Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

Parameter dependent H∞ control by finite dimensional LMI optimization: application to trade‐off dependent control

In this paper, we consider the design of an H_∞ trade‐off dependent controller, that is, a controller such that, for a given Linear Time‐Invariant plant, a set of performance trade‐offs parameterized by a scalar θ is satisfied. The controller state space matrices are explicit functions of θ. This new problem is a special case of the design of a parameter dependent controller for a parameter dependent plant, which has many application in Automatic Control. This last design problem can be naturally formulated as a convex but infinite dimensional optimization problem involving parameter dependent Linear Matrix Inequality (LMI) constraints. In this paper, we propose finite dimensional (parameter independent) LMI constraints which are equivalent to the parameter dependent LMI constraints. The parameter dependent controller design is then formulated as a convex finite dimensional LMI optimization problem. The obtained result is then applied to the trade‐off dependent controller design. A numerical example emphasizes the strong interest of our finite dimensional optimization problem with respect to the trade‐off dependent control application. Copyright © 2005 John Wiley & Sons, Ltd.     

Best achievable decentralized performance

In this paper, a novel parameterization of all decentralized stabilizing controllers is employed in mathematically formulating the best achievable decentralized performance problem as an infinite dimensional optimization problem, Finite dimensional optimization problems are then constructed that have values arbitrarily close to this infinite dimensional problem. An algorithm which identifies the best achievable performance over all linear time-invariant decentralized controllers is then presented. It employs a global optimization approach to the solution of these finite dimensional approximating problems     

H∞ control for discrete‐time nonlinear switching systems

This paper formulates and solves the robust H^∞ control problem for discrete‐time nonlinear switching systems. The H^∞ control problem is interpreted as the l₂ finite gain control problem and is studied using a dissipative systems theory for switched systems. Both state and measurement feedback control problems are formulated as dynamic games and solved using dynamic programming. The partially observed dynamic game corresponding to the measurement feedback control problem is solved by transforming into a completely observed, full state infinite‐dimensional game problem using information states. Our results are illustrated with an example. Copyright © 2007 John Wiley & Sons, Ltd.     

Control of nonstationary LPV systems

This paper considers control of nonstationary linear parameter-varying systems, and is motivated by interest in the control of nonlinear systems along prespecified trajectories. In the paper, synthesis conditions are derived for such systems using an operator theoretical framework with the ?₂ induced norm as the performance measure. These conditions are given in terms of structured operator inequalities. In general, evaluating the validity of these conditions is an infinite dimensional convex optimization problem; however, if the initial system is eventually periodic, they reduce to a finite dimensional semi-definite programming problem. The paper concludes with an in-depth example on the control of a two-thruster hovercraft along an eventually periodic trajectory.     

Best achievable performance: non-switching compensation for multiple models

This paper presents a solution to the best achievable performance problem for a family of process models that are simultaneously stabilized by a non-switching LTI compensator. Specifically, a method is presented that can quantify the best dynamic performance achievable by a dynamic feedback compensator, for a finite family of process models. Closed-loop dynamic performance is quantified through a new performance measure that guarantees performance with respect to all allowable disturbances and allows for closed-loop response shaping with respect to fixed disturbances. The simultaneous performance problem is then formulated as a quadratically constrained minimax optimization problem that is non-differentiable and infinite dimensional. It is shown that the simultaneous performance problem can be solved through the iterative solution of appropriately constructed finite-dimensional nonlinear programming problems. A method that identifies ϵ-globally optimal solutions to this type of problems is presented. Finally, the proposed approach is demonstrated through an illustrative numerical example problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Extended Kalman filter and adaptive backstepping for mean temperature control of a three‐way catalytic converter

This contribution is concerned with an adaptive control strategy for the mean temperature of an automotive three‐way catalytic converter. Tailored finite‐dimensional approximations of the complex infinite‐dimensional mathematical model of the catalytic converter serve as a basis for the design of an extended Kalman filter for state profile estimation and of an adaptive backstepping controller for the mean temperature. Although the model used for observer design relies on (semi‐)discretising the infinite‐dimensional model, a simple model for the mean temperature employing a phenomenological approach to describe the reaction heat is used for control design. The observer/controller is tested in simulation scenarios using a validated model of the three‐way catalytic converter. Copyright © 2013 John Wiley & Sons, Ltd.     

