Hossam Seddik Abbas

On the State-Space Realization of LPV Input-Output Models: Practical Approaches

A common problem in the context of linear parameter-varying (LPV) systems is how input-output (IO) models can be efficiently realized in terms of state-space (SS) representations. The problem originates from the fact that in the LPV literature discrete-time identification and modeling of LPV systems is often accomplished via IO model structures. However, to utilize these LPV-IO models for control synthesis, commonly it is required to transform them into an equivalent SS form. In general, such a transformation is complicated due to the phenomenon of dynamic dependence (dependence of the resulting representation on time-shifted versions of the scheduling signal). This conversion problem is revisited and practically applicable approaches are suggested which result in discrete-time SS representations that have only static dependence (dependence on the instantaneous value of the scheduling signal). To circumvent complexity, a criterion is also established to decide when an linear-time invariant (LTI)-type of realization approach can be used without introducing significant approximation error. To reduce the order of the resulting SS realization, an LPV Ho-Kalman-type of model reduction approach is introduced, which, besides its simplicity, is capable of reducing even non-stable plants. The proposed approaches are illustrated by application oriented examples.

A neural-network based technique for modelling and LPV control of an arm-driven inverted pendulum

This paper presents a generalization of a recurrent neural-networks (RNNs) approach which was proposed previously in [1], together with stability and identifiability proofs based on the contraction mapping theorem and the concept of sign-permutation equivalence, respectively. A slight simplification of the generalized RNN approach is also proposed that facilitates practical application. To use the RNN for linear parameter-varying (LPV) controller synthesis, a method is presented of transforming it into a discrete-time quasi LPV model in polytopic and linear fractional transformation (LFT) representations. A novel indirect technique for closed-loop identification with RNNs is proposed here to identify a black box model for an arm-driven inverted pendulum (ADIP). The identified RNN model is then transformed into a quasi-LPV model. Based on such LPV models, two discrete-time LPV controllers are synthesized to control the ADIP. The first one is a full-order standard polytopic LPV controller and the second one is a fixed-structure LPV controller in LFT form based on the quadratic separator concept. Experimental results illustrate the practicality of the proposed methods.

LPV modelling and control of a 2-DOF robotic manipulator using PCA-based parameter set mapping

This paper presents the construction of a realistic linear parameter-varying (LPV) model of a robotic manipulator using parameter set mapping, for the purpose of synthesizing an LPV gain-scheduled controller. A nonlinear dynamic model of the manipulator is obtained and a quasi-LPV model is derived. Since the quasi-LPV model has a large number of affine scheduling parameters and a large overbounding, parameter set mapping is used to reduce conservatism and complexity in controller design by finding tighter parameter regions with fewer scheduling parameters. Then, a polytopic LPV gain-scheduled controller is synthesized and implemented experimentally on an industrial robot for a trajectory tracking task. Comparison of results with a decentralized PD controller illustrates that the designed LPV controller improves the tracking error significantly. Moreover, it achieves a slightly better accuracy than a model-based inverse dynamics controller while being of lower complexity.

Polytopic Quasi-LPV Models Based on Neural State-Space Models and Application to Air Charge Control of a SI Engine

Abstract This paper is one of two joint papers, each presenting a different representation of a feedforward neural network. Here a discrete-time polytopic quasi linear parameter varying (LPV) model of a nonlinear system based on a neural state-space model is proposed, whereas in the joint paper (Abbas and Werner [2008]) a neural state-space model is transformed into a linear fractional transformation (LFT) representation to obtain a discrete-time quasi-LPV model of the nonlinear system. As a practical application, air charge control of a spark-ignition (SI) engine is used in both papers to illustrate two different synthesis methods for fixed structure low-order discrete-time LPV controllers. In the present paper, the synthesis of a fixed-structure low-order self-scheduled H ∞ controller is based on linear matrix inequality (LMIs) and evolutionary search. A controller is designed for the nonlinear system and its performance is compared with that achieved when a standard self-scheduled H ∞ controller is used.

