains for an unstable system as state cost is varied:

The gain converges to a minimum as approaches zero, or approaches infinity:

As k decreases, the closed-loop poles move nearer the stable open-loop poles and mirror images of unstable ones:

When approaches infinity, or approaches zero, the gain becomes unbounded:

As the gains increase, the states are penalized more, and their values become smaller:

The optimal cost-to-go is a Lyapunov function:

The state trajectory projected on the optimal cost surface asymptotically approaches the origin:

The optimal state trajectory for a system with one state:

The co-state trajectory:

The optimal input trajectory:

The optimal cost trajectory:

The optimal cost satisfies the infinite horizon Hamilton–Jacobi–Bellman equation:

The optimal input minimizes the Hamiltonian, thus satisfying :

The gain computations fail for an unstabilizable system:

The gain computations fail if the control weighting matrix is not positive definite:

Use a positive-definite control weighting matrix:

If has unobservable modes on the imaginary axis, there is no continuous-time solution:

The zero eigenvalue is unobservable:

If has unobservable modes on the unit circle, there is no discrete-time solution:

The eigenvalue 1 is unobservable: