Quantitative Robust Control Engineering: Theory and Applications

This paper presents a summary of the main concepts and references of the Quantitative Feedback Theory (QFT). It is a frequency domain engineering method to design robust controllers. It explicitly emphasises the use of feedback to simultaneously reduce the effects of model plant uncertainty and to satisfy performance specifications. QFT highlights the trade-off (quantification) among the simplicity of the controller structure, the minimization of the ‘cost of feedback’, the existing model uncertainty and the achievement of the desired performance specifications at every frequency of interest. The technique has been successfully applied to control a wide variety of physical systems. After a brief introduction about the essential aspects of the QFT design methodology, including a wide set of QFT references, this paper presents a new method to extend the classical diagonal QFT controller design method for MIMO plants with model uncertainty to a fully populated matrix controller design method. The paper simultaneously studies three cases: the reference tracking, the external disturbance rejection at plant input and the external disturbance rejection at plant output. The work ends showing several real-world examples where the controllers have been designed using QFT techniques: an industrial SCARA robot manipulator, a wastewater treatment plant, a variable speed wind turbine of 1.65 MW and an industrial furnace of 1 MW. 1.0 INTRODUCTION Much of the current interest in frequency domain robust stability and robust performance dates from the original works of H.W. Bode (1945) [1] and I. Horowitz (1963) [2]. Since then, and during the entire second half of the twentieth century, there has been a tremendous advance in the state-of-the-art of robust frequency domain methods. One of the main techniques, introduced by Prof. Isaac Horowitz in 1959 [24], which characterises closed loop performance specifications against parametric plant uncertainty, mapped into open loop design constraints, became known as Quantitative Feedback Theory (QFT) in the seventies [25-27]. This paper presents a summary of the main ideas and references of the QFT methodology. The method searches for a controller that guarantees the achievement of the desired performance specifications for every plant within the existing model uncertainty. QFT highlights the trade-off (quantification) among the simplicity of the controller structure, the minimization of the ‘cost of feedback’ (bandwidth), the model uncertainty (parametric and non-parametric) and the achievement of the desired performance specifications at every frequency of interest. Following this introduction, Section 2 presents a brief description of the essential aspects of the QFT methodology. Section 3 introduces a method to design non-diagonal QFT controllers for MIMO systems. Afterwards the paper describes some real-world applications of the technique, carried out by the author: an Garcia-Sanz, M. (2006) Quantitative Robust Control Engineering: Theory and Applications. In Achieving Successful Robust Integrated Control System Designs for 21st Century Military Applications – Part II (pp. 1-1 – 1-44). Educational Notes RTO-EN-SCI-166, Paper 1. Neuilly-sur-Seine, France: RTO. Available from: http://www.rto.nato.int/abstracts.asp. RTO-EN-SCI-166 1 1 Quantitative Robust Control Engineering: Theory and Applications industrial SCARA robot manipulator in Section 4, a wastewater treatment plant of 5000 m/hour in Section 5, a variable speed wind turbine of 1.65 MW in Section 6 and an industrial furnace of 40 metres and 1 MW in Section 7. The paper ends with a wide References Section that includes a representative collection of books and papers related with the theory and applications of QFT. 2.0 QUANTITATIVE FEEDBACK THEORY The Quantitative Feedback Theory (QFT), first introduced by Prof. Isaac Horowitz in 1959 [24], is an engineering method, which explicitly emphasises the use of feedback to simultaneously reduce the effects of plant uncertainty and satisfy performance specifications. Horowitz’s work is deeply rooted in classical frequency response analysis involving Bode diagrams, template manipulations and Nichols Charts (NC). It relies on the observation that the feedback is needed principally when the plant presents model uncertainty or when there are uncertain disturbances acting on the plant. Frequency