Output Of Dynamical System Research Articles

What Observables Are Needed for Precision Data-Enabled Learning of Inverse Operators?

Abstract The advent of easy access to large amount of data has sparked interest in directly developing the relationships between input and output of dynamic systems. A challenge is that in addition to the applied input and the measured output, the dynamics can also depend on hidden states that are not directly measured. In general, it is unclear what type of data, such as past input and or past output is needed, to learn inverse operators (that predict the input needed to track a desired output for control purposes) with a desired precision. The main contribution of this work is to show that, irrespective of the selected model, removing the hidden-state dependence and achieving a desired precision of inverse operators require (i) a sufficiently-long past history of the output and (ii) sufficiently-precise estimates of the output's instantaneous time derivatives that are necessary and sufficient for linear systems, and under some conditions, for nonlinear systems. This insight, about the required observables (output history and derivative) for removing the hidden-state dependence and achieving precision, is used to develop a data-enabled algorithm to learn the inverse operator for multi-input multi-output square systems. Simulation examples are used to illustrate that neural nets (with universal approximation property) can learn the inverse operator with sufficient precision only if the required observables, identified in this work, are included in training.

Precision Data-enabled Koopman-type Inverse Operators for Linear Systems

The advent of easy access to large amount of data has sparked interest in directly developing the relationships between input and output of dynamic systems. A challenge is that in addition to the applied input and the measured output, the dynamics can also depend on hidden states that are not directly measured. The main contribution of this work is to identify the information needed (in particular, the past history of the output) to remove the hidden-state dependence in Koopman-type inverse operators for linear systems. Additionally, it is shown that the time history of the output should be augmented with the instantaneous time derivatives of the output to achieve precision. This insight into the required output (history and instantaneous derivative) information, to remove the hidden-state dependence and improve the precision of data-enabled inverse operators, is illustrated with an example system.

State and unknown input observers for nonlinear systems with delayed measurements

The use of connected devices or humans-in-the-loop for measuring outputs of dynamical systems inevitably produces time-varying measurement delays. These delays can lead to instability or severe degradation of system performance. In this paper, linear matrix inequality-based sufficient conditions are proposed for the design of state and unknown input observers based on delayed measurements for a class of nonlinear systems, where the nonlinearities are characterized by incremental multiplier matrices. The proposed observer is guaranteed to perform at specified operational levels in the presence of unknown exogenous inputs acting on the states and measurement outputs. Sufficient conditions are also provided for the estimation of these unknown inputs to a specified degree of accuracy. The potential of the proposed approach is illustrated via estimation of enzyme kinetics.

Can Hamilton energy feedback suppress the chameleon chaotic flow?

The dynamical behaviors of nonlinear systems are much dependent on the parameter setting and nonlinear terms, and some controllable parameters can be adjusted to modulate the outputs of dynamical systems. This paper confirms that the dynamical behaviors of the chameleon chaotic flow can be regulated by using the scheme of the Hamilton energy feedback. The Hamilton energy function can be approached by using the Helmholtz’s theorem. The dynamical system is improved by adding one new variable associated with Hamilton energy, and the feedback gain for energy is adjusted to find target orbits. The Lyapunov exponent, which is used to discern the emergence of chaos when positive value is approached, is calculated when energy feedback is applied, and the phase portraits are also plotted to understand the stability of oscillation behaviors. It is found that setting appropriate positive feedback gain for Hamilton energy can suppress the chaos. It could be helpful for further stability control of other complex dynamical systems.

Incremental Design of Simplex Basis Function Model for Dynamic System Identification.

In this paper, we propose a novel adaptive piecewise linear model for dynamic system identification. It has four unique features. First, the model designs a new kind of basis function for function approximation. It maintains the uniform shape for each basis function, so as to achieve a satisfactory tradeoff between generalization ability and model complexity. Second, the model takes the structure of basis functions as decision variables to optimize the formulated identification problems instead of taking expansion coefficients as decision variables as proposed by many existing approaches. Third, we establish an incremental design strategy to solve the system identification problems. In each step of the identification, the selection of optimal basis function is a Lipschitz continuous optimization problem that is likely to be easily handled with some mature toolboxes. This incremental design strategy greatly reduces the estimation cost. Fourth, we introduce a smoothing mechanism to avoid overfitting, when the output of dynamic systems is disturbed by noise. Tests on several benchmark dynamic systems demonstrate the potential of the proposed model.

Some new results on time-varying Stratonovich and Itô bilinear stochastic systems

The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term, where the input is a random input and the state is the output of dynamical systems. Stochastically influenced bilinear systems are described via bilinear stochastic differential equations. In this paper, first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itô counterpart. Second, the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit. The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well. The theory of this paper hinges on the ‘Stratonovich calculus’, conversion of the Stratonovich integral into the Itô integral and characteristic function of the vector Brownian motion. The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.

Forecasting: Exercises to Enhance Learning from Business Simulations

ABSTRACTForecasting the outputs of dynamic systems develops a richer understanding of relevant inputs and their interrelationships than merely observing them ex post. Academic business simulations foster students’ development of this critical competency, but learning outcomes can be significantly augmented with relatively simple, complementary exercises in forecasting. We introduce an in‐class forecasting exercise that is easily added to simulations or in any course where students gain from untangling the determinants of quantitative outcomes. The exercise also enhances opportunities for team‐building and spirited competition. Implementation details, evidence of effectiveness, and implications are discussed.

A Globally Convergent Algorithm for the Run-to-Run Control of Systems with Sector Nonlinearities

Run-to-run control is a technique that exploits the repetitive nature of processes to iteratively adjust the inputs and drive the run-end outputs to their reference values. It can be used to control both static and finite-time dynamic systems. Although the run-end outputs of dynamic systems result from the integration of process dynamics during the run, the relationship between the input parameters p (fixed at the beginning of the run) and the run-end outputs z (available at the end of the run) can be seen as the static map z(p). Run-to-run control consists in computing the input parameters p* that lead to the reference values zref. Although a wide range of techniques have been reported, most of them do not guarantee global convergence, that is, convergence toward p* for all possible initial conditions. This paper presents a new algorithm that guarantees global convergence for the run-to-run control of both static and finite-time dynamic systems. Attention is restricted to sector nonlinearities, for which it is shown that a fixed-gain update can lead to global convergence. Furthermore, since convergence can be very slow, it is proposed to take advantage of the mathematical similarity between run-to-run control and the solution of nonlinear equations, and combine the fixed-gain algorithm with a faster variable-gain quasi-Newton algorithm. Global convergence of this hybrid scheme is proven. The potential of this algorithm in the context of run-to-run optimization of dynamic systems is illustrated via the simulation of an industrial batch polymerization reactor.

Capability of neural networks in computing the outputs of dynamic systems with inputs defined on the whole space

Strong theorems on equi-uniform approximation to functionals and operators in abstract spaces are given. As applications, theorems on system identification by neural networks, whose inputs are defined in the whole space $$\mathbb{R}^n $$ , are given.

Approximations of continuous functionals by neural networks with application to dynamic systems

The paper gives several strong results on neural network representation in an explicit form. Under very mild conditions a functional defined on a compact set in C[a, b] or L(p)[a, b], spaces of infinite dimensions, can be approximated arbitrarily well by a neural network with one hidden layer. The results are a significant development beyond earlier work, where theorems of approximating continuous functions defined on a finite-dimensional real space by neural networks with one hidden layer were given. All the results are shown to be applicable to the approximation of the output of dynamic systems at any particular time.