For continuous-time systems, the optimal gain matrix is
, where
is the solution of the continuous-time Riccati equation:
For discrete-time systems, the optimal gain matrix is
, where
is the solution of the discrete-time Riccati equation:
For an LQR design, the Nyquist plot of the open-loop transfer function always 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:
For stable open-loop systems, 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:
For unstable open-loop systems, the gain
converges to a minimum as
approaches zero or
approaches infinity:
As the gain decreases, the closed-loop poles move closer to any stable open-loop poles and the stable mirror images of the 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
, where
:
The optimal input minimizes the Hamiltonian, thus satisfying
:
, where a and b represent the state and input matrices in either a continuous-time or a discrete-time system:
in 
.
is equivalent to LQRegulatorGains[{ss, All}, {...}].
, the cross-coupling matrix p is assumed to be zero.
, where 
is the computed feedback gain matrix.
, where 
is the solution of the continuous Riccati equation 
, and 
is the submatrix of 
associated with the feedback inputs 
.
, where 
is the solution of the discrete Riccati equation 
.
is unique and stabilizing if 
is stabilizable, 
is detectable, 
, and 
.
has no unobservable modes on the imaginary axis:
has no unobservable modes on the unit circle centered at the origin:
EXAMPLES
Basic Examples 
Scope 