A linear approach to generalized minimum variance controller design for MIMO nonlinear systems

International audience

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Then, it is important to derive a good approximate sampled-data model because the exact sampled- data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate sampled-data model which includes extra zero dynamics, so-called the sampling zero dynamics, corresponding to the relative degree of the continuous-time nonlinear system. This paper shows that a more accurate sampled-data model is required for a controlled Van der Pol system with the relative degree two. The reason is that the closed-loop system becomes unstable when a controller design method based on cancellation of the zero dynamics is applied, and the phenomenon seems related to the instability of the sampling zero dynamics of the more accurate sampled-data model. Further, this paper derives a more accurate model than that of Yuz and Goodwin for continuous-time nonlinear systems with the relative degree two, and presents a condition which assures the stability of the sampling zero dynamics of the obtained model.

The work presented in this dissertation was conducted in an effort to achieve improvement in conventional and digital control methods applied to chemical processes. Emphasis was placed on simplicity and effectiveness in an attempt to provide control improvements which are easily implemented and yet efficient in their tasks. In these studies digital simulation was used exclusively utilizing the IBM 7040 and 360 and the XDS Sigma V computers. Initially the inaccuracies involved in obtaining a firstorder plus dead time model graphically from an open loop response curve are discussed. It is pointed out that fine tuning of two and three mode controllers based on these inaccurate models is necessary and presently must be done by trial and error. Modifications of existing controller tuning relations are presented which guide fine tuning attempts and eliminate much of the trial and error procedure. Second, the subject of linear discrete models applied to sampled-data systems is considered. A multiple linear least squares technique is applied to sampled system input/output records to identify parameters of a linear discrete model. The method is demonstrated to give good results when applied xii to linear and non-linear systems with and without dead time. The simplicity and flexibility of implementing this modeling technique are discussed. Application of this method to systems with measurement noise are shown to be improved when the data are filtered. Third, the implementation of linear discrete models in Deadbeat, Dahlin and Kalman controller synthesis methods are presented. It is demonstrated that the form of the models fits well into each of these controller design philosophies. The Dahlin algorithms which incorporate a tuning parameter give the most consistent performance for model orders and sampling times considered. In general, the performance of controllers based on low order models is seen to compare well with that of higher order controllers. Fourth, utilizing the speed of the multiple linear regression identification, the application of this technique in an adaptive control scheme is presented. Applying a recursive solution of the regression equation enables the updating of the linear discrete model at each sample instant and subsequently the adaptation of the control strategy. This scheme is demonstrated to provide closer control of a non linear, time-variant system than is possible with a stationary model. xiii CHAPTER I The form of the transfer function, HG(z), comprised of the digital hold device and the process itself, is such that multiple linear regression techniques can be applied to identify the model constants (parameters). Model identification is then based only on a linear regression of the sampled-data records of the system input and output. A modeling technique based on multiple linear regression only requires the solution of a set of normal equa tions and therefore is rapid enough to be applied on-line. Model order can also be changed with relative ease requiring only an other pass through the system input/output response data already collected. The effectiveness of this identification method is demonstrated on linear systems as well as on a simulated non linear plug flow reactor system. The effect of measurement noise on modeling accuracy is also considered, and comparison of identification using both filtered and unfiltered data are presented. The advantage of filtering data under these operating conditions is illustrated. Linear discrete models obtained for the simulated reactor were demdnstrated in their capacities as both one step ahead predictor models and as free running models. In these studies the higher order models provided significantly better results. Digital controller synthesis methods are the subject of Chapter IV. The discrete form of the model fits well into these design methods. Flexibility in varying model order is carried over to allow: for flexibility in controller order. The advantages of this flexibility are illustrated with the Deadbeat, Dahlin and Kalman controller design algorithms. Controllers based on each of these design methods are implemented in the plug flow reactor system and compared as to overall control performance and manipulate variable action. Comparisons are made also for both fast and slow sampling rates. Finally, Chapter V incorprates the multiple linear regres sion modeling technique with a direct controller synthesis method in an attempt to achieve an adaptive control scheme. The intent here is to update the process model with each new sample taken, and therefore adapt the controller, which is di rectly based on this model to any variation in the operating levels of the system. Adaptive control in this fashion is demonstrated in application to the non-linear plug flow reactor system with a slowly changing feed rate. The result is a demonstration of an adaptive control strategy which has the capability to function in many possible process applications.

