Linear State-space Systems Research Articles

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This paper presents a study on the driveline output torque estimation using a discrete Kalman filter. The in-situ output shaft torque is first measured by a non-contacting magneto-elastic torque transducer. The linear state-space system equations are first derived and the discrete Kalman filter is designed based on the Kalman filter theory to recover the driveline output torque contaminated by random noises. In addition to using torque measurement, the estimation of the output torque using two angular velocities: the output and wheel, is also conducted. The experimental results show that the discrete Kalman filter can be effective for not only removing the random noise in output torque but also estimating the output torque without torque measurement.

Covariance control for stochastic uncertain multivariable systems via sliding mode control strategy

The main idea proposed in this study is based on converting continuous-time covariance matrix differential equations of a multivariable system into a new uncertain deterministic linear system. The closed-loop form of the new state covariance equations is derived by converting the covariance matrix Riccati equation to a new linear deterministic vector state space system. Since the new covariance system is linear and deterministic, all conventional and well-defined control strategies can be applied to it. Using a sliding mode control strategy, the uncertain interconnection terms satisfying a matching condition are nullified and the closed-loop covariance system is asymptotically stable. This is accomplished by applying a control strategy composed of sliding mode and covariance feedback control for each local subsystem.

A Detailed Nonlinear Dynamic Model of a 3-DOF Laboratory Helicopter for Control Design

AbstractThis work investigates the model of a 3-DOF laboratory helicopter, which constitutes a highly nonlinear system. For this model, advanced control concepts are developed and applied. In the beginning, the equations of motion are derived from a physical modeling approach. The modeling procedure yields highly nonlinear differential equations, from which linear state space systems are derived for arbitrary operating points. The major part of this work consists of the development of different linear and non-linear control architectures. At first, a classic state vector feedback controller with integration of the control error is implemented. Based on the system parameters of the linearized model, a gain scheduling approach is developed using one of the degrees of freedom as scheduling parameter. The gain scheduling application uses both discrete operating points as well as one possible continuous scheduling through interpolation. Additionally, a flatness-based feedforward controller architecture is added for transient set point changes using linear and nonlinear inverse dynamics. The control performance is validated in its dynamical and steady-state behavior. Finally, the previously derived approaches are tested on the actual helicopter.

Using the "Chandrasekhar Recursions" for Likelihood Evaluation of DSGE Models

In likelihood-based estimation of linearized dynamic stochastic general equilibrium (DSGE) models, the evaluation of the Kalman filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the 'Chandrasekhar Recursions' developed by Morf (1974) and Morf, Sidhu, and Kalaith (1974) for evaluating the likelihood of a linear Gaussian state space system. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs -- no model modification is required. Moreover, the algorithm is complementary with other approaches.

Using the 'Chandrasekhar Recursions' for Likelihood Evaluation of DSGE Models

In likelihood-based estimation of linearized Dynamic Stochastic General Equilibrium (DSGE) models, the evaluation of the Kalman Filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the Chandrasekhar Recursions developed by Morf (1974) and Morf, Sidhu, and Kalaith (1974) for evaluating the likelihood of a Linear Gaussian State Space System. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs--no model modification is required. Moreover, the algorithm is complementary with other approaches.

Weighted Measurement Fusion Fractional Order Kalman Filter

A weighted measurement fusion fractional Kalman filter is presented for the multisensor linear discrete fractional state-space systems. It is more universal and practical, which is because the description of fractional order is more accurate than that given by integer order. Compared with centralized fusion Kalman filter, it can reduce the computational burden because of the lower dimension of the measurement vector. It is numerically identical to the centralized fusion Kalman filter, so that it has the global optimality. A simulation example shows its effectiveness.

Sensitivity Analysis of Flutter of A Two-Degree of Freedom Linear Aeroelastic System

This paper deals with the problem of the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by unsteady aerodynamic loads. A method is presented to model the unsteady lift and pitching moment acting on a two dimensional typical aerofoil section, operating under attached flow conditions in an incompressible flow. Starting from suitable generalisations and approximations to aerodynamic indicial functions, the unsteady loads due to an arbitrary forcing are represented in a state-space form. From the resulting equations of motion, the flutter speed is computed through stability analysis of a linear state-space system. The sensitivity analysis of the aeroelastic stability boundaries to the structural parameter is evaluated. The results show that the parameter with the greatest influence on flutter speed is the center of gravity.

