∞ filtering for time-invariant continuous–discrete linear systems
Section snippets
Continuous–discrete linear systems
Throughout this paper, we consider the filtering for the continuous–discrete linear systems. In the following, the structure and state-space model of the continuous–discrete linear systems are presented. In the literature, we can find several general definitions of the hybrid systems [21], [22]. Even though the system model considered in this paper is a special case of the general hybrid systems, it is remarkable that the continuous–discrete linear system models have a sufficient potential
Problem formulation
Let us consider an filter which is realized by the state-space equations as follows:where and are filter states which are represented by continuous and discrete-time dynamics respectively, and is the estimated state. To consider more general solutions of synthesis, we assume that the estimation affects the dynamics of the system as follows:
Closed-loop dynamics
Let us combine the system model (8) and the filter (5). By substituting the output y(t) in Eq. (8) into the input y(t) in Eq. (5), and using the performance measure in (7), the closed-loop dynamics from the disturbance to the estimation error is obtained as follows:whereNote that the closed-loop dynamics (11), (12) represents the system in Fig. 2. As a shorthand notation, the above closed-loop dynamics (11) can be rewritten as follows:where the
Numerical example
In this section, we apply the proposed filter design technique to the following continuous–discrete linear system. In the following, a step-by-step procedure for the filter design is given:
- Step 1:
Apply similarity transform to the system using the permutation matrix (35), then the matrices HC, PC, QC, , , HD, PD, and QD (see Eqs. (22), (23), (30)) are obtained with the transformed matrices (37).
- Step 2:
Using Lemma 3.3, compute , , , and . For illustration purposes, a solution is listed
Conclusion
In this paper, analytic filter design method for a class of LTI continuous–discrete linear systems has been studied. Using the well-known bounded real lemma, constraints and Lyapunov stability conditions have been formulated by the two LMIs. Even though we assumed a special case of the continuous–discrete linear systems, an analytic and numerical method to solve a convex sub-optimization problem with constraint has been made in the class of continuous–discrete linear systems. Using the
Acknowledgement
This research was supported by the National Research Foundation of Korea (NRF) (2013R1A2A2A01067449).
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