Convergent algorithms for frequency weighted L2 model reduction

This paper is concerned with computing an L₂-optimal reduced-order model for a given stable multivariable linear system in the presence of input and output frequency weightings. By parametrizing a class of reduced-order models in terms of an orthogonal projection and using manifold techniques as tools, both continuous and iterative algorithms are derived and their convergence properties are established. As an application, we show that an L₂ optimal reduced-order filter in the closed-loop sense can be computed using these algorithms.     

H ∞ sampled data control of systems with time-delays

Sampled-data H _∞ control of linear systems with constant state, control and measurement delays is considered. The sampling of the controlled input and of the measured output is not assumed to be uniform. The system is modelled as a continuous-time one, where the controlled input and the measurement output have piecewise-continuous delays. The input–output approach to stability and L ₂-gain analysis is applied to the resulting system. The discretised Lyapunov functional method is extended to the case of multiple delays, where the Lyapunov functional is complete in one of the delays (in the state) and is simple in the other delays (those in the input and in the output), which are constant. Solutions to the state-feedback and the output-feedback H _∞ control problems are derived in terms of linear matrix inequalities (LMIs).     

Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation — state feedback case

We consider the problem of simultaneous finite gain L_p-stabilization and internal stabilization of linear systems subject to input saturation via linear static state feedback. We show that bounded input finite-gain L_p-stabilization and local asymptotic stabilization can always be achieved simultaneously no matter where the poles of the open-loop system are, and the locations of these poles play a role only when bounded input finite gain L_p-stabilization and global or semi-global stabilization are required simultaneously.     

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.     

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.     

不确定串联非线性系统H∞鲁棒自适应控制

针对一类含有未知参数和干扰的非最小相位串联非线性系统,结合H_∞控制和自适应控制方法并利用李雅普诺夫函数递推设计方法设计了状态反馈H_∞自适应控制器,避免了求解Hamilton-Jacobi-Isaacs不等式设计控制器的困难.该控制器不仅保证闭环系统ISS(input-to-state)稳定,而且使得系统对于所有允许的参数不确定从干扰输入到可控输出的L₂增益不大于给定的值.最后,给出了一个仿真例子,仿真结果充分表明了所设计的控制器的可行性和有     

On globally stable adaptive control providing l 1 tracking performance for linear discrete-time systems

Existing adaptive control algorithms at best guarantee that the tracking error is a l ₂ sequence. This paper presents globally stable adaptive control algorithms for linear discrete-time systems providing l ₁ tracking performance. Two algorithms with different degree of complexity are proposed, one for the case of known control directions, and a separate algorithm for the case of unknown control directions. It is demonstrated that in both cases the tracking error is l ₁ sequence, while the input and output signals are uniformly bounded.     

Anti-disturbance control for time-varying delay Markovian jump nonlinear systems with multiple disturbances

In this paper, the problem of composite anti-disturbance resilient control is addressed for time-varying delay Markovian jump nonlinear systems with multiple disturbances. The disturbances are assumed to include two parts: the first one in the input channel is described by an external system with perturbations; the second one is supposed to be bounded H₂ norm. By combining disturbance observer and L₂–L_∞ control method, the disturbances are attenuated and rejected, simultaneously, and the desired dynamic performance can be obtained for time-varying delay Markovian jump nonlinear systems. Moreover, the gains of the resilient controller and the observer are acquired by applying linear matrix inequalities (LMIs) technology. Finally, an application example is presented to show the effectiveness of the proposed approach.     

Persistent inputs and the standard H 2 multivariable control problem

The H ₂ control problem is formulated with exogenous inputs having unstable shape-deterministic components. Such inputs are called persistent. Some issues in connection with past solutions of the H ₂ control problem in this case are carefully described and addressed. In particular, it is established that two- and three-degree-of freedom (2DOF and 3DOF) systems with persistent inputs cannot be treated within the framework of the standard configuration. In addition, past treatments of persistent inputs within the framework of the generalised 2DOF configuration focused on solutions which yielded stable error transforms without explicitly requiring the same for controller outputs. In this article, a more general configuration is treated and physical considerations are invoked to justify imposition of the requirement that both the regulated variable z(s) and the controller output u(s) be stable. A persistent input model for which there exists a stabilising controller that makes both z(s) and u(s) stable is called acceptable and the necessary and sufficient condition for such acceptability is determined. Also considered is a persistent input model for which there exists a stabilising controller that makes the controller output, the measured output and the regulated variable stable. Such a model is called strictly acceptable and the necessary and sufficient condition for strict acceptability is given. The subset of all stabilising controllers associated with an acceptable persistent input model is parameterised and this parameterisation is used to formulate an H ₂ optimisation problem with persistent inputs which can then be solved using standard procedures.     

