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Comparing with traditional stochastic differential equations (SDEs) involving white noise, random differential equations (RDEs) with colored noise are claimed to have more practical meaning. This paper considers the event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities. Combining a tracking error-based dynamic gain with a relative threshold event triggered control mechanism (ETCM), the tracking control problem for the random systems is solved without Zeno behavior. The tracking error can be rendered small enough by tuning design parameters. First, a series of adaptive control laws are designed by using backstepping technique. Then, two special cases are considered and the main results are extended to MIMO systems. Finally, a simulation example confirms the validity of the results. To the best of the authors knowledge, this paper serves as the first attempt of event-based control for RDE systems.

In this article, the performance limitations problem are investigated for a class of signal-input, signal-output networked control systems with repeated zeros and poles. The additive white noise is adopted in the communication channel. The tracking performance limitations are investigated based on a two-degree of freedom (2DOF) controller. The results demonstrate that the multiplicity of nonminimum phase (NMP) zeros and unstable poles of the plant can affect the tracking performance limitations. The explicit quantitative relationship is characterized for the tracking performance limitations, which is derived based on NMP zeros, unstable poles of the plant with corresponding multiplicity, as well as the statistical characteristics of the reference noise and communication noise. The admissible infimal signal-to-noise ratio (SNR) is also obtained for the tracking system by using a 2DOF controller structure, where the admissible infimal SNR can satisfy the stability conditions while achieving tracking performance limitations. Finally, an illustrative example is presented to validate the effectiveness of our proposed scheme.

This technical note proposes a new sufficient condition for local controllability of nonlinear control systems ensured by single inputs. The objective is achieved by allowing control inputs to have the limitation of the derivation of unbounded-variation functions as with white noises, provided that we just consider deterministic signals. The treatment of the signals is mathematically recognized via rough path analysis, and then the dynamics of the target systems are represented by rough differential equations as the extension of ordinary differential equations. Our sufficient condition is clarified by deterministic Wiener processes that produce an effect on the dynamics of the systems similar to Wiener processes despite the nonprobabilistic property. The novelty of our condition is confirmed by a single-input two-link underactuated robot arm moving on the horizontal plane.

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ($\dot x(t) = u(t)$) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schr\"odinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases., Comment: 25 pages, 13 figures

It is shown that the Sliding Mode Control (SMC) technique can be successfully applied to stochastic systems governed by the stochastic differential equations of the Ito type which contain additive stochastic unbounded white noise perturbations. The existence of a strong solution to the corresponding stochastic differential inclusion is discussed. To do this approach workable the gain control parameter is suggested to be done state-dependent on norms of system states. It is demonstrated that under such modification of the conventional SMC we can guarantee the exponential convergence of the averaged squared norm of the sliding variable to μ-zone (around the sliding surface) which is proportional to the diffusion parameter σ in the model description and inversely depending on the gain parameter k 0 .

In this note, the filtering problem for discrete-time linear systems with time-correlated multiplicative measurement noises is considered where the vector consisting of all the multiplicative measurement noises can be described by a linear system model with white noise. By introducing several new recursive terms, a novel algorithm for optimal filtering of the system under consideration is proposed in the sense of linear minimum mean-square error. The proposed algorithm is recursive and has time-independent complexity. Computer simulations are provided to illustrate the performance of the proposed algorithm.

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

In this paper, we study the limitations in stabilization and tracking of multi-input, multi-output (MIMO) networked feedback systems. We adopt a parallel additive white noise (AWN) model for MIMO communication channels, and consider as our performance measure the mean square error for a system's output to track in the mean square sense a random reference signal with finite power. We derive necessary and sufficient conditions for the system to be mean square stabilizable and obtain analytical expressions of the optimal performance achievable by linear time-invariant (LTI) controllers subject to channel input power constraint. We show that the AWN channel power constraint imposes fundamental limits on the system's stabilizability and tracking performance, which depend on the unstable poles and nonminimum phase zeros of the system. In particular, for MIMO systems, these limits are seen to be dependent on the directions of the unstable poles and nonminimum phase zeros, and especially in how these directions are aligned with noise power distribution; in order to achieve the optimal tracking performance, the channel input power must be allocated to individual channels in ways accounting for pole/zero directions, a scheme that departs from the Shannon's classical “water-filling” strategy. Channel scalings are investigated as a means of realigning pole/zero directions and redistributing the channel power, which are found to be capable of improving fundamentally a system's stabilizability and tracking performance.

An innovative procedure for designing fractional optimal control to asymptotically stabilize, with probability one, quasi integrable Hamiltonian systems with fractional derivative dampings is proposed. It is proved that the systems indeed can be stabilized by using the proposed procedure. An example is given to illustrate the procedure and its effectiveness.

In this technical brief, the problem of the estimation of a remote signal generated by a discrete-time dynamical system with periodic coefficients subject to multiplicative and additive white noise perturbations is investigated. To measure the quality of the estimation achieved by an admissible filter, we introduced a performance criterion described by the Cesaro limit of the mean square of the deviation between the estimated signal zF(t) and the remote signal z(t). The dimension of the state space of the admissible filters is not prefixed. The state-space representation of the optimal filter is constructed based on the unique periodic solution of a discrete-time linear equation together with the stabilizing solution of a suitable discrete-time Riccati equation with periodic coefficients.

