Gain vector

In summary, after each new measurement a cycle of the algorithm starts with the calculation of the new gain vector (eq. (41.4)). With this gain vector the variance-... 

By way of illustration, the regression parameters of a straight line with slope = 1 and intercept = 0 are recursively estimated. The results are presented in Table 41.1. For each step of the estimation cycle, we included the values of the innovation, variance-covariance matrix, gain vector and estimated parameters. The variance of the experimental error of all observations y is 25 10 absorbance units, which corresponds to r = 25 10 au for all j. The recursive estimation is started with a high value (10 ) on the diagonal elements of P and a low value (1) on its off-diagonal elements. 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

Influence of initial values of the diagonal values of the variance-covariance matrix P(0) and the variance of the experimental error on the gain vector and the Innovation sequence (see Table 41.1 for the experimental values, > )... 

Step 2. Update of the Kalman gain vector k(l) and variance-covariance matrix P(l)... 

Calculated gain vector, variance covariance matrix and estimated concentrations (see Table 41.5 for the starting conditions)... 

This is the result of full state feedback pole-placement design. If the system is completely state controllable, we can compute the state gain vector K to meet our selection of all the closed-loop poles (eigenvalues) through the coefficients a . 

Here, y = Cx has been used in writing the error in the estimation of the output, (y - y). The (/ x 1) observer gain vector Ke does a weighting on how the error affects each estimate. In the next two sections, we will apply the state estimator in (9-32) to a state feedback system, and see how we can formulate the problem such that the error (y - y) can become zero. 

Example 9.3 Consider the second order model in Example 9.1, which we have calculated the state feedback gains in Example 9.2. What is the observer gain vector Ke if we specify that the estimator error should have eigenvalues -9 repeated thrice ... 

Do the time response simulation in Example 7.5B. We found that the state space system has a steady state error. Implement integral control and find the new state feedback gain vector. Perform a time response simulation to confirm the result. 

If the estimated and observed values of v are identical, the value of b will be unchanged. If the observed value is more than the estimated value, it makes sense to increase the size of the coefficients to compensate. The column vector kt is called the gain vector. There are a number of ways of calculating this, but the larger it is the greater is the uncertainty in the data. 

A common (but complicated way) of calculating the gain vector is as follows. 

Equation (6) updates the current state vector by adding the correction factor, which is givCT by the difference between the actual measurranent z(k) and the predicted state vector, weighted by the Kalman gain vector. Similarly, the Kalman gain is used to estimate the updated current error covariance matrix, as in Eqn.(7). 

The product g<,gp is the open-loop vector. If the inner loop has been adjusted for 1. -ampIitude damping, its open-loop gain will be 0.5 at the period of oscillation. But the phase lag at the period of oscillation is 180°, which makes the gain vector 0.5, Z —180°, or -0.5. The closed-loop vector g a at the natural period is then... 

An appreciation for the dynamic effects that coupled closed loops have on one another can be gained by analyzing the block diagram shown in Fig. 7.6. One loop, comprised of a controller whose gain vector is gci and a process with a dynamic vector of gi and relative gain Xn, has a period of Toi. The loop is upset by a manipulated variable irti from the second closed loop, whose period is t 2. In the path of mt is the process vector g2 and the relative gain of Ci with respect to m2, that is, X12. 

---