Best domain for an elliptic problem in cartesian coordinates by means of shape‐measure

In (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143–155) we introduced a method to determine the optimal domains for elliptic optimal‐shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure‐theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new design of robust filters for uncertain systems

In this paper, a structured polynomial parameter-dependent approach is proposed for robust H₂ filtering of linear uncertain systems. Given a stable system with parameter uncertainties residing in a polytope with s vertices, the focus is on designing a robust filter such that the filtering error system is robustly asymptotically stable and has a guaranteed estimation error variance for the entire uncertainty domain. A new polynomial parameter-dependent idea is introduced to solve the robust H₂ filtering problem, which is different from the quadratic framework that entails fixed matrices for the entire uncertainty domain, or the linearly parameter-dependent framework that uses linear convex combinations of s matrices. This idea is realized by carefully selecting the structure of the matrices involved in the products with system matrices. Linear matrix inequality (LMI) conditions are obtained for the existence of admissible filters and based on these, the filter design is cast into a convex optimization problem, which can be readily solved via standard numerical software. Both continuous and discrete-time cases are considered. The merit of the methods presented in this paper lies in their less conservatism than the existing robust filter design methods, as shown both theoretically and through extensive numerical examples.     

Performance control for interconnection of identical systems: Application to PLL network design

In this paper, the problem of the control law design for interconnected identical systems ensuring the global stability and the global performance properties is under consideration. Inspired by the decentralized control law design methodology using the dissipativity input–output approach, the problem is reduced to the problem of satisfying two conditions: (i) the condition on the interconnection and (ii) the condition on the local subsystem dynamics. Both problems are efficiently solved applying a (quasi‐) convex LMI optimization and standard H_∞ synthesis. The proposed design methodology is applied to the control law design of a synchronous PLL network. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust model predictive control of a class of uncertain nonlinear systems with application to typical CSTR problems

This paper proposes a robust predictive control approach for additive discrete time uncertain nonlinear systems. The controller design is characterized as an optimization problem of the “worst-case” objective function over an infinite moving horizon. A sufficient state feedback synthesis condition is provided in the form of a linear matrix inequality (LMI) optimization and is solved online at each time step. A few simulation examples are exploited to illustrate the effectiveness of this method. Among them are two typical CSTR problems.     

Fault‐tolerant control synthesis for a class of nonlinear systems: Sum of squares optimization approach

This paper studies the fault‐tolerant control (FTC) problem for nonlinear systems, with guaranteed cost or H_∞ performance objective in the presence of actuator faults. The faulty mode is built as a multi‐model framework of the typical aberration in actuator effectiveness. The novelty of this paper is that the effect of the nonlinear terms is described as an index in order to transform the FTC design problem into a semi‐definite programming. The proposed optimization approach is to find zero optimum for this index. Combined with other performance indexes, the conceived multi‐objective optimization problem is solved by using sum of squares method in a reliable and efficient manner. Numerical examples are included to verify the applicability of this new approach for the nonlinear FTC synthesis. Copyright © 2008 John Wiley & Sons, Ltd.     

Dimensional analysis approach to dominant three-pole placement in delayed PID control loops

PID control loops with time delay are characterized by infinite number of poles but the pole assignment technique for adjusting the controller parameters can be applied to placing three poles only. The dominance of these poles is therefore an essential condition for this application. A novel approach to this problem involves applying dimensional analysis theory to obtain a generalized model of the control loop and then to perform a parameter tuning for its dimensionless representation. A one-row dimensional matrix results from the assumption of the usual dimensionless interpretation of both control error and actuating signals of the controller. Dimensionless similarity numbers of the so-called swingability and laggardness are introduced to specify the plant dynamics in the controller synthesis. A trio of numbers is assigned to become the dominant zeros of the characteristic quasi-polynomial of the control loop and the corresponding PID parameter adjustment is derived in the form of uniform formulae. The tuning of the proper damping and the real pole position ratios is provided by means of an IAE optimization technique. A dominance degree notion is introduced and an argument increment criterion is proposed to check the dominance of any of the pole placement cases. The quality of the disturbance rejection response is taken as the general criterion in the design of the time delay plant control.     