Rate control system algorithm developed in state space for models with parameter uncertainties

researching in weightlessness above the atmosphere needs a payload to carry the experiments. To achieve the weightlessness, the payload uses a rate control system (RCS) in order to reduce the centripetal acceleration within the payload. The rate control system normally has actuators that supply a constant force when they are turned on. The development of an algorithm control for this rate control system will be based on the minimum-time problem method in the state space to overcome the payload and actuators dynamics uncertainties of the parameters. This control algorithm uses the initial conditions of optimal trajectories to create intermediate points or to adjust existing points of a switching function. It associated with inequality constraint will form a decision function to turn on or off the actuators. This decision function, for linear time-invariant systems in state space, needs only to test the payload state variables instead of spent effort in solving differential equations and it will be tuned in real time to the payload dynamic. It will be shown, through simulations, the results obtained for some cases of parameters uncertainties that the rate control system algorithm reduced the payload centripetal acceleration below µg level and keep this way with no limit cycle.

Dual Adaptive Controllers based on Partial Certainty Equivalence

AbstractThe article deals with the optimal control of a linear discrete stochastic state space system with uncertain parameters. The solution of this optimization problem leads to design of controllers with dual features. Because the closed-form solution is hardly attainable a suitable suboptimal approaches has to be used. Many of the simpler approaches restrict the control horizon only to one step ahead and thus suffer from the myopic behavior. One way how to overcome this restriction is to use the partial certainty equivalence approximation of the joint probability density functions. The goal of this paper is to present a comparison of suboptimal adaptive dual control methods employing this approximation.

New scheme for discrete observer design with estimation error covariance assignment

This article presents an innovative method for solving an estimation error covariance assignment problem to design an observer for a stochastic linear system. In the proposed method, the covariance assignment problem is converted to the problem of finding an extra noise-like input to the observer. Using appropriate matrix manipulation, the Riccati equation of the estimation error covariance assignment problem, is converted to a new deterministic linear state-space model. Also, the extra noise-like input to the observer is modelled as an input to the new deterministic linear state-space model. Therefore, all the conventional and well-defined control strategies could be applied and there is no need to solve a complicated Riccati equation. Moreover, using the proposed method, a multi-objective estimation error covariance tracking problem would be easily converted to the problem of controlling a standard deterministic linear state-space system. Based on the integral control method, which is applied to the new state-space model, formulations for the proposed covariance feedback law are presented. The control law results in a stable closed-loop covariance system and assigns a pre-specified covariance matrix to the estimation errors.

On the closed-form model for state covariance assignment problem

In this study a closed-form innovative scheme for covariance assignment problems based on the original linear stochastic system is proposed. The closed-form of the state covariance equations is derived by converting the covariance matrix Riccati equation to a new linear deterministic vector state space system. As the main result of this study, according to the Kronecker product operator, the closed-form model of the new covariance system matrices is presented. In this study, the authors perform the covariance assignment problem, reformulated as a standard disturbance rejection problem. Since the new covariance system is linear and deterministic, all conventional and well-defined control strategies would be applied.

A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching

A unified approach to stability analysis of switched linear descriptor systems under arbitrary switchingWe establish a unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains. The approach is based on common quadratic Lyapunov functions incorporated with linear matrix inequalities (LMIs). We show that if there is a common quadratic Lyapunov function for the stability of all subsystems, then the switched system is stable under arbitrary switching. The analysis results are natural extensions of the existing results for switched linear state space systems.

Improved fractional Kalman filter and its application to estimation over lossy networks

This paper presents improvements on the fractional Kalman filter (FKF) based on the infinite dimensional form of a linear discrete fractional order state-space system. Furthermore, taking into account the considerable interest in estimation over networks with packet losses, the application and extension of the improved FKF are included. Some simulation cases are given in order to demonstrate the effectiveness of the proposed algorithms, with significant improvements in terms of estimation and smoothing results.

State covariance assignment problem

Covariance control theory provides a parameterisation of all controllers which assign a specified state covariance matrix to the closed-loop system according to different requirements on system performance. In this study, an innovative scheme for covariance system description based on the original linear stochastic system is proposed. The main idea of the proposed scheme is based on a special rearrangement of the corresponding covariance matrix Riccati equation to a new linear deterministic state space system. The variances of the states and cross-covariance between each two states of the original system would be the states of the new covariance system. By some mathematical manipulations, the covariance assignment problem is reformulated as a standard disturbance rejection problem. Since the new covariance system is linear and deterministic, all conventional and well-defined control strategies can be applied on it. Results are presented by a covariance feedback control law based on an integral control action applied to the new covariance system. The control law causes a stable closed-loop system and assigns a pre-specified state covariance matrix to the states of the closed-loop system.

