H ∞ structured model reduction algorithms for linear discrete systems via LMI-based optimisation

In this article, H _∞ structured model reduction is addressed for linear discrete systems. Two important classes of systems are considered for structured model reduction, i.e. Markov jump systems and uncertain systems. The problem we deal with is the development of algorithms with the flexibility to allow any structure in the reduced-order system design, such as the structure of an original system, decentralisation of a networked system, pole assignment of the reduced system, etc. The algorithms are derived such that an associated model reduction error guarantees to satisfy a prescribed H _∞ norm-bound constraint. A new condition for the existence of desired reduced-order models preserving a certain structure is presented in a set of linear matrix inequalities (LMI) and non-convex equality constraints. Effective computational algorithms involving LMI are suggested to solve the matrix inequalities characterising a solution of the structured model reduction problem. Numerical examples demonstrate the advantages of the proposed model reduction method.     

H ∞ model reduction for discrete time-delay systems: delay-independent and dependent approaches

This paper investigates the problem of H _∞ model reduction for linear discrete-time state-delay systems. For a given stable system, our attention is focused on the construction of reduced-order models, which guarantee the corresponding error system to be asymptotically stable and have a prescribed H _∞ error performance. Both delay-independent and dependent approaches are developed, with sufficient conditions obtained for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be readily solved in standard numerical software. In addition, the development of reduced-order models with special structures, such as delay-free models and zeroth-order models, is also addressed. The approximation methods presented in this paper can be further extended to cope with systems with uncertain parameters. Two numerical examples have been provided to show the effectiveness of the proposed theories.     

ℋ ∞ model reduction for continuous-time switched stochastic hybrid systems

This article deals with the problem of computing an approximation system for a continuous-time switched stochastic system, such that the ?_∞ gain of the error system is less than a prescribed scalar. By using the average dwell-time approach and the piecewise Lyapunov function technique, a sufficient condition is first proposed, which guarantees the error system to be mean-square exponentially stable with a weighted ?_∞ performance. Then, the model reduction is solved by using the projection approach, which casts the model reduction into a sequential minimisation problem subjected to linear matrix inequality constraints by employing the cone complementary linearisation algorithm. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.     

ℋ ∞ model reduction for linear parameter-varying systems with distributed delay

This paper is concerned with the ?_∞ model reduction for linear parameter-varying (LPV) systems with both discrete and distributed delays. For a given stable system, our attention is focused on the construction of reduced-order models, which approximate the original system well in an ?_∞ norm sense. First, a sufficient condition is proposed for the asymptotic stability with an ?_∞ performance of the error system by using the parameter-dependent Lyapunov functional method. Then, the decoupling technique is applied, such that there does not exist any product term between the Lyapunov matrices and the system matrices in the parametrised linear matrix inequality (PLMI) constraints; thus a new sufficient condition is obtained. Based on the new condition, two different approaches are developed to solve the model reduction problem. One is the convex linearisation approach and the other is the projection approach. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Frequency-limited H∞-controller order reduction for linear parameter-varying systems

In this paper, a controller order reduction method for linear parameter-varying systems is presented. The proposed method is based on the frequency-weighted balanced truncation technique, which has the advantage to reduce the order in a specific frequency range. The approach is discussed and is proved to preserve the closed-loop stability with a guaranteed upper error bound. Effectiveness and performance of the obtained reduced-order controller are investigated by applying it to an automotive semi-active suspension control. The obtained simulation results show that objectives such as the road handling and the passenger comfort realised with the reduced-order controller are kept in the same performance level as with the full-order controller. Moreover, a comparison with an other-order reduction method is shown and confirms the advantage of the developed method.     

H ∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities

In this article, the H _∞ model reduction problem for a class of discrete-time Markov jump linear systems (MJLS) with partially known transition probabilities is investigated. The proposed systems are more general, relaxing the traditional assumption in Markov jump systems that all the transition probabilities must be completely known. A reduced-order model is constructed and the LMI-based sufficient conditions of its existence are derived such that the corresponding model error system is internally stochastically stable and has a guaranteed H _∞ performance index. A numerical example is given to illustrate the effectiveness and potential of the developed theoretical results.     

Combined L2/H∞ model reduction

A model-reduction problem is considered which involves both L ₂ (quadratic) and H _∞ (worst-case frequency-domain) aspects. Specifically, the goal of the problem is to minimize an L ₂ model-reduction criterion subject to a prespecified H _∞ constraint on the model-reduction error. The principal result is a sufficient condition for characterizing reduced-order models with bounded L ₂ and H _∞ approximation error. The sufficient condition involves a system of modified Riccati equations coupled by an oblique projection, i.e. idempotent matrix. When the H _∞ constraint is absent, the sufficient condition specializes to the L ₂ model-reduction result given by Hyland and Bernstein (1985).     

