ECE 501: 2^nd Midterm: Solution - Fall 1998

The equation:

is the most general form of an algebraic Riccati equation (all matrices are assumed to be of sizes n×n.
We define the matrix:

as the generalized Hamiltonian matrix associated with this Riccati equation.
Let:

be the right model matrix of H, such that:

is its spectral decomposition, i.e.:

Show that, if V₁ is non-singular, then:

is a solution of the generalized matrix Riccati equation.
We start by plugging the proposed solution into the generalized matrix Riccati equation:

We decompose the eigenvector/eigenvalue matrix equation into its four component (still matrix) equations:

Let us solve the first of these equations for L₁:

Plugging this into the third equation, we obtain:

Multiplying this equation from the right with V₁^-1, one gets:

which is the same as above except for the ordering of the terms.
Given the system:

We often like to have this system together with the state feedback:

behave like the model:

where:

One way to reformulate this problem is:

Thus:

or:

We want to reformulate this performance index in terms of a problem that we already know:

Find:

Then develop a Matlab code:

implementing the model follower optimal control problem.
By differentiating the output equation, we find:

and therefore:

Plugging this expression into the performance index, one obtains:

The terms can be rearranged as follows:

and assuming that:

one can find:

This can now easily be programmed into Matlab:
```
          function [K,P] = modfol(A,B,C,Am,Q,R)
            H1 = C*A - Am*C;
            H2 = C*B;
            Qh = H1'*Q*H1;
            Nh = H1'*Q*H2;
            Rh = R + H2'*Q*H2;
            [K,P] = lqr(A,B,Qh,Rh,Nh);
          return
      
```
Given the system:

Design a functional observer for this system, such that the closed-loop poles are at the locations:

and the observer poles are at the locations:

Find a 2-input / 1-output realization of this system:

where:

The open-loop denominator polynomial is:

The numerator polynomial (both open- and closed-loop) is:

The observer denominator polynomial is:

and the desired closed-loop denominator polynomial is:

Therefore:

Hence, the functional observer equations can be computed from the matrix equation:

The resulting c-vector is:

Thus, the two feedback filter polynomials are:

and:

Thus, the feedback system can theoretically be realized as follows:

However, the gains are far too large for this system to be realizable in a meaningful fashion. The open-loop system has the poles so far away from the desired closed-loop poles, that the desired goals cannot be reached by realistic means.
Given the system:

Find the Gilbert realization of this system.
First, we need to separate the strictly proper subsystem from the direct input output gain:

Now, we can perform a partial fraction expansion on the strictly proper subsystem:

Thus, R₁ is:

and R₂ is:

This leads to the state-space description:

ECE 501: 2nd Midterm: Solution - Fall 1998

ECE 501: 2^nd Midterm: Solution - Fall 1998