An identification algorithm is proposed based on an extension of the results of H∞ filtering and ... more An identification algorithm is proposed based on an extension of the results of H∞ filtering and Kalman filtering theory. The objective is to minimize the H∞ norm of the map from exogenous inputs (noise) to the estimation error of the parameters of an autoregressive moving average with external variable (ARMAX) model. The technique can provide an improved fit of a low-order estimated model to be obtained, relative to the usual least squares based algorithms. The function γ which arises in H∞ filtering problems can be found by iteration, starting with a high initial value and then computing γ online until it converges to the optimal value. An online check on the a posteriori covariance matrix is necessary to make sure the solution remains valid. The proposed algorithm is straightforward to implement and has the potential to improve the robustness of self-tuning filtering and control algorithms
Robust Control of Linear Systems and Nonlinear Control, 1990
... 1.5) * z= 1" yy where Pn is a dynamic weighting term and O is the power spectral density... more ... 1.5) * z= 1" yy where Pn is a dynamic weighting term and O is the power spectral density matrix _ yy of the error signal y (t) Define the left coprime factorization Ajl [Cn, Cd]=[Wn, z^ MW](1.6) and the generalized spectral factor Y= Aj1 Df where Df, Df0 satisfy DfDf= Cdc3+ CnCJ+ ...
A game theory approach is presented to optimal state estimation. It is found that under certain c... more A game theory approach is presented to optimal state estimation. It is found that under certain conditions a min-max estimation is identical to the optimal estimation in the minimum H, -norm sense. These conditions are similar to those obtained in [13], where the relation between Kalman filtering and the min-max terminal state estimation has been explored. This new interpretation of the H,-optimal state estimation provides a better insight into the mechanism of the
Linear, state delayed, continuous-time systems are considered with both stochastic and norm-bound... more Linear, state delayed, continuous-time systems are considered with both stochastic and norm-bounded deterministic uncertainties in the state-space model. The problem of robust dynamic H<sub>∞</sub> output-feedback control is solved, for the stationary case, via the input-output approach where the system is replaced by a nonretarded system with additional deterministic norm-bounded uncertainties. In this problem, a cost function is defined which is the expected value of the standard H<sub>∞</sub> performance cost with respect to the stochastic parameters. An example is given that demonstrates the applicability of the theory.
Linear discrete-time systems with stochastic and deterministic polytopic-type uncertainties in th... more Linear discrete-time systems with stochastic and deterministic polytopic-type uncertainties in their state-space model are considered. A dynamic output-feedback controller is obtained via a new approach, which allows a derivation of a controller in spite of parameter uncertainty. In the proposed approach, the system is described via a difference equation and an augmented system is then used to obtain the output-feedback controller parameters. The controller is obtained without assuming a specific structure to the quadratic Lyapunov function. It is the first time that an output-feedback controller is obtained for robust state-multiplicative systems. The controller minimizes the stochastic \(l_2\)-gain of the closed-loop system, where the cost function is defined to be the expected value of the standard \(H_{\infty }\) performance index with respect to the stochastic uncertainty. An example is given that demonstrates the merit of the theory.
Abstract We consider linear state-multiplicative noisy, discrete-time systems with polytopic unce... more Abstract We consider linear state-multiplicative noisy, discrete-time systems with polytopic uncertainties. The stationary problems of H∞ state-feedback control and filtering are solved by applying a vertex dependent Lyapunov function. This enable us to reduces considerably the over-design that is associated with the classical design and that is based on a single Lyapunov function for the whole parameters range. The solutions to the robust problems of both: the state-feedback and the filtering problems are obtained via a set of LMIs which are readily applicable.
This paper addresses the stability, as well as an upper bound on the ℓ2-gain of linear switched s... more This paper addresses the stability, as well as an upper bound on the ℓ2-gain of linear switched state-multiplicative stochastic systems. The switching law is assumed to obey a dwell time constraint. The results are achieved by assigning each subsystem its own Lyapunov function. The Lyapunov function is time varying during the dwell time and becomes time invariant afterwards. The Lyapunov function is required to be non-increasing at the switching instances. If the dwell time approaches zero-the standard quadratic conditions are recovered. Stability and performance are guaranteed in the mean-square sense by requiring the expected value of the Lyapunov function to decrease, both for nominal systems and systems that entail polytopic-type parameter uncertainties. The theory is then extended to state-feedback controller design. The conditions are all given in terms of linear matrix inequalities.
