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Theorem Consider the switched linear system 1. Ifthere exist a set of matrices. Remark The nonincreasing and bounded properties with relationship between L2 induced gain and dwell time hold for continuous-time case through considering optimization problem. Furthermore, similar to the stability analysis case, the L related to the division of dwell time interval has to be prespecified, and, obviously, if a larger L is chosen which implies a denser division, a more precise characterization can be attained.
Block-Diagonal Matrix-Valued Lyapunov Function and Stability of an Uncertain Impulsive System
Furthermore, since Dr c DT. Noting the sufficient conditions in Theorems 6 and 10, the LMIs are all affine in system matrices. As a result, many other existing standard techniques can be easily employed into Theorems 6 and 10; for example, when the system matrices contain uncertainties, the standard techniques such as  dealing additive uncertainties or  for polytopic uncertainties can be straightforwardly used in Theorems 6 and Thus, the discretized Lyapunovfunction approach not only obtains results for stability and disturbance attenuation issues, but also provides a general convex tool for dwell-time constrained switched systems.
Consider the control synthesis problem, the following switched linear system with control input is given as. As for switched system 31 , the control synthesis problem basically involves controller and switching law design.
By the previous results for dwell-time constrained switched system, such as Lemma 3, due to its nonconvex nature, most of the control synthesis procedures only have to be executed undesirably by two separate steps:. The two-step design method abruptly splits the controller and switching law design into two independently parts, which ought to be related to each other, and thus it often leads to conservative results for control problems under the dwell time constraint.
In this section, we are going to establish a one-step design methodology, by which the controller can be determined with respect to each specific dwell time, or in other words, the controller and dwell time can be determined simultaneously. We note that the information of dwell time is involved within the LMI-based convex sufficient conditions proposed in. Hereby, we are going to formulate the controller of discretized time-scheduled structure; then, the controller and switching law can be obtained by one unified step which explicitly facilitates the control synthesis process.
Assuming the switching between system and the corresponding controller is synchronous, the mode-dependent time-scheduled state feedback controller which we are interested in is given in the form of. After substituting controller 32 into system 31 , the closed-loop system becomes. As far as we know, due to the unsolved open problem on L2 gain, few results about relevant HTO control problem under dwell time constraint have been reported so far, though there exist numerous papers which are concerned with the weighted HTO performance based on Lemma 3.
Some previous results [26, 36], claimed that the HTO control performance can be obtained by a straightforward extension from Lemma 3. Unfortunately, these results suffer from a mistake that the desired HTO performance shouldbe replaced by weighted HTO performance; readers can refer to the recent comment paper ; however, the nonweighted control problem performance related to dwell time still remains unsolved. In this section, inspired by the discretized Lyapunov function approach presented in previous section, the sufficient conditions for the existence of HTO controller of switched system Motivated by the discretized Lyapunov function technique in continuous-time case, we define the following two time-scheduled matrices S t t and X i t , i e I, which is discretized as follows:.
Then, with the aid of above discretized matrices, the following theorem can be obtained. Consider switched system 1 with dwell time t. The philosophy of one-step method is also applicable in continuous-time case with Theorem Once the parameter L is chosen, the controller gains are determined with respect to a specific dwell time by one step via solving LMIs 36 - And the controller with optimal HTO control performance can be computed by.
Similar to the discussion for disturbance attenuation analysis, a larger L will yield a less conservative design result. Example The switched system considered is composed of two subsystems. The HTO control problem will be considered in the following example involved. The computation on optimal HTO control performance versus dwell time with difference L is shown in Figure 1. The relationship between dwell time and HTO induced gain is characterized in Figure 1. From the numerical results in Figure 1, we see following facts:.
The stability and disturbance attenuation properties for switched system with dwell time constraint are investigated in this paper. By introducing the discretized Lyapunov function technique, sufficient convex conditions ensure that the switched system GUAS are derived for continuous-time case. Then, the idea of discretized Lyapunov function is further extended to disturbance attenuation performance analysis in the sense of L2 gain, which has been viewed as an open problem not completely solved so far.
Due to the convex feature of analysis results on the basis of discretized Lyapunov function, the HTO control synthesis problem is taken into consideration, and a one-step design method is proposed, which has obvious advantages over conventional two-step design method. By one-step design method, the controllers for subsystems and the switching law can be determined.
As what has been discussed in this paper, the discretized Lyapunov function technique can not only derive results for stability and disturbance attenuation issues, but also provide a general convex tool for dwell-time constrained switched systems, which can be used to studied other issues for switched systems in our future work. The authors declare that there is no conflict of interests regarding the publication of this paper. This work is supported by Sichuan Provincial Education Department no.
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