Robust H2/H∞-state estimation forsystems with error variance constraints: the continuous-time case

The paper is concerned with the state estimator design problem for perturbed linear continuous-time systems with H_∞ norm and variance constraints. The perturbation is assumed to be time-invariant and norm-bounded and enters into both the state and measurement matrices. The problem we address is to design a linear state estimator such that, for all admissible measurable perturbations, the variance of the estimation error of each state is not more than the individual prespecified value, and the transfer function from disturbances to error state outputs satisfies the prespecified H_∞ norm upper bound constraint, simultaneously. Existence conditions of the desired estimators are derived in terms of Riccati-type matrix inequalities, and the analytical expression of these estimators is also presented. A numerical example is provided to show the directness and effectiveness of the proposed design approach     

H∞ state estimation for linear periodic systems

This note deals with the H_∞ state estimation problem for linear periodic systems. The question addressed is the design of an unbiased linear periodic and asymptotically stable estimator that achieves a prescribed H_∞ performance on an infinite horizon. Necessary and sufficient conditions for the existence of a periodic estimator have been derived. Asymptotic properties of the finite horizon estimation problem when the time-horizon tends to infinity are also investigated     

H∞ optimization with time-domain constraints

Standard H_∞ optimization cannot handle specifications or constraints on the time response of a closed-loop system exactly. In this paper, the problem of H_∞ optimization subject to time-domain constraints over a finite horizon is considered. More specifically, given a set of fixed inputs wⁱ, it is required to find a controller such that a closed-loop transfer matrix has an H_∞-norm less than one, and the time responses yⁱ to the signals wⁱ belong to some prespecified sets Ωⁱ. First, the one-block constrained H_∞ optimal control problem is reduced to a finite dimensional, convex minimization problem and a standard H_∞ optimization problem. Then, the general four-block H_∞ optimal control problem is solved by reduction to the one-block case. The objective function is constructed via state-space methods, and some properties of H_∞ optimal constrained controllers are given. It is shown how satisfaction of the constraints over a finite horizon can imply good behavior overall. An efficient computational procedure based on the ellipsoid algorithm is also discussed     

Multiobjective H2/H∞ control

For a linear time-invariant system with several disturbance inputs and controlled outputs, we show how to minimize the nominal H₂-norm performance in one channel while keeping bounds on the H₂-norm or H_∞-norm performance (implying robust stability) in the other channels. This multiobjective H₂ /H_∞-problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems. We show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller. If it exists, we reveal how the novel trick of optimizing the trace norm of the Youla parameter over certain convex constraints allows one to design a nearly optimal controller whose Youla parameter is of the same order as the optimal one     

Robust H∞ control for linear systems withnorm-bounded time-varying uncertainty

Robust H_∞ control design for linear systems with uncertainty in both the state and input matrices is treated. A state feedback control design which stabilizes the plant and guarantees an H_∞-norm bound constraint on disturbance attenuation for all admissible uncertainties is presented. The robust H_∞ control problem is solved via the notion of quadratic stabilization with an H_∞ -norm bound. Necessary and sufficient conditions for quadratic stabilization with an H_∞-norm bound are derived. The results can be regarded as extensions of existing results on H_∞ control and robust stabilization of uncertain linear systems     

Hierarchical optimization in H∞

This paper considers a hierarchical optimal control problem which involves an optimal H_∞-norm cost in the primary problem and an H_∞-norm type or a quadratic secondary objective. Using allpass dilation techniques and results from superoptimal interpolation theory, it is shown that the problem can be reduced to a multidisk minimization in terms of a free parameter of reduced dimensions. Convex programming techniques may then be employed to obtain a numerical solution to the problem     

Robust state estimation for perturbed systems with error variance and circular pole constraints: The discrete-time case

This paper is concerned with the problem of robust state estimation for linear perturbed discrete-time systems with error variance and circular pole constraints. The goal of this problem addressed is the design of a linear state estimator such that, for all admissible uncertainties in both state and output equations, the following two performance requirements are simultaneously satisfied: (1) the poles of the filtering matrix are all constrained to lie inside a prespecified circular region; and (2) the steady-state variance of the estimation error for each state is not more than the individual prespecified value. It is shown that this problem can be converted to an auxiliary matrix assignment problem and solved by using an algebraic matrix equation/inequality approach. Specifically, the conditions for the existence of desired estimators are obtained and the explicit expression of these estimators is also derived. The main results are then extended to the case when an H performance requirement is added. Finally, a numerical example is presented to demonstrate the significance of the proposed technique.     

