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We consider the stability of the feedback interconnection of possibly unstable n-input n-output subsystems whose interconnection is described by e 1 = u 1 - y 2 , e 2 = u 2 + y 1 and y i = G i (e i ), i = 1,2. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on G 2 so that the stability of u 1 ↦ y 1 guarantees that of the feedback system. The other two theorems consider continuous-time linear time-invariant subsystems. It is noted that in the multivariable case, the stablity of u i ↦ y i , i = 1,2 is not sufficient to guarantee the stability of the feedback system, and Theorem 2 specifies some additional requited conditions. Theorem 3 shows that if G^ 2 and G^ 1 (I + G^ 2 G^ 1 )-1are in some special stable classes, so is the transfer function of the feedback system. In both theorems, corollaries specialize the results to lumped and single-input single-output cases. The paper ends by showing how these results can be translated for the discrete-time case.

Multiple loop linear time invariant feedback systems stability, discussing conditions, gain matrix and input-output properties

This paper considers reciprocal stationary systems. By requiring a strict convexity condition (condition (C) in the text), we can prove that six generic representations are globally equivalent. The key is the global inversion theorem of Palais. Examples are given in electric circuits and thermodynamic systems. Relations to geometric formulations are indicated.

This paper considers a nonlinear time-invariant network N (of order n+h+l) which contains, in addition to the usual elements, h stray elements (stray capacitances and lead inductances) and l sluggish elements (chokes and coupling capacitors). It is proved that the asymptotic stability of any equilibrium point of N is guaranteed once the simplified (i.e., with stray and sluggish elements neglected) linearized network and two other linear networks S H and S L are asymptotically stable. The networks S H and S L are obtained by both a physically intuitive argument and by a rigorous one. It is also proved that if any one of the three linear networks is exponentially unstable, then the equilibrium point of N is unstable. This theory explains the commonly occurring fact that N is unstable even though the simplified linearized network is asymptotically stable. An example illustrates the several possibilities. The asymptotic behavior of the natural frequencies of N (valid in a neighborhood of the equilibrium point) is obtained in the proof. It is shown in Appendix II how the natural modes of the three simple networks are related to those of the given network N.

This paper develops two theorems concerning the small-signal behavior of nonlinear time-varying networks whose state equations are of the form x·= f(x, u, t). The conclusions of the theorems are supported by experiments. The input is of the form U(t) + u(t), where the bias U(t) is allowed to be time-varying (typically, slowly varying) and u(t) is the small signal. The bias induces a moving operation point X(t). Given some simple assumptions concerning the linearized small-signal equivalent circuit it is shown that provided u(t) is sufficiently small on [0, ∞), the state trajectory about the operating point is bounded on [0, ∞) and tends to zero as u → 0. The method of proof also shows that this result applies to some distributed circuits. The second theorem shows that the push-pull connection reduces the distortion due to the nonlinearities of both resistors and energy storing elements. The third part of the paper describes numerical experiments that support the conclusions of the theory and a design procedure for nonlinear networks to be operated in the small-signal mode.

It is shown that in order that a Norton high-pass, low-pass complementary ladder network be a constant resistance, all its reactive elements must be linear. The proof requires a novel perturbational analysis of nonlinear networks which gives precise bounds, in terms of the input, of the effect of the nonlinearities. This type of perturbational analysis can be considered to be a contribution to the identification problem.

Optimal characteristics for single-input single- output memoryless time invariant nonlinear dynamic systems

The Fundamental stability theorem giving the input-output properties of linear time-invariant discrete systems is extended to include multiple poles on or outside the unit circle and to include properties of the error coefficients. The engineering significance of some of the assumptions is discussed.

Electrical networks are special types of irreversible thermodynamic systems. The underlying mathematical unity of the two fields allows many of the techniques of network analysis to be generalized, thus permitting a wide class of non-linear, non-steady-state irreversible processes to be formulated in a network theoretic framework. One of the most powerful of the network theorems is Tellegen's theorem. Most, if not all, of the energy distribution theorems and extremum principles can be derived from it. In the following presentation we will derive several thermodynamic inequalities as well as explicit conditions guaranteeing the stability or the unstability of an equilibrium point, or a steady state.

If all eigenvalues of all matrices Ai (i=0, 1, 2, ...) lie in a disc of radius

Re represents a linear time-invariant system of order n + h. When e → 0, Re, reduces to R0 of order n ≤ n + h. Conditions are given under which the bounded-input bounded-state stability of R0 implies that of Re and we show that, under these conditions, as e → 0, the state trajectories of Re approach those of R0 for any continuous input.