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This paper considers a time-division multiple-access system that is designed for voice services and analyzes the effects of adding data services with a given bandwidth allocation protocol. This protocol stipulates priority for voice and movable traffic boundaries, and is unique in that it allocates slots at designated update-interval times. We develop a Markovian analytical model and use it to derive relevant system performance characteristics for voice and data traffic. In the analytical model, reserved and shared slots are separated, and therefore, frame compaction is not applied. We study the effects of reserved data slots on performance and determine the bandwidth available for the data, given a specified level of voice service. Additionally, we explore the optimum update interval that maximizes the data-capacity subject to the specified voice-traffic characteristics. Our analytical model provides a means for studying mixed traffic system dynamics, as well as the effects of different traffic types on the others' performance. It provides an approximation to the bandwidth allocation protocol that allows us to select system parameters without the need for lengthy simulation runs.

This paper considers a feedback control system where a plant is actuated by a quantized control. The measurements are assumed to be disturbed by an additive wideband Gaussian noise in the feedback loop. The nonlinear control in the presence of the noise may cause the system to be variable structure. Variable structure systems disturbed by noise exhibit sliding modes on the switching manifold. The performance and behavior of the system in the steady state are examined and compared via a quadratic performance index. In order to improve the performance and reduce the noise effect, three types of quantizers are employed to represent different level of optimality.

This paper develops methodologies for the averaging, aggregation and optimal control of stochastic hybrid systems whose state equations depend on continuoustime, nearly completely decomposable finite state Markov chains. Such Markovian switching models are assumed to have several identified groups of strongly interacting states. The random sample solution process of the system state can be well approximated by a deterministic trajectory, for the duration of an interval of which the switching process sojourns in a group. Aggregated models are obtained by utilizing the aggregation method over the infinite time interval. By using the perturbation approach, necessary and sufficient conditions of stochastic stabilizability and controllability proposed by Ji and Chizeck (1990) are presented for the systems. With the aggregation models and the known results of stochastic stabilizability, near-optimum finite time and infinite time Markovian jump linear quadratic control laws are also investigated. The limiting be...

The hybrid system is an extension of the state-space system model which allows for random and abrupt changes in the model parameters. In this paper we use the hybrid system to model burst type interference in a direct-sequence spread-spectrum (DS/SS) communication system. The resulting optimum receiver is easily derived but too computationally complex to be practical. Two suboptimal but more practical receivers are presented. In an example, the hybrid system is used to model pulsed narrowband interference. The performance of each suboptimal receiver is evaluated using computer simulations. >

Hybrid systems arc used to represent systems that switch their dynamic behavior inresponse to failures or external and internal variations. This paper considers stochastic hybrid systems in continuous-time which switch among a finite number of known stochastic models based on an underlying finite-state Markov process. The transition probabilities of the process are assumed to depend on the system states. The representation of the optimal filtering equations for the system observed in white Gaussian noise is derived. Approaches to suboptimal implementation of the filters are then considered.

Smoothing for a linear system driven by a marked point process with a rate that depends on the state of the system is considered. A linear combination of the states is observed in additive white Gaussian noise. A smoother that uses estimation and detection is simulated on a computer and its mean squared error (MSE) is compared with the theoretical MSE of the optimal linear smoother and filter. The falso alarm rate is shown to depend strongly on the region of support of the mark distribution. When false alarms are low, the estimation/detection scheme has lower MSE than the optimal linear smoother.

This paper considers a singularly perturbed stochastic hybrid system whose state equations are governed by a stochastic switching process, which is modelled as a nearly completely decomposable continuous time finite state Markov chain. Approximate models for the slow mode subsystem are derived over the intervals of rapid switching and their accuracy (the mean-squared errors between the approximate models and the original systems) is quantified. Aggregate models for the subsystem are obtained over a longer period. The almost surely exponential stability of the subsystem is studied. The behaviour of the fast subsystem which depends on the relative size of the perturbation parameters is analysed. The decomposition of the overall system and the switching process together into slow and fast subsystems is investigated. The results derived in the paper are shown to hold when the switching process is stationary and irreducible, and each group of strongly interacting states is irreducible and time reversible. Fina...

The stabilization of hybrid systems with a non switching gain is cheaper and simpler to implement than the switching one. One approach to the design of a non-switching gain is based on the averaging of the hybrid system. For obvious reasons, the non-switching gain exists if the average system is controllable. In this paper, the minimality of the average system is investigated and a sufficient criterion is derived. Furthermore, these results also shed some light on the topology of minimal LTI systems in parameter space.

