INTRODUCTION
Eigenstructure assignment one of the most powerful methods in multivariable
control systems design. According to the results of Konstantopoulos
and Antsaklis (1996) and degrees of freedom in eigenstructure assignment
method by using output feedback (Duan, 2003), in recent
two decades using eigenstructure assignment has been developed in all field
of multivariable control systems. Reconfigurable control design (Konstantopoulos
and Antsaklis, 1996), Helicopter control systems (Manness
and Smiths, 1992), Control reconfiguration in second order dynamic systems
(Wang et al., 2005), missile control systems
(Sobel and Cloutier, 1992), robust control design in
electrical induction machine (Duval et al., 2006)
are examples of using eigenstructure assignment.
Most of the methods that have used eigenstructure assignment for designing
robust control, only consider eigenstructure assignment sensitivity and they
haven’t combined eigenstructure assignment with other robust control performances.
In recent years, some researches have been performed for such problems. Apkarian
et al. (2001) a new method for combining eigenstructure assignment
with H_{2} constraint has been suggested. He et
al. (2004) a new method for designing state feedback control to satisfy
multiobjective controller design (specially H_{∞} constraint)
has been proposed. However, this method can be used when all states are available.
In this study, combining eigenstructure assignment problem and robust H_{∞}
control design is investigated while states aren’t measurable. According
to parametric eigenstructure assignment (Duan, 2003)
characteristics of enhanced LMI (Shimomura et al.,
2001), a method is proposed. Based on this method the output feedback controller
will be designed such that robust H_{∞} control performance is
satisfied. Also, the method is developed for dynamical output feedback. By developing
this method for dynamical output feedback, degrees of freedom in dynamical feedback
can be used.
In this study M* is transpose conjugate of M.M>0 means M is positive definite matrix. H_{∞} is used for Hermitian matrix. Diag(.) is used for diagonal matrix.
PARAMETRIC EIGENSTRUCTURE ASSIGNMENT VIA OUTPUT FEEDBACK
Consider the following LTI system:
where, x∈R^{n}, u∈R^{r}, y∈R^{m} are, respectively, state vector, input vector, output vector. A, B and C are known matrices with appropriate dimensions. By applying the static output feedback of the form:
In Eq. 1, the closed loop matrix is obtained as follow:
Based on eigenstructure assignment definition, we have:
where, V is right eigenvectors matrix and J represents the Jordan form of the
closed loop matrix (3). We know that the eigenvalues of nondefective matrix
are distinct and they have less sensitivity to parameter changes (Duan,
2003). Assume that the closed loop matrix A_{cl} is nondefective
then, the Jordan form of A_{cl} is diagonal. The Jordan form of A_{cl}
is:
where, s_{i} (i = 1,2,...,n) are desired closed loop eigenvalues. Also s_{i} (i = 1,2,...,n) are self conjugate complex numbers. Assume (A,B) is controllable. By applying SVD to the matrix [B s_{i}IA], we have:
where, Ψ^{c}_{i}, n^{c}_{i} are orthogonal matrices with appropriate dimensions and Σ^{c}_{i} is nonsingular diagonal matrix that diagonal elements are the singular values of [B s_{i}IA]. Partition φ^{c}_{i} as:
According to the above preliminaries and the results of Duan
(2003), the following theorem for parametric eigenstructure assignment by
using output feedback is described (Duan, 2003).
Theorem 1: Consider (A,B) is controllable and rank (C) = m then here exist matrices V∈C^{nxn}, det(V) ≠ 0 and K∈R^{rxm} satisfying (4), iff there exists parameter vectors f_{i}∈C^{r} which is satisfying the following constraints:
If the above conditions are satisfied, all matrices K∈R^{rxm} and V∈C^{nxn} (det(V) ≠0 are obtained as follow:
where, N_{i}, D_{i} are determined by Eq. 7. In special case rank(C) = n the output feedback is given by:
PROBLEM FORMULATION
Consider the following LTI systems:
where, x∈R^{n}, w∈L^{q}_{2}[0,∞), u∈R^{r},
z∈R^{1}, y∈R^{m}, respectively are state vector, exogenous
disturbance vector, input vector, state combination (objective functional signal)
and output vector. A, B_{1}, B_{2}, C_{1}, C_{2}
and D_{12} are known matrices with appropriate dimensions. Assume that
(A,B_{2}) is controllable and D*_{12} D_{12}>0. By
applying the static output feedback of the following form to Eq.
11:
The closed loop system is obtained as:
Based on Enhanced LMI characterizations that have been proposed (Shimomura
et al., 2001), we combine eigenstructure assignment problem with
H_{∞} control design. Consider the closed loop system Eq.
13. There exist positive definite Y∈H_{n} and S∈H_{n}
such that:
For some ε>0, iff T_{zw}_{∞} <γ_{∞}. Note that S is variable which is represented by the Enhanced LMI characterizations.
According the previous results and theorem (1) we state the main problem of
this study.
1. 
Main problem: Consider the system (13) and a set of
desired closed loop eigenvalues M = {S_{i}, i = 1,2,...,n}. We want
to determine the output feedback of the from Eq. 12 such
that H_{∞} control performance ε>0, T_{zw}_{∞},
<γ_{∞} or based on theorem 1 the LMI (14) is satisfied. 
According to theorem 1 and Enhanced LMI (14) the following theorem is represented to solve the main problem.
Theorem 2: Consider the system (1) that (A,B_{2}) is controllable.
There exist the output feedback of the form Eq. 12 for the
main problem, if there exist 0<ε<1 and matrices Y>0, M such that:
where, J = diag(s_{1}, s_{2},...,s_{n} ), also the following conditions are satisfied:
If the above conditions are met, the output feedback can be calculated as:
Proof: Based on the above definition we can rewrite (16) as follows:
The output feedback is obtained as:
By substituting V instead of S in Eq. 14, the closed loop system is satisfied the H_{∞} control performance if the following LMI is feasible:
According to Eq. 4 and C_{c1} = C_{1}+D_{12}KC_{2 }the following LMI is obtained:
If we substituted C_{2} instead of C in (8), where W = KC_{2}V and also by substituting V,W from (17) the following equation is obtained:
And the proof is completed. According to the above theorem, we can combine the parametric eigenstructure assignment problem with H_{∞} control problem. If the LMI (21) is feasible, then the main problem has a solution. Based on the method that is proposed in this note, we design the parametric eigenstructure assignment elements f_{1} such that, in addition to satisfy H_{∞} control performance, other performances are met too. Note that for solving main problem, satisfying all of the following conditions are important:
DYNAMICAL CONTROLLER DESIGN
For controllable system (8) if suppose that B,C are full rank then static output
feedback can assign max (m,r) eigenvalues and corresponding eigenvectors (Konstantopoulos
and Antsaklis, 1996). In general, we may desire to exercise some control
over more than max(m,r) closed loop eigenvalues. So we generalize this method
by using dynamical feedback of the form:
where, .
If combine the above controller with (1), then closed loop system is obtained:
Assume that:
Based on (24), Eq. 23 changes as follows:
According to the above equations is clear that dynamical output feedback controller can be translated as static output feedback. A number of eigenvalues of this system (25) is n+n_{c}. By choosing appropriate n_{c}, all of the eigenvalues can be assigned. We can use theses degrees of freedom in dynamical feedback, for satisfying other control objectives.
For solving the main problem in dynamical output feedback case, By combining
Eq. 11 and 22 the following equations
are obtained:
By substituting
instead of A,B_{1},....., respectively, the LMI (21) can be written
for new augmented system (26). Based on the new augmented system (26) and using
dynamical system, Eq. 21 can be represented as:
where,
are obtained from the similar Eq. 17, for augmented system
(26).
According to degrees of freedom in dynamical controller (by selection n_{c}), the controller can be determined such that in addition to satisfy the main problem, satisfying other control objective.
NUMERICAL EXAMPLE
In order to investigate the proposed method the following example is stated.
Consider the simplified third order linearized dynamics of a small aircraft
witch is given by (Satoh and Sugimoto, 2004). We assume
that the system has 40% uncertainty at A(1,1), A(2,2)and A(3,3). According to
(Satoh and Sugimoto, 2004) this structured uncertainty
can be described as follows:
and
where, p and r are roll rate (rad/s) and yaw rate, respectively. β is
the side slip angle (rad) and s_{a}, s_{r} are the aileron deflection
and rudder deflection angle(rad). Our objective is to design the output feedback
controller K such that the H_{∞} performance T_{zw}_{∞}
<1 is satisfied. The matrices in Eq. 11 are defined as:

