Distributed H ∞ consensus of multi-agent systems: a performance region-based approach

This article addresses the distributed H _∞ consensus problem of multi-agent systems with general linear node dynamics using relative output measurements. The notion of H _∞ consensus performance region is first introduced and then analysed as a basis for the protocol design. A new kind of distributed observer-type H _∞ protocols is further proposed. Theoretical analysis indicates that the distributed H _∞ consensus problem can be solved if and only if the coupling strength of the protocol belongs to the H _∞ performance region of the closed-loop network. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.     

Internal positivity preserved model reduction

This article studies model reduction of continuous-time stable positive linear systems under the Hankel norm, H _∞ norm and H ₂ norm performance. The reduced-order systems preserve the stability as well as the positivity of the original systems. This is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the Lyapunov matrices are decoupled with the system matrices. In this way, the positivity constraints in the reduced-order model can be imposed in a natural way. As the model reduction performances are expressed in linear matrix inequalities with equality constraints, the desired reduced-order positive models can be obtained by using the cone complementarity linearisation iterative algorithm. A numerical example is presented to illustrate the effectiveness of the given methods.     

Implementation of FIR control for H ∞ output feedback stabilisation of linear systems

In this article, finite impulse response (FIR) control is addressed for H _∞ output feedback stabilisation of linear systems. The problem we deal with is the construction of an output feedback controller with a certain FIR structure such that the resulting closed-loop system is asymptotically stable and a prescribed H _∞ norm bound constraint is guaranteed. Some solvability conditions are suggested in this article. Sufficient conditions are derived to obtain a suboptimal solution of the H _∞ FIR control problem via convex optimisation. Also, an equivalent condition for the existence of H _∞ FIR control is presented in the set of linear matrix inequalities (LMIs) and a reciprocal matrices equality constraint. An effective computational algorithm involving LMIs is suggested to solve a concave minimisation problem characterising a local optimal solution of the H _∞ FIR control problem. Numerical examples demonstrate the validity of the proposed H _∞ FIR control and the numerical efficiency of the proposed algorithm for FIR control.     

Optimal discrete-time H∞/γ0 filtering and control under unknown covariances

New stochastic γ₀ and mixed H_∞/γ₀ filtering and control problems for discrete-time systems under completely unknown covariances are introduced and solved. The performance measure γ₀ is the worst-case steady-state averaged variance of the error signal in response to the stationary Gaussian white zero-mean disturbance with unknown covariance and identity variance. The performance measure H_∞/γ₀ is the worst-case power norm of the error signal in response to two input disturbances in different channels, one of which is the deterministic signal with a bounded energy and the other is the stationary Gaussian white zero-mean signal with a bounded variance provided the weighting sum of disturbance powers equals one. In this framework, it is possible to consider at the same time both deterministic and stochastic disturbances highlighting their mutual effects. Our main results provide the complete characterisations of the above performance measures in terms of linear matrix inequalities and therefore both the γ₀ and H_∞/γ₀ optimal filters and controllers can be computed by convex programming. H_∞/γ₀ optimal solution is shown to be actually a trade-off between optimal solutions to the H_∞ and γ₀ problems for the corresponding channels.     

H ∞ consensus control of multi-agent systems with switching topology: a dynamic output feedback protocol

This article is devoted to the consensus control for switching networks of multiple agents with linear coupling dynamics and subject to external disturbances, which is transformed into an H _∞ control problem by defining an appropriate controlled output. On this basis, a distributed dynamic output feedback protocol is proposed with an undetermined system matrix, and a condition in terms of linear matrix inequalities (LMIs) is derived to ensure consensus of the multi-agent system with a prescribed H _∞ level. Furthermore, system matrix of the protocol is designed by solving two LMIs. A numerical example is included to illustrate the effectiveness of the proposed consensus protocol.     

