Dario Calderon-Alvarez

Central suboptimal H ∞ filter design for linear time-varying systems with state and measurement delays

This article presents the central finite-dimensional H ∞ filters for linear systems with state and measurement delay that are suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the results previously obtained for linear time delay systems, this article reduces the original H ∞ filtering problem to H 2 (optimal mean-square) filtering problem using the technique proposed in Doyle, Glover, Khargonekar, and Francis (1989 ‘State-space Solutions to Standard H 2 and H ∞ Control Problems’, IEEE Transactions on Automatic Control, 34, 831–847). Application of the reduction technique becomes possible, since the optimal closed-form filtering equations solving the H 2 (mean-square) filtering problem have been obtained for linear systems with state and measurement delays. This article first presents the central suboptimal H ∞ filter for linear systems with state and measurement delays, based on the optimal H 2 filter from Basin, Alcorta-Garcia, and Rodriguez-Gonzalez (2005, ‘Optimal Filtering for Linear Systems with State and Observation Delays’, International Journal of Robust and Nonlinear Control, 15, 859–871), which consists, in the general case, of an infinite set of differential equations. Then, the finite-dimensional central suboptimal H ∞ filter is designed in case of linear systems with commensurable state and measurement delays, which contains a finite number of equations for any fixed filtering horizon; however, this number still grows unboundedly as time goes to infinity. To overcome that difficulty, the alternative central suboptimal H ∞ filter is designed for linear systems with state and measurement delays, which is based on the alternative optimal H 2 filter from Basin, Perez, and Martinez-Zuniga (2006, ‘Alternative Optimal Filter for Linear State Delay Systmes’, International Journal of Adaptive Control and Signal Processing, 20, 509–517). In all cases, the standard H ∞ filtering conditions of stabilisability, detectability and noise orthonormality are assumed. Finally, to relax the standard conditions, this article presents the generalised versions of the designed H ∞ filters in the absence of the noise orthonormality. The proposed H ∞ filtering algorithms provide direct methods to calculate the minimum achievable values of the threshold γ, based on the existence properties for a bounded solution of the gain matrix equation. Numerical simulations are conducted to verify the performance of the designed central suboptimal filters for linear systems with state and measurement delays against the central suboptimal H ∞ filter available for linear systems without delays. The simulation results show a definite advantage in the values of the noise-output transfer function H ∞ norms in favour of the designed filters.

Joint state filtering and parameter estimation for linear stochastic time-delay systems

This paper presents the joint state filtering and parameter estimation problem for linear stochastic time-delay systems with unknown parameters. The original problem is reduced to the mean-square filtering problem for incompletely measured bilinear time-delay system states over linear observations. The unknown parameters are considered standard Wiener processes and incorporated as additional states in the extended state vector. To deal with the new filtering problem, the paper designs the mean-square finite-dimensional filter for incompletely measured bilinear time-delay system states over linear observations. A closed system of the filtering equations is then derived for a bilinear time-delay state over linear observations. Finally, the paper solves the original joint estimation problem. The obtained solution is based on the designed mean-square filter for incompletely measured bilinear time-delay states over linear observations, taking into account that the filter for the extended state vector also serves as the identifier for the unknown parameters. In the example, performance of the designed state filter and parameter identifier is verified for a linear time-delay system with an unknown multiplicative parameter over linear observations.

Central Suboptimal H ∞ Filter Design for Linear Time-Varying Systems with State or Measurement Delay

This paper presents central finite-dimensional H∞ filters for linear systems with state or measurement delay that are suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including an attenuation control term with opposite sign. In contrast to the results previously obtained for linear time-delay systems, the paper reduces the original H∞ filtering problems to H2 (optimal mean-square) filtering problems, using the technique proposed in Doyle et al. (IEEE Trans. Automat. Contr. AC-34:831–847, 1989). The paper first presents a central suboptimal H∞ filter for linear systems with state delay, based on the optimal H2 filter from Basin et al. (IEEE Trans. Automat. Contr. AC-50:684–690, 2005), which contains a finite number of filtering equations for any fixed filtering horizon, but this number grows unboundedly as time goes to infinity. To overcome that difficulty, an alternative central suboptimal H∞ filter is designed for linear systems with state delay, which is based on the alternative optimal H2 filter from Basin et al. (Int. J. Adapt. Control Signal Process. 20(10):509–517, 2006). Then, the paper presents a central suboptimal H∞ filter for linear systems with measurement delay, based on the optimal H2 filter from Basin and Martinez-Zuniga (Int. J. Robust Nonlinear Control 14(8):685–696, 2004). Numerical simulations are conducted to verify the performance of the designed three central suboptimal filters for linear systems with state or measurement delay against the central suboptimal H∞ filter available for linear systems without delays.

Alternative optimal filter for linear systems with multiple state and observation delays

In this paper, the optimal filtering problem for linear systems with multiple state and observation delays is treated using the optimal estimate of the state transition matrix. As a result, the alternative optimal filter is derived in the form similar to the traditional Kalman-Bucy one, i.e., consists of only two equations, for the optimal estimate and the estimation error variance. This presents a significant advantage in comparison to the previously obtained optimal filter [1], which includes infinite or variable number of covariance equations, unboundedly growing as the filtering horizon tends to infinity. Performances of the two optimal filters are compared in example; the obtained results are discussed.

Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems

This paper presents the optimal control problem for a nonlinear polynomial system with respect to a Bolza-Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional polynomial-quadratic regulator does not lead to a causal solution and, therefore, fails. Performance of the obtained optimal controller is verified in the illustrative example against the conventional polynomial-quadratic regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.

Central suboptimal H∞ filter design for nonlinear polynomial systems

This paper presents the central finite-dimensional H<inf>∞</inf> filter for nonlinear polynomial systems, that is suboptimal for a given threshold γ with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the paper reduces the original H<inf>∞</inf> filtering problem to the corresponding optimal H<inf>2</inf> filtering problem, using the technique proposed in [1]. The paper designs the central suboptimal H<inf>∞</inf> filter for the general case of nonlinear polynomial systems, based on the optimal H<inf>2</inf> filter given in [24]. The central suboptimal H<inf>∞</inf> filter is also derived in a closed finite-dimensional form for third (and less) degree polynomial system states. Numerical simulations are conducted to verify performance of the designed central suboptimal filter for nonlinear polynomial systems against the central suboptimal H<inf>∞</inf> filter available for the corresponding linearized system.

Sliding mode regulator as solution to optimal control problem

This paper presents the optimal control problem for a linear system with respect to a Bolza-Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional linear feedback control fails to provide a causal solution. Performance of the obtained optimal controller is verified in the illustrative example against the conventional LQ regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.

Optimal LQG controller for linear stochastic systems with unknown parameters

This paper presents the optimal LQG controller for linear systems with unknown parameters. The optimal controller equations are obtained using the separation principle, whose applicability to the considered problem is substantiated. Performance of the obtained optimal controller is verified in the illustrative example against the conventional LQG controller that is optimal for linear systems with known parameters. Simulation graphs verifying overall performance and computational accuracy of the designed optimal controller are included.

Approximate finite-dimensional filtering for polynomial states over polynomial observations

In this article, the mean-square filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, this article deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third-order sensor filtering problem for a quadratic state, assuming a conditionally Gaussian initial condition for the extended third-order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman–Bucy filter.