In this paper, we investigate the observability and detectability of stochastic discrete-time hybrid switching systems. The observability test is based on a mutual information measure between the system state and the observations. Monte Carlo results and an upper bound are presented.

This paper develops methodologies for the averaging, aggregation and optimal control of stochastic hybrid systems whose state equations depend on continuous-time and nearly completely decomposable finite state Markov chains. Such Markovian switching models consist of several identified groups of strongly interacting states. The random sample solution processes of the system state can be well approximated by deterministic trajectories, for the duration of intervals of which the switching process takes values in different groups. Aggregated models are derived, by utilizing averaging and aggregation ideas, to describe the global behavior for the original systems. By modifying concepts of stochastic stabilizability and controllability, necessary and sufficient conditions are established by the perturbation approach. Based on these properties, near-optimum finite and infinite time Markovian jump linear quadratic control problems are explored as well and corresponding averaged cost functions are studied. All the results are shown to hold when the system state and the group index can be exactly measured. Finally, an illustrative example is provided to demonstrate the aforementioned techniques.

A jump linear system (JLS) is a linear differential equation with N possible sets of randomly jumping (switching) coefficients. The JLS model has many real-world physical and conceptual (e.g. economic) systems. The bulk of the JLS-research effort concerns pure-Markov JLS (MJLS). In this paper we consider the more general semi-Markov JLS (SMJLS). This paper also shows a new sub-optimal but tractable and parsimonious control design method for SMJLS, and a closed-loop stability test.

A fundamental matrix inequality is used to improve upon existing stability bounds for linear periodic switched systems. The inequality is further used to derive an upper stability bound for the singular perturbation parameter in the case of linear periodic singularly perturbed switched systems.

This paper presents an algorithm to compute the upper-stability bound for linear singularly-perturbed discrete-time systems, specifically those formulated using the delta operator. Similar results already exist for continuous-time systems and for shift-operator-formulated discrete-time systems. In comparison to the shift-operator formulation the algorithm presented here leads to simpler criteria and is more robust.

The authors analyze the error introduced by the averaging of hybrid systems. These systems involve linear systems which can take a number of different realizations based on the state of an underlying finite state process. The averaging technique (based on a formula from Lie algebras known as the Backer-Campbell-Hausdorff (BCH) formula) provides a single system matrix as an approximation to the hybrid system. The two errors discussed are: (1) the error induced by the truncation of the BCH series expansion and (2) the error between the actual hybrid system and its average. A simple sufficient stability test is proposed to check the asymptotic behavior of this error. In addition, conditions are derived that allow the use of state feedback instead of averaging to arrive at a time-invariant system matrix. >

A sequential detection scheme is used to determine the approximate occurrence times of impulses in a self-excited point process which drives a scalar linear system. Observations of the state are corrupted by additive white Gaussian noise. The state estimate is constructed on the basis of the detected impulses. The approach was chosen because of its explicit dependence on a priori statistics. This dependence is highly desirable, given the elaborate input model assumed. >

A binary detection problem in which the observation noise is modeled as a hybrid system, i.e., a system characterized by abruptly changing parameters, is formulated. The optimal receiver for such a system, though conceptually straightforward, is too computationally complex to be practical. A suboptimal but practical receiver is presented and applied to the specific problem of direct sequence spread spectrum (DS/SS) communications in the presence of pulsed narrowband interference.

Jump-linear systems are dynamic systems with abrupt switches among several linear models, conditioned on an underlying finite state Markov process. This paper is concerned with optimal control of jump-linear systems when the discrete Markov process is not directly observable. Necessary conditions for optimality are found and a local algorithm to obtain such solutions is derived. >

The authors address a stabilization scheme for a hybrid system whose state equations are governed by a stochastic switching process, which is modeled as a continuous-time finite-state Markov chain. The proposed linear control law switches its gain based on the imperfectly detected states of the Markov chain. It is assumed that the correct detection probability is the same for all Markov states. The range of the correct detection probability of the detector is computed by solving the conditions for the almost sure sample sure sample path stability of the closed-loop system. Two simple examples are provided to illustrate the technique. >

This paper considers a feedback control system where a plant is actuated by a quantized control. The measurements are assumed to be disturbed by an additive wideband noise in the feedback loop. The nonlinear control in the presence of the noise may cause the system to be variable structure. Variable structure systems (VSS) disturbed by noise exhibit sliding modes on the switching manifold. If the quantizer causes a high gain in the system due to quantization of a small measurement for a fixed quantization level, the quantized control systems can be represented by a singularly perturbed model and analyzed by two approximate reduced-order models. It is shown that the slow model approximates the system in the sliding mode with an error of an order of the estimated singular perturbation parameter.

