bilinear systems

We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network. In particular, the H2-norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges are most sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the H2-norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulation.

In this paper, we investigate the problem of finding a sparse sensor and actuator (S/A) schedule that minimizes the approximation error between the input-output behavior of a fully sensed/actuated bilinear system and the system with the scheduling. The quality of this approximation is measuredby an H2-like metric, which is defined for a bilinear (time-varying) system with S/A scheduling based on the discrete Laplace transform of its Volterra kernels. First, we discuss the difficulties of designing S/A schedules for bilinear systems, which prevented us from finding a polynomial time algorithmfor solving the problem. We then propose a polynomial-time S/A scheduling heuristic that selects a fraction of sensors and node actuators at each time step while maintaining a small approximation error between the input-output behavior of thefully sensed/actuated system and the one with S/A scheduling in this H2-based sense. Numerical experiments illustrate the good approximation quality of our proposed methods.

When measuring importance of nodes in a network, the interconnections and dynamics are often supposed to be perfectly known. In this paper, we consider networks of agents with both uncertain couplings and dynamics. Network uncertainty is modeled by structured additive stochastic disturbances on each agent's update dynamics and coupling weights. We then study how these uncertainties change the network's centralities. Disturbances on the couplings between agents resul in bilinear dynamics, and classical centrality indices from linear network theory need to be redefined. To do that, we first show that, similarly to its linear counterpart, the squared H2 norm of bilinear systems measures the trace of the steady-state error covariance matrix subject to stochastic disturbances. This makes the H2 norm a natural candidate for a performance metric of the system. We propose a centrality index for the agents based on the H2 norm, and show how it depends on the network topology and the noise structure. Finally, we simulate a few graphs to illustrate how uncertainties on different couplings affect the agents' centrality rankings compared to a linearized model of the same system.

In large-scale networks, agents and links are often vulnerable to attacks. This paper focuses on continuous-time bilinear networks, where additive disturbances model attacks or uncertainties on agents/states (node disturbances), and multiplicative disturbances model attacks or uncertainties on couplings between agents/states (link disturbances). It investigates network robustness notion in terms of the underlying digraph of the network, and structure of exogenous uncertainties and attacks. Specifically, it defines a robustness measure using the H_2-norm of the network and calculates it in terms of the reachability Gramian of the bilinear system. The main result is that under certain conditions, the measure is supermodular over the set of all possible attacked links. The supermodular property facilitates the efficient solution finding of the optimization problem. Examples illustrate how different structures can make the system more or less vulnerable to malicious attacks on links.

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.

This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L2 stability, in much the same way that ISS generalizes L-infinity stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL is proved as well.

This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context.

For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for discrete time.

The present article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations of controllability have proved elusive. We show in particular that for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.

This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case, and which, for observability, results in the classical Nyquist theorem.

Explicit equations are given for the moduli space of framed instantons as a quasi-affine variety, based on the representation theory of noncommutative power series, or equivalently, the minimal realization theory of bilinear systems.

Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case.

A state-space realization theory is presented for a wide class of discrete time input/output behaviors. Although In many ways restricted, this class does include as particular cases those treated in the literature (linear, multilinear, internally bilinear, homogeneous), as well as certain nonanalytic nonlinearities. The theory is conceptually simple, and matrix-theoretic algorithms are straightforward. Finite-realizability of these behaviors by state-affine systems is shown to be equivalent both to the existence of high-order input/output equations and to realizability by more general types of systems.