Linear Discrete State-space Models Research Articles

Robust Linearly Constrained Kalman Filter for General Mismatched Linear State-Space Models

It is well known that Wiener filter and Kalman filter (KF) like techniques are sensitive to misspecified covariances, uncertainties in the system matrices, filter initialization, or unwanted system behaviors. A possible solution to robustify these estimation techniques is to impose linear constraints (LCs). In this article: 1) we introduce a general class of linearly constrained KF (LCKF), where a set of nonstationary LCs can be set at every time step; 2) explore the use of such LCs to mitigate modeling errors in general mismatched linear discrete state-space models; and 3) provide the theoretical formulation to show that the gain-constrained KF is a particular instance of the proposed LCKF. Because such LCs can be taken into account in any KF generalization, this sets the basis for a new robust filtering framework. An illustrative example is provided to support the discussion.

Design and Implementation of Takagi-Sugeno Fuzzy Model Based Control Scheme for the Continuous Stirred Tank Reactor

In this work, the authors have designed and implemented a Takagi-Sugeno fuzzy model based control scheme (TS-MBC) for the continuous stirred tank reactor (CSTR) process. The proposed control scheme can be viewed as an alternative to the nonlinear model based control scheme proposed by Panda & Prakash1. In this work the nonlinear process is represented using a bank of linear discrete state space models combined using fuzzy membership function. The output of the proposed control scheme is computed in two steps. The bank of linear models weighted using fuzzy membership function is used to compute the value of the controller output in the first step. In the second step, the controller output computed in the first step is updated based on the measurement. The performance of the proposed control scheme has been compared with that of a nonlinear model based control scheme (NMBC) and nonlinear PID (NPID).

Recursive linearly constrained Wiener filter for robust multi-channel signal processing

This article introduces a new class of recursive linearly constrained minimum variance estimators (LCMVEs) that provides additional robustness to modeling errors. To achieve that robustness, a set of non-stationary linear constraints are added to the standard LCMVE that allow for a closed form solution that becomes appealing in sequential implementations of the estimator. Indeed, a key point of such recursive LCMVE is to be fully adaptive in the context of sequential estimation as it allows optional constraints addition that can be triggered by a preprocessing of each new observation or external information on the environment. This methodology has significance in the popular problem of linear regression among others. Particularly, this article considers the general class of partially coherent signal (PCS) sources, which encompasses the case of fully coherent signal (FCS) sources. The article derivates the recursive LCMVE for this type of problems and investigates, analytically and through simulations, its robustness against mismatches on linear discrete state-space models. Both errors on system matrices and noise statistics uncertainty are considered. An illustrative multi-channel array processing example is treated to support the discussion, where results in different model mismatched scenarios are provided with respect to the standard case with only FCS sources.

On LMVDR Estimators for LDSS Models: Conditions for Existence and Further Applications

For linear discrete state-space models, under certain conditions, the linear least mean squares (LLMS) filter estimate has a recursive format, a.k.a. the Kalman filter (KF). Interestingly, the linear minimum variance distortionless response (LMVDR) filter, when it exists, shares exactly the same recursion as the KF, except for the initialization. If LMVDR estimators are suboptimal in mean-squared error sense, they do not depend on the prior knowledge on the initial state. Thus, the LMVDR estimators may outperform the usual LLMS estimators in case of misspecification of the prior knowledge on the initial state. In this perspective, we establish the general conditions under which existence of the LMVDRF is guaranteed. An immediate benefit is the introduction of LMVDR fixed-point and fixed-lag smoothers (and possibly other smoothers or predictors), which has not been possible so far. Indeed, the LMVDR fixed-point smoother can be used to compute recursively the solution of a generalization of the deterministic least-squares problem.

