Linear Algebra Methods for the Control of Multidimensional Systems

The purpose of the thesis is to develop a comprehensive theory of the geometric control for N-dimensional systems. Two possible representations and their structural invariance properties of 2-D systems will be considered and generalised to the N-dimensional case: the Fornasini-Marchesini first order model and Fornasini-Marchesini second order model. In addition, necessary and sufficient conditions for the existence of solutions for the implicit 2-D Fornasini-Marchesini models will be provided, and generalised to the N-dimensional case

Model based fault detection for two-dimensional systems

Fault detection and isolation (FDI) are essential in ensuring safe and reliable operations in industrial systems. Extensive research has been carried out on FDI for one dimensional (1-D) systems, where variables vary only with time. The existing FDI strategies are mainly focussed on 1-D systems and can generally be classified as model based and process history data based methods. In many industrial systems, the state variables change with space and time (e.g., sheet forming, fixed bed reactors, and furnaces). These systems are termed as distributed parameter systems (DPS) or two dimensional (2-D) systems. 2-D systems have been commonly represented by the Roesser Model and the F-M model. Fault detection and isolation for 2-D systems represent a great challenge in both theoretical development and applications and only limited research results are available. In this thesis, model based fault detection strategies for 2-D systems have been investigated based on the F-M and the Roesser models. A dead-beat observer based fault detection has been available for the F-M model. In this work, an observer based fault detection strategy is investigated for systems modelled by the Roesser model. Using the 2-D polynomial matrix technique, a dead-beat observer is developed and the state estimate from the observer is then input to a residual generator to monitor occurrence of faults. An enhanced realization technique is combined to achieve efficient fault detection with reduced computations. Simulation results indicate that the proposed method is effective in detecting faults for systems without disturbances as well as those affected by unknown disturbances.The dead-beat observer based fault detection has been shown to be effective for 2-D systems but strict conditions are required in order for an observer and a residual generator to exist. These strict conditions may not be satisfied for some systems. The effect of process noises are also not considered in the observer based fault detection approaches for 2-D systems. To overcome the disadvantages, 2-D Kalman filter based fault detection algorithms are proposed in the thesis. A recursive 2-D Kalman filter is applied to obtain state estimate minimizing the estimation error variances. Based on the state estimate from the Kalman filter, a residual is generated reflecting fault information. A model is formulated for the relation of the residual with faults over a moving evaluation window. Simulations are performed on two F-M models and results indicate that faults can be detected effectively and efficiently using the Kalman filter based fault detection. In the observer based and Kalman filter based fault detection approaches, the residual signals are used to determine whether a fault occurs. For systems with complicated fault information and/or noises, it is necessary to evaluate the residual signals using statistical techniques. Fault detection of 2-D systems is proposed with the residuals evaluated using dynamic principal component analysis (DPCA). Based on historical data, the reference residuals are first generated using either the observer or the Kalman filter based approach. Based on the residual time-lagged data matrices for the reference data, the principal components are calculated and the threshold value obtained. In online applications, the T2 value of the residual signals are compared with the threshold value to determine fault occurrence. Simulation results show that applying DPCA to evaluation of 2-D residuals is effective.Doctoral These

Filtering for uncertain 2-D discrete systems with state delays

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.This paper is concerned with the problem of robust H∞ filtering for two-dimensional (2-D) discrete systems with time-delays in states. The 2-D systems under consideration are described in terms of the well-known Fornasini–Marchesini local state-space (FMLSS) models with time-delays. Our attention is focused on the design of a full-order filter such that the filtering error system is guaranteed to be asymptotically stable with a prescribed H∞ disturbance attenuation performance. Sufficient conditions for the existence of desired filters are established by using a linear matrix inequality (LMI) approach, and the corresponding filter design problem is then cast into a convex optimization problem that can be efficiently solved by resorting to some standard numerical software. Furthermore, the obtained results are extended to more general cases where the system matrices contain either polytopic or norm-bounded parameter uncertainties. A simulation example is provided to illustrate the effectiveness of the proposed design method.This work was partially supported by the National Natural Science Foundation of China (60504008), Program for New Century Excellent Talents in University of China and the Postdoctoral Science Foundation of China (20060390231)

A first cubic upper bound on the local reachability index for some positive 2-D systems

