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Controllability metrics based on the controllability Gramian have been widely used in linear control theory, and have recently seen renewed interests in the study of complex networks of dynamical systems. For example, the minimum eigenvalue and the trace of the Gramian are related to the worst-case and average minimum input energy, respectively, to steer the state from the origin to a target state. This paper explores similar questions that remain unanswered for bilinear control systems. In the context of complex networks, bilinear systems characterize scenarios where an actuator not only can affect the state of a node, but also can affect the strength of the interconnections among some neighboring nodes. Under the assumption that the infinity norm of the input is bounded by some function of the network dynamic matrices, we derive a lower bound on the minimum input energy to steer the state of a bilinear network from the origin to any reachable target state based on the generalized reachability Gramian of bilinear systems. We also provide a lower bound on the average minimum input energy over all target states on the unit hypersphere in the state space. Based on the reachability metrics proposed, we propose an actuator selection method that provides guaranteed minimum average input energy.

The study of methods that involve rational interpolation is of interest. We provide an overview of the Loewner framework which is interpolatory but uses measured or computed data (e.g. measurements of the frequency response of a to-be approximated system) instead of the system matrices, and constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. We concentrate on generalizing the already existing framework for linear to bilinear and quadratic bilinear systems. The primary reason behind using quadratic bilinear systems is that they represent the bridge between linear and nonlinear systems. For certain types of nonlinear systems, we can always find an equivalent quadratic bilinear model without performing any approximation.

We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted.

This paper studies near-controllability of discrete-time upper-triangular bilinear systems. By using a recent result, the near-controllability problem of discrete-time upper-triangular bilinear systems can be reduced to the near-controllability problem of a special class of discrete-time upper-triangular bilinear systems, which would make the near-controllability problem easier to deal with, especially if the system dimension is high. An example is given to demonstrate the statement.

A number of important inverse problems in signal processing, including blind deconvolution, dictionary learning and matrix factorization, are instances of bilinear inverse problems. This paper shows that bilinear inverse problems are identifiable with probability close to one for random inputs provided that the number of rank-2 matrices in the null space grows as o(mn) for key applications.

Model order reduction of bilinear time invariant systems based on H∞ norm or least upper bound of difference equation is presented in this paper. The difference equation of the bilinear time invariant system is presented as error transfer function between full order and reduced order of the bilinear time invariant system. The proposed method is graphically easier than using alteration of Hankel singular values. The least upper bound of the error transfer function and H∞ norm of difference bilinear system are a function of controllability gramian. In this paper, the reduced bilinear system is carried out by using singular perturbation method. The numerical simulation results are given to clarify the proposed method for selection of reduced order bilinear time invariant system.

In this paper, the stabilization problems of two-dimensional symmetric bilinear systems by constant controls are considered. Both the continuous-time case and the discrete-time case are studied and necessary and sufficient conditions for stabilization of the systems are presented. It is shown that if the control is multiple then the continuous-time system is stabilizable if and only if its discrete-time counterpart is stabilizable.

Stated in a few word, the aim of this paper is to give evidence to the following author's conjecture: ‘any’ nonlinear control system is equivalent to a (larger dimensioned) bilinear fractional differential system. The main purpose is to motivate a new approach in nonlinear control, and for this reason, in this paper, some simple examples, nevertheless yet meaningful, are given, where the above conjecture holds. Starting with a simple scalar example, in order to present the basic feature of the method, the paper is endowed with a case consisting in a two dimensional control system, which is nevertheless amenable to be readily generalized to a general state-space dimension. A subresult of this paper is interesting by itself: for classical polynomial systems, where just positive integers powers are involved, the result holds always, and the equivalent system result in an ordinary (non-fractional) bilinear system.

In this paper, we present a novel bilinear invariant representation for video classification and retrieval. We rely on the kernel space in functional analysis to formulate a general invariants theory. We show that null-space invariants is a special case of the general theory when the transformation is linear. Subsequently, we derive an invariant basis representation for bilinear transformations. We also extend the basis representation to tensor bilinear invariants. We demonstrate that the proposed bilinear invariant basis provides a much more powerful tool than null-space invariants for video classification and retrieval when the different data elements undergo distinct transformations. Simulation results illustrate the superior performance of the proposed bilinear invariant basis representation compared to traditional approaches to invariant video classification and retrieval.

A special kind of port-controlled bilinear Hamilton system is researched. The controllability distribution and observability codistribution of the kind of port-controlled bilinear Hamilton systems are disused by using differential geometric of nonlinear control system. Lie brackets between input vector field and control vector field, Lie derivative of output function along input vector field and control vector field are researed. At the same time, the controllability and observability of the port-controlled bilinear Hamilton systems are disused Lie brackets between input vector field and control vector field. Then the relation between the controllability and observability is studied. The equivalence condition of controllability and observability is given.

In this paper, saddle point equilibrium problem of the bilinear quadratic performance index dynamic game will be discussed from the basic saddle point equilibrium strategy of the game theory, by introducing a proper transformation, transforming the bilinear systems of nonlinear two points boundary value problem into a "separation" form of the linear two-point boundary, and then finding the saddle-point iteration method using equilibrium strategies.

The paper discusses differential games of quadratic performance indices in continuous time bilinear systems in the finite time. Based on the optimal control method, the problem of saddle-point equilibrium strategies in differential games of continuous time bilinear systems is transformed into a sequence of linear differential equations and a Riccati matrix equation. And a practical, effective, uniformly convergent iterative algorithm is further present.