Algorithm for Estimating the Parameters of Sea Pitching for Increasing the Accuracy of a Ship-Borne Radar Beam Positioning

The sea waves cause the effect of a rotational motion (i.e., pitch, roll and heading angle) of ships. The ship rotational motions result in beam pointing error of the phased array radar aboard the ship. The antenna stabilization aimed to achieve a beam pointing accuracy over a long dwell time is an important problem for the shipborne phased array radars. Due to the dynamic and stochastic nature of the sea environment, the shipborne phased array radar must be able to compensate for the ship's motion adaptively. In this paper, the linear discrete Kalman filter is proposed as a predictor for the shipborne phased array radar, which can compensate for the beam pointing error and track the air and sea surface targets. The pitch, roll and heading angles and the its velocities data are measured on-line from a gyroscope of the sea vehicle and used from the ship motion mathematical models for their prediction. It is proved, that the pitch, roll, and heading motion models may be considered as independent. All these models are presented as the same second order linear differential equations with different parameters. Besides, equivalent linear discrete state space models for the angles' changes are constructed in the paper. The estimation accuracy of the Kalman filter (predictor) depends on the values of different parameters, such as the parameters of the ship's motion mathematical model, measurement error covariance matrices, etc. The sea environments are very dynamic, hence, there is a need for an adaptive system for the controlling and compensating devices, operating regardless of the ship's motion. Continuous monitoring of the environment and adapting filter ensure parameters with less computational burden needed for a real time application. The paper describes a technique for identification of such parameters by the measured correlation functions of the pitch, roll, and heading angles. Finally, it is proved by the computer simulations that the proposed compensations technique is a real time applicable algorithm for a shipborne phased array radar. The computer simulation was performed with the following parameters: the measurement frequency for the gyroscopes was 100 Hz, the prediction times of the Kalman filter were 0.01 s and 0.1 s. The following two cases were considered. In the first case, only the predicted angles were measured with gyroscopes. In the second case, the angles and the rates of their change were measured. The simulations demonstrated that the presented prediction algorithm ensured higher accuracy (less root mean square errors of the predicted angles) than the initial accuracy of the gyroscopes measurements.

Minimum Variance Distortionless Response Estimators for Linear Discrete State-Space Models

For linear discrete state-space models, under certain conditions, the linear least-mean-squares filter estimate has a convenient recursive predictor/corrector format, aka the Kalman filter. The purpose of this paper is to show that the linear minimum variance distortionless response (MVDR) filter shares exactly the same recursion, except for the initialization which is based on a weighted least-squares estimator. If the MVDR filter is suboptimal in mean-squared error sense, it is an infinite impulse response distortionless filter (a deconvolver) which does not depend on the prior knowledge (first- and second-order statistics) on the initial state. In other words, the MVDR filter can be pre-computed and its behaviour can be assessed in advance independently of the prior knowledge on the initial state.

Autonomous UAV controlled collision landing via eye-in-hand visual servoing

Catastrophic damage to fixed-wing unmanned aerial vehicles (UAV) is most likely to occur during the landing stage. Vision-based autonomous landing methods for small UAV have been proposed in literature to address the issue. This paper improves on previous work on vision-based landing via a controlled collision with a large inflatable airbag. In the proposed method the UAV steers towards a large, multi-coloured, dome-shaped airbag using visual feedback obtained from a UAV-mounted camera. A multi-colour airbag is proposed to facilitate a robust and efficient colour-based detection method for real-time detection of the dome in images. A proposed visual servoing controller translates the dome location into desired pitch and yaw rate commands to achieve an exponential decay of the image error. A model predictive UAV controller that achieves desired roll and yaw rates is designed and tested in simulations. The lateral and longitudinal UAV dynamics were decoupled and a linear discrete state space model of the dynamics was obtained via system identification. The proposed dome landing method has been implemented on the FlightGear flight simulator and has presented successful landings in simulations over multiple trials.

Strict System Equivalence of 2D Linear Discrete State Space Models

The connection between the polynomial matrix descriptions (PMDs) of the well-known regular and singular 2D linear discrete state space models is considered. It is shown that the transformation of strict system equivalence in the sense of Fuhrmann provides the basis for this connection. The exact form of the transformation is established for both the regular and singular cases.

Time scaling internal state predictive control of a solar plant

The control of a distributed collector solar field is addressed in this work, exploiting the plant's transport characteristic. The plant is modeled by a hyperbolic type partial differential equation (PDE) where the transport speed is the manipulated flow, i.e. the controller output. The model has an external distributed source, which is the solar radiation captured along the collector, approximated to depend only of time. From the solution of the PDE, a linear discrete state space model is obtained by using time-scaling and the redefinition of the control input. This method allows overcoming the dependency of the time constants with the operating point. A model-based predictive adaptive controller is derived with the internal temperature distribution estimated with a state observer. Experimental results at the solar power plant are presented, illustrating the advantages of the approach under consideration.