Alena Kozáková

Design of robust decentralised controllers using the M-Δ structure robust stability conditions

This article proposes a new frequency domain approach to the design of robust decentralised controllers (DC) for continuous-time systems described by a set of transfer function matrices. Nominal stability, required nominal performance and robust stability (RS) are guaranteed using the independent design approach applied for nominal system, adapted so as to satisfy the M−Δ structure-based RS conditions modified for the closed-loop system under a decentralised controller. The proposed design technique enables us to include nominal interactions in the nominal model thus relaxing the conservatism of the RS conditions.

Robust Decentralized PID Controller Design

Robust stability of uncertain dynamic systems has major importance when real world system models are considered. A realistic approach has to consider uncertainties of various kinds in the system model. Uncertainties due to inherent modelling/identification inaccuracies in any physical plant model specify a certain uncertainty domain, e.g. as a set of linearized models obtained in different working points of the plant considered. Thus, a basic required property of the system is its stability within the whole uncertainty domain denoted as robust stability. Robust control theory provides analysis and synthesis approaches and tools applicable for various kinds of processes, including multi input – multi output (MIMO) dynamic systems. To reduce multivariable control problem complexity, MIMO systems are often considered as interconnection of a finite number of subsystems. This approach enables to employ decentralized control structure with subsystems having their local control loops. Compared with centralized MIMO controller systems, decentralized control structure brings about certain performance deterioration, however weighted against by important benefits, such as design simplicity, hardware, operation and reliability improvement. Robustness is one of attractive qualities of a decentralized control scheme, since such control structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections. Considerable effort has been made to enhance robustness in decentralized control structure and decentralized control design schemes and various approaches have been developed in this field both in time and frequency domains (Gyurkovics & Takacs, 2000; Zecevic & Siljak, 2004; Stankovic et al., 2007). Recently, the algebraic approach has gained considerable interest in robust control, (Boyd et al., 1994; Crusius & Trofino, 1999; de Oliveira et al., 1999; Ming Ge et al., 2002; Grman et al., 2005; Henrion et al., 2002). Algebraic approach is based on the fact that many different problems in control reduce to an equivalent linear algebra problem (Skelton et al., 1998). By algebraic approach, robust control problem is formulated in algebraic framework and solved as an optimization problem, preferably in the form of Linear Matrix Inequalities (LMI). LMI techniques enable to solve a large set of convex problems in polynomial time (see Boyd et al., 1994). This approach is directly applicable when control problems for linear uncertain systems with a convex uncertainty domain are solved. Still, many important control problems even for linear systems have been proven as NP hard, including structured linear control problems such as decentralized control and simultaneous static output feedback (SOF) designs. In these cases the prescribed structure of control feedback matrix (block diagonal for decentralized control) results in nonconvex problem formulation. There

PID Controller Design for Specified Performance

„How can proper controller adjustments be quickly determined on any control application?” The question posed by authors of the first published PID tuning method J.G.Ziegler and N.B.Nichols in 1942 is still topical and challenging for control engineering community. The reason is clear: just every fifth controller implemented is tuned properly but in fact:  30% of improper performance is due to inadequate selection of controller design method,  30% of improper performance is due to neglected nonlinearities in the control loop,  20% of improper closed-loop dynamics is due to poorly selected sampling period. Although there are 408 various sources of PID controller tuning methods (O Dwyer, 2006), 30% of controllers permanently operate in manual mode and 25% use factory-tuning without any up-date with respect to the given plant (Yu, 2006). Hence, there is natural need for effective PID controller design algorithms enabling not only to modify the controlled variable but also achieve specified performance (Kozakova et al., 2010), (Osuský et al., 2010). The chapter provides a survey of 51 existing practice-oriented methods of PID controller design for specified performance. Various options for design strategy and controller structure selection are presented along with PID controller design objectives and performance measures. Industrial controllers from ABB, AllenB these methods are based on minimum information about the plant obtained by the well-known relay experiment. Model-based PID controller tuning techniques acquire plant parameters from a step-test; useful tuning formulae are provided for commonly used system models (FOPDT – first-order plus dead time, IPDT – integrator plus dead time, FOLIPDT – first-order lag and integrator plus dead time and SOPDT – second-order plus dead time). Optimization-based PID tuning approaches, tuning methods for unstable plants, and design techniques based on a tuning parameter to continuously modify closed-loop performance are investigated. Finally, a novel advanced design technique based on closed-loop step response shaping is presented and discussed on illustrative examples.

A frequency domain design technique for robust decentralized controllers

Abstract The paper proposes a new frequency domain approach to the design of robust decentralized controllers (DC) for continuous-time systems described by a set of transfer function matrices. To guarantee the nominal stability and the prespecified nominal performance, the recently developed DC design technique (Kozakova and Veselý, 2003) has been applied, adapted so as to guarantee the robust M -δ structure based stability conditions modified for the closed-loop system under decentralized controller as well. Unlike the standard robust approaches to the DC, this technique allows the inclusion of the nominal interactions into the nominal model; thus the conservativeness of the robust stability conditions is relaxed.

