Logarithm of a matrix

Matrix Polynomials and Functions of Square Matrices

Published in Sohail A. Dianat, Eli S. Saber, ®, 2017

Compute the natural logarithm of the matrix C; that is, identify the matrix B = ln(C) that satisfies the equation C = eB. Is the assumption of nonsingularity of C really needed for the solution of this problem?

A design technique for fast sampled-data nonlinear model predictive control with convergence and stability results

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Journal Information

Published in International Journal of Control, 2020

Andreas Steinboeck, Martin Guay, Andreas Kugi

We consider a piecewise constant input parameterisation of the form: This means that κ can be designed using a zero-order hold. We use the linear feedback law with the constant gain K. Here, xk = x(Tk) and k is the index of the time grid Tk. Inspired by the approach reported in Chen and Allgöwer (1998b) for continuous-time NMPC, we suggest a different systematic sampled-data control design based on the locally linearised system. Consider the Jacobian linearisation: of the system (1) with Ac = ∂f/∂x|x = 0, u = 0, and Bc = ∂f/∂u|x = 0, u = 0. Its discrete-time counterpart found by zero-order hold reads as with A = exp (κTAc) and B = ∫κT0exp (tAc)dtBc and is assumed to be stabilisable in . Using the steady-state LQR design method with the positive definite objective function and the solution P of the corresponding algebraic Riccati equation, we obtain Based on the correspondence with (Franklin, Powell, & Workman, 1997; Van Loan, 1978), the equivalent objective function of the sampled-data LQR design problem reads as In Equation (17), ln ( · ) returns the logarithm of a matrix. This operation is only defined for invertible matrices.

Logarithm of a matrix

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Matrix Polynomials and Functions of Square Matrices

A design technique for fast sampled-data nonlinear model predictive control with convergence and stability results