
this purpose, it will be presented a simple derivation of the interactor matrix and preferable solution of the Diophantine equation. Both methods are based on the state space representation for the (estimated) transfer function matrix of the plant. Unlike direct calculation of polynomial matrices, it is preferable to compute via the state space representation. This chapter is organized as follows. In the next section, the basic controller design with known parameters will be shown. The derivation of the interactor matrix and the solution to the Diophantine equation will be also presented. Then, the indirect MRACS will be shown in section 3 using the results in the previous sections. Some simulation results will be presented in section 4 to confirm the validity of the proposed method. Concluding remarks will be presented in section 5.

Notations (See Wolovich, 1974 The control input to achieve the above objective is given by

satisfy the following Diophantine equation:

satisfies the following relation: 

Thus from assumption 1, the purpose will be achieved if 

If we set

If eqn. (5) 

and define the integer w by the least integer satisfying the above equation. Thus, using

Moore-Penrose pseudo-inverse

, L can be calculated if and only if

All of L that satisfy eqn. (11) are coefficient matrices of the identity interactor, which form a subset of coefficient matrices of the interactor defined by Mutoh & Ortega (1993) . In the paper, a certain calculating algorithm was used to obtain L and to assign stable zeros of the interactor as well. But, for the identity interactor, as shown above, it is quite natural to solve eqn. (11) using Moore-Penrose pseudo-inverse, because

is a fixed matrix and the pseudo-inverse can be calculated easily using some standard softwares in these days. Then, since we need a stable interactor in control design problems, the remaining problem is to check the location of zeros of the identity interactor given by eqn.(12). For

following Lemma holds. Lemma 1. For the integer 1 − ≥ w k , the following equation holds:

Substituting the above equation into eqn. (15), 

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From the existence of k P , 0 ) (
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holds. Using the singular value decomposition,
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for some unitary matrices
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post-multiplying the above equation by
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is obtained from eqns. (18) and (19). Using a free parameter matrix
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, the general solution of eqn. (17) is given by ( )
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Finally, by choosing k Z as in eqn. (14), eqn. (13) is obtained from eqns. (21) and (23). It is easy to verify that the above
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T satisfies the rest of conditions for the pseudo-inverse, i.e.
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( )
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Therefore the Lemma has been proved. ∇∇∇ In MRACS case, it can not be assumed to know the exact value of w in eqn.(11). Lemma 1 shows that non-zero parameters in the interactor are not changed, if the upper bound of w is known. This is a nice property of the proposed method for MRACS. The following Theorem shows that all zeros of the interactor by proposed method lie at the origin. So, the stability of the interactor is clear although it does not have a lower triangular form. Moreover, the proposed interactor is optimal for the LQ cost with singular weightings. See Kase et al. (2004) for the proof. Theorem 1. If the interactor is given by ), ( ) (
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then the following properties hold:
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where + L is the pseudo-inverse of L , and (Kase, 1999; 2008) There are many methods to solve eqn.(4). In this subsection, a method using state space parameters is presented. First, the following lemma holds.
A Solution of Diophantine Equation
are given by the following equations:
A Solution of Diophantine Equation
where f denotes the degree of
A Solution of Diophantine Equation
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Proof. Since ( ) 
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where Y is defined by [ ] 
Indirect Adaptive Controller Design
Using some suitable parameter estimation algorithm, such as the least squares algorithm, obtain the estimated values of D and Ñ . Then, obtain the observability canonical In the adaptive controller design, it will be carried out by the recursive calculation of the previous section. That is, based on the ) ( and
Indirect Adaptive Controller Design
using pseudo-inverse of ) (35) , the method employing the pseudo-inverse is effective for the plant with measurement noise (Kase & Mutoh, 2000) . It may be also useful for the improvement of the transit response of MRACS. 
Indirect Adaptive Controller Design
Then, using ) ( and
Indirect Adaptive Controller Design
, the adaptive control input is given by
Indirect Adaptive Controller Design
The global stability of the over-all system may be proved under the assumption 5. However, the details are under studying.
Numerical Examples
Consider the following plant: 
Numerical Examples
For the above three cases, the interactor can be obtained by solving
Numerical Examples
can be obtained by solving
Numerical Examples
Since C -matrix does not depend on t in this example and ) , ( A C is observable independent on t, ) ( 1 t O always has column-full rank and thus the above equation has a solution.
Numerical Examples
Finally, ) ( t X can be calculated by
Numerical Examples
The reference signal vector was given by 
Conclusion
In this chapter, an indirect MIMO MRACS with structural estimation of the interactor was proposed. By using indirect method, unreasonable assumptions such as assuming the diagonal degrees of interactor can be avoided. Since the controller parameters are calculated based on the observability canonical realization of the estimated values, the proposed method is suitable for on-line calculations. In the proposed method, the degree of the controllers do not depend on the estimated structure of the interactor. The global stability of the overall system is under studying. 

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