domain specifications and desired time-domain responses translated into frequency domain tolerances, lead to the so-called Horowitz-Sidi bounds (or constraints). These bounds serve as a guide for shaping the nominal loop transfer function L(s) = G(s) P(s), which involves the manipulation of gain, poles and zeros on the controller G(s). On the whole, the QFT main objective is to synthesize (loop-shape) a simple, low-order controller with minimum bandwidth, which satisfies the desired specifications and tackles feedback control problems with robust performance objectives. In the last few decades QFT has been successfully applied to many control problems. A wide collection of books and papers about the main aspects of the QFT methodology, theory and applications, is included in the references section: controller loop-shaping [41-44], existence conditions for controllers [45-47], multiinput multi-output MIMO systems [48-63], time-delay systems [64], digital QFT [65-66], distributed parameter systems [67-74], non-minimum phase systems [75-80], multi-loop systems [81-83], non-linear systems [84-92], linear time variant systems LTV [93-94], QFT software packages [95-102], real-world applications [103,127]. A detailed study about the history of QFT can be found in the papers written by Horowitz [19-21], Houpis [22] and Garcia-Sanz [23]. In 1992, Houpis and Chandler organized in Wright-Patterson (Dayton, Ohio) the first International QFT Symposium [10]. Since then, and with the continuous support of Prof. Houpis, the Symposia have been organized every two years: Indiana-USA-1995 [11], Glasgow-UK-1997 [12], Durban South Africa 1999 [13], Pamplona-Spain-2001 [14], and Cape Town South Africa -2003 [15]. The next one will be in Kansas-USA-2005. To go into the QFT theoretical aspects in depth, check the excellent tutorials written by Horowitz [16-17] and Houpis [18]. In addition, a major analysis can be found in the books written by Horowitz [3], Houpis, Rasmussen and Garcia-Sanz [4], Yaniv [5] and Sidi [6]. Finally, three special issues of the International Journal of Robust and Nonlinear Control (Wiley) describe some of the more significant advances of QFT: Houpis 1997 [7], Eitelberg 2001-2002 [8] and Garcia-Sanz 2003 [9]. The QFT design methodology is quite transparent, allowing the designer to see the necessary trade-offs to achieve the closed-loop system specifications. The basic steps of the procedure (see also Fig. 4) are presented in the following sub-sections. They are: • Plant model (with uncertainty), Templates generation and nominal plant selection Po(jω). • Performance Specifications. • QFT Bounds B(jω). • Loop-shaping the controller G(jω). • Pre-filter synthesis F(jω). • Simulation and Design Validation. 1 2 RTO-EN-SCI-166 Quantitative Robust Control Engineering: Theory and Applications 2.1 Plant Model and Templates Generation The plant dynamics to be controlled may be described by frequency response data, or by linear or nonlinear transfer functions with mixed (parametric and non-parametric) uncertainty models. It can be defined taking into account the parameter uncertainty of the process at every frequency of interest (ωi), that is to say the plant uncertainty templates, so that IP(jωi)={P(jωi), ωi∈∪Ωk}. The templates are sets of complex numbers representing the frequency response of the family of uncertain plants at a fixed frequency IP(jωi), i.e. a template is a projection of the n-dimensional parameter space onto the Nichols Chart. Fig.1 represents the QFT-template of the plant ) n n 2 2 )/( exp( ) ( ω + ω + − = s s s s P ζ τ with three parameters, two with uncertainty (ζ = 0.02, ωn = [0.7, 1.2], τ = [0, 2]), at ω = 1 rad/s. For more information about the QFT templates see [28-34]. Figure 1: Template of the plant at ω = 1 rad/s 2.2 Performance specifications The standard two degree of freedom system which best exemplifies the feedback problem considered in QFT is shown in Fig. 2. It includes the set of uncertain plants, -IP(jωi)={P(jωi), ωi∈∪Ωk}-, the loop controller -Gand the pre-filter -F-, both to be design, and the sensor dynamics -H-. On the other hand, R, E, U, Y and N are vectors representing respectively: the reference input, the error signal, the controller output, the plant output and the sensor noise input. W, D1 and D2 are the external disturbance inputs. From the structure we can define the Eqs. (1) to (3), Figure 2: Standard two-degree-of-freedom feedback structure F