A robustness design of fuzzy control is proposed in this paper to overcome the effect of modeling error between nonlinear system and Takagi-Sugeno (T-S) fuzzy model. In terms of Lyapunov's direct method, a stability criterion is derived to guarantee the asymptotic stability of nonlinear interconnected systems. Based on the decentralized control scheme and this criterion, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear interconnected systems. Finally, an example is given to illustrate the concepts discussed throughout the paper.

Numerous papers were published during the last decade on the normal forms of nonlinear control systems with applications in bifurcation and its control. The approach is motivated by Poincare’s theory of normal forms for classical dynamical systems using homogeneous transformations. In this paper, we summarize a variety of control system normal forms published in the literature so that the normal forms are derived in a same framework with consistent notations. Before we get into technical details, the rest of the introduction is a review of existing results on some related topics. It is well known that there are several normal forms for a linear control system. If the system is controllable then the system can be transformed into controllable or controller normal form. If the system has a linear output map and is observable then it can be transformed into observable or observer form. The nonlinear generalization of the linear controller normal forms were extensively studied during 1980’s, for instance, Krener [23], Hunt-Su [11], JackubczykRespondek [10], and Brocket [3], etc. If a nonlinear control system admits a controller normal form, it can be transformed into a linear system by a change of coordinates and feedback. Therefore, the design of a locally stabilizing state feedback control law is a straightforward task. In such a case, we say the system is feedback linearizable. On the other hand, most nonlinear systems do

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

Nonlinear [formula omitted]-gain verification for nonlinear systems

The current theory of nonlinear systems is still not perfect. The modeling and control of nonlinear system problem has always been the difficulty. In a variety of methods of its study, fuzzy system theory because of having the language descriptive way similar to the human mind, can obtain and deal with the qualitative information intelligently. The theory itself also has non-linear characteristics. Therefore the use of fuzzy systems theory to establish the fuzzy model of nonlinear system can well describe the nonlinear characteristics. T-S fuzzy systems, due to the combination of the good performance of the fuzzy system to deal with nonlinear problems with the simple linear expressions, are not only suitable for modeling the nonlinear system, but also use T-S fuzzy model and the linear control theory method to design the controller. So it has been widely used in nonlinear system control problems, and has also greatly developed the T-S fuzzy system theory, appearing a lot of methods of structural and parameter identification. However, this study of T-S fuzzy rules makes us have to face the difference of different ways to select the number of rules as well as online self-adaptability of the number of rules which off-line method lacks when using T-S fuzzy model to deal with nonlinear system modeling and control problem. In view of this, this paper researches on modeling and controlling of complex nonlinear systems based on TS model from different perspectives.

During the 20th century, the electronics industry has evolved from radio, television, and audio systems to computer systems. The trace of evolution is clearly from linear systems to nonlinear systems. This is by no means accidental, and is due to the fact that nonlinear systems are more versatile than linear systems. As of now, the successful commercial optical instruments are all linear systems. From the entropy point of view, as the knowledge of human beings further expands, we need more powerful computing capabilities such as memory association and self organization. Based on the fundamental differences between the electron and the photon, this paper presents the argument that optical computing will provide memory association and self organization ability (which compliments the current electronic logical computing). Thus, one can infer that the main future thrust of the optical industry will be in nonlinear optical systems and hybrid combinations with nonlinear electronic systems in order to provide both self organization and logical computing (to emulate, more or less, the computing power of the human brain). Currently, the stat of the ar has already demonstrated successful combinations of linear optical system with nonlinear electronic systems (computers). These kinds of hybrid systems are called electro-photonic systems in the following curriculum. Based on the above vision for the future and taking into account the need for immediate employment, the following curriculum is designed to provide a core training to the students who are interested in choosing optics or electro-photonics as their career. This curriculum is being offered in the upper years at the Physics Department of Chung Yuan University. Electronics, Optics, Electro-photonic System Design and Analysis, Optical System Design and Analysis, Introduction to Optical Computing, Holography, Nonlinear Optics, The Experimental Technology for Electro- photonics (I), The Experimental Technology for Electro-photonics (II). The main thrust among all these courses is the system modeling and analysis. In this paper, the design of this curriculum will be elaborated and the topics for each course will be listed in detail.