Finite-time H∞ control of periodic piecewise linear systems

In this paper, the finite-time stability, stabilisation, L₂-gain and H_∞ control problems for a class of continuous-time periodic piecewise linear systems are addressed. By employing a time-varying control scheme in which the time interval of each subsystem constitutes a number of basic time segments, the finite-time controllers can be developed with periodically time-varying control gains. Based on a piecewise time-varying Lyapunov-like function, a sufficient condition of finite-time stability and the relevant time-varying controller are proposed. Considering the finite-time boundedness of the closed-loop periodic system, the L₂-gain criterion with continuous time-varying Lyapunov-like matrix function is studied. A finite-time H_∞ controller is proposed based on the L₂-gain analysis. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed criteria.     

H∞ control problem of linear periodic piecewise time-delay systems

This paper investigates the H_∞ control problem based on exponential stability and weighted L₂-gain analyses for a class of continuous-time linear periodic piecewise systems with time delay. A periodic piecewise Lyapunov–Krasovskii functional is developed by integrating a discontinuous time-varying matrix function with two global terms. By applying the improved constraints to the stability and L₂-gain analyses, sufficient delay-dependent exponential stability and weighted L₂-gain criteria are proposed for the periodic piecewise time-delay system. Based on these analyses, an H_∞ control scheme is designed under the considerations of periodic state feedback control input and iterative optimisation. Finally, numerical examples are presented to illustrate the effectiveness of our proposed conditions.     

Adaptive H2/H∞ tracking control for a class of uncertain robotic systems

This paper addresses the problem of designing mixed H₂/H_∞ tracking control for a large class of uncertain robotic systems. Nonlinear H_∞ control theory, H₂ control theory and intelligent adaptive control algorithm are combined to construct a hybrid adaptive/robust H₂/H_∞ tracking control scheme. One adaptive neural network system is constructed to approximate the behaviour of uncertain robot dynamics, and the other adaptive control algorithm is designed to estimate the behaviour of the modelled disturbance. Moreover, a robust H_∞ control algorithm is designed to attenuate the effects of the unmodelled disturbance. Only a set of algebraic matrix Riccati-like equations is required to implement the proposed mixed H₂/H_∞ tracking controller, and so an explicit and closed-form solution is obtained. Consequently, the mixed H₂/H_∞ adaptive/robust tracking controller developed here can be analytically computed and easily implemented. Finally, simulations are presented to illustrate the effectiveness of the proposed control algorithm.     

H∞ control for a class of structured time-delay systems

In this paper, we design an H_∞ controller for a class of lower-triangular time-delay systems. Backstepping is applied to construct an explicit feedback controller, and the closed-loop system maintains internal stability and an L₂-gain from the disturbance input to the output. The design is delay-dependent. Simulations on an example system demonstrate the good performance of the proposed design.     

Stabilisation and L 2-gain analysis for a class of uncertain switched non-linear systems

This article is concerned with the problem of stabilisation and L ₂-gain analysis for a class of switched non-linear systems with norm-bounded time-varying uncertainties. A system in this class is composed of two parts: an uncertain linear switched part and a non-linear part, which are also switched systems. When all the subsystems are stabilisable and have an L ₂-gain, the switched feedback control law and the switching law are designed respectively using average dwell-time method such that the corresponding closed-loop switched system is exponentially stable and achieves a weighted L ₂-gain.     

Exponential stability and L 2-gain analysis for a class of faulty systems

This article is concerned with the exponential stability and L ₂-gain problems for a class of systems with failures. Possible failures of sensors, actuators and/or controllers are considered simultaneously. Inspired by a switching control scheme, we develop an idea that is turning the class of faulty systems into a switched system composed of stable and unstable subsystems. Based on an average dwell‐time method, an activation time ratio scheme between stable and unstable subsystems is derived. It is shown that, under the proposed scheme, the faulty systems ensure exponential stability as well as a desire weighted L ₂-gain performance level. A numerical example and a practical simulation example on the flight control system are given to illustrate the practicability of the proposed method.     

