Gain matrix

In equation (8.93), r(t) is a vector of desired state variables and K is referred to as the state feedback gain matrix. Equations (8.92) and (8.93) are represented in state variable block diagram form in Figure 8.7. 

Wlienxi =yandx2 = Li, express the state equation in the eontrollable eanonieal form. Evaluate the eoeffieients of the state feedbaek gain matrix using ... 

The value of the observer gain matrix Kg ean be ealeulated direetly using... 

N is of rank 2 and therefore non-singular, henee the system is eompletely observable and the ealeulation of an appropriate observer gain matrix Kg realizable. 

The redueed-order observer gain matrix Ke ean also be obtained using appropriate substitutions into equations mentioned earlier. For example, equation (8.132) beeomes... 

In reverse-time, starting with P(A ) = 0 at NT = 20 seconds, compute the state feedback gain matrix K(kT) and Riccati matrix P(kT) using equations (9.29) and (9.30). Aiso in reverse time, use the desired state vector r(/c7 ) to drive the tracking equation (9.53) with the boundary condition s(N) = 0 and hence compute the command vector y kT). 

The general form of the Kalman filter usually eontains a diserete model of the system together with a set of reeursive equations that eontinuously update the Kalman gain matrix K and the system eovarianee matrix P. 

The Kalman gain matrix K is obtained from a set of reeursive equations that eommenee from some initial eovarianee matrix P(/c//c)... 

The reeursive equations (9.74)-(9.76) that ealeulate the Kalman gain matrix and eovarianee matrix for a Kalman filter are similar to equations (9.29) and (9.30) that... 

Before equations (9.99) can be run, and initial value of P(/c//c) is required. Ideally, they should not be close to the final value, so that convergence can be seen to have taken place. In this instance, P(/c//c) was set to an identity matrix. Figure 9.16 shows the diagonal elements of the Kalman gain matrix during the first 20 steps of the recursive equation (9.99). 

Fig. 9.16 Convergence of diagonal elements of Kalman gain matrix.

The final values of the Kalman Gain matrix K and eovarianee matrix P were... 

Calculate feedback gain matrix using Ackermann s formula K=acker(A,B,desiredpoles)... 

Disturbance noise covariance matrix %Kalman gain matrix... 

This is the inverted pendulum eontrol problem and, as a benehmark, is initially solved as a multivariable eontrol problem, using pole plaeement (Aekermann s formula) to ealeulate the feedbaek gain matrix in examplOS.m. 

Thus, in this problem, the process transfer function matrix Eq. (10-27) can be written in terms of the steady state gain matrix ... 

We may question other obvious scenarios of the process gain matrix. The sweetest is an identity matrix, meaning no interaction among the manipulated and controlled variables. A quick summary of several simple possibilities 10... 

You may not find observing the process gain matrix satisfactory. That takes us to the relative gain array (RGA), which can provide for a more quantitative assessment of the effect of changing a manipulated variable on different controlled variables. We start with the blending problem before coming back to the general definition. 

The relative gain array can be derived in terms of the process steady state gains. Making use of the gain matrix equation (10-32), we can find (not that hard see Review Problems)... 

For your information, relative gain array can be computed as the so-called Hadamard product, Ay = KjjKrH, which is the element-by-element product of the gain matrix K and the transpose of its inverse. You can confirm this by repeating the examples with MATLAB calculations. 

Before we design the MIMO system, we need to check the paring of variables. The steady state gain matrix is... 

Find the gain matrix and show that the relative gain parameter is 1. Show how this partially decoupling scheme can be implemented as analogous to Fig. 10.13. 

If the pairing had been reversed, the steadystate gain matrix would be... 

---