We obtain an intrinsic vector-valued symmetric bilinear form that can be associate with an underactuated simple mechanical control system. We determine properties of the form which serve as necessary conditions for driving underactuated simple mechanical control systems to rest. We also determine properties of the form that serve as sufficient conditions for driving a simple mechanical systems underactuated by one control to an epsiv-neighborhood of rest from an arbitrary initial configuration and velocity. These conditions are computable and coordinate invariant. We focus on the case where the symmetric form is real-valued and indefinite on the entire configuration manifold. Our technical results give rise to a nonlinear control law that drives these systems to an epsiv-neighborhood of rest given an arbitrary initial configuration, velocity and epsiv > 0.

This paper studies the realizability property of continuous-time bilinear i/o equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order.

The control of smooth input-affine nonlinear systems can be performed using exact linearization techniques which are based on the existence of a fictitious output map producing a (nonlinear) relative degree equal to the system order. A method is proposed on how to explicitly construct a map having the desired properties for a special class of nonlinear systems. This class is obtained by taking into account only the linear and quadratic part of the drift vector and the constant and linear part of the input vector of a general nonlinear system and can be thought of as a better approximation of that system than the usual linear or bilinear approaches. One important fact presented is that under some additional assumptions on the higher-order terms more general results are obtainable than those obtained by the well-known matching conditions. >

The authors derive a simple sufficient condition for the output of a discrete-time, time-invariant bilinear system to be bounded whenever the input signal to the system is bounded by a finite constant. >

This paper studies the realizability property of bilinear input-output (i/o) models in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The complete list of 2nd and 3rd order realizable input-output bilinear models together with the corresponding state equations is given. In the general case some subclasses of realizable bilinear models together with their state-space realizations are presented, including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model.

We show that QLS (quadratic linear systems) yield an exact representation of nonlinear, quadratic integrable input affine system as bilinear models do. However, the QLS representation is shown to be finite dimensional in most cases, while the bilinear system usually requires an infinite dimension. The dimension of the bilinear model is never lower than the dimension of the QLS model. This paper gives sufficient conditions for a finite dimensional representation.

Stability of discrete, time-varying, stochastic, bilinear systems is studied. Bilinear systems with output feedback are included. Mean-square stability conditions are derived for stochastic models without the assumption of stationarity for the random noise. The feedback function includes a larger class of functions than the class of linear functions or functions satisfying the Lipschitz condition. The sufficient stabilizing conditions depend only on the coefficient matrices of the bilinear system. >

A bilinear approximation technique for identification of a class of nonlinear dynamical models is presented, It is shown that for bilinear systems with a stable drift term the natural space of input-output models is that of causal Fock functionals. The functional spline interpolators hence obtained are found to have time-varying but asymptotically stable and decoupled bilinear realizations. Motivation is given by the need for systems order reduction and input-output analysis of Carleman bilinearization, with application to the attitude dynamics of a rigid spacecraft.

A nonlinear system identification technique based on the functional spline interpolation for dealing with high-dimensional bilinear dynamical models is described. At first, the nonlinear dynamics of a given system are transformed through Carleman bilinearization into a bilinear form. Decoupled bilinear models are then constructed, with input-output mappings expressible in closed form and with dimension determined by the number of training signals used in a prior learning stage. The motivation is given by the need for order reduction and input-output analysis of Carleman bilinearization. For illustration purposes, the technique is then applied to a forced version of Duffing's equation. >

For a general class of bilinear control systems on vector bundles the exponential growth rates and corresponding decompositions into subbundles are described.

This paper derives some optimization results for bilinear systems using higher-order method by characterizing them over matrix Lie groups. In the derivation of the results, a bilinear system is first transformed to a left-invariant system on matrix Lie groups. The product of exponential representation is then used to express this system in a canonical form. The conditions for optimality are then obtained by the principles of variational calculus. It is demonstrated that closed-form analytical solutions exist for classes of bilinear systems whose Lie algebra is nilpotent.

By using shift operators on linear spaces a theory of bilinear systems is presented. A restricted backward shift realization (RBSR) of a Volterra series input-output map is introduced. The RBSR is used to solve the bilinear realization problem. The RBSR is used to develop a theory of minimality for bilinear systems. Minimal bilinear systems are shown to be equivalent. The RBSR is used to develop a theory of reachability and observability for bilinear systems. An algorithm that computes the minimal bilinear realization of a Volterra series input-output map is given.

It is shown in this paper that for bilinear systems with a stable drift term the natural space of input-output models is that of Bochner-integrable causal Fock functionals, the latter being a causal version of the states of non-self-interacting quanta fields. Moreover, the functional spline interpolators or smoothers thereby obtained are found to have time-varying but asymptotically stable and decoupled bilinear realizations, with dimension equal to the number of test signals used for model matching. Motivation is given by the need for order reduction and input-output analysis of Carleman bilinearization, but the results apply to general bilinear systems.

From the external descriptions algorithms are obtained for the construction of discrete-time, internally bilinear state-space models with unknown initial states. Both the homogeneous and the inhomogeneous bilinear systems are treated. In both cases explicit existence criteria for minimal realization as well as the state space isomorphism theorems are given. One unusual result is that for a given input/output map there exist minimal realizations which are not equivalent under basis transformations. In the special case of zero initial state the algorithm can be reduced to the one previously obtained by Isidori, and if the input-output map is linear, the algorithm reduces to that given by Ho and Kalman for linear systems. Finally, the abstract realization theory is given. We use the affine tensor product as a basic tool to describe both homogeneous and inhomogeneous bilinear systems. The state space is constructed via Nerode equivalence. This forms a theoretical basis for the realization algorithms obtained in this paper.