**1. Introduction**

The study of optimal control for time-varying systems involves, in general, the solutions of Riccati differential equations (RDEs) and computations of the time-varying correction gains [1– 4]. It is noted that the system is typically computer-implemented system, upon which the RDE and correction gains are calculated. The computations, however, induces unavoidable time

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delay. Although the time-delayed control has been considered, it leads to two disadvantages complication of control mechanism and bulk of the control board. For some systems, for example, hard disk drives (a typically time-varying system) can only tolerate no delayed or very limited time delay control [5, 6] and use very small compartment. Hence, many literatures have focused on the static gain control of time-varying systems or the systems with time-varying or nonlin‐ ear uncertainties [7, 8]. It represents the simplest closed-loop control form but still encounters problems. One should aware that static output control is nonconvex, in which iterative linear matrix inequality approaches are exploited after it is expressed as a bilinear matrix inequality formulation( see [9–12]). As a result, it cannot be easily implemented in controlling the timevarying system and time delay problems remain.

It is a great challenge problem to design a linear continuous time-invariant observer with constant correction gains that regulate linear continuous time-varying plants. Although the vast majority of continuous TV control applications are implemented in digital computers [6, 13, 14], there are still opportunities to implement control with Kalman observer in continuous time (i.e., in analog circuits) [Hug88]. In particular, those control systems requiring fast response ask no or little delay effects. The difficulties for setting up those boards are because the algorithm of the design is too complex to implement in board level design, too expensive which can only be realized in a laboratory, or digital computation time induced unsatisfactory delay. It should be noticed that to realize the Kalman observer involves the computation of Riccati differential equations and inversion of matrices, which cause the obstacles of the board level design. A survey of linear and nonlinear observer design for control systems has been conducted in the literatures [15–18] and references therein. For controlling an linear timeinvariant (LTI) system, the Lungerber observer [19] design with constant correction gain is straightforward and can be implemented on a circuit board with ease.

Many practical control systems implement time-invariant controllers with observers in the feedback loop, which can be easily realized not only in the laboratory but also in the industrial merchandize [20]. The advantages of realization for the time-invariant controllers and observers are due to the constant parameters, which can be easily assembled by using resistors and other analog integrated elements in circuits board. The use of observers is also essential in industrial controls due to, in some cases, the states can be either not reachable or expensive to be sensed. Therefore, the use of observers are undoubtedly required to estimate unmeasured states since not merely full-state feedback control can be easily implemented but unmeasured states can be monitored [21–25].

With the aforementioned disadvantages and advantages, the control of time-varying systems is naturally arisen by designing a time-invariant observer-based controller that stabilizes, in particular exponentially, this time-varying plant. It is believed that this is a great challenge problem since we found no literatures tackling this problem. In what follows time-varying system control is first reviewed for laying the foundation of the robust control of the system with optimality property.

The feedback control of linear time-varying system has been extensively studied [1, 6, 7, 26– 31]. The key observation of early works for exponential stability of time-varying systems requires that the time-dependent matrix-valued functions be bounded and piecewise contin‐ uous satisfying Lyapunov quadratic stability [29, 31]. In this regard, many, but not all, of them can be translated to robust control framework since time-dependent matrices are essentially bounded and are treated as uncertainties [8, 32]. This gives an opportunity for the control system design without solving RDEs, although what we pay for the avoiding solving RDEs is the conservative of control. The conservativeness comes from two reasons—solutions of RDEs are avoided and admit fast varying parameters. This, however, can be reduced by designing parameter-dependent type of criterion or introducing slack variables such that reduces the tightness of dependent variables( see, e.g., [33] and reference therein).

This paper is organized as follows. The following section, Section 2, sets up the time-varying systems to be tackled, time-invariant observer to be built, feedback control problems to be solved, and system properties (assumptions) to be with the systems. Section 3 gives the main results for solving the feedback control problems, in which LMIs characterize the quadratically stability of the closed-loop system with L2-gain of the closed-loop system is preserved; in addition, least-squares algorithm is suggested to drive the time-varying observer such that the time-varying plant states can be estimated asymptotically. Consequently, Section 4 demon‐ strates the synthesis of the static gains of control input and correction gains of observer. To verify the effectiveness, illustrative applications are to test the overall design of the feedback closed-loop system. The last section, Section 5, concludes the overall paper.