Abstract Two methods for soft controller switching are introduced. They both treat static output-... more Abstract Two methods for soft controller switching are introduced. They both treat static output-feedback control and achieve a prescribed bound on the L2 -gain of the closed-loop over the entire uncertainty range. This L2 -gain can then be used to guarantee tracking performance, disturbance rejection, and stability margins over the entire uncertainty range. The soft switching is performed by continuously changing the controller gain from one of the controllers to a second one over the switching interval. The first method treats the case where the gain may follow any continuous trajectory during the switching interval. The second method treats the case where the soft switching is performed by a linear interpolation between the two control gains over the switching interval. The designer may choose to prescribe any, all or none of the controllers gains. The problem is then formulated as a set of linear matrix inequalities. The conditions are applied to a z-axis control of a quadcopter.
ABSTRACT Xiefu Jiang et al. point out some some misleading statements in the above mentioned pape... more ABSTRACT Xiefu Jiang et al. point out some some misleading statements in the above mentioned paper (Shaked et al., 1998) and relation results are given. Also Wen-Shyong Yu points out serious mistakes in the same paper and also untenable main results.
The problem of robust H∞ control of uncertain linear switched system with dwell time is addressed... more The problem of robust H∞ control of uncertain linear switched system with dwell time is addressed. A Lyapunov function in a quadratic form, which is non-increasing at the switching instances, is assigned to each subsystem. This function is used to determine the stability and the L2-gain of the switched system, where a bounded real lemma (BRL) is derived that provides an upper-bound to the system L2-gain over the uncertainty polytope. The obtained results are used to solve the corresponding problem of robust H∞ state-feedback control for cases where the switching signal is either measured online, or is unknown. A numerical example is given which demonstrates the applicability of the theory developed to the control of switched systems.
An identification algorithm is proposed based on an extension of the results of H∞ filtering and ... more An identification algorithm is proposed based on an extension of the results of H∞ filtering and Kalman filtering theory. The objective is to minimize the H∞ norm of the map from exogenous inputs (noise) to the estimation error of the parameters of an autoregressive moving average with external variable (ARMAX) model. The technique can provide an improved fit of a low-order estimated model to be obtained, relative to the usual least squares based algorithms. The function γ which arises in H∞ filtering problems can be found by iteration, starting with a high initial value and then computing γ online until it converges to the optimal value. An online check on the a posteriori covariance matrix is necessary to make sure the solution remains valid. The proposed algorithm is straightforward to implement and has the potential to improve the robustness of self-tuning filtering and control algorithms
Robust Control of Linear Systems and Nonlinear Control, 1990
... 1.5) * z= 1" yy where Pn is a dynamic weighting term and O is the power spectral density... more ... 1.5) * z= 1" yy where Pn is a dynamic weighting term and O is the power spectral density matrix _ yy of the error signal y (t) Define the left coprime factorization Ajl [Cn, Cd]=[Wn, z^ MW](1.6) and the generalized spectral factor Y= Aj1 Df where Df, Df0 satisfy DfDf= Cdc3+ CnCJ+ ...
A game theory approach is presented to optimal state estimation. It is found that under certain c... more A game theory approach is presented to optimal state estimation. It is found that under certain conditions a min-max estimation is identical to the optimal estimation in the minimum H, -norm sense. These conditions are similar to those obtained in [13], where the relation between Kalman filtering and the min-max terminal state estimation has been explored. This new interpretation of the H,-optimal state estimation provides a better insight into the mechanism of the
Linear, state delayed, continuous-time systems are considered with both stochastic and norm-bound... more Linear, state delayed, continuous-time systems are considered with both stochastic and norm-bounded deterministic uncertainties in the state-space model. The problem of robust dynamic H<sub>∞</sub> output-feedback control is solved, for the stationary case, via the input-output approach where the system is replaced by a nonretarded system with additional deterministic norm-bounded uncertainties. In this problem, a cost function is defined which is the expected value of the standard H<sub>∞</sub> performance cost with respect to the stochastic parameters. An example is given that demonstrates the applicability of the theory.
