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| function [ Qd, Rd, Sd ] = c2d_weights( sys, Qc, Rc, Ts ) | ||
| %C2D_WEIGHTS Discretize the weights for the optimal control problem | ||
| % | ||
| % This function will compute the optimal discrete-time weights that make | ||
| % the discrete-time LQR cost function equivalent to the continuous-time LQR | ||
| % cost function. | ||
| % | ||
| % The algorithm used in this function is based on the matrix-exponential | ||
| % method presented in Section 9.3.4 of | ||
| % G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of | ||
| % Dynamic Systems, Third. Menlo Park, CA, USA: Addison-Wesley, 1998. | ||
| % | ||
| % | ||
| % Usage: | ||
| % [ Qd, Rd, Sd ] = C2D_WEIGHTS( sys, Qc, Rc, Ts); | ||
| % | ||
| % Inputs: | ||
| % sys - The continuous-time system | ||
| % Qc - The continuous-time state weighting matrix | ||
| % Rc - The continuous-time input weighting matrix | ||
| % Ts - The sampling time | ||
| % | ||
| % Output: | ||
| % Qd - The discrete-time state weighting matrix | ||
| % Rd - The discrete-time input weighting matrix | ||
| % Sd - The discrete-time cross-term weights | ||
| % | ||
| % | ||
| % Created by: Ian McInerney | ||
| % Created on: September 18, 2018 | ||
| % Version: 1.0 | ||
| % Last Modified: September 18, 2018 | ||
| % | ||
| % Revision History | ||
| % 1.0 - Initial release | ||
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| %% Check to see if a discrete-time system was supplied | ||
| if ( sys.Ts ~= 0 ) | ||
| error('Must supply the system in continuous-time.'); | ||
| end | ||
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| %% Extract the system matrices | ||
| A = sys.A; | ||
| B = sys.B; | ||
| [n, m] = size(B); | ||
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| %% Compute the matrix exponential | ||
| zn = zeros(n); | ||
| zm = zeros(m); | ||
| znm = zeros(n, m); | ||
| zmn = znm'; | ||
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| mmat = [-A', znm, Qc, znm; | ||
| -B', zm, zmn, Rc; | ||
| zn, znm, A, B; | ||
| zmn, zm, zmn, zm]; | ||
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| ex = expm(mmat.*Ts); | ||
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| %% Extract the weights | ||
| phi11 = ex( 1:(n+m), 1:(n+m)); | ||
| phi12 = ex( 1:(n+m), (n+m+1):end); | ||
| phi22 = ex((n+m+1):end, (n+m+1):end); | ||
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| phi = phi22'*phi12; | ||
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| Qd = phi( 1:n, 1:n); | ||
| Sd = phi( 1:n, (n+1):end); | ||
| Rd = phi((n+1):end, (n+1):end); | ||
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| end | ||
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