Easy Tuning of PID Controllers for Specified Performance

Abstract The presented method allows achieving maximum overshoot and specified settling time of the closed-loop step response. It provides a simple way to control linear stable SISO systems even if the mathematical model is unknown. Tuning rule parameters are based on one suitably chosen point of the plant frequency response obtained by sine-wave signal with specified excitation frequency, and the required phase margin. The main result provided is construction of empirical charts used to convert time-domain performance specifications (maximum overshoot and settling time) into frequency domain performance measure (phase margin). The method is applicable for systematic shaping of the closed-loop response of the plant. The new approach has been verified on a set of benchmark examples and on a real plant as well.

Decentralized Digital PID Controller Design for Performance

Abstract The paper deals with the frequency domain decentralized discrete-time controller design methodology to guarantee specified performance of the overall system. The underlying theory evolves from the stability conditions developed in the “Equivalent Subsystems Method” approach and applies closed-loop performance specification based on the relationship between phase margins of equivalent subsystems and maximum overshoot of the full system. To design local PI controllers for specified phase margin, Bode plots of discrete equivalent subsystems are used. Designed local continuous controllers are then converted to their discrete versions and implemented for the controlled plant. The design procedure is illustrated on the case study.

Robust Controller Design: New Approaches in the Time and the Frequency Domains

Robust stability and robust control belong to fundamental problems in control theory and practice; various approaches have been proposed to cope with uncertainties that always appear in real plants as a result of identification /modelling errors, e.g. due to linearization and approximation, etc. A control system is robust if it is insensitive to differences between the actual plant and its model used to design the controller. To deal with an uncertain plant a suitable uncertainty model is to be selected and instead of a single model, behaviour of a whole class of models is to be considered. Robust control theory provides analysis and design approaches based upon an incomplete description of the controlled process applicable in the areas of non-linear and time-varying processes, including multi input – multi output (MIMO) dynamic systems. MIMO systems usually arise as interconnection of a finite number of subsystems, and in general, multivariable centralized controllers are used to control them. However, practical reasons often make restrictions on controller structure necessary or reasonable. In an extreme case, the controller is split into several local feedbacks and becomes a decentralized controller. Compared to centralized full-controller systems such a control structure brings about certain performance deterioration; however, this drawback is weighted against important benefits, e.g. hardware, operation and design simplicity, and reliability improvement. Robust approach is one of useful ways to address the decentralized control problem (Boyd et al., 1994; Henrion et al., 2002; de Oliveira et al., 1999; Gyurkovics & Takacs, 2000; Ming Ge et al., 2002; Skogestad & Postlethwaite, 2005; Kozakova and Veselý, 2008; Kozakova et al., 2009a). In this chapter two robust controller design approaches are presented: in the time domain the approach based on Linear (Bilinear) matrix inequality (LMI, BMI), and in the frequency domain the recently developed Equivalent Subsystem Method (ESM) (Kozakova et al., 2009b). As proportional-integral-derivative (PID) controllers are the most widely used in industrial control systems, this chapter focuses on the timeand frequency domain PID controller design techniques resulting from both approaches. The development of Linear Matrix Inequality (LMI) computational techniques has provided an efficient tool to solve a large set of convex problems in polynomial time (e.g. Boyd et al., 1994). Significant effort has been therefore made to formulate crucial control problems in

Direct Design of Robust Decentralized Controllers

Abstract The paper introduces an important enhancement to the robust decentralized controller design for multi input/multi output uncertain systems within the setting of the Equivalent Subsystems Method (ESM). The class of design problems considered includes guaranteeing robust stability and nominal performance in terms of maximum overshoot of the full system achieved through specified phase margins in equivalent subsystems. The developed design procedure is illustrated by an example.

Improved tuning technique for robust decentralized PID controllers

Abstract The paper presents a further improvement to the decentralized controller design technique for continuous-time uncertain systems first proposed in (Kozakova and Veselý, 2003) and further developed towards securing robust stability and required nominal performance of the uncertain system (e.g. Kozakova and Veselý, 2005; 2006). The proposed approach evolves from the generalized Nyquist stability criterion and the M-Δ structure robust stability conditions adapted for the decentralized control structure, the innovation consists in their interpretation and implementation in the tuning procedure. Robust decentralized controllers designed for two real plants show practical applicability of the proposed design philosophy.

Robust Decentralized Control of MIMO Systems in the Frequency Domain

Abstract The paper presents an application-oriented robust decentralized control design technique based upon the sufficient condition for robust stability (SCRS) of systems under decentralized control (DC) which is fulfilled using parameter tuning of fixed-structure local controllers. Graphical interpretation of the SCRS provides two useful stability criteria used to test stability of the overall system as well as stability of subsystems. Theoretical results are illustrated in a case study.