Publications of Eduardo D. Sontag jointly with Y. Wang |
Articles in journal or book chapters |
This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification. |
A result is presented showing the existence of inputs universal for observability, uniformly with respect to the class of all continuous-time analytic systems. This represents an ultimate generalization of a 1977 theorem, for bilinear systems, due to Alberto Isidori and Osvaldo Grasselli. |
The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y, then either Omega(x) < Omega (y), or Omega(x) = Omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems. |
This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem. |
This paper develops characterizations of various uniform stability properties of switched systems described by differential inclusions, and whose switchings are governed by a digraph. These characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of Lyapunov functions. |
We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of "separation principles" for input-output-to-state stability , characterizations of the property in terms of weaker stability notions. When applied to the closely related notion of integral ISS, these characterizations yield analogous results. |
The stability of differential inclusions defined by locally Lipschitz compact valued maps is addressed. It is shown that if such a differential inclusion is globally asymptotically stable, then in fact it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact (not necessarily convex) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability. |
A new notion of input-to-state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized. An example shows that this notion of stability is indeed weaker than the usual ISS. Applications to the study of global asymptotic stability of cascaded nonlinear systems are discussed. |
The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition. |
We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed. |
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators", and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates. |
Just as input to state stability (ISS) generalizes the idea of finite gains with respect to supremum norms, the new notion of integral input to state stability (IISS) generalizes the concept of finite gain when using an integral norm on inputs. In this paper, we obtain a necessary and sufficient characterization of the IISS property, expressed in terms of dissipation inequalities. |
This paper continues the study of the integral input-to-state stability (IISS) property. It is shown that the IISS property is equivalent to one which arises from the consideration of mixed norms on states and inputs, as well as to the superposition of a ``bounded energy bounded state'' requirement and the global asymptotic stability of the unforced system. A semiglobal version of IISS is shown to imply the global version, though a counterexample shows that the analogous fact fails for input to state stability (ISS). The results in this note complete the basic theoretical picture regarding IISS and ISS. |
This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated. |
This paper deals with several related notions of output stability with respect to inputs (which may be thought of as disturbances). The main such notion is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. For systems with no inputs, IOS provides a generalization of the classical concept of partial stability. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions |
The notion of input-to-state stability (ISS) has proved to be useful in nonlinear systems analysis. This paper discusses a dual notion, output-to-state stability (OSS). A characterization is provided in terms of a dissipation inequality involving storage (Lyapunov) functions. Combining ISS and OSS there results the notion of input/output-to-state stability (IOSS), which is also studied and related to the notion of detectability, the existence of observers, and output injection. |
This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets. |
We present new characterizations of the Input to State Stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature. |
This paper studies various stability issues for parameterized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state space stability to ``input to state'' (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analyzed. |
We show that the well-known Lyapunov sufficient condition for input-to-state stability is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided. |
This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems. |
It is shown that realizability of an input/output operators by a finite-dimensional continuous-time rational control system is equivalent to the existence of a high-order algebraic differential equation satisfied by the corresponding input/output pairs ("behavior"). This generalizes, to nonlinear systems, the classical equivalence between autoregressive representations and finite dimensional linear realizability. |
This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations. |
We prove that for any family of n-dimensional controllable linear systems, continuously parameterized by up to three parameters, and for any continuous selection of n eigenvalues (in complex conjugate pairs), there is some dynamic controller of dimension 3n which is itself continuously parameterized and for which the closed-loop eigenvalues are these same eigenvalues, each counted 4 times. An analogous result holds also for smooth parameterizations. |
This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls. |
Conference articles |
The fundamental Filippov--Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In a recent paper of ours, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties. |
For systems whose output is to be kept small (thought of as an error output), the notion of input to output stability (IOS) arises. Alternatively, when considering a system whose output is meant to provide information about the state (i.e. a measurement output), one arrives at the detectability notion of output to state stability (OSS). Combining these concepts, one may consider a system with two types of outputs, an error and a measurement. This leads naturally to a notion of partial detectability which we call measurement to error stability (MES). This property characterizes systems in which the error signal is detectable through the measurement signal. This paper provides a partial Lyapunov characterization of the MES property. A closely related property of stability in three measures (SIT) is introduced, which characterizes systems for which the error decays whenever it dominates the measurement. The SIT property is shown to imply MES, and the two are shown to be equivalent under an additional boundedness assumption. A nonsmooth Lyapunov characterization of the SIT property is provided, which yields the partial characterization of MES. The analysis is carried out on systems described by differential inclusions -- implicitly incorporating a disturbance input with compact value-set. |
This paper studies the input-to-state stability (ISS) property for discrete-time nonlinear systems. We show that many standard ISS results may be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties, as well as provide a small gain theorem. |
This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case. |
This paper deals with a notion of "input to output stability (IOS)", which formalizes the idea that outputs depend in an "aymptotically stable" manner on inputs, while internal signals remain bounded. When the output equals the complete state, one recovers the property of input to state stability (ISS). When there are no inputs, one has a generalization of the classical concept of partial stability. The main results provide Lyapunov-function characterizations of IOS. |
Contains a proof of a technical step, which was omitted from the journal paper due to space constraints |
Previous characterizations of ISS-stability are shown to generalize without change to the case of stability with respect to sets. Some results on ISS-stabilizability are mentioned as well. |
This paper studies various types of input/output representations for nonlinear continuous time systems. The algebraic and analytic i/o equations studied in previous papers by the authors are generalized to integral and integro-differential equations, and an abstract notion is also considered. New results are given on generic observability, and these results are then applied to give conditions under which that the minimal order of an equation equals the minimal possible dimension of a realization, just as with linear systems but in contrast to the discrete time nonlinear theory. |
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