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The concept of the “cone Ẋ(x) of all realizable state-velocity values” ẋ at each x in state-space, is used to characterize important geometric properties of state-trajectories x(t) for controlled dynamical systems. The results are useful in discovering invisible “barriers” in state-space across which controlled state-trajectories cannot pass. The new concept of hyper-controllability is introduced...
The ability of a dynamical system to remain stable in the face of perturbations in values of the system's parameters/coefficients is an important safety-attribute in control-system analysis and design and is referred-to as "stability-robustness". The most fundamental question in the study of stability-robustness is to identify, or safely-approximate, the "extent/range" of parameter-variations...
In this paper, two new spectra?? canonical relizations are developed for the class of time-varying Scalar Linear Dynamical Systems (SLDS) of the form: y(n)+??n(t)y(n-1)+...+??2(t)??+ ??1(t)y=??(t)u(n)+??n(t)u(n-1)+...+ ??2(t)u??+??1(t)u, and the class of completely controllable time-varying Vector Linear Dynamical Systems (VLDS) of the form: x??=A(t)x+b(t)u, y=C(t)x+d(t)u, where A(t), b(t), C(t) and...
In a series of recent papers [11], [12], the new notions of ED-eigenvalues, D-spectra, D-characteristic equation, and D-similarity transformations were introduced for (time-varying) Vector Linear Dynamical Systems (VLDS) of the form ??= A(t)x and Scalar LDS (SLDS) of the form y(n) + ??nk=1 ??k(t)y(k-1)=0. The present paper establishes several fundamental features of D-similarity classes of (time-varying)...
Let A(t) be a real-valued matrix function on [0, + ??]. A (time-varying) Linear Dynamical Systems (LDS) of the form x=A(t)x is said to be well-defined if A(t) is Lebesgue integrable on every finite subinterval of [0, + ??], and is called proper if A(t)=f(t,G) for some constant generating matrix G and scalar primitive function f(t, ??) (see [13], [15]). According to a recent result obtained in [17],...
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