Jump-linear systems are dynamic systems with abrupt switches among several linear models, conditioned on an underlying finite state Markov process. This paper is concerned with implementable control schemes for jump-linear systems when the discrete state is not directly observable. A stabilizability condition is discussed. Also, the feasibility of improving the control cost through more thorough probing of the system is considered.

Considers a TDMA system designed for voice traffic, and analyzes the effects of adding data services with a specific bandwidth allocation protocol. This protocol stipulates priority for voice and movable traffic boundaries, and allocates slots at designated update interval times. A controlled Markovian analytical model is developed and used to derive relevant system performance characteristics for voice and data traffic. Additionally, the model is used to explore the optimization of system parameters to maximize data capacity subject to specified voice traffic characteristics.

The bunch train cavity interaction is an accelerator physics problem, for which a system-theoretic model is lacking. A form of modal analysis has been used to characterize the system dynamics, and this model has been used for control system design. Correspondingly, control design has been done using input-output classical frequency-domain-based control, including several nested loops. Several shortcomings of this control-design method are highlighted, all of which are remedied by a new time-domain, system-theoretic model developed herein. The new formulation is a periodic, discrete-time system, amenable to state-space control-design methods.

This paper considers an optimal control approach to the channel allocation problem in a multimedia wireless system. The system accommodates several sources with variable bit rates (VBR). The channel allocation paradigm is based on the queue length of the data of the various sources. The objective is to maintain the queue lengths of all the sources and the overall queue length at or near desired levels thus minimizing delays and avoiding data losses while maximizing bandwidth utilization. Since the system equations for the queue lengths are linear, a quadratic performance index can be utilized for the solution of the control problem, even though the uncertainties may not be Gaussian. The performance of the system is also considered in the case when the allowable source rates are discrete, and in the presence of control delay. The results are compared with simulations to indicate the improved performance of the system.

This paper considers a feedback control system where a plant is actuated by a quantized control. The control is obtained by the quantization of measurements which are assumed to be disturbed by an additive wideband Gaussian noise in the feedback loop. In the continuous-time case, the nonlinear control may cause the system to behave as a variable structure system (VSS) which exhibits a sliding mode on the sliding surface. In the discrete-time case, however, the system may or may not yield an associated sliding mode by discretizing the controller in the continuous-time case. It provides the motivation to consider the quantum levels in a uniform quantizer so that the discrete-time sliding mode has the same behavior as the continuous-time sliding mode. Thus, the sliding controller design scheme for continuous-time quantized control systems may be extended to discrete-time systems.

This paper considers a feedback control system where a plant is actuated by a quantized control. The measurements are assumed to be disturbed by an additive wideband Gaussian noise. The nonlinear control is obtained by the quantization of the measurements and it may cause the system to be variable structure in the presence of the noise. Variable structure systems disturbed by the noise exhibit sliding modes on the switching manifold. When the system is unstable, linear optimal controls using linear-quadratic-Gaussian (LQG) formulation are investigated to stabilize the system in a region around the origin while the quantized control guides the system to the region. The performance and behavior of the stabilized systems are examined via a quadratic performance index in the steady state.

This paper considers a singularly perturbed hybrid system whose state equations are governed by a stochastic switching process, which is singularly perturbed and is modeled as a near decomposable continuous time finite state Markov chain (FSMC). The decomposition of the system and the switching process together into slow and fast subsystems is investigated. An approximate model for the slow subsystem over the interval of fast transitions within each group is derived and the mean-squared error between the model and the actual subsystem is quantified. The stability of the slow mode subsystem is studied and two stability criteria are introduced. The behavior of the fast subsystem depending on the relative size of perturbation parameters is analyzed. Finally an example is used to illustrate the aforementioned techniques.

The Kalman filter provides optimal state estimates for completely known linear systems. Unfortunately, many physical systems are neither exactly known, nonlinear. Numerous filtering schemes for nonlinear systems have been introduced over the years: general theories for nonlinear systems tend to be complex, and, due to their generality, are of little practical use to the design engineer. On the other hand, solutions for specific nonlinearites usually apply only to a single nonlinearity, and thus are limited in their applications. This paper, however, presents a methodology whereby the nonlinearity is first approximated by a piecewise linear model, and then a common filtering scheme is applied. The efficacy of this approach is that the same filtering algorithm may be applied to a broad class of nonlinear stochastic systems.

in the xixs plane, and assume throughout that fi , fi E Co in Q and fi , fi E Cl in QO. Notice that every trajectory of (1) starting in Q at t = 0 lies entirely either in Q” or on one of the positive semiaxes. Indeed, if x(t) = (xl(t), x2(t)) is a solution of (1) originating in Q” and if t, > 0 were the time of its first intersection with one of the axes, then f,(x(t)) and f,(x(t)) would be bounded for 0 0, m2 > 0); thus 3ii 3 -m,x, and 2s > -msxa , and therefore xl(t) > xi(O) e-“lt and x2(t) 3 x,(O) emzt for all 0 < t < to . A similar argument holds for to < 0, and hence a trajectory of (1) starting in QO cannot reach the x1 or xs axis in finite time. For the same reason, a trajectory originating on one of the positive semiaxes cannot enter Q”. Finally, we assume below conditions on fi and fi that eliminate the possibility of finite escape times.

