lecture10过程控制讲义10.ppt

Linear Regression: Objective Linear Regression: Mathematical Formulation Linear Regression: Least Squares Fit Linear Regression: Least Squares Fit Linear Regression: Least Squares Fit Observation General Linear Least Squares General Linear Least Squares General Linear Least Squares: Algorithm General Linear Least Squares: Example ARX Parameter Estimation * Fit a straight line to a set of paired observations: (x1, y1), (x2, y2), …, (xn, yn). P1 P2 P3 P4 P5 P6 P7 y=a0+a1x Mathematical representation for the straight line: For each given point (xi, yi), there is a discrepancy between its true value of yi and the approximate value a0+a1xi: ei: Residual Error Since and are unknown, we need to find the values of these two variables so that will be minimized. This can be done by letting: Reorganize the equations in terms of two unknowns: a0 and a1 Solving these two equations with two unknown variables gives: Chapter 7 Least squares estimation can be extended to more generalmodels with: More than one input or output variable. Functionals of the input variables x, such as poly-nomials and exponentials, as long as the unknown parameters appear linearly. Extensions of the Least Squares Approach where are called basis functions Examples: ? and is the modelled value of y. A general nonlinear steady-state model which is linear in the parameters has the form, Suppose that the number of data points is n. Step 1: Construct [Z] and {Y}. Step 2: form the system of equations Step 3: Solve {A}. A curve is described by y=1.7+cos(4.189t+1.0472). Generate 10 discrete values for this curve at intervals of ?t=0.15 for the range t=0 to 1.35. Use this information to evaluate the coefficients of the following model by a least square fit: y=A0+A1cos(?0t)+B1sin(?0t)+e -9-1 3.0925641e-1 1 2.6782004 1.20 -5-1 8.0918343e-1 1 2.6134292 1.35 -9-1 -3.0880748e-1 1 2.3692961 1.05 -5-1 -8.0890599e-1 1 1.8047186 0.90 -1.5734640e-4 -9.999