Full-block Multipliers Research Articles

On Robust Stabilization of Discrete-Time LPV/LFR Systems

This technical article presents novel robust stabilization conditions for discrete-time linear parameter-varying (LPV) systems with linear fractional representation (LFR). The proposed conditions rely on the use of slack variables and decision matrices associated with the LFR approach to provide new controller designs. In addition, we address parameter-dependent Lyapunov functions and full-block multipliers to obtain less conservative synthesis conditions for discrete-time LPV/LFR systems. Design conditions are cast in the form of linear matrix inequalities to generate robust state-feedback and output-feedback controllers. Numerical examples demonstrate the effectiveness of the proposed method.

Improved ℋ2 and ℒ2 control designs for discrete-time LPV/LFR systems

This paper addresses novel and norm-based controller designs for discrete-time linear parameter-varying (LPV) systems with linear fractional representation (LFR). Despite the importance of LPV/LFR systems to the control literature, there is a lack of works on this topic. The distinctive formulation of the proposed controller designs is given by using parameter-dependent Lyapunov functions in combination with slack variables and general full-block multipliers. Such proposed formulation leads to less conservative synthesis conditions characterised in terms of a finite number of linear matrix inequalities for both state-feedback and static output-feedback controllers subject to and performances for discrete-time LPV/LFR systems. Improvements obtained by the proposed controller design conditions over existing methods in the area are assessed by numerical examples.

Mixed-sensitivity [formula omitted] controller synthesis for discrete-time LPV/LFR systems

This paper presents novel control procedure to obtain a mixed-sensitivity L2 controller for discrete-time linear parameter-varying (LPV) systems using a linear fractional representation (LFR). The proposed method makes use of parameter-dependent Lyapunov functions in combination with static full-block multipliers and slack variables to provide less conservative results than other existing control strategies. The distinctive controller is established from an observer-based control structure, where observer and state-feedback synthesis are derived. Moreover, both H2 and L2 performances indexes are addressed and some insights about factorization contractiveness are explored. The output feedback controller is given through a two-step synthesis and its design conditions are addressed in terms of a finite number of linear matrix inequalities. Finally, two numerical examples are carried out to show the effectiveness of the proposed control method.

Robust Gain-Scheduling Controller for Airbreathing Hypersonic Flight Vehicle

This paper addresses issues related to robust control for an airbreathing hypersonic flight vehicle. Owing to aero-propulsion couplings caused by the unique structure shape, the model of the vehicle is greatly nonlinear and complex, which presents an enormous technical challenge for control. The nonlinear model is transformed into a linear fractional transformation (LFT) model, and a robust gain-scheduling controller based on linear parameter-varying control (LPV) with full block multipliers is obtained. Simulations illustrate great improvements of the dynamic performance in closed-loop system.

Linear Parameter-Varying Control of Complex Mechanical Systems

In standard linear fractional representation (LFR)-based linear parameter-varying (LPV) modeling the size of the (diagonal) scheduling block depends on the number of scheduling parameters and their repetitions, which in turn influences both the complexity of synthesis conditions and the computational load during online implementation of LPV controllers. A modeling framework motivated by, but not limited to, mechanical systems is proposed, where the size of the scheduling block depends on the system's physical degrees-of-freedom. The scheduling block then turns out block-diagonal and can be parameterized in an affine or rational manner. This parameterization yields less complex LFRs when considering the example of a three degrees-of-freedom robotic manipulator, for which then full-block multipliers are tractable and also necessary in synthesis. Synthesis and both simulation and experimental implementation results indicate that the novel rational LPV controller provides improved performance at both reduced implementation and synthesis complexity as compared to an affine LPV controller.

A hybrid approach to the synthesis of simply structured robust and gain-scheduled controllers

Abstract Frequently in engineering applications, simply structured controllers are required for plants whose dynamics are either not fully known or varies with time. The problem of designing such structurally constrained controllers is known to be NP-hard. This paper presents a method for designing such controllers for both continuous-time and discrete-time uncertain or linear parameter varying (LPV) systems. The method relies on sufficient linear matrix inequalities (LMI) conditions also presented here. These LMIs are used to evaluate performance of a given controller in terms of the worst-case induced L 2 -norm for complete operational envelope. The strength of this LMI formulation lies in the fact that the robust performance evaluation condition is less conservative than standard results because it is based on a parameter-dependent Lypunov function and full block multipliers which allows to consider a bound on the rate of change in parameters. The LMI conditions are used in a hybrid evolutionary-LMI algorithm. The proposed algorithm employs evolutionary search to span the solution space and LMI solvers to find the worst case performance. The algorithm is independent of the controller structure, which allows to design simply structured controllers for complex systems in a systematic way. To illustrate the approach, two applications are presented: a distillation column and a spark ignition (SI) engine. For the distillation column, simple PI controllers with first order pre-filters are designed to achieve time and frequency domain control objectives in the presence of 20% input uncertainty. For the SI engine example, gain-scheduled PID controllers are designed and tuned. these examples illustrate that the proposed approach can be used to systematically design simply structured low-order controllers, while satisfying all performance requirements in terms of the worst-case induced L 2 -norm.

Applied LPV Control with Full Block Multipliers and Regional Pole Placement

A formulation of an LPV control problem with regional pole placement constraints is presented, which is suitable for the application of a Full Block S-Procedure. It is demonstrated that improved bounds can be obtained on the inducedL2norm of closed loop systems, while satisfying pole placement constraints. An application consisting in the 6 degrees of freedom (DOF) control of a space vehicle is developed as an example, with hardware in the loop (HIL) simulation. This shows that the method is appealing from the practical point of view, considering that the synthesized control law can be implemented satisfactorily in standard flight control systems. Conclusions with remarks towards the practical use of the method are presented as well.

