The Use of Filters in the Integrated Characteristic Locus Multivariable Root Locus Design Method

The two stage design provides an effective way for reaching a compromise between the requirements for stability and accuracy. It does this by pre-conditioning the system during the inner loop stage, improving its stability margins and subsequently injecting gain into the CL of the outer loop system. Thus, given a system with state space matrices (A,B,C), depending on the stability margins it may or may not be necessary to include an inner loop. The corresponding configurations are depicted in figures 3 and 4 where u^, y^, and denote the input, output and state vectors of S^, respectively.

Note that figure 3 refers to the situation where the entire state vector is available for the purposes of feedback.In general, however, only a limited number of measurements may be available and this will restrict the design freedom available during the inner loop stage. The question then arises as to whether observers may be used to obtain estimates of the required number of states for the purposes of the inner loop design. The answer to this is affirmative and recent work [118] has shown that from the MRL design point of view it is possible to invoke a form of the separation principle to separate the design of the operator F from the design of the observer.

A more challenging situation arises when apart from the unavailability of excess measurements the outputs themselves may not be measured directly due to the presence of various disturbances and noise. A solution to this problem which is often prefered by practising engineers is to use notch filters to remove the effects of

disturbances at known frequencies. An alternative approach would be to introduce into the feedback loop a Kalman filter which would make available estimates of the system states as well as the system outputs and thus would enable the formation of both the inner and outer loops. The feedback configuration resulting from the use of Kalman filter and notch filters are depicted in figures 6 and 5" respectively

In the absence of noise and disturbance the Kalman filter would be designed as a simple observer and by the separation principle one may use exactly the same controllers for the configurations of both figures 3 and 5 [ie: K*(s)=K(s), k f=k and F f=F], It is of interest to know whether the same applies in the presence of noise and disturbances.

Consider the Kalman filtering configuration of figure 2 and assume that the measurement vector z is the sum of the actual output y^ of the given plant model S^=(A^B^,C^), and an output of a dynamical system SgsCA^B^C^) which describes the effect of disturbances and a vector v of white measurement noise. Assume also that an uncorrelated white noise input vector w, enters the system S^. The situation certainly corresponds to the Dynamic Ship Positioning application but is also widely encountered in other industrial applications. Then the state space equation for the Kalman filter may be written as:

x 1=A1x 1+Kl(z-y)+B.Ju 1 (5.24)

x^A^Xj+K-gCz-y) (5.25)

y=ci M 2x2 (5-26>

where to obtain the matrices K^ and one needs to specify the noise covariance matrices. The estimate of the state vector of is

KF

Figure 5 Feedback Configuration with Kalman Filter

NF

Figure 6 Feedback Configuration with Notch Filter

provided by the component x.j of the filter state vector and the estimate for the output vector is given by the vector

The description of the transference from the system input u^ to the output of the Kalman filter y«j is obtained by manipulation of the state space equations of the system and is given by the augmented state space model:

-Sl°2 £ 1 _{A l} 0 • £ — 1 = 0 Ai - K1C1 4 A 0 - K2C1 A [ C l C i 0] *1

A2 ” K2C2

where 6^ is the estimation error x^-x^. Clearly then the filter modes are uncontrollable from u^ and as such do not affect the transference from u.j to y^ which is given by:

^ = C l(sI-Air 1B lu 1 (5.29)

Thus, the transference from u^ to y^ and u^ to y^ are identical.

The same result can be obtained for the transferance of the inner loop of figure 5 if the matrix in 5.24 is replaced by F. We may state therefore that the introduction of the Kalman filter in figure 3 does not affect the MRL properties of the inner loop and the CL properties of the outer loop system. As a consequence, a form of the separation principle may be invoked and we may introduce the same inner and outer loop controllers that we would use, had there been an excess of measurements and no noise and disturbances (see also section 5.5.3).

In contrast to the situation above the same does not apply to the configuration with the notch filter (figure 6). Notch filters do affect the system loop transference and will normally result in a certain deterioration of the system stability margins and speed of response.

5.6 The Charaoteristic Locus and Multivariable Root Locus design