In this note, we propose a design for a robust finite-horizon Kalman filtering for discrete-time systems suffering from uncertainties in the modeling parameters and uncertainties in the observations process (missing measurements). The system parameter uncertainties are expected in the state, output and white noise covariance matrices. We find the upper-bound on the estimation error covariance and we minimize the proposed upper-bound.

In this technical note, we fully characterize the dynamic response of a saturating static feedback-controlled single integrator driven by Gaussian white noise by solving the derived Fokker-Planck equation.

Linear, continuous-time systems with stochastic uncertainties in their state-space model are considered. The problems of induced L∞ state-feedback control and filtering are solved, for the stationary case. In both problems, the cost function is defined to be the expected value of the standard induced L∞ performance index with respect to the uncertain parameters. An example that demonstrates the applicability of the theory is given.

The problem of estimating the parameters in a continuous-time ARX process from unevenly sampled data is studied. A solution where the differentiation operator is replaced by a difference operator is suggested. In the paper, results are given for how the difference operator should be chosen in order to obtain consistent parameter estimates. The proposed method is considerably faster than conventional methods, such as the maximum likelihood method. The Crameacuter-Rao bound for estimation of the parameters is computed. In the derivation, the Slepian-Bangs formula is used together with a state-space framework, resulting in a closed form expression for the Crameacuter-Rao bound. Numerical studies indicate that the Crameacuter-Rao bound is reached by the proposed method

When a covariance matrix with a Toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity, the singular summand corresponds to the covariance of a purely deterministic component of a time-series whereas the identity corresponds to white noise-this is the Caratheacuteodory-Fejeacuter-Pisarenko (CFP) decomposition. In the present paper we study multivariable analogs for block-Toeplitz matrices as well as for matrices with the structure of state-covariances of finite-dimensional linear systems (which include block-Toeplitz ones). To this end, we develop theory which addresses questions of existence, uniqueness and realization of multivariable power spectra, possibly having deterministic components. We characterize state-covariances which admit only a deterministic input power spectrum, and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part. We then show that multivariable decomposition of a state-covariance in accordance with a "deterministic component + white noise" hypothesis for the input does not exist in general. We finally reinterpret the CFP-dictum and consider replacing the "scalar multiple of the identity" by a covariance of maximal trace which is admissible as a summand. The summand can be either (block-)diagonal corresponding to white noise or have a "short-range correlation structure" corresponding to a moving average component. The trace represents the maximal variance/energy that can be accounted for by a process at the input (e.g., noise) with the aforementioned structure, and this maximal solution can be computed via convex optimization. The decomposition of covariances and spectra according to the range of their time-domain correlations is an alternative to the CFP-dictum with potentially great practical significance

This note presents a noise analysis for an algorithm to identify the uncertain frequency of periodic signals or disturbances. This algorithm is based on the time-varying states of an internal model principle controller which can be mapped nonlinearly to the frequency and the magnitude or energy of the periodic signal or disturbance. This note provides an analysis of the 'measurement' of this frequency in the presence of white noise. In the case of an additive white noise, we develop some formulas to calculate the means and variances of the measured difference between the true frequency and nominal frequency for high and low signal-to-noise ratio (SNR). When an integral controller is used to eliminate this difference, we prove that this frequency estimation is unbiased. The formulae to calculate the mean and variance are also given for the output of the integral controller. The simulations verify the validity of approximations used in our noise analysis.

In this paper, the linear quadratic optimization problem for a class of linear stochastic systems subject both to multiplicative white noise and Markovian jumping is investigated. Two classes of admissible controls are considered. One of these classes contains controls with additional property that corresponding trajectories tend to zero (in mean square) when tends to /spl infin/, while concerning the controls contained in the second class of admissible controls there is not any stability assumption. In the optimization problem over the first class of admissible controls, the cost functional could have indefinite sign of weights matrices. An iterative procedure to compute the maximal solution of the systems of generalized Riccati equations is provided. A numerical example to illustrate the applicability of the iterative procedure is given.

This note presents a sinusoid estimation algorithm, which will converge to the spectrum of the sinusoid process in nonstationary additive noise. The algorithm is in the framework of the tangential Nevanlinna-Pick setting. The procedure is a generalization of Capon's maximum likelihood estimate and some classical limit theorem for orthogonal polynomials. The method is used in an experimental example to find the natural frequencies of a lightly damped structure.

In this note, the problem of the frequency estimation of a sinusoid embedded in white noise is considered. The approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisy sinusoid. In particular, this note focuses on closed-form expressions of the frequency estimate, which can be obtained using notch filters having an all-zeros finite-impulse response (FIR) structure. The results presented in this note are as follows: 1) it is shown that the FIR notch filters obtained from standard second-order infinite-impulse response (IIR) filters are inadequate; 2) a new second-order IIR notch filter is proposed, which provides an unbiased estimate of the frequency; 3) the FIR filter obtained from the new IIR filter provides a closed-form unbiased frequency estimate; and 4) the closed-form frequency estimate obtained using the new FIR notch filter asymptotically converges toward the Pisarenko harmonic decomposition estimator and the Yule-Walker estimator.