A remark on abstract multiplier conditions for robustness problems

This paper presents a generalization of recent results of Açıkmeşe and Corless [B. Açıkmeşe, M. Corless, Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions, Systems Control Letters 57 (2008) 78–94] concerning multiplier conditions of quadratic stability of uncertain/nonlinear systems. Abstract full block S-procedure results are formulated, extending the results of Scherer [C.W. Scherer, LPV control and full block multipliers, Automatica 37 (2001) 361–375] for such cases when the sets representing the uncertainties do not necessarily have subspace structure. The main contribution of the present work is the investigation of the conditions under which the results of Scherer and Açıkmeşe and Corless can be formulated in a unified framework.

Formula omitted] and [formula omitted] performance analysis and gain-scheduling synthesis for parameter-dependent systems

This paper focuses on quadratic performance analysis and output feedback gain-scheduling synthesis for parameter-dependent systems with linear fractional representations (LFRs). New quadratic H 2 and L 2 performance analysis results based on full-block multipliers are given. A novel gain-scheduling output feedback synthesis, which can be regarded as a natural extension of the loop shaping design procedure of McFarlane and Glover to parameter-dependent systems, is proposed. This gain-scheduling technique is computationally attractive and yields a simple and transparent control structure that can be easily implemented. Its effectiveness is illustrated on a benchmark missile autopilot example.

Some Results on Guaranteeing Cost Strategies: Full-block Multipliers and Fictitious Games

Guaranteeing cost robust minimax controls are determined for a discrete-time linear system with uncertainty of linear fractional form over infinite horizon by means of the full-block multiplier technique. It is highlighted in what sense this technique can be considered as the extension of other approaches applying a single adjustable parameter in the derivation of the linear matrix inequalities to be solved. Moreover, it is shown how to construct a fictitious game with completely known dynamics and with a cost criterion parametrized by the full-block multipliers in such a way that the minimax strategy and the upper value for it turn out to be the robust minimax strategy and the guaranteed cost for the original uncertain system.

Predictive LPV control of a liquid–gas separation process

The problem of controlling a liquid–gas separation process is approached by using LPV control techniques. An LPV model is derived from a nonlinear model of the process using differential inclusion techniques. Once an LPV model is available, an LPV controller can be synthesized. The authors present a predictive LPV controller based on the GPC controller [Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part I. Automatica 1987;23(2):137–48; Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part II. Extensions and interpretations. Automatica 1987;23(2):149–60]. The resulting controller is denoted as GPC–LPV. This one shows the same structure as a general LPV controller [El Gahoui L, Scorletti G. Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 1996;32(9):1273–84; Scorletti G, El Ghaoui L. Improved LMI conditions for gain scheduling and related control problems. International Journal of Robust Nonlinear Control 1998;8:845–77; Apkarian P, Tuan HD. Parametrized LMIs in control theory. In: Proceedings of the 37th IEEE conference on decision and control; 1998. p. 152–7; Scherer CW. LPV control and full block multipliers. Automatica 2001;37:361–75], which presents a linear fractional dependence on the process signal measurements. Therefore, this controller has the ability of modifying its dynamics depending on measurements leading to a possibly nonlinear controller. That controller is designed in two steps. First, for a given steady state point is obtained a linear GPC using a linear local model of the nonlinear system around that operating point. And second, using bilinear and linear matrix inequalities (BMIs/LMIs) the remaining matrices of GPC–LPV are selected in order to achieve some closed loop properties: stability in some operation zone, norm bounding of some input/output channels, maximum settling time, maximum overshoot, etc., given some LPV model for the nonlinear system. As an application, a GPC–LPV is designed for the derived LPV model of the liquid–gas separation process. This methodology can be applied to any nonlinear system which can be embedded in an LPV system using differential inclusion techniques.

Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions

In this paper, we propose a new control design approach for linear fractional transformation (LFT) systems using parameter-dependent Lyapunov functions. Instead of assuming parameter dependency in LFT fashion, we consider general parameter-dependent controllers to achieve better closed-loop performance. Using full-block multipliers, new LPV synthesis conditions have been derived in terms of finite number of linear matrix inequalities (LMIs). Both continuous- and discrete-time cases are discussed. A ship steering example has been used to demonstrate advantages and benefits of the proposed approach.

Robust quadratic performance for time‐delayed uncertain linear systems

AbstractIn this paper, the analysis and control synthesis problems were studied for a general class of uncertain linear systems with variable time delay. It is assumed that the structured time‐varying parametric uncertainties enter the system state‐space description in a linear fractional fashion. The generic quadratic performance metric encompasses many types of dynamic system performance measure. In the context of delay‐independent stability, it was shown that the analysis and state‐feedback synthesis problems for such time‐delayed uncertain systems can be formulated equivalently as linear matrix inequality (LMI) optimization problems using the mechanism of full block multipliers. However, the solvability condition to output‐feedback problem was given as bilinear matrix inequality (BMI), which leads to a non‐convex optimization problem. A numerical example is provided to demonstrate the advantages of newly proposed control synthesis condition for time‐delayed uncertain systems over existing approaches. Copyright © 2002 John Wiley & Sons, Ltd.

LPV control and full block multipliers

For systems affected by time-varying parameters, we provide a novel construction of controllers that take on-line measurements of these parameters into account to achieve a certain desired performance specification. Since the controller construction is based on multipliers or scalings that are not, as typical in many robust control problems, structured but that are full and do not need to obey any inertia properties, our results permit to reduce the conservatism if allowing for arbitrary fast parameter variations to an arbitrary degree. The paper contains results for parameterized linear matrix inequalities (full block S-procedure) and dualization as well as elimination results for quadratic matrix inequalities that are of independent interest.