Fig. 1: 
First initial output response 

Fig. 2: 
Second initial output response 

Fig. 3: 
Third initial output response 
Note that all of the calculation have been performed by MATLAB, Sedumi 1.1
and YALMIP 2.4. (Lofberg, 2004) Also the desired eigenvalues
are selected as:
By applying theorem 2 for ε = 0.01 and γ = 1 the output feedback and parameter matrix are obtained as:
Based on the above controller the eigenvalues are given by:
λ_{1} 
= 
1.0009 
λ_{2} 
= 
5.0002 
λ_{3} 
= 
10.0011 
The responses of the system (28) for initial condition [0.04 1 0] are shown
in Fig. 13. Figure 1
demonstrates the initial response of the first output, Fig. 2
shows the second initial response and Fig. 3 demonstrates
the third initial response.
In Fig. 13, the H_{∞} control design of uncertain and nominal system are shown. The uncertain system with robust H_{∞} output feedback controller is calculated from theorem 2. It can be seen that the proposed method is more effective for designing robust H_{∞} output feedback controller. It has been seen that the proposed controller can tolerate in the presence of uncertainties.
CONCLUSION
In this study robust H_{∞} control design via eigenstructure assignment is considered. According to enhanced LMI and parametric eigenstructure a method is proposed such that H_{∞} control performance is satisfied. Also based on degrees of freedom in dynamical output feedback the proposed method is developed. In order to demonstrate the effectiveness of proposed method the example is considered and from this example we saw that the proposed robust controller can deal with the uncertainty. Also, when we want to satisfy other control objective in addition to robust performance we can use the dynamical controller. By using dynamical controller, degrees of freedom in dynamical controller can be used.