Existence of realizations of nonlinear analytic input-output maps

We are interested in the existence of realizations of nonlinear input-output maps when they are defined for positive times, and when the set of control values is finite. First, we give examples that point out the nature of the problems which may appear in the attempt to find global results in realization theory. Then, we state a necessary and sufficient condition for the existence of realizations and we give a sketch of the proof.     

Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.     

Minimal-order realizations for continuous-time two-power input-output maps

This paper considers two-power input-output maps which are continuous-time, impulsive, and time-invariant. Realizability, by a very general class of nonlinear differential systems, is characterized simply in terms of the symmetric Volterra transfer function. The characterization leads directly to a family of minimal-order realizations having the (feedforward) structure described by Crouch [6]. Members of the family have states which are related by quadratic homeomorphisms. An especially simple member of the family serves as a canonical form. The development is self-contained and unlike Crouch's requires only elementary mathematical tools.     

Minimality of dynamic input-output decoupling for nonlinear systems

In this note we study the strong dynamic input-output decoupling problem for nonlinear systems. Using an algebraic theory for nonlinear control systems, we obtain for a dynamic input-output decouplable nonlinear system a compensator of minimal dimension that solves the decoupling problem.     

Linearization and input-output decoupling for general nonlinear systems

Necessary and sufficient conditions are obtained for the linearization and input decoupling (by state feedback) of general nonlinear systems. It is shown how these conditions can be derived from the already known conditions for affine nonlinear systems, thereby also elucidating the existing theory for affine systems.     

Some integral properties for balanced realizations of scalar systems

Certain structural properties for balanced minimal realizations of scalar transfer functions are used to develop closed forms for the semi-infinite integral of the impulse response and of the square of the impulse response.     

Dynamic input-output decoupling of nonlinear control systems

The problem of dynamic input-output decoupling of nonlinear control systems is studied. Based on an analytic algorithm the authors obtain necessary and sufficient conditions for the solvability of this problem. The solution of the problem is constructed by applying a series of simple precompensations and linking maps. Some interesting connections with other approaches in nonlinear control theory are discussed. The authors also give a few (simple) examples to illustrate the methods used     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

Distributed estimation algorithms for nonlinear systems

In this paper, we consider the problem of combining the local conditional distributions of a random variable which have been generated by local observers having access to their private information. Sufficient statistics for the local distributions are communicated to a coordinator, who attempts to reconstruct the global centralized distribution using only the communicated statistics. We obtain a distributed processing algorithm which recovers exactly the centralized conditional distribution. The results can be applied in designing distributed hypothesis-testing algorithms for event-driven systems.     

Existence and uniqueness of minimal realizations of nonlinear systems

In the theory of finite dimensional linear systems, it is well known that every input-output map that can be realized by one such system can also be realized by a system which is minimal, i.e. both controllable and observable. Moreover, the minimal realization of a given map is unique up to isomorphism. It is shown here that similar results hold for the class of all systems whose state space is a real analytic manifold, whose dynamics is given by a family of complete real analytic vector fields, and whose output is an arbitrary real analytic function on the state space.Research partially supported by NSF Grant GP-37488.     

Smoothing algorithms for nonlinear distributed parameter systems

Fixed interval and fixed-lag smoothing algorithms are developed for a class of noisy nonlinear distributed parameter systems with unknown volume disturbances. Using a least-squares estimation criterion a two point boundary value problem is obtained. The solution of this boundary value problem is obtained by converting it into an initial value problem using a Riccati transformation. Using a Gaussian assumption for the disturbances, approximate error covariance expressions are developed. The possible instability of the fixed-lag smoothing algorithm is noted. An alternate fixed-interval algorithm is developed to obviate storage requirements. The applicability of these algorithms is illustrated with an example from nuclear reactor kinetics.     

GPU-based parallel algorithms for sparse nonlinear systems

In this work we describe some parallel algorithms for solving nonlinear systems using CUDA (Compute Unified Device Architecture) over a GPU (Graphics Processing Unit). The proposed algorithms are based on both the Fletcher–Reeves version of the nonlinear conjugate gradient method and a polynomial preconditioner type based on block two-stage methods. Several strategies of parallelization and different storage formats for sparse matrices are discussed. The reported numerical experiments analyze the behavior of these algorithms working in a fine grain parallel environment compared with a thread-based environment.     

Consensus problems for multi-agent systems with nonlinear algorithms

The paper investigates consensus problems for multi-agent systems with nonlinear algorithms. Group consensus algorithms with actuator saturation for the first-order and second-order multi-agent systems are proposed. In addition, the adaptive consensus algorithm with nonlinear dynamic is also given. By applying the graph theory, Lyapunov function, and LaSalle’s invariance principle, consensus conditions for multi-agent systems are derived. Finally, three simulation examples are provided to denote the effectiveness of obtained theoretical results.     

Input identification to a class of nonlinear input-output causal systems

In this paper, we adapt Lavrent'ev method so to obtain a reconstruction procedure for the input u to a nonlinear input-output system described by a Volterra integral equation. The proof is based on a monotonicity assumption which does not imply the monotonicity of the operators appearing in the Volterra equation.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

Robust adaptive control of nonlinear systems represented by input-output models

A robust adaptive control scheme is proposed for a class of nonlinear systems represented by input-output models with unmodeled dynamics. The scheme does not require the unknown parameters to satisfy the linear dependence condition and parameter estimation is not needed. With the proposed control scheme, all the variables in the closed-loop system are bounded in the presence of unmodeled dynamics and bounded disturbances. Moreover, the mean-square tracking error can be made arbitrarily small by choosing some design parameters appropriately.     

Feedback passivity of nonlinear discrete-time systems with direct input-output link

This paper is devoted to the study of the feedback passivity property in nonlinear discrete-time systems. The relative degree and zero dynamics of the non-passive system are related to the feedback passivity of the system. Two main results are presented. First, some relative degree-related properties of passive systems in general form are stated. Second, sufficient conditions in order to render a multiple-input multiple-output (MIMO) system passive by means of a static state feedback control law are obtained.     

Adaptive output feedback control of nonlinear systems representedby input-output models

We consider a single-input-single-output nonlinear system which can be represented globally by an input-output model. The system is input-output linearizable by feedback and is required to satisfy a minimum phase condition. The nonlinearities are not required to satisfy any global growth condition. The model depends linearly on unknown parameters which belong to a known compact convex set. We design a semiglobal adaptive output feedback controller which ensures that the output of the system tracks any given reference signal which is bounded and has bounded derivatives up to the nth order, where n is the order of the system. The reference signal and its derivatives are assumed to belong to a known compact set. It is also assumed to be sufficiently rich to satisfy a persistence of excitation condition. The design process is simple. First we assume that the output and its derivatives are available for feedback and design the adaptive controller as a state feedback controller in appropriate coordinates. Then we saturate the controller outside a domain of interest and use a high-gain observer to estimate the derivatives of the output. We prove, via asymptotic analysis, that when the speed of the high-gain observer is sufficiently high, the adaptive output feedback controller recovers the performance achieved under the state feedback one