Theory of nonlinear control

<p>Whilst a lot of our strategic focus in the public sector is on linear policy approaches, many systems/ phenomena of importance are defined as non-linear or far from equilibrium. Traditional approaches to linear forecasting have not proved effective for non-linear systems, since non-linear systems follow a different set of rules. Historically, non-linear systems were too hard to forecast, but over recent decades some rules and approaches are starting to emerge. One important and clearly defined category of non-linear systems are those that follow a ‘power-law’ distribution rather than the ‘normal’ distribution, which is often associated with linear systems or systems in equilibrium. My research collects, analyses, and does a comparative analysis of the different power law populations, as well as the main strategic forecasting techniques that can be applied to those populations/ systems. Overall Conclusions and observations. Just as in science and mathematics, there is now a clearly defined separation and understanding of linear and non-linear systems and the rules that apply to each. My thesis has as its central theme, the idea that strategy as a subject also fits this same philosophical separation of approaches, which I have called the strategic planning versus the strategic thinking divide. Strategic planning is essentially the linear approach – being rational and assuming relatively stable conditions. Strategic thinking assumes the world is effectively non-linear and ‘far from equilibrium’. Non-linear approaches mean acknowledging concepts like; punctuated equilibrium, power law ‘log-log’ graphs, ‘scale-free’ characteristics, ‘self organising criticality’, accepting only pattern prediction (including 1/f formulas) and not precise prediction etc. Understanding non-linearity is essential to understand such things as ‘Black Swans’. Luck, serendipity and ‘bounded rationality’ are always involved in non-linear complex adaptive systems, whereas linear systems tend to comply with the so called ‘rational’ traditions in science and economics. Power law statistical distributions can be seen in a wide variety of non-linear natural and man-made phenomena, from earthquakes and solar flares to populations of cities and sales of books. This sheer diversity of effects that have power law distributions is actually an amazing fact that has only become evident over the last decade or so. Since the world contains aspects that are clearly linear and other aspects that are clearly non-linear, it is essential for someone interested in strategy to be able to understand both systems and be able to apply the correct techniques to each approach. The two parts of ‘punctuated equilibrium’ effectively link the two strategic approaches together as there is only one world and not two separate realities. It therefore follows that a strategist needs a good understanding of both strategic planning and strategic thinking, since both are needed for different phases or periods, and perhaps both are needed for any period when you can't tell what phase you are in, which can also happen. I suggest that under a linear phase, the strategic planning approach should be dominant, but supported by strategic thinking (since you never know when events will turn abruptly); whereas in a turbulent non-linear period the strategic thinking approach should be dominant, but supported by strategic planning (since you know that great turbulence will not last). This is a sort of a swapping dominant/ recessive situation, which has a loose parallel in the theory of the left/ right brain split, where it is not wise to use only one style of thinking, since there are two styles which suit different situations. The key is to pick the right thinking style for the right situation. Just as we have one brain, but two thinking styles, so in the strategy toolbox we also have two valid, useful and complimentary general strategic approaches. However for this thesis, I have focused on the non-linear power law aspects of life which have strong implications for strategic thinking, since that is the new area for me as well as one of the new knowledge frontiers for strategy as a subject (and for leadership, politics and many other areas).</p>

Controllability of linear and nonlinear systems governed by Stieltjes differential equations

This paper is concerned with data-driven methods for virtual reference controller design of high-order nonlinear systems via neural network. Virtual reference feedback tuning (VRFT) is a one-shot direct data-based method to design controller of linear or nonlinear systems. In this paper, we recall the model reference control problem of high-order nonlinear systems and design a new objective function of VRFT. In ideal conditions, the two problems are demonstrated to have the same solution. For the first time, we prove that the value of the optimization problem for model reference control is bounded by that of the objective function of VRFT. A three-layer neural network is employed as a general approximator of the designed controller and two simulations are given to verify the validity of our method.

NONLINEAR SELFTUNING CONTROLLER BASED UPON LAGUERRE SERIES REPRESENTATION

This paper presents a method of designing optimal dynamic compensator for a class of typical chemical reaction process-CSTR systems. Firstly, the model of nonlinear CSTR system is provided, Secondly, by using feedback linearization method, the nonlinear system is converted into a linear system, then according to the LQR theory, the methods of designing dynamic compensator is designed, and we analyze the asymptotic stability of the optimal regulator which has been compensated. At last, the simulation results show the effectiveness of the method.