Modelling flow control problems under diverse operating conditions by identification and blending of multiple systems

In this paper, a dynamical modelling procedure for fluid flow control problems is proposed. The resulting model is simple in that it consists of a number of linear time-invariant systems, but powerful in that it is capable of representing diverse operating conditions within a given flow envelope. The procedure makes use of snapshots of the flow process, which are obtained from experiments or computational fluid dynamics simulations. A proper orthogonal decomposition expansion of the flow is computed from snapshots, and the time coefficients of the expansion are coupled with the input values to form the estimation data. A linear state-space system representing the time coefficients is obtained using subspace system-identification methods. The procedure is repeated for a number of operating points, called breakpoints, which are characterized by one or more flow parameters of interest. The dynamical models obtained at the breakpoints are fused using the output-blending technique. The modelling procedure is illustrated with a flow control case study, where the flow dynamics are governed by the Navier—Stokes equations, the flow parameter of interest is the kinematic viscosity, and the control goal is to regulate the velocity of a given point inside the flow domain.

Dipteran insect flight dynamics. Part 1 Longitudinal motion about hover

This paper presents a reduced-order model of longitudinal hovering flight dynamics for dipteran insects. The quasi-steady wing aerodynamics model is extended by including perturbation states from equilibrium and paired with rigid body equations of motion to create a nonlinear simulation of a Drosophila-like insect. Frequency-based system identification tools are used to identify the transfer functions from biologically inspired control inputs to rigid body states. Stability derivatives and a state space linear system describing the dynamics are also identified. The vehicle control requirements are quantified with respect to traditional human pilot handling qualities specification. The heave dynamics are found to be decoupled from the pitch/fore/aft dynamics. The haltere-on system revealed a stabilized system with a slow (heave) and fast subsidence mode, and a stable oscillatory mode. The haltere-off (bare airframe) system revealed a slow (heave) and fast subsidence mode and an unstable oscillatory mode, a modal structure in agreement with CFD studies. The analysis indicates that passive aerodynamic mechanisms contribute to stability, which may help explain how insects are able to achieve stable locomotion on a very small computational budget.

Model predictive control of relative blood volume and heart rate during hemodialysis

To maintain the hemodynamic stability of patient undergoing hemodialysis, this article proposes a novel model-based control methodology to regulate the changes in relative blood volume (RBV) and percentage change in heart rate (DeltaHR(%)) during hemodialysis by adjusting the ultrafiltration rate (UFR). The control algorithm uses model predictive control (MPC) to account for system variability and to explicitly handle the constraints on UFR. Linear state-space system with time-varying parameters is introduced to model the RBV and DeltaHR. MPC was used to track the change in RBV and DeltaHR to pre-defined reference trajectories. At each sampling instant, the system parameters are updated to get the best fitting into the parameterized model. Simulation results demonstrate that the system is able to regulate RBV and DeltaHR to the reference by adjusting UFR while keeping it within practically realizable bounds. The results show that adjusting UFR may improve the stability of patient during dialysis when compared to conventional hemodialysis with constant UFR.

Non-myopic Innovations Dual Controller

Optimal control of a linear discrete stochastic state space system with uncertain parameters is treated. The problem statement leads to design of a dual controllers. Unfortunately, except for few special cases it is not possible to obtain closed-form solution for such controller. Many suboptimal approaches were proposed to overcome this obstacle, however, mostly they restrict the control horizon only to one step ahead and thus suffer from myopic behaviour. This paper presents technique that makes it possible to derive suboptimal dual controller in closed-form for arbitrarily long control horizon. The design of the presented controller is based on the innovations dual controller and use of the partial certainty equivalence principle. The proposed dual controller is compared to other non-dual controllers in a numerical example.

LQR 제어 기법을 적용한 수면 근처에서의 수중운동체 심도 제어

The submerged body near the free surface is disturbed by the 1st and 2nd order wave forces, which results in unstable movements when no control is applied. In this paper, the vertical motions of the submerged body are analyzed, and the time-variant nonlinear system for the vertical motions of the submerged body is transformed to the time-invariant linear system in state space. Next, depth controller of the submerged body is designed by using LQR control, one of the modern optimal control technique. Numerical simulation shows that effective depth controls can be achieved by LQR control.

QZ-Based Algorithm for System Pole,Transmission Zero, and Residue Derivatives

The QZ algorithm gives a robust way of computing solutions to the generalized eigenvalue problem. The generalized eigenvalue problem is used in linear control theory to find solutions to Ricatti equations, as well as to determine system transmission zeros. In state-space linear system analysis, the system poles and transmission zeros are particularly important for determining system time and frequency response. Here, we embed calculation of the eigenvalue derivatives in the QZ algorithm such that the derivatives of system poles and transmission zeros are computed simultaneously with the poles and zeros themselves. The resulting method is further exercised in finding generalized eigenvalues and their sensitivities required for finding the derivatives of system residues. This technique should open the door to solutions of problems of interest by unconstrained gradient-based methods. Typical numerical results are presented.