![]() ![]() Optimizing the cost function and solving for involves solving a Riccati differential equation.would be large if it is expensive to apply control. is a qxq matrix (where q is the size of control vector ) that specifies the penalty for using up control effort.For instance: for a state vector, if it has a matrix of then it means it gives a penalty of for reaching state parameters and slowly, a penalty of for reaching state parameter slowly and a penalty of for reaching state parameter slowly. is an nxn matrix that expresses the penalty for the speed at which each of the parameters in the state vector should be reached.The LQR algorithm chooses the best eigenvalues by optimizing (minimizing) a cost function whose parameters are specified by the user.So how to we choose the perfect eigenvalues that lie at the sweet spot between too fast and too slow? Use LQRs!.And this might not be ideal in some cases. ![]() Small negative parts of eigenvalues also cause the system to very slowly approach stability. This is because, large negative parts of eigenvalues causes instability because the system tends to ‘shoot’ too quickly towards stability, making it overshoot in some cases. Choosing the good_eigens vector manually is tricky.More on how the place function works in Matlab can be found here Where good_eigens is a vector of eigenvalues with negative real parts. This can easily be done in Matlab, using the comand.So naturally, since is the only changeable variable in the matrix, you’d want to choose a that makes the eigenvalues of the matrix have negative real parts.If the real parts of all the eigenvalues of the matrix are negative, then the system is stable.If the real part of any of the eigenvalues of the matrix is positive, then the system is unstable.An interesting finding is that, the stability of the system can be determined by the nature of the eigenvalues of the matrix.Substituting into the state-space representation of the linear system, we end up with the equation.Where is the proportional factor that must be chosen to regulate ![]()
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