Controller synthesis for input-output LPV models

This paper considers the synthesis of linear parameter-varying (LPV) controllers for plant models given in input-output LPV form. For SISO systems, a method for synthesizing LPV gain-scheduled controllers in input-output form has been proposed recently, where the a priori choice of a central polynomial plays a critical role, and the synthesis problem is solved using a sum-of-squares relaxation. In this paper we propose a way of simplifying this design procedure, by replacing the sum-of-squares approach by representing the closed-loop model in polytopic input-output LPV form and then using a gradient-based optimization to solve the synthesis BMI. In this procedure the central polynomial is tuned while the closed-loop performance index is minimized over the decision variables, which include the controller parameters. The proposed method is illustrated with simulation examples.

Fixed-structure LPV controller synthesis based on implicit input output representations

In this paper a novel LPV controller synthesis approach to design fixed-structure LPV controllers in input output (IO) form is presented. The LPV-IO model and the LPV-IO controller are assumed to depend affinely as well as statically on the scheduling variable. By using an implicit representation of the system model and the controller, an exact representation of the closed-loop behavior is achieved. Using Finslers Lemma, novel stability conditions are derived in the form of linear matrix inequalities (LMIs). Based on these conditions a quadratic performance synthesis approach is introduced in form of bilinear matrix inequalities (BMIs) and solved using a DK-iteration based approach.

Synthesis of LPV Controllers With Low Implementation Complexity Based on a Reduced Parameter Set

A major difficulty encountered in the application of linear parameter-varying (LPV) control is the complexity of synthesis and implementation when the number of scheduling parameters is large. Often heuristic solutions involve neglecting individual scheduling parameters, such that standard LPV controller synthesis methods become applicable. However, stability and performance guarantees are rendered void, if controller designs based on an approximate model are implemented on the original plant. In this brief, a synthesis method for LPV controllers that achieves reduced implementation complexity is proposed. The method is comprised of first synthesizing an initial controller based on a reduced parameter set. Then closed-loop stability and performance guarantees are recovered with respect to the original plant, which is considered to be accurately modeled. Iteratively solving a nonconvex bilinear matrix inequality may further improve performance. A two-degrees-of-freedom (2-DOF) and three-degrees-of-freedom robotic manipulator is considered as an illustrative example, for which experimental results indicate a good performance for controllers of reduced scheduling order. Furthermore, in the 2-DOF case, controller performance has been significantly improved.

Benchmark problem — nonlinear control of a 3-DOF robotic manipulator

This document proposes the nonlinear control of an industrial three-degrees-of-freedom (3-DOF) robotic manipulator as a benchmark problem for controller synthesis methods, that are applicable to complex plants, but can provide implementation with low complexity. Full details on the nonlinear model of the industrial robotic manipulator Thermo CRS A465 are provided. Furthermore, a solution is presented by considering linear parameter-varying (LPV) controller synthesis based on a reduced parameter set. Stability and performance guarantees are rendered void as the plants parameter dependency is first approximated by means of principle component analysis. The guarantees are recovered by tools which have previously been reported.

Frequency-Weighted Discrete-Time LPV Model Reduction Using Structurally Balanced Truncation

This paper proposes a method for frequency weighted discrete-time linear parameter-varying (LPV) model reduction with bounded rate of parameter variation, using structurally balanced truncation with a priori (nontight) upper error bounds for each fixed parameter. For systems with both input and output weighting filters, guaranteed stability of the reduced-order model is proved as well as the existence of solutions, provided that the full-order model is stable. A technique based on cone complementarity linearization is proposed to solve the associated linear matrix inequality (LMI) problem. Application to the model of a gantry robot illustrates the effectiveness of the approach. Moreover, a method is proposed to make the reduced order model suitable for practical LPV controller synthesis.

A Hybrid Gradient-LMI Algorithm for Solving BMIs in Control Design Problems

Abstract This paper presents an algorithm for solving optimization problems with bilinear matrix inequality constraints. The algorithm is based on a combination of gradient-based optimization and LMIs, which makes it fast and enables it to handle a large number of decision variables. It is applied to two controller synthesis problems: static output feedback controller synthesis and robust controller synthesis for linear parameter varying (LPV) systems using the idea of quadratic separation. Since the second problem has large number of decision variables, a hybrid approach is applied, in which LMI solvers are used for the evaluation of the cost function. The algorithm is applied to two examples, and results are compared with some existing approaches.