H 2 input load disturbance rejection controller design for synchronised output regulation of time-delayed multi-agent systems with frequency domain method

An analytical H ₂ input load disturbance rejection (ILDR) controller design approach is presented for the synchronised output regulation (SOR) of homogeneous time-delayed multi-agent systems. First, a closed-loop multi-input multi-output framework with transfer functions is introduced, and an SOR condition is given. Second, the decomposition method is utilised to simplify the analysis of internal stability and H ₂ performance index of the whole system to a set of independent optimisation problems with respect to the eigenvalues of the topology matrix. Finally, for each decomposed subsystem, the H ₂ optimal ILDR controller can be computed from all the stabilising controllers. By comparison with the conventional given-structured controllers, the contributions of the new approach are that the design procedure is calculated analytically for multi-agent systems with input disturbances, and a simple quantitative tuning way is developed to trade-off the nominal performance and robustness. The simulation example shows the effective ILDR capability of the proposed control strategy.     

Robust L 2 − L ∞ consensus control for uncertain high-order multi-agent systems with time-delay

This article studies that the robust consensus problem for high-order multi-agent systems with time-delay, which are subjected to external disturbances and communication uncertainties. An approach based on the weighting matrix is introduced to meet the requirement that the peak value of controlled output is often required into a certain range. Then, a dynamic model of high-order multi-agent systems is established. Using a feedback controller and interactions from the neighbours including uncertainties, a linear control protocol is proposed for the systems with networks under fixed or switching topologies. Sufficient conditions in terms of linear matrix inequalities are derived to make all the agents reach robust consensus with the prescribed L ₂???L _∞ performance. A numerical simulation is provided to demonstrate the effectiveness of the proposed protocol.     

Robust L2 − L∞ filtering for a class of dynamical systems with nonhomogeneous Markov jump process

This paper investigates the problem of robust L₂ ? L_∞ filtering for a class of dynamical systems with nonhomogeneous Markov jump process. The time-varying transition probabilities which evolve as a nonhomogeneous jump process are described by a polytope, and parameter-dependent and mode-dependent Lyapunov function is constructed for such system, and then a robust L₂ ? L_∞ filter is designed which guarantees that the resulting error dynamic system is robustly stochastically stable and satisfies a prescribed L₂ ? L_∞ performance index. A numerical example is given to illustrate the effectiveness of the developed techniques.     

Adaptive nonlinear excitation control with L2 disturbance attenuation for power systems

This paper presents a design approach to nonlinear feedback excitation control of power systems with unknown disturbance and unknown parameters. It is shown that the stabilizing control law with desired L₂ gain from the disturbance to a penalty signal can be designed by a recursive way without linearization. A state feedback law is presented for the case of the system with known parameters, and then the control law is extended to adaptive controller for the case when the parameters of the electrical dynamics of the power system are unknown. Simulation results demonstrate that the proposed controllers guarantee transient stability of the system regardless of the system parameters and faults.     

Further results on the L1 analysis of sampled-data systems via kernel approximation approach

This paper gives two methods for the L₁ analysis of sampled-data systems, by which we mean computing the L_∞-induced norm of sampled-data systems. This is achieved by developing what we call the kernel approximation approach in the setting of sampled-data systems. We first consider the lifting treatment of sampled-data systems and give an operator theoretic representation of their input/output relation. We further apply the fast-lifting technique by which the sampling interval [0, h) is divided into M subintervals with an equal width, and provide methods for computing the L_∞-induced norm. In contrast to a similar approach developed earlier called the input approximation approach, we use an idea of kernel approximation, in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions. Furthermore, it is shown that the approximation errors in the piecewise constant approximation or piecewise linear approximation scheme converge to 0 at the rate of 1/M or 1/M², respectively. In comparison with the existing input approximation approach, in which the input function (rather than the kernel function) of the input operator is approximated by piecewise constant or piecewise linear functions, we show that the kernel approximation approach gives improved computation results. More precisely, even though the convergence rates in the kernel approximation approach remain qualitatively the same as those in the input approximation approach, the newly developed former approach could lead to quantitatively improved approximation errors than the latter approach particularly when the piecewise linear approximation scheme is taken. Finally, a numerical example is given to demonstrate the effectiveness of the kernel approximation approach with this scheme.