Elsevier

Automatica

Volume 50, Issue 5, May 2014, Pages 1349-1359
Automatica

Dynamic generalized controllability and observability functions with applications to model reduction and sensor deployment

https://doi.org/10.1016/j.automatica.2014.02.041Get rights and content

Abstract

In this paper we introduce the notion of Dynamic Generalized Controllability and Observability functions for nonlinear systems. These functions are called dynamic and generalized since they make use of additional states (dynamic extension) and are such that partial differential inequalities are solved in place of equations. The presence of the dynamic extension permits the construction of classes of canonical controllability and observability functions without relying on the solution of any partial differential equation or inequality. The effectiveness of the proposed concept is validated by means of two applications: the model reduction problem via balancing and the sensor deployment problem in a continuous stirred tank reactor (CSTR).

Introduction

Minimal realization theory for linear systems provides the tool to determine a state-space description of minimal order, the impulse response of which matches the impulse response of the original system. In situations in which this approach cannot be pursued for practical or theoretical reasons, one may still be interested in finding a lower-order representation that approximates the behavior of the original model, thus solving the so-called model reduction problem (Antoulas et al., 2001, Glover, 1984, Moore, 1981, Pernebo and Silverman, 1982, Scherpen, 1993, Scherpen and Van Der Schaft, 1994).

The first step towards the solution of the model reduction problem is the characterization of a measure of importance of the state components according to some desired criterion. Controllability and observability Gramians are well-known mathematical tools to describe linear systems in terms of their input/output behavior. In the case of locally asymptotically stable systems, this concept permits an ordering of the states with respect to their input/output energy, namely taking into account the control effort necessary to steer the system from a specific state to the origin in infinite time and the energy released by the output of the system initialized at a particular state, respectively.

In the linear framework, the controllability and observability functions are determined from the solutions of Lyapunov matrix equations and are related to the controllability and observability Gramians of the system. Mimicking the above ideas, controllability and observability functions have been defined also for general nonlinear systems (Scherpen, 1993). The controllability and observability functions of nonlinear systems are the solution of first-order partial differential equations, which may be hard or impossible to determine analytically. This aspect represents a serious drawback to the applicability of model reduction by balancing (of the controllability and the observability functions) to practical cases.

Once these functions have been determined, there exists a (nonlinear) change of coordinates such that, in the new coordinates, the functions are in the so-called balanced form (Scherpen, 1993). After the coordinates transformation the components of the state can be ordered according to the energy required to steer the state and the output energy released by the corresponding initial condition. Therefore, if a state demands a large amount of energy to be controlled, on one hand, and it is hardly observable (in terms of output energy), on the other hand, then clearly the contribution of the aforementioned state to the input/output behavior of the system is negligible and could be ignored in a lower-order approximation.

In addition, in the case of unstable nonlinear systems several techniques have been proposed, such as LQG, HJB or H balancing. The latter has been introduced for linear systems in Mustafa and Glover (1991) and subsequently extended to the nonlinear case in  Scherpen (1996). Moreover, it is shown in Scherpen and Van Der Schaft (1994) that the HJB singular value functions, obtained from the past and future energy functions, are strongly interconnected with the Graph Hankel singular value functions, derived by balancing the controllability and observability functions of the normalized coprime factorizations of the nonlinear system. As a result, the reduced order models obtained with the two different approaches coincide. Finally, the assumption of zero-state observability is relaxed in  Gray and Mesko (1999) allowing for non-zero inputs in the definition of the observability function.

The main contribution of the paper consists in the definition of the notion of Dynamic Generalized Controllability and Observability functions. They are said to be generalized and dynamic since partial differential inequalities are solved, in place of equations, in an extended state-space, respectively. In fact, the additional state may be considered as a dynamic extension introducing auxiliary dynamics to combine with a positive definite function. While the notion of generalized Gramians has been introduced in the literature, see for instance (Prajna and Sandberg, 2005, Sandberg, 2010, Sandberg and Rantzer, 2004), the idea of considering dynamic Gramians is new. Interestingly, the main advantage of these functions over the classical controllability and observability functions is that the former can be constructively defined avoiding the explicit solution of any partial differential equation or inequality. Preliminary results about the notion of Dynamic Generalized Controllability and Observability function and its construction are reported in Sassano and Astolfi (2012b). Different from  Sassano and Astolfi (2012b) an alternative notion of (matrix) algebraic P̄ solution is considered here, which leads to significant simplifications in the construction of such functions and, moreover, in the application of the latter functions to the model reduction problem via optimal balancing. In addition, a detailed discussion about the point-wise minimization of the approximation error, intrinsically introduced by the proposed functions, and a description of the application of the latter functions to the problem of sensor deployment are reported.

The rest of the paper is organized as follows. Section  2 introduces the notion of Dynamic Generalized Controllability and Observability functions which can be constructed, as shown in Section  3, without relying on the solution of any partial differential equation or inequality. A class of Dynamic Generalized Controllability and Observability functions is constructively defined in Section  3. Finally, the effectiveness of the proposed notion is validated by means of two different examples in Section  4. To begin with, the application of the Dynamic Generalized Controllability and Observability functions to the problem of balancing and model reduction for nonlinear systems is discussed. Once the functions have been constructed, a nonlinear system can be transformed into a dynamically balanced form by means of a change of coordinates. Then, the optimal sensor deployment problem is discussed and a dynamic observability function is employed to determine whether it is more informative to measure the substrate or the biomass concentration in a continuous stirred tank reactor (CSTR). Conclusions are drawn and future work is outlined in Section  5.