We study a linear discrete-time partially observed system perturbed by white noises. The observations are transmitted to the controller via communication channels with irregular transmission times. Various measurement signals and even parts of a given sensor output may incur independent delays; messages transferred via the channels may be lost or corrupted. The minimum variance state estimation problem is solved. It is shown that the proposed state estimator is exponentially stable under natural assumptions.

This paper considers the problem of identifying linear systems, where the input is observed in white noise but the output is observed in colored noise which also includes process disturbances. An efficient method is developed, which can perform unbiased parameter estimation without utilizing a prefilter. The developed method is characterized by attractive features: direct use of the observed data without prefiltering; no need to evaluate autocorrelation functions for the input noise; no need to identify a high-order augmented system; and provision of a direct unbiased estimate of the system parameters without parameter extraction. Computer simulations are presented to illustrate its superior performance, including its significantly reduced computational complexity.

In systems with sensing cost, an active sensing policy is needed to determine when to collect sensing observations. This note presents an active sensing policy for systems with additive and parametric white noise. The policy uses an open-loop estimator between sensings and a Kalman filter when observations are requested. We present two active sensing policies. The goal of the first policy is to maintain the uncertainty (variance) of the state estimate below a given threshold. Sufficient conditions are presented that guarantee that this goal is achievable and will be met. The second policy senses when needed to distinguish discrete state regions for control. Sufficient conditions are presented that show within any specified probability, the control under the active sensing will be identical to the control under conventional sensing. Experiments demonstrate that sensing and sensing communications can be significantly reduced with active sensing policies, while still meeting control objectives.

The concept of wide-band noise is analyzed via a certain integral representation. It is proved that there are infinitely many wide-band noise processes represented in integral form which correspond to the same autocovariance function. Based on this integral representation, a technique of reduction of a wide-band noise driven system to a white noise driven system is presented. This technique is used to modify the separation principle and the Kalman-Bucy filtering to wide-band noise driven systems.

In this note, the delay-independent stability of delay systems is studied. It is shown that the strong delay-independent stability is equivalent to the feasibility of certain linear matrix inequality (LMI), that is to the existence of a quadratic Lyapunov-Krasovskii functional, independent of the (nonnegative) value of the delay. This constitutes the analogue of some well-known properties of finite-dimensional systems. This result is then applied to study delay-independent stability of systems with polytopic uncertainties.

We develop a realization theory for a class of multiscale stochastic processes having white-noise driven, scale-recursive dynamics that are indexed by the nodes of a tree. Given the correlation structure of a 1-D or 2-D random process, our methods provide a systematic way to realize the given correlation as the finest scale of a multiscale process. Motivated by Akaike's use of canonical correlation analysis to develop both exact and reduced-order models for time-series, we too harness this tool to develop multiscale models. We apply our realization scheme to build reduced-order multiscale models for two applications, namely linear least-squares estimation and generation of random-field sample paths. For the numerical examples considered, least-squares estimates are obtained having nearly optimal mean-square errors, even with multiscale models of low order. Although both field estimates and field sample paths exhibit a visually distracting blockiness, this blockiness is not an important issue in many applications. For such applications, our approach to multiscale stochastic realization holds promise as a valuable, general tool.

Expressions for the variance of an estimated frequency function are necessary for many issues in model validation and experiment design. A general result is that a simple expression for this variance can be obtained asymptotically as the model order tends to infinity. This expression shows that the variance is inversely proportional to the signal-to-noise ratio frequency by frequency. Still, for low order models the actual variance may be quite different. We derive an exact expression for the variance, which is not asymptotic in the model order. This expression applies to a restricted class of models: AR-models, as well as fixed pole models with a polynomial noise model. It brings out the character of the simple approximation and the convergence rate to the limit as the model order increases. It also provides nonasymptotic lower bounds for the general case. The calculations are illustrated by numerical examples.

We describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steady-state probability density.

Scales are used to reduce the conservatism encountered in most multiobjective approaches to control design. The most general case (i.e., matrix scales) results in a nonconvex problem, though the use of scalar scales leads to convex searches in the analysis and state feedback problems. Output feedback synthesis and other extensions are discussed. Numerical examples are provided to show the effectiveness of the suggested approach. The article considers, in particular, H/sup /spl infin// control.

A method for evaluating the H/sub 2/ performance robustness of an LTI SISO feedback system with respect to parametric uncertainties in the disturbance model is developed. The disturbance is assumed to be colored noise obtained by passing standard white noise through a linear finite-dimensional filter with uncertain coefficients. The method is based on the converse of a spectral factorization problem. The main result obtained is an explicit formula for the output variance in terms of the uncertain parameters of the coloring filter. This result is illustrated on a ship roll stabilization problem.