Linear time invariant minimax filtering

The problem of filtering a signal from a linear time invariant system with white Gaussian observation and unknown driving noise bounded at each instant of time is considered. We review the minimax filter of Johansen and Berkovitz–Pollard for the double integrator. While their solution is very elegant, the optimal filter is infinite dimensional. In a previous paper we showed that nearly the same performance can be achieved by a two dimensional filter and we generalized their approach to other linear time invariant systems. In this paper we show how to design nearly optimal filters for any linear time invariant system.     

A Sequential Design Approach for Switching ℌ∞ LPV Control

This paper proposes a sequential design scheme for switching ℌ_∞ LPV (Linear Parameter-Varying) control, aiming to reduce the computational complexity of the associated optimization problem. Different from the traditional approach that simultaneously designs switching LPV controllers and solves a high-dimensional optimization problem, the proposed sequential design approach renders a bundle of low-dimensional optimization problems to be solved iteratively. Individual ℌ_∞ LPV controller for each subregion is synthesized by independent PLMIs (Parametric Linear Matrix Inequalities) to guarantee ℌ_∞ performance, and controller variables are interpolated on the overlapped subregions such that the ℌ_∞ performance is also guaranteed on the overlapped subregion. Numerical examples are used to demonstrate the effectiveness of this method to reduce the computational load in each design iteration and improved ℌ_∞ performance over the conventional simultaneous design method with well-tuned interpolation coefficient.

     

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

In this paper, the ??_∞ and l₂–l_∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ??_∞ and l₂–l_∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ??_∞ and l₂–l_∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.     

A design of disturbance observer in standard H∞ control framework

The disturbance observer (DOB)‐based controller is widely used to estimate and suppress disturbance in motion control system. Because the low‐pass filter (Q‐filter) in DOB decides the performances of disturbance suppression, noise rejection, and robust stability against system uncertainties, design of Q‐filter is the principal task in DOB construction. This paper presents a systematic scheme for Q‐filter design based on H_∞ norm optimization. Cost function for optimization is proposed by considering performance and relative order condition of the Q‐filter. The norm minimization problem is then transformed to a standard H_∞ control problem. Furthermore, the relationship between performance and frequency weighting functions is investigated based on which selection of weighting functions is presented. Simulation results validate the global optimality and systematicness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Improved Parameter-dependent H∞ Filtering for Polytopic Delta Operator Systems

The problem of H∞ filtering for polytopic Delta operator linear systems is investigated. An improved H∞ performance criterion is presented based on the bounded real lemma. Upon the improved performance criterion, a sufficient condition for the existence of parameter-dependent H∞ filtering is derived in terms of linear matrix inequalities. The designed filter can be obtained from the solution of a convex optimization problem. The filter design makes full use of the parameter-dependent approach, which leads to a less conservative result than conventional design methods. A numerical example is given to illustrate the effectiveness of the proposed approach.     

H∞ optimization with plant uncertainty and semidefinite programming

The fundamental H^∞ problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H^∞ problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H^∞ optimization problem of control can be naturally treated with modern primal–dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H^∞ optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H^∞ optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. © 1998 John Wiley & Sons, Ltd.     

Monotone Smoothing Splines using General Linear Systems

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second‐order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval which has the cardinality of the continuum. To solve this problem, we first formulate the problem as a semi‐infinite quadratic programming problem, and then we adopt a discretization technique to obtain a finite‐dimensional quadratic programming problem. It is shown that the solution of the finite‐dimensional problem always satisfies the infinite‐dimensional monotonicity constraint. It is also proved that the approximated solution converges to the exact solution as the discretization grid‐size tends to zero. An example is presented to show the effectiveness of the proposed method.