انفعال از سیستم های خطی انتخاب شده : تجزیه و تحلیل و طراحی کنترل

A passive system with positive definite storage function is not only stable but is intrinsically robustly stable with respect to a wide class of feedback disturbances. For linear time invariant systems, passivity can be characterized either in time domain or in frequency domain from positive realness. This paper aims to generalize this concept to continuous-time switched linear systems. Analysis is performed by taking into account state dependent and arbitrary time dependent switching functions with a prescribed dwell time. A control design problem related to the determination of a switching strategy, based upon output measurements, that renders a switched linear system passive is also considered. The methods introduced in the paper can be effectively applied to the control of the duty cycle and passivation of switched circuits.

Recently, much attention has been paid in the circuit and control communities to switched dynamical systems, where the continuous-time dynamics is subject to abrupt changes according to a switching signal, either exogenous or controlled. Switched circuits and power converters are natural examples of interesting applications of switched systems theory. The stability analysis of continuous-time switched linear systems has been addressed by many authors, [1], [2], [3], [4], [5] and [6]. General results on this topic are presented in the book [7] and in the survey papers [8] and [9]. The reader is requested to see [10] and [11] for complete reviews on stability of continuous-time switched linear systems. Passivity is one of the main concepts at the intersection of circuit and control theory, [12] and [13]. Roughly speaking, a system is passive if it dissipates energy without generating its own. This notion encompasses the property of stability, and positive storage functions that characterize passivity can be qualified as Lyapunov functions. In this framework, passivity is intimately related to stability robustness of the system under negative feedback perturbations belonging to certain classes, including sector static nonlinearities and passive dynamical systems, [14]. The passivity concept of general nonlinear switched systems was originally considered in [15] where multiple positive definite storage functions have been successfully used for stability analysis and for the design of a state dependent switching law that renders the closed-loop system passive, see also [16]. In switched circuits, the passivity property proved to be a powerful tool for the control of power converters, [17] and [18], and fault tolerant control of RLC circuits, [19]. The present paper is focused on the class of switched linear systems, whose passivity properties are investigated by resorting to suitable linear matrix inequality (LMI) conditions, similarly to what is usually done for linear time-invariant systems (LTI). Differently from the latter case, our analysis calls for piecewise quadratic positive storage functions. The paper can be seen as an extension of the conference paper [20]. Considering a switched linear system with input w(⋅)∈Rmw(⋅)∈Rm and output z(⋅)∈Rmz(⋅)∈Rm, the main goal is to design a switching signal such that the closed-loop system satisfies equation(1) View the MathML source∫0∞z(t)′w(t)dt≥0,∀w∈L2. Turn MathJax on To this end two classes of design problems are considered. The first one, called exogenous switching , is characterized by switching signals σ(⋅)σ(⋅) with a given dwell time T>0T>0, that is σ∈DTσ∈DT. An upper bound of the minimum dwell time T∗T∗ is determined preserving the validity of (1) for all σ∈DT∗σ∈DT∗. The complete solution to this problem is extremely difficult to obtain and, to our best knowledge, only few references are available up to now in the literature, see [15] and [16] and the related papers on RMS gain [21], [22] and [23]. The design problem, converted into an optimal control problem is characterized through the Hamilton–Jacobi-Bellman equation which is virtually impossible to solve due to the algebraic structure of the set DTDT for T>0T>0 given. Refs. [24] and [25] give an idea of the difficulties if one want to solve this kind of optimal control problem. Hence, for the moment, suboptimal solutions easier to calculate are acceptable. Notice that the knowledge of a dwell time guaranteeing passivity can be useful in the design of the duty-cycle of switched circuits. The second class, called controlled switching , is characterized by assuming that the switching signal is of the form σ(t)=u(x(t))σ(t)=u(x(t)) where u(⋅):Rn→K={1,2,…,N}u(⋅):Rn→K={1,2,…,N} is a state-dependent switching function to be determined in order to preserve the inequality (1). As an important generalization we analyze the possibility to define the switching function using only partial information provided by a measured output. This control strategy can be beneficial for passivation via switching of nonpassive circuits. The paper is organized as follows. After some preliminaries presented in Section 2, the main results for LTI systems concerning passivity are recalled in Section 3. The dwell-time analysis problem with exogenous switchings is treated in Section 4 whereas the controlled switching design is dealt with in Section 5. The paper ends with Section 6 where two simple examples are discussed. The first concerns the computation of the dwell-time preserving passivity of a switched system composed by two passive subsystems. The second puts in evidence the usefulness of the design results in providing switching strategies that are able to orchestrate two LTI non passive subsystems to become a passive switched linear system. The notation used throughout is standard. For real matrices or vectors (′′) indicates the transpose. The symbol (•)(•) denotes the symmetric blocks of a symmetric matrix. The set McMc denotes the set of Metzler matrices Π∈RN×NΠ∈RN×N with nonnegative off diagonal elements satisfying the normalization constraints View the MathML source∑j=1Nπji=0 for all i=1,…,Ni=1,…,N. Finally, the square norm of a trajectory s(t)s(t) defined for all t≥0t≥0, denoted by View the MathML source‖s‖22, equals View the MathML source‖s‖22=∫0∞s(t)′s(t)dt. All trajectories with bounded norm constitute the set L2L2. The set of all nonnegative integers is denoted by NN.

This paper presents new results on switched linear systems passivity. For switching functions considered as exogenous perturbations, the determination of an upper bound to the minimum dwell time that preserves passivity is investigated by using a multiple positive definite storage function that solve the HJB-inequality. On the other hand, the design problem of imposing passivity by means of a state or output dependent switching function is treated from a min-type positive definite storage function. In both cases, numerical solutions are found (if any) by solving LMIs together with a line search, in the first design problem.