feat: exercise-9 (WIP)

.gitignore

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+
+exercise-1/SystemID_HS2022_Exercise1.pdf
+exercise-1/SystemID_HS2022_Solution1.pdf
+exercise-2/SysID_lecture03_small.pdf
+exercise-2/SystemID_HS2022_Exercise02.pdf
+exercise-3/SystemID_HS2022_Exercise03.pdf
+exercise-3/SystemID_HS2022_Solution03.pdf
+exercise-4/SystemID_HS2022_Solution4.pdf
+exercise-5/Exercise5_SysID_2022.pdf
+exercise-5/Exercise5_SysID_2022_Solution.pdf
+exercise-8/HS2022_Exercise8_Solution.pdf
+exercise-9/exercise.asv
+exercise-8/exercise.asv

exercise-9/exercise.m

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+clear, close, clc
+
+B = [0.0, 0.3, -1.5];
+A = [1.0, 1.1, 0.7];
+ts = 0.1;
+
+%% Exercise 9.1
+
+N = 100;
+e_v = 0.5;
+u_v = 0.5;
+
+rng(1);
+u = randn(N, 1) * sqrt(e_v);
+rng(2);
+e = randn(N, 1) * sqrt(u_v);
+
+B_A = tf(B, A, ts);
+
+y = lsim(B_A, u) + e;
+
+%% Exercise 9.2
+
+ns = [1, 2, 3, 2, 3];
+ms = [2, 2, 2, 3, 3];
+
+% TODO error is a bit lower than solution
+mse_ls = zeros(length(ns), 1);
+systems = cell(5, 1);
+for i=1:5
+    [A_ls, B_ls] = ls_estimate(y, u, ns(i), ms(i));
+
+    systems{i} = estimate_to_sys(A_ls, B_ls, ts);
+    y_pred = lsim(systems{i}, u);
+    mse_ls(i) = mean((y_pred - y).^2);
+end
+
+%% Exercise 9.3
+
+u_instr = randn(N, 5) * sqrt(u_v);
+y_instr = zeros(N, 5);
+
+rng(3)
+u_instr(:, 2) = randn(N, 1) * sqrt(u_v);
+
+mse_iv = zeros(5, 1);
+inst_systems = cell(5, 1);
+for i=1:5
+    y_instr(:, i) = lsim(systems{i}, u_instr(:, i));
+
+    [A, B] = inst_estimate(y, y_instr(:, i), u_instr(:, i), ns(i), ms(i));
+    inst_systems{i} = estimate_to_sys(A, B, ts);
+
+    y_pred = lsim(inst_systems{i}, u_instr(:, i));
+    mse_iv(i) = mean((y_pred - y_instr(:, i)) .^ 2);
+end
+[mse_ls mse_iv]
+
+
+%% Common function
+
+function [A, B]=inst_estimate(y, y_inst, u, na, nb)
+    phi_u_F = toeplitz(u(1:end-1), zeros(nb, 1));
+    phi_y_F = toeplitz(-y(1:end-1), zeros(na, 1));
+    phi = [phi_y_F, phi_u_F];
+
+    theta = (y_inst(2:end).' *  phi) \ (y_inst(2:end).' * y(2:end));
+
+    A = theta(1:na).';
+    B = theta(na+1:end).';
+end
+
+function [A,B]=ls_estimate(y, u, na, nb)
+    phi_u_F = toeplitz(u(1:end-1), zeros(nb, 1));
+    phi_y_F = toeplitz(-y(1:end-1), zeros(na, 1));
+    phi = [phi_y_F, phi_u_F];
+
+    theta = phi \ y(2:end);
+
+    A = theta(1:na).';
+    B = theta(na+1:end).';
+end
+
+function sys=estimate_to_sys(A, B, ts)
+    A = [A zeros(1, max(0, length(B) - length(A)))];
+    B = [B zeros(1, max(0, length(A) - length(B)))];
+
+    sys = tf(B, [1 A], ts);
+end