On The Design Of Output Feedback Controllers For Continuous-Time LTI Systems Over Fading Channels

This paper considers continuous-time linear timeinvariant (LTI) systems over fading channels, and addresses the design of output feedback controllers that stabilize the closed-loop system in the mean square sense. It is shown that a sufficient and necessary condition for the existence of such controllers can be obtained by solving a convex optimization problem in the form of a semidefinite program (SDP). This condition is obtained by introducing a modified Hurwitz stability criterion and by exploiting polynomials that can be written as sums of squares of polynomials.

Recently, the work (9) has investigated networked control systems with fading channels.Here the connections between plant and controller are affected by multiplicative white noise processes.Sufficient and necessary conditions for stability in the mean square sense are provided.Also, sufficient and necessary conditions for the existence of stabilizing state feedback controllers are provided.
This paper aims at generalizing the results provided in the work (9) by considering the design of output feedback controllers for continuous-time LTI systems over fading channels.Specifically, a continuous-time LTI plant is controlled in closed-loop by an output feedback controller, which is connected through fading channels modeled as multiplicative white noise processes.The problem consists of establishing the existence of controllers in a desired semialgebraic set such that the closed-loop system is stable in the mean square sense.First, an equivalent reformulation of mean square stability is given as asymtotical stability of a suitable matrix.Second, an equivalent reformulation of Hurwitz matrx is derived based on a modified Hurwitz stability criterion.Third, a sufficient and necessary condition for the existence of stabilizing output feedback controllers is obtained based on the solution of a convex optimization problem in the form of an SDP by exploiting polynomials that can be written as sums of squares of polynomials.
The paper is organized as follows.Section 2 introduces some preliminaries.Section 3 describes the proposed results.Section 4 presents the illustrative example.Section 5 reports the conclusions.

Preliminaries
The notation used in the paper is as follows.The sets of real numbers and complex numbers are denoted by  and .The notation ) (  denotes the real part of a complex number  .The inverse and the transpose of a matrix A are denoted by 1 A  and ' A .The notation det( ) A denotes the determinant of a matrix A .The spectrum of a matrix A , i.e., the set of eigenvalues of A , is denoted by spec( ) A .The vector obtained by stacking all the columns of a matrix A into one column vector is denoted by ) ( vec A .

Problem Formulation
Let us consider the situation depicted in Figure 1 where a plant is controlled in closed-loop by an output feedback controller over fading channels.
The plant is described by where is the plant input, and The controller is initially supposed static for clarity of presentation 1 , and is described by where m v  is the controller output and 1 As it will be explained in Remark 2, the proposed methodology can be used also to design dynamic output feedback controllers.
where The closed-loop system obtained by connecting the output of the controller (2) to the input of the unstable plant (1) over the fading channels (4) can be described by Let us further define Definition 1 (12) The closed-loop system (10) is said to be stable in the mean square sense if () Xt is well-defined for all 0 t  and ( ) = 0 (0) .lim The basic problem addressed in this paper is as follows.
Problem 1 Design a static output controller K in the set such that the closed-loop system (10) is stable in the mean square sense.

Modified Routh-Hurwitz Table
The Routh-Hurwitz stability criterion provides a necessary and sufficient condition for establishing whether the roots of a univariate polynomial with real coefficients have negative real parts (13) .The entries of this table are rational functions of the coefficients of the polynomial.Hereafter, we introduce a modified Routh-Hurwitz table where the entries are polynomial functions of these coefficients instead of rational functions (14) .

(s v
By multiplying each entry by their denominator, one obtains the modified Routh-Hurwitz table defined as where the number of rows is It can be easily shown that all the roots of ) (s v have negative real parts if and only if the first column of the modified Routh-Hurwitz table contains positive entries only.

SOS Polynomials
Here we briefly introduce the class of sums of squares (SOS) polynomials (15) :

Stability Analysis
In this subsection, we consider the stability analysis of the closed-loop systems (10) where Proof.The closed-loop system can be rewritten into the o Ît form as follows: is defined in (12).Hence, it can be observed that )), ( is a symmetric matrix, one can remove the replicated entries using (25).Then, it can be obtained that )), where G is a full column rank constant matrix.After pre- multiplying the above equation by , which follows that the closed-loop system (10) is stable in the mean square sense if and only if where  is defined as ( 26)- Let us observe that () K  can be considered as a matrix alternative to for describing the closed-loop system.

Controller Synthesis
Let us start by defining the new variable and let us define 1), ( 2 which is the size of the square matrix  For   , let us define the polynomial


Since Assumption 1 holds, it follows from the work (16) .
. Thus, the optimal solution of *  in (36) should be less than or equal to 0 , contradicting (35), which completes the proof.
Theorem 3 states that one can establish whether 0 > r by solving the optimization problem (36), which is an SDP.In particular, this theorem provides a sufficient condition for any chosen degrees of the polynomials Moreover, this condition is also necessary when these degrees are large enough.
The following theorem explains how one can use the SDP (36) to determine a controller that solves Problem 1.

Theorem 4 Suppose that
Proof."  " Suppose (37) holds.Then, the first and the third conditions in (38) hold with the maximizer * K of (32).It follows that

Examples
In this section, we present a numerical example to illustrate the proposed results.The computations are done by Matlab with the toolboxes SeDuMi (18)  and SOSTOOLS (19)  .
The set  ˆ is defined with

Conclusion
This paper has considered the design of stabilizing output feedback controllers for continuous-tme LTI systems over fading channels.It has been shown that a sufficient and necessary condition for the existence of such controllers can be obtained by solving a convex optimization problem in the form of an SDP.
The notation I denotes the identity matrix with the size DOI: 10.12792/icisip2016.067specified in the context.For scalars is denoted by deg( ( )) p  .The symbol  denotes the Kronecker product.The operator ) determined in the semi-algebraic set
containing all the monomials of degree less than or equal to d in s and , satisfied for typical semi-algebraic set such as hyper-ellipsoids, hyper-rectangles, etc.

Figure 2 .
Figure 2. Next, we solve the SDP (36) finding the upper bound of r given by .
over fading channels.

.
Next, let us exploit the modified Routh-Hurwitz table to derive an equivalent condition of Theorem 1.Let T be the modified Routh-Hurwitz table of the characteristic polynomial (30).As defined in 2.2, one can observe that all the entries of the modified table are polynomials with respect  can be simply chosen a priori as the smallest positive number allowed by the available computing platform.