Automatica 46 (2010) 815–822 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Unknown-state, unknown-input reconstruction in discrete-time nonminimum-phase systems: Geometric methods ✩ Giovanni Marro, Elena Zattoni ∗ Department of Electronics, Computer Science and Systems, University of Bologna, 40136 Bologna, Italy article info Article history: Received 23 June 2009 Received in revised form 26 November 2009 Accepted 7 February 2010 Available online 12 March 2010 Keywords: Nonminimum-phase systems Geometric methods Invariant zeros State and input reconstruction abstract The complete solution to the unknown-state, unknown-input reconstruction problem in systems with invariant zeros is inherently conditioned by the fact that, for any invariant zero, at least one initial state exists, such that the output is not affected when the mode of the invariant zero is properly injected into the system. Despite this intrinsic limitation, the problem of reconstructing the initial state and the inaccessible inputs from the available measurements has recently attracted remarkable interest, owing to its impact on the synthesis of enhanced-reliability control systems. This contribution consists of a geometric method which solves the unknown-state, unknown-input reconstruction problem in discrete-time systems with invariant zeros anywhere in the complex plane, except the unit circumference. The case of systems with the invariant zeros in the open set outside the unit disc is regarded as the basic one. The difficulties related to the presence of those invariant zeros are overcome by accepting a reconstruction delay commensurate to the invariant zero time constants and the accuracy required for reconstruction. The solution devised for that case also applies to systems without invariant zeros. However, in this case, reconstruction is exact and the delay depends on the number of iterations needed for a certain conditioned invariant algorithm to converge. Finally, the more general case of systems with invariant zeros lying anywhere in the complex plane, with the sole exception of the unit circumference, is reduced to the fundamental one through the synthesis of an appropriate filter. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Unknown-state, unknown-input reconstruction has been the object of a fair amount of research effort in the last two decades, mainly in connection with the increasing demand for sophisti- cated, enhanced-reliability control systems in safety-critical fields like aerospace, underwater robotics, power plants and chemi- cal or petrochemical plants. A direct approach to unknown-state, unknown-input reconstruction can be found in Basile and Marro (1992), where it is shown that a continuous-time multivariable lin- ear system is completely unknown-state, unknown-input recon- structable if and only if the maximal controlled invariant subspace contained in the null space of the output is the origin of the state space. Under the assumption of left invertibility, that condition, which also holds for discrete-time systems, is equivalent to the ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo (System and Control Theory). ∗ Corresponding address: Department of Electronics, Computer Science and Systems, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. Tel.: +39 051 2093023; fax: +39 051 2093073. E-mail addresses: giovanni.marro@unibo.it (G. Marro), elena.zattoni@unibo.it (E. Zattoni). absence of invariant zeros. Nevertheless, with regard to conti- nuous-time systems with invariant zeros, the way to determine the combinations of invariant zero modes and initial states resulting in zero output can also be found in Basile and Marro (1992). Conver- sion of that reasoning into the discrete time is straightforward. Since then, several works have been directed to further inves- tigation of the subject, particularly in systems with invariant ze- ros, and often referring to specific applications. In Hou and Patton (1998), necessary and sufficient conditions for complete unknown- state, unknown-input reconstruction are stated in terms of normal rank of matrix pencils. Numerical reliability of algorithms for in- put reconstruction is also discussed. The impact on system supervi- sion, fault diagnosis and fault tolerant control, filtering and coding theory motivates the study. In Corless and Tu (1998), it is shown how an arbitrarily accurate, asymptotic estimate of state and in- put can be obtained without measurement differentiation, which, on the contrary, is required to achieve exact asymptotic estimation in Hou and Patton (1998). The authors refer to the estimation of the cutting force exerted by a machine tool as a typical problem solv- able by their method. In Xiong and Saif (2003), Corless and Tu’s algorithm is revisited and a variant which also applies to a spe- cial class of nonminimum-phase systems is proposed. Xiong and Saif apply their technique to observation of unknown inputs in a double-effect pilot plant evaporator. 0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.02.012