Decentralised two-time-scale motions control based on generalised sampling R. Becerril-Arreola, A.G. Aghdam and V.D. Yurkevich Abstract: A design method for the decentralised time-varying discrete-time output-feedback control of linear time-invariant plants with unstable unstructured decentralised fixed modes (UDFM) is introduced. The design method uses generalised sampled-data hold functions to elim- inate the UDFMs and to decouple the discrete-time equivalent model of the plant into independent input – output channels. Through this structural change, the plant becomes suitable for a stabilising high-sampling-rate controller that induces two-time-scale motions (TTSM) in the closed-loop system. As a result, the discrete-time controller is likewise decoupled into distinct local agents and the TTSM closed-loop system is decentralised. 1 Introduction Most of the existing results on the output feedback control of decentralised linear time-invariant (LTI) systems focus on the properties of the system such as stabilisability by means of LTI or linear time-varying (LTV) controllers [1, 2] and design techniques based on such properties. All of these existing methods consider the interconnected sub- systems as one big system and then propose the design of local controllers for each subsystem by taking the effect of interconnections into account. Consequently, most of these methods involve complex algorithms and long itera- tive procedures [3, 4]. Even of more importance, some of them have limited applications because they cannot over- come the limitations imposed by the structural properties of the plant [2]. The most significant obstacle in the design of decentra- lised output-feedback controllers for certain class of LTI plants is the presence of unstable decentralised fixed mode (DFM), which LTI controllers cannot stabilise [2, 5]. DFMs can be classified as being either ‘structured’ or ‘unstructured’ [2, 5]. Structured DFMs are the modes that remain ‘fixed’ under any type of decentralised output- feedback control, including nonlinear and time-varying control. Therefore systems with unstable structured DFMs cannot be stabilised by any type of nonlinear or time- varying decentralised controller. Unstructured DFMs (UDFM), on the other hand, are the modes that can be elimi- nated through appropriate time-varying control laws. Since the combination of an ideal sampler, a discrete-time LTI controller and a hold operator acts as an LTV controller for the original continuous-time system, sampling can remove the non-zero and distinct UDFMs for almost all sampling rates [5]. Besides removing UDFMs, sampling can also modify the structure of the digraph of a plant. This property of sampling can be used to simplify the decentralised control design. For example, one can use generalised sampled-data hold func- tions (GSHF) to change the structure of the system to a hier- archical form [6]. The equivalent discrete-time system can then be stabilised by a series of smaller decentralised con- trollers that one can design by using centralised control design methods for each subsystem, independently. An approach is proposed in [7] to design a near-optimal GSHF for decentralised control systems. The approach uses a finite set of basis functions for constructing the desired hold function. This method is further developed in [8] to design a high-performance decentralised simultane- ous stabiliser for a finite set of continuous-time systems using the linear matrix inequality (LMI) technique. The robustness of control via GSHF has been previously studied [9–11]. Of particular interest are the results in [12], which address the robustness of zero-shifting via GSHFs. These results are of special interest because DFMs are, in fact, transmission zeros of the system and a set of its subsys- tems [3]. Therefore the conditions stated in [12] directly apply to the decentralised control problem studied here. One centralised control design method that can be effec- tively combined with GSHF-based structural modification to achieve decentralised output-feedback control is the two-time-scale motions (TTSM) control method [13]. The main advantages of this method are its robustness to par- ameter variations and its disturbance rejection capability [13–15]. Elaborating on the results introduced in [16], the present development incorporates generalised sampling into the original TTSM control method to provide a simple solu- tion to the decentralised output-feedback stabilisation problem for multi-input multi-output (MIMO) linear plants. Such a solution encompasses the design of GSHFs for the structural modification of the system, as well as the design of a controller whose dynamics are much faster than those of the plant. The structural modification has two purposes: (i) to eliminate unstable UDFMs from the discrete-time equivalent system so that it complies with the requirements of TTSM control; and # The Institution of Engineering and Technology 2007 doi:10.1049/iet-cta:20070020 Paper first received 25th January 2007 R. Becerril-Arreola is with the Department of Computing and Decision Sciences, Lingan University, 8 Castle Peak Road, Tuen Mun, Hong Kong A.G. Aghdam is with the Department of Electrical and Computer Engineering, Concordia University, 1455 de Maisonneuve Blvd. W., EV005-139, Montreal, Quebec, Canada H3G 1M8 V.D. Yurkevich is with the Automation Department, Novosibirsk State Technical University, Novosibirsk 630092, Russia E-mail: becerril@ece.concordia.ca IET Control Theory Appl., 2007, 1, (5), pp. 1477–1486 1477 IET Control Theory and Applications, 2007, 1 (5), pp. 1477-1486.