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     

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     

Nonlinear H∞ control and its application to rigidspacecraft

The problem of H_∞ control of a rigid spacecraft with three control torques and disturbances is addressed. The Hamilton-Jacobi-Isaacs inequality for H_∞ feedback design is solved. An H_∞ suboptimal feedback is given, and the stability properties of the closed-loop system are studied     

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     

On robust H∞ control for nonlinear discrete andsampled-data systems

The linearization technique for H_∞ control for nonlinear discrete and sampled-data systems is developed in a unified framework under minimal assumptions. It is shown directly, without using any Riccati or Hamilton-Jacobi-Isaacs equation, that H_∞ control for a linearized system is also H_∞ control for a nonlinear original one     

H∞ strong stabilization

This note presents a technique for designing stable H_∞ controllers. Similar to some methods in the existing literature, the proposed method also uses the parameterization of all suboptimal H _∞ controllers so that the stable H_∞ design problem can be (conservatively) converted into another 2-block standard H_∞ problem. However, a weighting function is introduced in this method to alleviate the conservativeness of the previous formulations. It is further shown that the resulting high-order controller can be significantly reduced by a two-step reduction algorithm. Numerical examples are presented to demonstrate the effectiveness of the proposed method     

On the parametric H∞ problems of weighted intervalplants

In this paper, we consider the parametric H_∞ problems of weighted interval plants. A sufficient condition will be given on weighting function W(s) such that for any interval plant P(s, q, r) the maximal H_∞ norm of W(s)P(s, q, r) is achieved at one of the 16 so-called weighted Kharitonov plants. It is shown that for first-order and some high-order stable weighting functions, the extremality property is enjoyed generally by the maximal H_∞ norm of weighted interval plants     

H∞ multiple objective robust controllers forinfinite-horizon single measurement single control input problems

A graphical method is introduced that solves the robust infinite horizon H_∞ multiple-objective control problem for single measurement, single control input systems. The solution is obtained by describing boundaries on the Nichols chart. Each boundary defines the set of all admissible gain and phase values for the loop transmission at a given frequency. These boundaries are obtained by using the well-known parameterization of all the solutions for a single objective H_∞ control problem. The new method links between the theories of H_∞ and quantitative feedback theory (QFT). It can be used to design robust H_∞ controllers with almost no overdesign, and it provides a convenient solution of H_∞ multiple-objective problems that are difficult to solve by the standard four-block setting. It also extends the methods of SISO QFT to deal with a vector of disturbances. The latter may affect the controlled plant through any input coupling matrix and not necessarily through the controller input, or the measurement output     

Time-domain H∞ identification

The problem of parameter identification, for single-input, single-output ARX systems, is considered. Recent results in H_∞-nonlinear filtering are used to formulate a nonlinear H_∞ time-domain prediction-error-modeling (PEM) identification method. The performance of the new method is guaranteed by a preassigned bound on the ratio between the energy of the prediction error of the obtained model and the energy of the exogenous disturbances. The potential usefulness of the H_∞ time-domain identification method is illustrated by a numerical example     

Nonlinear H2 and H∞ optimal controllersfor current-fed induction motors

This paper presents a nonlinear control design for both the H₂ and H_∞ optimal control for current-fed induction motor drives. These controllers are derived using analytical stationary solutions that minimize a generalized convex energy cost function including the stored magnetic energy and the coil losses, while satisfying torque regulation control objectives. Explicit control expressions for both the H₂ and H_∞ optimal design are given. Furthermore, the optimal attenuation factor, i.e., the optimal H_∞ norm and the corresponding worst case disturbance, are both computed explicitly     

Multiobjective PID control design in uncertain robotic systemsusing neural network elimination scheme

A PID-type controller incorporating neural network elimination scheme and sliding-mode control action for different objectives including H₂ tracking performance, H_∞ tracking performance, and regional pole constraints is developed in robotic systems under plant uncertainties and external disturbances. The adaptive neural networks are used to compensate the plant uncertainties. The sliding-mode control action is included to eliminate the effect of approximation error via neural network approximation. The sufficient conditions are developed for different objectives in terms of linear matrix inequality (LMI) formulations. The interesting combinations of different objectives are considered in this paper, which include H_∞ PID tracking control design with regional pole constraints and mixed H₂/H_∞ PID tracking control design with regional pole constraints. These multiobjective PID control problems are characterized in terms of eigenvalue problem (EVP). The EVP can be efficiently solved by the LMI toolbox in Matlab. The proposed methods are simple and the PID control gain for different objectives can be obtained systematically     