The effect of random inputs on a continuous piecewise-linear singularly perturbed system is investigated in this paper. Reduced-order models are developed for a second-order system (one fast and one slow variable) which has a random input. It is shown that the solutions of the reduced-order models approximate the actual solution with differences in probability density functions of order 0 (in a distributional sense). For the special case of a system which is linear in the fast variable, it is shown that the mean-squared error between the approximate and actual solutions in the fast time scale is of order 0(μ). An outline is provided for the extehsion of the results to the vector variable case

An estimation-detection scheme is proposed for the estimation of signals with unknown parameters. The scheme is illustrated by considering the case of a signal with unknown but bounded variance. Two sets of bounds are used to determine two regions of signal uncertainty, and different linear estimators are used for each region. A detector is then used to select the appropriate estimator. Since no a priori information is assumed on the unknown parameters a weighted minimax criterion is used to optimize the detector and the estimators.

A special class of hybrid systems is considered, whose state space is a cross-product space of a euclidean space and a finite-state space. Such models may be used to represent systems subject to known abrupt parameter variations, such as commu- tated networks. They may also be used to approximate some types of time-varying systems. Controllability, observability and stability of hybrid systems are inves- tigated. In particular, a necessary and sufficient algebraic condition, a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived. Moreover, a simple sufficient stability condition is provided.

A receiver is investigated with one-bit memory for communication channels with memory. The memory of the channel is characterized by the statistical correlation of the received bits due to fading. An optimal solution to the problem is considered but is found to be quite complex to implement. Therefore, a suboptimal receiver having a decision-feedback structure is examined. The performance of the receiver is shown to be better than a receiver without memory. Binary on-off keying system is considered as an example and numerical results are obtained.

A combined detection-estimation scheme for the estimation of the state of a linear discrete-time system with unknown noise covariances is considered. It is a heuristic extension of the standard minimax scheme to the case when multiple bounds on the unknown parameters are available. The convergence of the scheme is discussed and an example is considered for illustration.

The state estimation problem for a linear system driven by homogeneous Poisson processes of low intensities in the presence of white Gaussian noise is considered. A sequential estimation-detection scheme is developed as a suboptimal solution to the problem. The asymptotic performance for the time-invariant case as the intensity becomes small is derived and numerically simulated on a digital computer. The two results agree extremely closely, indicating that the scheme performs considerably better than the linear optimal filter in the sense of MSE when the Poisson intensity is small.

The asymptotic behavior of a Bayes optimal adaptive estimation scheme for a linear, discrete-time system with interrupted observations is investigated. The interrupted observation mechanism is expressed in terms of a stationary two-state Markov chain. The transition probability matrix is unknown and can take values only from a finite set.

Minimax state estimation for uncertain systems is discussed. The conservative performance of the standard minimax estimator in the absence of an intelligent adversary is reduced by a combined detectorestimator structure and an incremental mean-squared error (IMSE) performance criterion. The optimal structure is defined for a wide class of linear and nonlinear systems whose uncertain parameters are elements of some known compact space and is also obtained for convex parameter spaces. Since the complete specification of the optimal estimator detector is problem dependent, a computational procedure is outlined. In an example, the resulting combined detector-estimator is shown to increase the estimation accuracy in the incremental minimax sense by a factor of two over the standard minimax estimator.

A two time-scale lower order design is proposed as a near optimal solution to the fixed-interval smoothing problem for systems with slow and fast modes. The slow mode smoothing solution, in the limit as the perturbation parameter @m -> 0, tends to that of the reduced-order problem; and the near optimal fast mode smoother is simply a weighted sum of lower order two time-scale filters. While the near optimal solution affords a significant reduction in computational complexity, the performance degradation, as illustrated in an example, is typically negligible.

Predictors are derived which minimize the maximum possible mean-squared prediction error for signals observed in white noise and having a bounded kth derivative. Expressions are given for the resulting worst-case error and a suboptimal solution is presented which, for the case k = 2, performs nearly as well as the optimal and is far easier to implement.