Robust H ∞ model predictive control for uncertain systems using relaxation matrices

In this paper, a robust H _∞ model predictive control (MPC) technique is proposed for time-varying uncertain discrete-time systems in the presence of input constraints and disturbances. We formulate a minimization problem of the upper bound of finite horizon cost function subject to the terminal inequality for an induced l ₂-norm bound. In order to improve system performance, we propose an LMI condition for the terminal inequality by using relaxation matrices. The LMI condition guarantees induced l ₂-norm bounds of the system despite system uncertainty and disturbance. A numerical example shows the effectiveness of the proposed method.     

Decentralised H ∞ control for singular systems

In this article, considering the design problem of decentralised H _∞ controller of singular systems, the two cases of controllers via measurement feedback are designed: one is precise controller, and the other is additive controller gain variation. The design procedures of the two cases of controllers are presented in terms of the solutions to generalised algebraic Riccati inequalities. The designed controllers in each case guarantee that closed-loop singular systems are admissible and with H _∞-norm bound on disturbance attenuation. Finally, a numerical example to demonstrate the validity of the proposed approach is given.     

Finite-time H∞ filtering for non-linear stochastic systems

This paper describes the robust H_∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H_∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton–Jacobi inequality. Based on this result, the existence of a finite-time H_∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.     

Quantised H ∞ filter design for discrete-time systems

This article is concerned with the quantised H _∞ filtering problem for discrete-time systems. The quantiser considered here is dynamic and composed of a dynamic scaling and a static quantiser. Motivated by practical transmission channels requirements, the static quantiser ranges are fully considered in this article. A quantised H _∞ filter design strategy is proposed, taking quantiser errors into account, where a convex optimisation method is developed to minimise static quantiser ranges with meeting H _∞ performance requirement for quantised augmented systems. A numerical example is given to illustrate the effectiveness of the proposed filter design method.     

H ∞ filtering for Markovian jump linear systems

The problem of H X filtering for continuous-time linear systems with Markovian jump is investigated. It was assumed that the jumping parameter was available. This paper develops necessary and sufficient conditions for designing a Markovian jump linear filter that ensures a prescribed bound on the L 2 -induced gain from the noise signals to the estimation error. The main result is tailored via linear matrix inequalities.     

H∞ control for discrete-time Takagi-Sugeno fuzzy systems

We consider the design problem of H X controllers for discrete-time fuzzy systems. We introduce two Riccati inequalities from the standard H X theory, one for the state feedback controller, the other for the filter gain. We rewrite the second inequality and obtain an equivalent inequality for the inverse. We then rewrite this using LMI to obtain the final form of the inequality and show that the proposed output feedback controller is n -suboptimal. We give two examples and construct n -suboptimal controllers for them.     

On H ∞-consensus filtering for multi-agent systems

This article investigates the H _∞-consensus filtering problem for multi-agent systems. The purpose of the problem addressed herein is to design decentralised full-order filters such that the filtering error dynamics achieve consensus and the prescribed noise attenuation level in H _∞ sense. Based on the state decomposition and the Lyapunov function method, sufficient conditions expressed in terms of linear matrix inequalities are derived for existence of such filters. Finally, a simulation example is utilised to demonstrate the usefulness of the developed theoretical results.     

Quantized H ∞ fault-tolerant control for networked control systems

Quantized H_∞ fault-tolerant control for networked control systems (NCSs) with partial actuator fault with respect to actuators is concerned in this paper. Considering transmission delay, packet dropout and quantization, a synthesis model with partial actuator fault is established. The piecewise constant controller is adopted to model NCS with the transmission delay and packet dropout. Due to data transmitted in practical NCSs should be quantized before they are sent to the next network node, the logarithmic static and time-invariant quantizers at the sensor and controller sides are proposed in the paper. For the established model, an appropriate type of Lyapunov functions is provided to investigate the delay-dependent H_∞ control problem. According to an optimal problem, the controller that makes the system achieve the best performance is designed. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed approach.     

Robust H ∞ filter design for affine fuzzy systems

This paper investigates the problem of robust H _∞ filtering for nonlinear systems, which are described by affine fuzzy parts with norm-bounded uncertainties. The system outputs have been chosen as premise variables, which can guarantee the plant and the filter always switch to the same region. By using a piecewise Lyapunov function and adding slack matrix variables, a fuzzy-basis-dependent H _∞ filter design method is obtained in the formulation of linear matrix inequalities (LMIs), which can be efficiently solved numerically. Compared with the existing results, the proposed method needs less LMI constraints and leads to less conservatism. Finally, numerical examples illustrate the effectiveness of the new results.     

Event-based H2/H∞ controllers for networked control systems

This paper is concerned with event-based H₂/H_∞ control design for networked systems with interval time-varying delays. The contributions are twofold. First, conditions for uniform ultimately bounded stability are provided in the H₂/H_∞ event-based context. The relation between the boundedness of the stability region and the threshold that triggers the events is studied. Second, a practical design procedure for event-based H₂/H_∞ control is provided. The method makes use of Lyapunov–Krasovskii functionals (LKFs) and it is characterised by its generality, as only mild assumptions are imposed on the structures of the LKF and the cost functional. The robustness and performance of the proposed technique is showed through numerical simulations.