H2 control of discrete-time periodic systems with Markovian jumps and multiplicative noise

This paper addresses the problem of optimal and robust H₂ control for discrete-time periodic systems with Markov jump parameters and multiplicative noise. To analyse the system performance in the presence of exogenous random disturbance, an H₂ norm is firstly established on the basis of Gramian matrices. Further, under the condition of exact observability, a necessary and sufficient condition is presented for the solvability of H₂ optimal control problem by means of a generalised Riccati equation. When the transition probabilities of jump parameter are incompletely measurable, an H₂-guaranteed cost norm is exploited and the robust H₂ controller is designed through a linear matrix inequality (LMI) optimisation approach. An example of a networked control system is supplied to illustrate the proposed results.     

Relating H2 and H∞ bounds for finite-dimensional systems

For a linear time invariant system, the infinity-norm of the transfer function can be used as a measure of the gain of the system. This notion of system gain is ideally suited to the frequency domain design techniques such as H_∞ optimal control. Another measure of the gain of a system is the H₂ norm, which is often associated with the LQG optimal control problem. The only known connection between these two norms is that, for discrete time transfer functions, the H₂ norm is bounded by the H_∞ norm. It is shown in this paper that, given precise or certain partial knowledge of the poles of the transfer function, it is possible to obtain an upper bound of the H_∞ norm as a function of the H₂ norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H₂ norm can be bounded by a function of the H_∞ norm and the bandwidth of the system.     

Robust H∞ controller design for dynamic consensus networks

This paper studies an H_∞ suboptimal control problems of consensus networks whereby the weights of network edges are no longer static gains, but instead are dynamic systems, leading to the notion of dynamic consensus networks. We apply model, orthogonal and diagonal transformations to a dynamic consensus network in order to reduce the overall system into N ? 1 independent subsystems. We then establish a generalised methodology for designing a controller for a dynamic consensus network in the presence of external disturbances, focusing especially on using decentralised controllers that achieve consensus in the absence of disturbances and attenuation of disturbances to a prescribed H_∞ performance level. A design example is given to illustrate our results.     

Extensive theoretical/numerical comparative studies on H2 and generalised H2 norms in sampled-data systems

This paper is concerned with linear time-invariant (LTI) sampled-data systems (by which we mean sampled-data systems with LTI generalised plants and LTI controllers) and studies their H₂ norms from the viewpoint of impulse responses and generalised H₂ norms from the viewpoint of the induced norms from L₂ to L_∞. A new definition of the H₂ norm of LTI sampled-data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions. We then establish unified treatment of the three definitions of the H₂ norm through a matrix function G(τ) defined on the sampling interval [0, h). This paper next considers the generalised H₂ norms, in which two types of the L_∞ norm of the output are considered as the temporal supremum magnitude under the spatial 2-norm and ∞-norm of a vector-valued function. We further give unified treatment of the generalised H₂ norms through another matrix function F(θ) which is also defined on [0, h). Through a close connection between G(τ) and F(θ), some theoretical relationships between the H₂ and generalised H₂ norms are provided. Furthermore, appropriate extensions associated with the treatment of G(τ) and F(θ) to the closed interval [0, h] are discussed to facilitate numerical computations and comparisons of the H₂ and generalised H₂ norms. Through theoretical and numerical studies, it is shown that the two generalised H₂ norms coincide with neither of the three H₂ norms of LTI sampled-data systems even though all the five definitions coincide with each other when single-output continuous-time LTI systems are considered as a special class of LTI sampled-data systems. To summarise, this paper clarifies that the five control performance measures are mutually related with each other but they are also intrinsically different from each other.     

Neural Net-Based H ∞ Control for a Class of Nonlinear Systems

A novel neural net-based approach for H _∞ control design of a class of nonlinear continuous-time systems is presented. In the proposed frameworks, the nonlinear system models are approximated by multilayer neural networks. The neural networks are piecewisely interpolated to generate a linear differential inclusion models by which a linear state feedback H _∞ control law can be constructed. It is shown that finding the permissible control gain matrices can be transformed to a standard linear matrix inequality problem and solved using the available computer software. This revised version was published online in June 2006 with corrections to the Cover Date.     

