The discrete-time tracking problem with H∞ model matching approach plus integral control

minimize the H∞ norm of the closed-loop transfer function and to maximize the closed-loop performance by introducing the model transfer matrix. In following, the discrete-time H∞ MMP via LMI approach is derived as the main result. The controller construction procedure is implemented by using a well-known toolbox to improve the usability of the presented results. Finally, some conclusions are given.


Introduction
The model matching problem has attracted a lot of attention in the control theory [13][14]. If Gm(z) and G(z) are the model and the system matrices, respectively, the discrete-time H∞ model matching problem (MMP) is introduced to derive a controller transfer matrix R(z) that minimizes the H∞ norm of Gm(z)-G(z)R(z). The model transfer matrix Gm(z) has the desired performance specifications defined by its poles and zeros. Moreover, Gm(z) and G(z)R(z) are stable and proper transfer matrices, that is Gm(z) and G(z)R(z)RH∞. The closedloop performance G(z)R(z) is considered to be approximated by the desired performance Gm(z) such that, H∞ MMP is elaborated in [5,8,9]. In these studies, the dynamic precompansator R(s) is obtained and then it is implemented by dynamic state feedback, [13]. Formerly, continuous-time H∞ MMP with one degree of freedom (1 DOF) static state feedback is derived in [1], the discretetime H∞ MMP with 1 DOF static output feedback and the continuous-time H∞ MMP with 2 DOF static output feedback is presented in [2][3], respectively. On the other hand, the integral control structure subject to the existence of state feedback is firstly used in [4].
In this paper, the discrete-time H∞ MMP with integral control is proposed by using a 2 DOF static output feedback. Both the solution of the discrete-time static H∞ optimal control problem (OCP) and discrete-time H∞ MMP is revisited toward the solution of our presented problem, whereas discrete-time H∞ MMP can be completely solved by the LMI-based numerical optimization. This paper is organized as follows: In Section 2, a special formulation for the discrete-time H∞ MMP by a 2 DOF static output feedback with integral control in linear matrix inequalities (LMIs) is elaborated. In Section 3, the main result is given by a theorem that provides two existence conditions of the solution. In Section 4, we construct the 2 DOF static output feedback with integral control by using this theorem. Some conclusions are finally given in Section 5.

R
The set of real numbers. C The set of complex numbers. R nxm The set of nxm real matrices.

Re(α)
The real part of αC.

L∞
The functions bounded on Re(s)=0 including at ∞.

H∞
The set of L∞ functions analytic in Re(s)>0. In An identy matrix of nxn dimension. 0n A zero matrix of nxn dimension. 0nxm A zero matrix of nxm dimension. KerM The kernel space the linear operator M. ImM The image space of the linear operator M. N T The tranpose of the matrix N. P>0 P positive definite matrix. dim(U) The dimension of the linear space U.

max(A)
The largest eigenvalue of the matrix A.

max(A)
The largest singular value of the matrix A defined as In order to present a synthesis theorem on the LMIbased characterization of the discrete-time H∞ model matching problem with integral control, the following lemmas are given. The first lemma is The Bounded Real Lemma and it is used to turn the discrete-time H∞ optimal control problem into an linear matrix inequality (LMI): Lemma 1.1 Consider a discrete transfer matrix T(z) of (not necessarily minimal) realization T(z)=D+C(zI-A) -1 B. The following statements are equivalent: i) ||T(z)=D+C(zI-A) -1 B||∞<γ and the matrix A is Schur (|λi(A)|<1, i=1,…,n). ii) There is a solution X>0 to the LMI: Proof: See [10].  Lemma 1.2 Suppose P, Q and H are matrices and that H is symmetric. The matrices NP and NQ are full rank matrices satifying ImNP=KerP and ImNQ=KerQ. Then there is a matrix J such that, if and only is the inequalities NP T HNP<0 and NQ T HNQ<0 (4) are both satisfied.
Proof: See [10].  In the sequel, P-MN -1 M T is referred to as the Schur complement of N.

The discrete-time H∞ mmp by dof static output feedback with integral control in lmi optimization
Toward the solution of the discrete-time H∞ MMP via LMI approach, the problem should be reformulated as standard discrete-time H∞ OCP. The state-space representation of the system G(z) and the model system Gm(z) is given: is the model state, v(k), w(k), ys(k) and ym(k)R m . We take that the given system is strictly proper in order to simplify the solution of the problem. The integral control is modelled by a serie integrator: and u(k)R m . The control input u(k) is generated by a two degrees of freedom static output feedback controller: The block diagram of a discrete-time H∞ MMP by a static output feedback with integral control is illustrated in Figure 1. In this formulation, the steady-output value ys(k) will follow a step function input with zero error. We will use a 2 DOF feedback control structure which is defined in the control theory, [12]:  2 The generalized plant P(z) shown in Figure 1 can be modelled as, Matrices are defined as follows : The above formulation concludes that the discrete-time H∞ model matching problem plus integral control with the two degrees of freedom static output feedback is equivalent to the discrete-time H∞ optimal control problem. This equivalency is drawn in Figure 2: A synthesis theorem on the LMI-based solution of the problem is presented in the following section.

Main result
We can now present a synthesis theorem on the LMIbased solution of the discrete H∞ model matching problem with integral control by two degrees of freedom static output feedback:

CONTROLLER CONSTRUCTION
Although Theorem 3.1 is about the solvability conditions of the discrete-time H∞ MMP by the 2 DOF static output feedback with integral control, it also provides a controller construction procedure. Moreover The MATLAB LMI Control Toolbox [9] can be used to solve LMIs. The controller construction procedure can be summarized as follows: Step 1: Find a solution X>0 to the LMIs (25) and (26) for opt which is the minimal of . Step

Conclusions
In this paper, we have studied the discrete-time H∞ model matching problem with two degrees of freedom static output feedback. We have induced integral controller to this classical problem. The introduction of integral type of controller to this configuration naturally forces the steadyoutput error to zero. Moreover, the nearly proposed block diagram reduces the problem to an H∞ optimal control problem and a theorem is proposed which provides a procedure to design the controller. However, we suppose that the two LMI conditions provided in the Theorem 3.1 can be simplified in future works.