[EN] The calculation of the smallest number of steps needed to deterministically reach all local states of an nth-order positive 2-D system, which is called local reachability index (ILR) of that system, was recently tackled bymeans of the use of a suitable composition table. The greatest index ILR obtained in the previous literature was n+3 ([n/2]) 2 for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on ILR depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order n = 6 consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first n(1) vertices and is connected to the another subdigraph through the 1-arc (2, n(1) +n(2)), being the natural numbers n(1) and n(2) such that n(1) > n(2) = 2 and n-n(1)-n(2) = 1. The second one has two main cycles, a cycle where only the remaining vertices n(1)+1,..., n appear and a cycle containing only the vertices n(1)+1, n(1)+n(2)-1. Moreover, the last vertices are connected through the 2-arc (n(1) +n(2)-1, n). Furthermore, if n > 12 and is a multiple of 3, for appropriate n(1) and n(2), the ILR of that family is at least cubic, exactly, it must be n(3)+9n(2)+45n+108/27, which shows that some local states can be deterministically reached much further than initially proposed in the literature.We are gratefully thankful to the reviewers for their valuable remarks. This work has been partially supported by the European Union [FEDER funds] and Ministerio de Ciencia e Innovacion through Grants MTM-2013-43678-P and DPI2016-78831-C2-1-R.Bailo Ballarín, E.; Gelonch, J.; Romero Vivó, S. (2019). A first cubic upper bound on the local reachability index for some positive 2-D systems. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3767-3784. https://doi.org/10.1007/s13398-019-00699-0S376737841134Bailo, E., Gelonch, J., Romero-Vivo, S.: An upper bound on the reachability index for a special class of positive 2-D systems. Electron. J. Linear Algebra 18, 1–12 (2009)Bailo, E., Gelonch, J., Romero-Vivo, S.: Advances on the reachability index of positive 2-D systems. IEEE Trans. Autom. Control 59(8), 2248–2251 (2014)Bartosiewicz, Z.: Reachability and observability graphs for linear positive systems on time scales. IFAC Proc. Vol. 47(3), 3803–3808 (2014)Benvenuti, L., De Santis, A., Farina, L. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 294. Springer, Berlin (2003)Benvenuti, L.: On the reachable set for third-order linear discrete-time systems with positive control. Syst. Control Lett. 60(9), 690–698 (2011)Benvenuti, L.: On the reachable set for third-order linear discrete-time systems with positive control: the case of complex eigenvalues. Syst. Control Lett. 60(9), 1000–1008 (2011)Benzaouia, A., Hmamed, A., Tadeo, F.: Two-dimensional systems: from introduction to state of the art. In: Studies in Systems, Decision and Control (Vol. 28). Springer, Switzerland, (2016). https://doi.org/10.1007/978-3-319-20116-0Bru, R., Romero-Vivo, S. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 389. Springer, Berlin (2009)Bru, R., Bailo, E., Gelonch, J., Romero, S.: On the reachability index of positive 2-d systems. IEEE Trans. Circ. Syst. II: Express Brief 53(10), 997–1001 (2006)Bru, R., Coll, C., Romero, S., Sánchez, E.: Reachability indices of positive linear systems. Electron. J. Linear Algebra 11, 88–102 (2004)Bru, R., Romero-Vivó, S., Sánchez, E.: Reachability indices of periodic positive systems via positive shift-similarity. Linear Algebra Appl. 429, 1288–1301 (2008)Bru, R., Cacceta, L., Rumchev, V.G.: Monomial subdigraphs of reachable and controllable positive discrete-time systems. Int. J. Appl. Math. Comput. Sci. 15(1), 159–166 (2005)Bru, R., Romero, S., Sánchez, E.: Canonical forms of reachability and controllability of positive discrete-time control systems. Linear Algebra Appl. 310, 49–71 (2000)Cacace, F., Farina, L., Setola, R., Germani, A. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 471. Springer, Berlin (2017)Cantó, B., Coll, C., Sánchez, E.: On stability and reachability of perturbed positive systems. Adv. Differ. Equ. 2014(1), 296 (2014). https://doi.org/10.1186/1687-1847-2014-296Coll, C., Fullana, M., Sánchez, E.: Reachability and observability indices of a discrete-time periodic descriptor system. Appl. Math. Comput. 153, 485–496 (2004)Commault, C.: A simple graph theoretic characterization of reachability for positive linear systems. Syst. Control Lett. 52(3–4), 275–282 (2004)Commault, C.: On the reachability in any fixed time for positive continuous-time linear systems. Syst. Control Lett. 56(4), 272–276 (2007)Commault, C., Marchand, N. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 341. Springer, Berlin (2006)Coxson, P.G., Shapiro, H.: Positive reachability and controllability of positive systems. Linear Algebra Appl. 94, 35–53 (1987)Coxson, P.G., Larson, L.C., Schneider, H.: Monomial patterns in the sequence $${A}^kb$$. Linear Algebra Appl. 94, 89–101 (1987)De la Sen, M.: On the reachability and controllability of positive linear time-invariant dynamic systems with internal and external incommensurate point delays. Rocky Mt J Math 40(1), 177–207 (2010)Fanti, M.P., Maione, B., Turchiano, B.: Controllability of linear single-input positive discrete time systems. Int J Control 50, 2523–2542 (1989)Fanti, M.P., Maione, B., Turchiano, B.: Controllability of multi-input positive discrete time systems. Int J Control 51, 1295–1308 (1990)Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Pure and Applied Mathematics. Wiley, New York (2000)Fornasini, E., Marchesini, G.: State-Space realization of two-dimensional filters. IEEE Trans. Autom. Control AC–21(4), 484–491 (1976)Fornasini, E., Marchesini, G.: Doubly indexed dynamical systems. Math. Syst. Theory 12, 59–72 (1978)Fornasini, E., Valcher, M.E.: On the positive reachability of 2D positive systems. In: Farina, L., Benvenuti, L., De Santis, A. (eds.) Positive Systems. Lecture Notes in Control and Information Sciences, pp. 297–304. Springer, Berlin (2003)Fornasini, E., Valcher, M.E.: Controllability and reachability of 2-d positive systems: a graph theoretic approach. IEEE Trans. Circuits Syst. I Regul. Pap. 52(3), 576–585 (2005)Hrynlów, K., Markowski, K.A.: Experimental evaluation of upper bounds of reachability index for set of solutions of 2-D positive system. In: 17th International Carpathian Control Conference (ICCC), Tatranska Lomnica, pp. 248–252 (2016). https://doi.org/10.1109/CarpathianCC.2016.7501103Kaczorek, T.: Reachability and controllability of 2D positive linear systems with state feedback. Control Cybern. 29(1), 141–151 (2000)Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)Kaczorek, T.: Reachability and minimum energy control of positive 2D systems with delays. Control Cybern 34(2), 411–423 (2005)Kaczorek, T.: New reachability and observability tests for positive linear discrete-time systems. Bull. Polish Acad. Sci. Tech. Sci. 55(1), 19–21 (2007)Kaczorek, T.: Reachability of linear hybrid systems described by the general model. J. Arch. Control Sci. 20(2), 199–207 (2010)Kaczorek, T., Borawski, K.: Existence of reachable pairs (A, B) of discrete-time linear systems. In: Proceedings of 21st International Conference on Methods and Models in Automation and Robotics (MMAR). IEEE, pp. 702–707 (2016). https://doi.org/10.1109/MMAR.2016.7575222Kostova, S.P.: A PLDS model of pollution in connected water reservoirs. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems. Lecture Notes in Control and Information Science, vol. 294, pp. 257–263. Springer, Berlin (2003)Marszalek, W.: Two-dimensional state space discrete models for hyperbolic partial differential equations. Appl. Math. Model. 8(1), 11–14 (1984)Markowski, K.A.: Determination reachability index space of positive two-dimensional linear system using digraph-based theory. In: 19th International Conference on System Theory, Control and Computing (ICSTCC), Cheile Gradistei, pp. 533–538 (2015). https://doi.org/10.1109/ICSTCC.2015.7321348Moysis, L., Mishra, V.: Existence of reachable and observable triples of linear discrete-time descriptor systems. Circ. Syst. Signal Process. 3, 1–13 (2018). https://doi.org/10.1007/s00034-018-0922-5Pereira, R., Rocha, P., Simões, R.: Characterizations of global reachability of 2D structured systems. Multidimens. Syst. Signal Process. 24, 1–14 (2011). https://doi.org/10.1007/s11045-011-0154-3Rumchev, V.G., James, D.J.G.: Reachability and controllability of time-variant discrete-time positive linear systems. Control Cybern. 33(1), 87–93 (2004)Rumchev, V., Chotijah, S.: The minimum energy problem for positive discrete-time linear systems with fixed final state. In: Bru, R., Romero-Vivo, S. (eds.) Positive Systems. Lecture notes in control and information sciences, vol. 389, pp. 141–149. Springer, Berlin (2009)Valcher, M.E.: Controllability and reachability criteria for discrete time positive systems. Int. J. Control 65(3), 511–536 (1996)Valcher, M.E.: Reachability properties of continuous-time positive systems. IEEE Trans. Autom. Control 54(7), 1586–1590 (2009)Valcher, M.E.: Reachability analysis for different classes of positive systems. In: Bru, R., Romero-Vivo, S. (eds.) Positive Systems. Lecture notes in control and information sciences, vol. 389, pp. 29–41. Springer, Berlin (2009