LMI-based H ∞-optimal control with transients

In this article, the worst-case norm of the regulated output over all exogenous signals and initial states as a performance measure of the system is characterised in terms of linear matrix inequalities (LMIs). Optimal time-invariant state- and output-feedback controllers are synthesised as minimising this performance measure. The essential role in this synthesis plays a weighting matrix reflecting the relative importance of the uncertainty in the initial state contrary to the uncertainty in the exogenous signal. H _∞-optimal control with transients is shown to be actually a trade-off between H _∞-control, being optimal under unknown exogenous disturbances and zero initial state, and γ-control, being optimal under zero exogenous signal and unknown initial conditions, if and only if the weighting matrix satisfies a fundamental inequality. If this inequality is met, the performance measure is achieved and the explicit formulae for the worst-case disturbance and initial state are provided. If this inequality fails, the performance measure coincides with the H _∞-norm and the trade-off gets broken.     

Neural network solution for finite-horizon H ∞ state feedback control of nonlinear systems

In this article, neural networks are used to approximately solve the finite-horizon optimal H _∞ state feedback control problem. The method is based on solving a related Hamilton–Jacobi–Isaacs equation of the corresponding finite-horizon zero-sum game. The neural network approximates the corresponding game value function on a certain domain of the state-space and results in a control computed as the output of a neural network. It is shown that the neural network approximation converges uniformly to the game-value function and the resulting controller provides closed-loop stability and bounded L ₂ gain. The result is a nearly exact H _∞ feedback controller with time-varying coefficients that is solved a priori offline. The results of this article are applied to the rotational/translational actuator benchmark nonlinear control problem.     

Optimal time-weighted H 2 model reduction for Markovian jump systems

This article addresses the optimal time-weighted H ₂ model reduction problem for Markovian jump linear systems. That is, for a given mean square stable Markovian jump system, our aim is to find a mean square stable jump system of lower order such that the time-weighted H ₂ norm of the corresponding error system is minimised. The time-weighted H ₂ norm of the system is first defined, and then a computational method is constructed. The computation requires the solution of two sets of recursive Lyapunov-type linear matrix equations associated with the Markovian jump system. To solve the optimal time-weighted H ₂ model reduction problem, we propose a gradient flow method for its solution. A necessary condition for minimality is derived, and a computational procedure is provided to obtain the minimising reduced-order model. The necessary condition generalises the standard result for systems when Markov jumps and the time-weighting term do not appear. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed approach.     

Design of H 2 Controllers for Sampled-Data Systems with Input Time Delays

This paper presents a general framework based on lifting technique for sampled-data systems with input time delays. By analyzing the properties of operator-valued matrices of lifted systems with input time delays, an extended lifting technique is obtained. It is then shown that, with the proposed lifting technique, the complex behavior of the system can be illustrated by two simple lifted systems, which construct the extended lifted system. The extended lifted system has the same induced norm as that of the original system with an input time delay, since the proposed lifting technique is an isometric isomorphism. Through applying the proposed lifting technique to sampled-data systems with input time delays, the time-invariant discrete-time system with infinite-dimensional input and output spaces is obtained. The equivalent discrete-time system, which is derived from the extended lifted system, can satisfy the problem of H ₂ sampled-data control systems with input time delays. Simulation results are given to show that the proposed method can guarantee a more stable system response than the conventional H ₂ sampled-data controller for the sampled-data systems with the various input time delays.     

An integrated plant/control design method and application in hard disk drives

One approach in servo control to achieve a high track density in hard disk drives is to minimise the H₂ norm from disturbances to position error signal. The H₂ performance optimisation is then deemed as a matter of great significance. This paper presents an integrated design method involving plant modification and controller design sequentially to achieve the H₂ performance requirement. A linear matrix inequality-based approach is developed for the plant damping ratio modification using the plant output. The proposed model modification method is then applied to the voice coil motor plant in hard disk drives, followed by the optimal H₂ controller design using the Riccati equation method with the modified plant. It turns out that the modified plant leads to better H₂ performance, stability margins than the original plant.     