Robust H∞ performance for lineardelay-differential systems with time-varying uncertainties

The present paper concerns robust H∞ performance for linear delay-differential systems which involve an uncertain time delay and time-varying norm-bounded parameter uncertainties. Based on the Lyapunov functional, a simple criterion is proposed which assures the pseudo-quadratic stability as a well as an H_∞-norm bound. The criterion is given in the form of a linear matrix inequality which is affine or convex in additional scalar parameters. A simple criterion is presented to evaluate the extent of the performance robustness     

A curve-shaping approach for determining bounds on H∞ performance under hard constraints

Consider an H_∞-norm optimization problem where the goal is to minimize the H_∞-norm of some transfer functions while subject to constraints on, for example, system bandwidth, peak sensitivity, or output variance. This paper presents a method of computing how these constraints limit the achievable performance. The approach involves optimally shaping the sensitivity function magnitude Bode plot     

Mixed l1/H∞ control of MIMO systems viaconvex optimization

We present a methodology for designing mixed l₁/H_∞ controllers for MIMO systems. These controllers allow for minimizing the worst case peak output due to persistent disturbances, while at the same time satisfying an H_∞-norm constraint upon a given closed loop transfer function. Therefore, they are of particular interest for applications dealing with multiple performance specifications given in terms of the worst case peak values, both in the time and frequency domains. The main results of the paper show that: 1) contrary to the H₂/H_∞ case, the l₁/H_∞ problem admits a solution in l₁; and 2) rational suboptimal controllers can be obtained by solving a sequence of problems, each one consisting of a finite-dimensional convex optimization and a four-block H_∞ problem. Moreover, this sequence of controllers converges in the l₁ topology to an optimum     

丢包网络化控制系统非脆弱量化H∞控制

针对非线性网络化控制（NCS）系统中控制器参数存在摄动的问题,考虑传感器—控制器和控制器—执行器均存在随机丢包和量化误差,提出了一种加性非脆弱量化H_∞控制器的设计方法．利用李亚普诺夫稳定性理论和线性矩阵不等式（LMI）方法,将该问题转化为线性矩阵不等式约束和线性目标函数的凸优化问题进行求解,给出了丢包下的非脆弱量化H_∞控制器存在的充分条件．所设计的控制器在容许的参数摄动、丢包概率和量化密度条件下,不仅能保证闭环NCS的稳定性和性能要求,而且是非脆弱的．数值仿真验证了所提方法的有效性．     

H∞-norm approximation of systems by constantmatrices and related results

In this work the H_∞-norm approximation of a given stable proper rational transfer function by a constant matrix is considered. A new improved bound on H_∞-norm of a given stable transfer function is derived. Optimal approximation for relaxation systems is given     

Mixed H2/H∞ fuzzy output feedbackcontrol design for nonlinear dynamic systems: an LMI approach

This study introduces a mixed H₂/H_∞ fuzzy output feedback control design method for nonlinear systems with guaranteed control performance. First, the Takagi-Sugeno fuzzy model is employed to approximate a nonlinear system. Next, based on the fuzzy model, a fuzzy observer-based mixed H₂/H_∞ controller is developed to achieve the suboptimal H₂ control performance with a desired H_∞ disturbance rejection constraint. A robust stabilization technique is also proposed to override the effect of approximation error in the fuzzy approximation procedure. By the proposed decoupling technique and two-stage procedure, the outcome of the fuzzy observer-based mixed H₂/H_∞ control problem is parametrized in terms of the two eigenvalue problems (EVPs): one for observer and the other for controller. The EVPs can be solved very efficiently using the linear matrix inequality (LMI) optimization techniques. A simulation example is given to illustrate the design procedures and performances of the proposed method     

H∞-control design for systems with multiple delays

In this paper, H_∞-control design is developed for a class of linear dynamical systems with multiple state and input delays. Both the asymptotic stability conditions and the H_∞-norm bound of the closed-loop transfer function are established. Feedback control synthesis is developed using state feedback. In addition, when the class of linear systems with multiple state and input delays undergoes norm-bound uncertainties, H_∞-control synthesis is established. The developed results are conveniently cast into linear matrix inequality (LMI) framework. Simulation examples are given to illustrate the theoretical developments.     