Realization of 2D convolutional codes of rate 1/n by separable Roesser models

In this paper, two-dimensional convolutional codes constituted by sequences in where is a finite field, are considered. In particular, we restrict to codes with rate and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized

Structural invariants of two-dimensional systems

In this paper, some fundamental structural properties of two-dimensional (2-D) systems which remain invariant under feedback and output-injection transformation groups are identified and investigated for the first time. As is well known, structural invariants that follow from the definition of controlled and conditioned invariance, output-nulling, input-containing, self-bounded and self-hidden subspaces play pivotal roles in many theoretical studies of systems theory and in the solution of several control/estimation problems. These concepts are developed and studied within a 2-D context in this paper

A new approach to applying discrete sliding mode control to 2D systems

Sliding mode control has been applied previously to a specific form of 2D systems (Roesser model). In this paper a new approach (ID vectorial form) is introduced for this problem. Using ID form to represent 2D systems can be used as an alternative strategy to reduce the inherent complexity of 2D systems and their applications. Unlike Wave Advanced Model (WAM) form (proposed by Porter and Aravena), the suggested ID vectorial form, in this paper, has invariable dimension and consequently can be converted to regular form for sliding mode control (SMC). In this paper, the first Fornasini and Marchesini (FM) model of 2D systems which is a second order recursive form is considered. Meantime, the suggested method can be simply deployed to other first or second order 2D models. ©2013 IEEE

Convolutional Neural Networks as 2-D systems

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible