An Iterative Approach to the Fixed-Order Robust H Control Problem Using a Sequence of Infeasible Controllers

It is well known that the robust disturbance attenuation against uncertainties can be achieved by the robust H controllers and some practical situations make us use the fixedorder controllers. These facts imply that the fixed-order robust H controllers are important for practical control problems. However it is difficult to design such robust controllers, because the robust H control problems include an infinite number of matrix inequality constraints, in other words, they are described by Robust Semi-Definite Programming (RSDP) problems. For obtaining a feasible solution of the RSDP problems coming from the robust control problems with state feedback controllers or full-order controllers, many numerical methods have been proposed. Classically, the quadratic stability theory, i.e. a common constant Lyapunov function for the entire uncertain set is used for reducing the infinite constraints to the finite ones at the expense of conservatism (Boyd et al. 1994). Recently, parameter dependent Lyapunov functions are used to improve the conservatism (Chesi et al. 2005) (Ichihara et al. 2003), (Kami et al. 2009) (Shaked 2001), (Xie 2008) and some one-shot type approaches using extended LMI conditions, which allows to use the affine parameter dependent Lyapunov functions, have been proposed (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008). However these methods can not always produce the robust controller, because common additional variables are required and these methods can not be used for designing fixed-order controllers. In this sense, an iterative type approach may be useful to the problems such that these one-shot type approaches can not be applied. In the field of the numerical optimization, there are two types of iterative approaches for finding feasible or locally optimal solutions of the optimization problems: one is an interiorpoint approach which needs an initial feasible solution to be carried out and the other is an exterior-point approach which does not need it. From these facts, exterior-point approach can be efficient for obtaining the solutions of the problems such that feasible solutions are difficult to be found. However, there are no exterior-point approaches except those in (Iwasaki & Skelton 1995), (Kami & Nobuyama 2004), (Kami et al. 2009), (Vanbierviet 2009) for control problems to our knowledge.


Introduction
It is well known that the robust disturbance attenuation against uncertainties can be achieved by the robust H  controllers and some practical situations make us use the fixedorder controllers.These facts imply that the fixed-order robust H  controllers are important for practical control problems.However it is difficult to design such robust controllers, because the robust H  control problems include an infinite number of matrix inequality constraints, in other words, they are described by Robust Semi-Definite Programming (RSDP) problems.For obtaining a feasible solution of the RSDP problems coming from the robust control problems with state feedback controllers or full-order controllers, many numerical methods have been proposed.Classically, the quadratic stability theory, i.e. a common constant Lyapunov function for the entire uncertain set is used for reducing the infinite constraints to the finite ones at the expense of conservatism (Boyd et al. 1994).Recently, parameter dependent Lyapunov functions are used to improve the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008) and some one-shot type approaches using extended LMI conditions, which allows to use the affine parameter dependent Lyapunov functions, have been proposed (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008).However these methods can not always produce the robust controller, because common additional variables are required and these methods can not be used for designing fixed-order controllers.In this sense, an iterative type approach may be useful to the problems such that these one-shot type approaches can not be applied.In the field of the numerical optimization, there are two types of iterative approaches for finding feasible or locally optimal solutions of the optimization problems: one is an interiorpoint approach which needs an initial feasible solution to be carried out and the other is an exterior-point approach which does not need it.From these facts, exterior-point approach can be efficient for obtaining the solutions of the problems such that feasible solutions are difficult to be found.However, there are no exterior-point approaches except those in (Iwasaki & Skelton 1995), (Kami & Nobuyama 2004), (Kami et al. 2009), (Vanbierviet 2009) for control problems to our knowledge.
In this paper, we deal with the fixed-order robust H  controller synthesis problem against time invariant polytopic uncertainties, which can be described by parameter dependent bilinear matrix inequality (PDBMI) problems.The purpose of this paper is to propose an iterative approach which is like an exterior-point one.To do that, we introduce an `axisshifted system' which is obtained by shifting the imaginary axis of the complex plane so that all perturbing closed-poles are included in the LHS of the shifted imaginary axis.Our approach constructs a sequence of infeasible controller variables on which the shifted imaginary axis returns to the original position while the H  norm of the axis-shifted system is less than the prescribed H  norm bound.The advantage of our approach is to be able to use any controller variables as an initial point.The efficiency of our approach is shown by a numerical example.In this paper, the following notations are used.

Problem formulation
In this paper, we consider the following plant () where () xt is the plant state, () wt is any exogenous input, () ut is the control input, () zt is the performance output, ()  y t is the measurement output and is an uncertain parameter vector whose elements satisfy ,1 , , .
For the controller d  () xt and the coefficient matrices in (7) are given by 0 1 () 0 () , ( ) , ( 0 , 1 , , ) , () 0 0 For the controller s  () cl xt and the coefficient matrices are given by () () , ( ) ( ) , , , , , For the closed-loop system (6) the control problem to be solved in this paper is defined as follows: Robust H  synthesis problem: Given an () : 0 This lemma implies that the robust H  synthesis problem (11) can be described as PDBMI problem, which has an infinite number of BMI constraints corresponding to all points on  .Hence it is difficult to obtain the feasible controller variables K achieving (13).One well known classical method for obtaining s  in the case that EI  is to use quadratic (parameter independent constant) Lyapunov functions (Boyd et al. 1994).i.e., defining to get the controller variables from where X and W are the solutions of the next inequalities: However the quadratic Lyapunov functions X do not always exist and even if they exist the obtained controller includes a high conservatism.Moreover, this method can be only used in the case that EI  .Recently, various studies with parameter dependent Lyapunov functions have been reported to reduce the conservatism (Chesi et al. 2005) - (Ichihara et al. 2003), (Kami et al. 2009) - (Shaked 2001), (Xie 2008).Especially, some interesting one-shot type approaches for designing static state feedback controllers or full-order controllers with extended matrix inequality conditions have been proposed (Pipeleers et. al., 2009), (Shaked 2001), (Xie 2008).However these methods do not always produce the feasible controllers in some cases, because some additional common matrix variables are required and this method can not be used in the case that EI  .In this paper, we propose an iterative approach to the fixedorder robust H  synthesis problem, which can be used if EI  .The features of our approach are to constructs a controller sequence from the infeasible region to the feasible one and to be able to use any matrix as an initial point.

Multi-convex relaxation method
In this section, let us consider the next PDMI problem where M z are symmetric matrices with appropriate sizes.It is well known that feasible solutions of the PDMI problem ( 17) are difficult to be obtained, because this problem has an infinite number of constraints corresponding to all points on  .In this section, we show the multi-convex relaxation method (Ichihara et al. 2003) which is used for reducing the infinitely constrained problem to a finitely constrained one for obtaining a feasible solution of (17).

Multi-convex function
In this subsection, we review the definition and the properties of the multi-convex function.
From the definition the multi-convex function has the next properties: Lemma 2 The next statements hold: 1.The function () f q is the multi-convex function if and only if () The maximum of the multi-convex function () f q on q ÎW is on the vertex of W (See Fig. 2).Using these properties the relaxation method for obtaining the feasible solution of ( 17) is shown in the next subsection.

Multi-convex relaxation
In this subsection, we show a relaxation method with multi-convex function (Ichihara et al. 2003) which is needed to derive our approach.The key idea of this method is to make the multi-convex upper bound of (, ) M z  .

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The multi-convex relaxation method can be described as the next lemma: Lemma 3 z is a feasible solution of the PDMI problem (17) if there exist z , Now, let us define 1 (, )   fz  and 2 () respectively, where )0 , : 1 , which is the necessary and sufficient condition for 1 (, )   fz  to be multi-convex function with respect to  .Then the function (, ) fz given by ( 24) becomes a multi-convex upper bound function of   2 1 () : .
Then, from the property of the multi-convex functions (, ) 0 fz  holds iff we have Therefore, z is a feasible solution of (,) 0 Mz   if there exist z , i  and i R which achieve (28) for all vert   and replacing ii R  by i Q in (28) we have ( 18).Using this lemma the problem ( 17) with an infinite number of constraints can be reduced into that with a finite number of constraints.

Iterative approach to the robust H  synthesis problems
In this section, we propose an iterative approach to the robust H  control problem (11) using Lemma 3. To do that, we introduce an `axis-shifted system' which is obtained by shifting the imaginary axis so that all perturbing poles are located in the LHS of the imaginary axis.The key idea of our approach is to return the shifted imaginary axis to the original position while the H  norm of the axis-shifted system is less than p  .The feature of our approach is to be able to use any controller variables as an initial point.Firstly, we add the practical assumption for the closed-loop system (6) such that the poles of the system (6) do not exist infinitely far from the imaginary axis on the RHS of the complex plane, i.e., there always exists a finite scalar  which achieves: and we introduce the following system using  , which is needed to derive our iterative approach: This system has the next property.Lemma 4 The system (30) is robustly stable for the parameter    .
Proof It is obvious from (29).Remark 1.In this paper, we interpret the meaning of " (,)  " as shifting the imaginary axis of the complex plane to the right by  (See Fig. 3).In this sense, the system (30) is called as `axis-shifted system' in this paper.Now, letting (,,) () 0  , , ,, ) : 0 Moreover, we have the next lemma with respect to the existence of  which achieves , (,,) Lemma 7 For a given k K achieving the condition (29) there always exists  achieving , (,,) Proof: Let us consider the next matrix: Then, from (29), we can choose  which is larger than the maximum eigenvalue of the next symmetric matrix: which implies that there exists  which achieves This inequality can be transformed into the next inequality: Using Lemmas 6 and 7, we propose the following iterative approach to obtain a feasible solution of the problem (11):

Algorithm
Step 1: Find any 1 K and let 1  and 1  be scalars which achieve and go to the next step.

Remark 1:
The key idea of our approach is to decrease k  so as to approach k  to 0, i.e., the shifted imaginary axis approach the original position while the H  norm constraint (, ,) ,     is achieved(See Fig. 4).This fact implies that the controller k K is updated from a non robust H  controller for the original system to a robust H  one as k increases.In this sense, this approach can be an exterior-point approach.Remark 2: Unfortunately, our approach can not always produce a robust H  controller, in other words, there does not exist the efficient ways of choices of 1 K , 1  , 1  and  so that a feasible robust controller is always obtained.Hence a condition for detecting an infeasibility for obtaining a robust feasible H  controller may be needed.Moreover, 0

Numerical example
To demonstrate the efficiency of our approach let us consider the following matrices:

Conclusions
In this paper, we have considered the robust H  control problem against time invariant uncertainties.Firstly, we show the relaxation method for obtaining a feasible solution of the PDMI problem with multi-convex functions.Secondly, we introduce the axis-shifted system and show that this system can be constructed so as to achieve the H  norm constraint.
Next, we propose an iterative approach using the axis-shifted system and multi-convex relaxation method for obtaining the robust H  controllers.The property of our approach is to construct a controller sequence on which the shifted imaginary axis approaches the original position with the H  norm constraint achieved and to be able to choose any controller variables as an initial point.Finally we have given a numerical example which shows the efficiency of our approach.

(Fig. 3 .
Fig.3.Concept of complex plane of the axis-shifted system. Hence we may also need a efficient criterion for k K to be a feasible solution of the problem (11).

Fig
Fig. 7.The contour plot of

Fig. 8 .
Fig. 8. Behaviours of k  and k  . 34) (Shaked 2001) al. 2009)type methods(Pipeleers et al. 2009),(Shaked 2001), (Xie 2008) can not use for designing the robust H  controller because of EI  .For this numerical example, we set the initial condition for carrying out our approach as follows:   , i.e., the perturbations of poles of the uncertain closed-loop system via initial controller variables 1 K .This figure shows that 1 K is not a robust stabilizing controller.After 10 iterations the next controller variables are given from our approach: *