Section snippets

Dynamic generalized controllability and observability functions

Consider a nonlinear system described by equations of the form ẋ=f(x)+g(x)u,y=h(x), where x(t)Rn denotes the state of the system, u(t)Rm the input, and y(t)Rp the output. The mappings f:RnRn, g:RnRn×m and h:RnRp are assumed to be sufficiently smooth and such that f(0)=0 and h(0)=0, without loss of generality. By the latter assumption, there exist, possibly not unique, continuous matrix-valued functions F:RnRn×n and H:RnRp×n such that f(x)=F(x)x and h(x)=H(x)x, respectively, for all xRn

A class of dynamic generalized controllability and observability functions

The notion of Dynamic Generalized Controllability and Observability functions is motivated by the fact that the differential equation Dc (Do, respectively) and the function Lc (Lo, respectively) can be explicitly constructed in a neighborhood of the origin. This construction hinges upon the solution of algebraic inequalities and it does not involve the solution of any pde or pdi.

Model reduction

Once the controllability and the observability functions have been obtained, standard model reduction techniques for nonlinear systems rely upon the definition of a sequence of changes of coordinates such that, in the new coordinates, the functions are in the so-called balanced form. Often these transformations are not constructive but exploit, for instance, Morse Lemma.

The main advantage of the notion of Dynamic Generalized Controllability and Observability functions over controllability and

Conclusions

The notion of Dynamic Generalized Controllability and Observability functions has been introduced and discussed. These functions are called dynamic and generalized since they rely on the definition of a dynamic extension and partial differential inequalities are solved in place of pde, respectively. In particular, it is shown that the dynamic extension permits the construction of a class of Dynamic Generalized Controllability and Observability functions without relying on the closed-form

Mario Sassano was born in Rome, Italy, in 1985. He received the B.S. degree in Automation Systems Engineering and the M.S. degree in Systems and Control Engineering from the University of Rome “La Sapienza”, Italy, in 2006 and 2008, respectively. In 2012 he was awarded a Ph.D. degree by Imperial College London, UK, where he had been a research assistant at the Department of Electrical and Electronic Engineering since 2009. Currently he is a postdoctoral research assistant at the University of

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    Mario Sassano was born in Rome, Italy, in 1985. He received the B.S. degree in Automation Systems Engineering and the M.S. degree in Systems and Control Engineering from the University of Rome “La Sapienza”, Italy, in 2006 and 2008, respectively. In 2012 he was awarded a Ph.D. degree by Imperial College London, UK, where he had been a research assistant at the Department of Electrical and Electronic Engineering since 2009. Currently he is a postdoctoral research assistant at the University of Rome “Tor Vergata”, Italy. His research interests are focused on nonlinear observer design, optimal control theory with applications to mechatronical systems and image processing.

    Alessandro Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering from the University of Rome in 1991. In 1992 he joined ETH-Zurich where he obtained an M.Sc. in Information Theory in 1995 and the Ph.D. degree with Medal of Honour in 1995 with a thesis on discontinuous stabilization of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome “La Sapienza” for his work on nonlinear robust control. Since 1996 he is with the Electrical and Electronic Engineering Department of Imperial College, London (UK), where he is currently Professor in Nonlinear Control Theory. From 1998 to 2003 he was also an Associate Professor at the Department of Electronics and Information of the Politecnico of Milano. Since 2005 he is also Professor at Dipartimento di Informatica, Sistemi e Produzione, University of Rome Tor Vergata. He has been visiting lecturer in “Nonlinear Control” in several universities, including ETH-Zurich (1995–1996); Terza University of Rome (1996); Rice University, Houston (1999); Kepler University, Linz (2000); SUPELEC, Paris (2001). His research interests are focused on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilization, robust stabilization, robust control and adaptive control. He is author of more than 70 journal papers, of 20 book chapters and of over 160 papers in refereed conference proceedings. He is author (with D. Karagiannis and R. Ortega) of the monograph “Nonlinear and Adaptive Control with Applications” (Springer Verlag). He is Associate Editor of Systems and Control Letters, Automatica, IEEE Trans. Automatic Control, the International Journal of Control, the European Journal of Control, the Journal of the Franklin Institute, and the International Journal of Adaptive Control and Signal Processing. He has also served in the IPC of various international conferences.

    This work is partially supported by the EPSRC Programme Grant Control For Energy and Sustainability   EP/G066477 and by the MIUR under PRIN Project Advanced Methods for Feedback Control of Uncertain Nonlinear Systems. The material in this paper was partially presented at the 52st IEEE Conference on Decision and Control (CDC), December 10–13, 2012, Maui, Hawaii, USA. This paper was recommended for publication in revised form by Associate Editor Nathan Van De Wouw under the direction of Editor Andrew R. Teel.

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