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LQRegulatorGains

LQRegulatorGains
gives the optimal state feedback gain matrix for the StateSpaceModel object ss and the quadratic cost function with state and control weighting matrices q and r.
LQRegulatorGains
includes the state-control cross-coupling matrix p in the cost function.
LQRegulatorGains
specifies finputs as the feedback inputs of ss.
  • The state-space model ss can be given as StateSpaceModel, where a and b represent the state and input matrices in either a continuous-time or a discrete-time system:
continuous-time system
discrete-time system
  • The argument finputs is a list of integers specifying the positions of the feedback inputs in .
  • The cost function is:
continuous-time system
discrete-time system
  • The optimal control is given by , where is the computed feedback gain matrix.
  • For continuous-time systems, the optimal feedback gain is computed as , where is the solution of the continuous Riccati equation , and is the submatrix of associated with the feedback inputs .
  • For discrete-time systems, the optimal feedback gain is computed as , where is the solution of the discrete Riccati equation .
  • The optimal control is unique and stabilizing if is stabilizable, is detectable, , and .
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 optimum feedback gain matrix for a continuous-time system:
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Calculate the optimal control gains for an unstable system:
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Compare the open- and closed-loop poles:
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Compute the optimal state-feedback gain matrix for a discrete-time system:
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Calculate the feedback gains for controlling a two-input system using the first input:
In[1]:=
Click for copyable input
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A set of feedback gains for a stabilizable but uncontrollable system:
In[1]:=
Click for copyable input
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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 in which the first two inputs are feedback inputs, and a cost function with cross-coupling:
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:
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 :
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:
The continuous-time optimal regulator problem has a solution if and only if the pair has no unobservable modes on the imaginary axis:
The zero eigenvalue is unobservable:
The discrete-time optimal regulator problem has a solution if and only if the pair has no unobservable modes on the unit circle centered at the origin:
The eigenvalue 1 is unobservable:
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