H∞-differential Riccati equations: convergenceproperties and finite escape phenomena

The paper studies the transient and asymptotic behavior of the solution of the sign-indefinite differential Riccati equation (DRE) arising in finite-horizon H_∞-filtering and control problems. Differently from the sign-definite H_∞-DRE, the solution of the H_∞-DRE can have finite escape times even for nonnegative initial conditions. Sufficient and necessary conditions for boundedness and convergence are derived in correspondence to a fixed value of the H_∞-norm attenuation level γ. Finally, a stepwise γ-switching strategy is devised to guarantee boundedness as well as asymptotic performance     

Controller approximation: approaches for preserving H∞ performance

Investigates the design of reduced-order controllers using an H _∞ framework. Given a stabilizing controller which satisfies a prespecified level of closed-loop H_∞ performance, sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of H_∞, performance. Such controllers are said to be (P,γ)-admissible, where P is the model of the plant under consideration and γ is the required level of prespecified H_∞ performance. The conditions are expressed as norm bounds on particular frequency-weighted errors, where the weights are selected to make a specific transfer function a contraction. The design of reduced-order (P,γ)-admissible controllers is then formulated as a frequency-weighted model reduction problem. It is advantageous for the required weights to be large in some sense. Solutions which minimize either the trace, or the determinant, of the inverse weights are characterized. We show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible. To conclude, we demonstrate by way of a numerical example, that when used in conjunction with a combined model reduction/convex optimization scheme, the proposed design procedures are effective in substantially reducing controller complexity     

Robust H∞ filtering of stationary continuous-timelinear systems with stochastic uncertainties

The problem of applying H_∞ filters on stationary, continuous-time, linear systems with stochastic uncertainties in the state-space signal model is addressed. These uncertainties are modeled via white noise processes. The relevant cost function is the expected value of the standard H_∞ performance index with respect to the uncertain parameters. The solution is obtained via a stochastic bounded real lemma that results in a modified Riccati inequality. This inequality is expressed in the form of a linear matrix inequality whose solution provides the filter parameters. The method proposed is also applied to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂ /H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a practical example     

On nonlinear H∞ control under sampled measurements

This paper deals with the nonlinear H_∞ control problem with sampled measurement feedback. This problem has already been studied in Suzuki et al. (1995), where, using a certainty equivalence principle, a control solution involving a state estimator with a linear injection gain is proposed. Using the same general framework, we propose a more refined estimator with a nonlinear injection gain. This gain is shown to be connected to a periodic solution of a Hamilton-Jacobi inequality with jumps     

Estimation variance and H∞-errorminimization of stationary process with perfect measurements

The singular estimation problem of linear continuous-time stationary processes is solved both in the minimum error variance and the minimum H_∞-norm sense. Simple explicit expressions for the resulting estimators and their corresponding minimum criteria are derived in terms of the system structure parameters. These expressions enable an easy comparison between the two minimization procedures. The theory is demonstrated by a simple example of fourth order which is solved by both minimization methods     

Nondefinite least squares and its relation toH∞-minimum error state estimation

The problem of recursive nondefinite least-squares state estimation of continuous-time stationary processes is solved, by applying Pontryagin's maximum principle. A comparison of the derived solution to the result that is obtained for the H_∞ -minimum error estimation suggests a new interpretation for the H_∞-optimal estimation mechanism. According to this interpretation, the estimator tries to optimally estimate the required combination of the states, in the l₂-norm sense, against the worst disturbance signal that stems from a fictitious measurement of this combination