Linear discrete-time systems with stochastic and deterministic polytopic-type uncertainties in th... more Linear discrete-time systems with stochastic and deterministic polytopic-type uncertainties in their state-space model are considered. A dynamic output-feedback controller is obtained via a new approach, which allows a derivation of a controller in spite of parameter uncertainty. In the proposed approach, the system is described via a difference equation and an augmented system is then used to obtain the output-feedback controller parameters. The controller is obtained without assuming a specific structure to the quadratic Lyapunov function. It is the first time that an output-feedback controller is obtained for robust state-multiplicative systems. The controller minimizes the stochastic \(l_2\)-gain of the closed-loop system, where the cost function is defined to be the expected value of the standard \(H_{\infty }\) performance index with respect to the stochastic uncertainty. An example is given that demonstrates the merit of the theory.
Abstract We consider linear state-multiplicative noisy, discrete-time systems with polytopic unce... more Abstract We consider linear state-multiplicative noisy, discrete-time systems with polytopic uncertainties. The stationary problems of H∞ state-feedback control and filtering are solved by applying a vertex dependent Lyapunov function. This enable us to reduces considerably the over-design that is associated with the classical design and that is based on a single Lyapunov function for the whole parameters range. The solutions to the robust problems of both: the state-feedback and the filtering problems are obtained via a set of LMIs which are readily applicable.
This paper addresses the stability, as well as an upper bound on the ℓ2-gain of linear switched s... more This paper addresses the stability, as well as an upper bound on the ℓ2-gain of linear switched state-multiplicative stochastic systems. The switching law is assumed to obey a dwell time constraint. The results are achieved by assigning each subsystem its own Lyapunov function. The Lyapunov function is time varying during the dwell time and becomes time invariant afterwards. The Lyapunov function is required to be non-increasing at the switching instances. If the dwell time approaches zero-the standard quadratic conditions are recovered. Stability and performance are guaranteed in the mean-square sense by requiring the expected value of the Lyapunov function to decrease, both for nominal systems and systems that entail polytopic-type parameter uncertainties. The theory is then extended to state-feedback controller design. The conditions are all given in terms of linear matrix inequalities.
Abstract Two methods for soft controller switching are introduced. They both treat static output-... more Abstract Two methods for soft controller switching are introduced. They both treat static output-feedback control and achieve a prescribed bound on the L2 -gain of the closed-loop over the entire uncertainty range. This L2 -gain can then be used to guarantee tracking performance, disturbance rejection, and stability margins over the entire uncertainty range. The soft switching is performed by continuously changing the controller gain from one of the controllers to a second one over the switching interval. The first method treats the case where the gain may follow any continuous trajectory during the switching interval. The second method treats the case where the soft switching is performed by a linear interpolation between the two control gains over the switching interval. The designer may choose to prescribe any, all or none of the controllers gains. The problem is then formulated as a set of linear matrix inequalities. The conditions are applied to a z-axis control of a quadcopter.
ABSTRACT Xiefu Jiang et al. point out some some misleading statements in the above mentioned pape... more ABSTRACT Xiefu Jiang et al. point out some some misleading statements in the above mentioned paper (Shaked et al., 1998) and relation results are given. Also Wen-Shyong Yu points out serious mistakes in the same paper and also untenable main results.
The problem of robust H∞ control of uncertain linear switched system with dwell time is addressed... more The problem of robust H∞ control of uncertain linear switched system with dwell time is addressed. A Lyapunov function in a quadratic form, which is non-increasing at the switching instances, is assigned to each subsystem. This function is used to determine the stability and the L2-gain of the switched system, where a bounded real lemma (BRL) is derived that provides an upper-bound to the system L2-gain over the uncertainty polytope. The obtained results are used to solve the corresponding problem of robust H∞ state-feedback control for cases where the switching signal is either measured online, or is unknown. A numerical example is given which demonstrates the applicability of the theory developed to the control of switched systems.
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