State Space Simulation in Matlab

Since matlab has first-class support for matrices and vectors, it is quite
simple to implement a state-space model in matlab using no support functions
whatsoever, *e.g.*,

% Define the state-space system parameters: A = [0 1; -1 0]; % State transition matrix B = [0; 1]; C = [0 1]; D = 0; % Input, output, feed-around % Set up the input signal, initial conditions, etc. x0 = [0;0]; % Initial state (N=2) Ns = 10; % Number of sample times to simulate u = [1, zeros(1,Ns-1)]; % Input signal (an impulse at time 0) y = zeros(Ns,1); % Preallocate output signal for n=0:Ns-1 % Perform the system simulation: x = x0; % Set initial state for n=1:Ns-1 % Iterate through time y(n) = C*x + D*u(n); % Output for time n-1 x = A*x + B*u(n); % State transitions to time n end y' % print the output y (transposed) % ans = % 0 1 0 -1 0 1 0 -1 0 0The restriction to indexes beginning at 1 is unwieldy here, because we want to include time in the input and output. It can be readily checked that the above examples has the transfer function

so that the following matlab checks the above output using the built-in

NUM = [0 1]; DEN = [1 0 1]; y = filter(NUM,DEN,u) % y = % 0 1 0 -1 0 1 0 -1 0 1To eliminate the unit-sample delay,

[A,B,C,D] = tf2ss([1 0 0], [1 0 1]) % A = % 0 1 % -1 -0 % % B = % 0 % 1 % % C = % -1 0 % % D = 1 x = x0; % Reset to initial state for n=1:Ns-1 y(n) = C*x + D*u(n); x = A*x + B*u(n); end y' % ans = % 1 0 -1 0 1 0 -1 0 1 0Note the use of trailing zeros in the first argument of

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