Robust control for synchronization of singular complex delayed networks with stochastic switched coupling

This paper investigates the robust H _∞ control of synchronization for uncertain singular complex delayed networks with stochastic switched coupling. The switched coupling is dependent on a continuous time Markovian process. Based on Lyapunov–Krasovskii stability theory, sufficient conditions for the existence of the H _∞ control laws are given in terms of linear matrix inequalities, which ensure robustly global mean-square synchronization and a prescribed H _∞ disturbance attenuation level for the resulting closed-loop complex networks. Numerical examples are given to illustrate the effectiveness of the proposed method.     

Robust H∞ performance for discrete time T-S fuzzy switched memristive stochasticneural networks with mixed time-varying delays

ABSTRACT

In this paper, we study the robust H _∞ performance for discrete-time T-S fuzzy switched memristive stochastic neural networks with mixed time-varying delays and switching signal design. The neural network under consideration is subject to time-varying and norm bounded parameter uncertainties. Decomposing of the delay interval approach is employed in both the discrete delays and distributed delays. By constructing a proper Lyapunov-Krasovskii functional (LKF) with triple summation terms and using an improved summation inequality techniques. Sufficient conditions are derived in terms of linear matrix inequalities (LMIs) to guarantee the considered discrete-time neural networks to be exponentially stable. Finally, numerical examples with simulation results are given to illustrate the effectiveness of the developed theoretical results.     

H 2 optimal control for a wide class of discrete-time linear stochastic systems

In this article, the problem of H ₂-control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Different H ₂ performance criteria (often called H ₂-norms) are introduced and characterised via solutions of some suitable linear equations on certain spaces of symmetric matrices. Some aspects specific to the discrete-time framework are revealed. The problem of optimisation of H ₂-norms is solved under the assumption that full state vector is available for measurements. One shows that among all stabilising controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilising solution of a system of discrete-time generalised Riccati equations.     

Modified fast-sample/fast-hold approximation and γ-independent H ∞-discretisation for general sampled-data systems by fast-lifting

This article is concerned with the fast-lifting approach to H _∞ analysis and design of sampled-data systems, and extends our preceding study on modified fast-sample/fast-hold (FSFH) approximation, in which the direct feedthrough matrix D ₁₁ from the disturbance w to the controlled output z was assumed to be zero. More precisely, this article removes this assumption and shows that a γ-independent H _∞ discretisation is still possible in a nontrivial fashion by applying what we call quasi-finite-rank approximation of an infinite-rank operator and then the loop-shifting technique. As in the case of D ₁₁ = 0, the modified FSFH approach retains the feature that both the upper and lower bounds of the H _∞-norm or the frequency response gain can be computed, where the gap between the upper and lower bounds can be bounded with the approximation parameter N and is independent of the discrete-time controller. This feature is significant in applying the new method especially to control system design, and this study indeed has a very close relationship to the recent progress in the study of control system analysis/design via noncausal linear periodically time-varying scaling. The significance of a key lemma pertinent to the fast-lifting approach is suggested in connection with such a relationship, and also with its application to time-delay systems.     

A probabilistic solution of robust H∞ control problem with scaled matrices

This paper addresses the robust H_∞ control problem with scaled matrices. It is difficult to find a global optimal solution for this non-convex optimisation problem. A probabilistic solution, which can achieve globally optimal robust performance within any pre-specified tolerance, is obtained by using the proposed method based on randomised algorithm. In the proposed method, the scaled H_∞ control problem is divided into two parts: (1) assume the scaled matrices be random variables, the scaled H_∞ control problem is converted to a convex optimisation problem for the fixed sample of the scaled matrix and a optimal solution corresponding to the fixed sample is obtained; (2) a probabilistic optimal solution is obtained by using the randomised algorithm based on a finite number N optimal solutions, which are obtained in part (1). The analysis shows that the worst case complexity of proposed method is a polynomial.