H∞ tuning for task-space feedback control of robotwith uncertain Jacobian matrix

The H_∞ tuning problem for task-space feedback control of a robot with uncertain kinematics and Jacobian matrix from joint space to task space is formulated and solved. H_∞ tuning methods are presented for two task-space feedback controllers when the systems are subject to external disturbance. It is shown that the conditions for the establishment of H_∞ tuning can be obtained in simple forms without solving the Hamilton-Jacobi inequalities even in the presence of kinematics uncertainty     

Relating H2 and H∞-norm bounds forsampled-data systems

We relate the H_∞ and H₂ norms for multi-input/multi-output sampled-data feedback control systems, where a continuous-time plant is controlled by a digital compensator with hold and sampler. Upper bounds on both H₂ and H_∞ norms are obtained based on fundamental relations derived by two different approaches, namely the hybrid state-space approach and the fast sampling and lifting approach     

Output feedback control of Markov jump linear systems incontinuous-time

This paper addresses the dynamic output feedback control problem of continuous-time Markovian jump linear systems. The fundamental point in the analysis is an LMI characterization, comprising all dynamical compensators that stabilize the closed-loop system in the mean square sense. The H₂ and H_∞-norm control problems are studied, and the H₂ and H_∞ filtering problems are solved as a by product     

H∞ type problem for sampled-data control systems-asolution via minimum energy characterization

This paper aims at deriving a solution for H_∞ type problem for sampled-data control systems. The solution is given in terms of an equivalent discrete-time H_∞ problem. The reduction procedure is viewed and characterized from the viewpoint of minimum energy principle and J-unitary transformations     

Performance preserving controller reduction via additiveperturbation of the closed-loop transfer function

An H_∞ performance preserving controller order reduction method is proposed. Here performance preservation indicates that the H_∞ norm bound of the closed loop transfer function with reduced-order controller is not greater than the H_∞ norm bound of the closed loop transfer function with full order controller. We assume additive perturbations to the closed-loop transfer function and obtain a sufficient condition for performance preservation. Two kinds of useful weightings are derived, and the controller reduction problem is solved via a frequency weighted model reduction problem     

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     

H∞ and positive-real control for linear neutraldelay systems

This note is concerned with the H_∞ and positive-real control problems for linear neutral delay systems. The purpose of H_∞ control is the design of a memoryless state feedback controller which stabilizes the neutral delay system and reduces the H_∞ norm of the closed-loop transfer function from the disturbance to the controlled output to a prescribed level, while the purpose of positive-real control is to design a memoryless state feedback controller such that the resulting closed-loop system is stable and the closed-loop transfer function is extended strictly positive real. Sufficient conditions for the existence of the desired controllers are given in terms of a linear matrix inequality (LMI). When this LMI is feasible, the expected memoryless state feedback controllers can be easily constructed via convex optimization     

All fixed-order H∞ controllers: observer-basedstructure and covariance bounds

This note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant, which yields for the closed-loop transfer matrix a specified H_∞ norm bound. The algebraic results of covariance control are applied to the H _∞ control problem to yield a parameterization in terms of the Lyapunov matrix, which carries many system properties (such as H ₂ performance, covariance bounds, system entropy at infinity, etc.). All low-order H_∞ controllers are shown to have observer-based structure for “reduced-order models” of the plant and are characterized by two Riccati equations with a coupling condition     

Comments on “Mixed L∞/H∞suboptimal controllers for SISO continuous time systems”

The authors state that they have no serious criticism of the original paper but make certain remarks. An example in the original illustrates a method to do tradeoffs between two design goals: H_∞ and L^∞. To make a fair comparison, they authors say that the optimal H_∞ norm should have been presented. The also point out misprints, and make remarks on the cancellation of the integrator pole and the tending to ∞ of the settling time