A singular perturbation technique is developed that allows for a decoupling of a continuous piecewise-linear system into slow and fast subsystems. Under the assumption of asymptotic stability, the fast variable is found to decay in the boundary layer to its quasi-steady-state solution, which is given by a continuous implicit function of the slow variable. The solution is found using a finite step algorithm. Sufficient conditions for the approximation to be accurate to an order of O( mu ), where mu is a parameter of the system, are given. The technique is illustrated by a numerical example. >

The effect of a scalar quantized control input on an otherwise linear, singularly perturbed control system is analyzed. The cases of open loop and closed loop control are studied. Analytical techniques are derived that allow the decoupling of such systems into fast and slow subsystems for analysis and design. The results of a numerical example are given to demonstrate these techniques.

This short paper considers the properties of singularly perturbed systems with white noise input. The theory of singular perturbations is then applied to the filtering problem for such systems. The resulting filter may be approximately decomposed into two filters in different time scales yielding the estimates of the slow-mode and the fast-mode states.

A combined detection-estimation scheme is proposed for state estimation in linear systems with random Markovian noise statistics. The optimal MMSE estimator requires exponentially increasing memory and computations with time. The proposed approach is an attempt to circumvent this problem. Simulation results are presented which show the advantages of the proposed scheme over some of the existing suboptimal approaches.

This paper is concerned with the properties of piecewise linear discrete-time dynamic systems driven by white Gaussian noise. The properties of the deterministic system are explored, and condition for the existence of invariant distributions are derived. The existence of an invariant distribution was then used to justify the approximation of the stochastic system by a switched Markov linear model if the piecewise linear regions are large “contracting” ones or small “expanding” ones relative to the input noise variance. The approach is expected to be useful for constructing approximate nonlinear filtering schemes for such systems

This paper applies singular perturbation theory to the stochastic control for the Linear-Quadratic-Gaussian (L-Q-G) problem for systems with fast and slow modes. The limiting behavior of the optimal control and the performance index is investigated. It is shown that the optimal control can be approximated by a near optimal control which is obtained as a combination of a slow control and a fast control computed in separate time scales.

The determination of minimum-mean-squared-error (MMSE) nonlinear filters usually involves formidable mathematical difficulties. These difficulties may be bypassed by restricting attention to special classes of filters or special processes. One such class is Zadeh's class n_{1} , which for the general case also involves mathematical difficulties. In this work two realizations of class n_{1} are used for the MMSE reconstruction and filtering of a sampled signal. The cases where the filter reduces to a zero-memory nonlinearity followed by a linear filter are discussed. A suboptimum scheme composed of a zero-memory nonlinearity followed by a linear filter is considered for the reconstruction and filtering of a subclass of the separable process.

A nonlinear filtering scheme with noiseless feedback is presented, based on a consideration of the minimum-mean-squared error filtering of independent signal samples corrupted by additive noise. The explicit solution for the general case is very complex. However, if the signal-to-noise ratio is assumed to be large and the nonlinear estimating filter has zero memory, the problem may be simplified by reducing it to the zero-memory prefiltering problem combined with predictive feedback. The improvement over the linear case without feedback is shown to be the product of the improvements due to the zero-memory non-linearities and the feedback. An example is considered to illustrate the improvements in the error

An algorithm is presented which provides a maximum-likelihood estimate for an unknown parameter contained in a linear dynamic system driven by white, Gaussian noise. Taylor series expansions are used to develop approximations to the estimation equations. These approximations are recursive and can be calculated iteratively. The algorithm can be realized either as an analog or as a digital system and is shown to compare favorably with existing techniques in a simple example.

A system is proposed which combines hypothesis testing and prediction to estimate the future value of a stochastic process whose statistics are known only to belong to some finite set of possible hypotheses. Bayes optimization of the individual components is performed, and system performance is discussed for a modified version of the usual mean-squared error predictor cost functions. An example is given illustrating various features of the system's performance for a specific choice of input hypotheses.

In this paper a method for designing control systems with random parameters and random initial state is presented. The observed signal available for feedback is a function of the state and system parameters corrupted by additive white Gaussian noise with zero mean. The proposed control consists of an open-loop term, which is the optimal control for the parameters and initial state equal to their mean values, plus a feedback correction term. The feedback correction term is a linear function of the estimated values of the deviation in initial state and system parameters from their mean values. The control tends to the optimally sensitive control when the measurement noise tends to zero, and the feedback correction tends to zero when the measurement noise tends to infinity. A numerical experiment with a simplified model for nuclear reactor control shows that in spite of fairly large measurement noise, significant improvement in performance can be achieved using the proposed control compared to open-loop control.