This paper describes the theory and algorithm allowing one to tune a multi-exciter system in order to obtain specified temporal and spatial structural response properties. Considerable effort is being put upon the desire to overcome practical difficulties and limitations as found in real-world systems. The main application that was envisaged for this algorithm is the creation of travelling vibration waves in structures. Such waves may be useful in testing and diagnostic applications or in ultrasonic motors for generating motion. The proposed method adaptively modifies a set of perturbations applied to the model so that an increasing amount of information is extracted from the system. The algorithm strives to overcome the following difficulties: (a) singular model inversion, (b) poor signal to noise ratio, (c) feedback, and (d) certain types of non-linear behaviour. High response levels, exciter–structure coupling and the inherent feedback existing in electro-mechanical systems are demonstrated to cause singularity, poor signal to noise levels and, to some extent, non linear behaviour. These phenomena pose some difficulties under operating conditions commonly encountered during dynamic testing of structures. The tuning of the multi-shaker system is approached in this work, as a non-linear optimisation problem where insight into the physical behaviour is emphasised in choosing the algorithmic strategy. The system's unknown model is inverted in an implicit manner using an automatic orthogonal and adaptive search direction. This adaptation uses the measured responses and forces at each step in order to determine the direction of progression during the tuning process. The non-linear behaviour of the exciters is compensated, in this work, by identification of the high-order (Volterra-like) transfer functions. This high-order model is than inverted allowing one to create a signal that cancels the unwanted harmonics. The proposed approach is analytically shown to converge and the necessary magnitude of perturbations to assure this convergence is derived. A simple case study is used to illustrate the proposed method, while an experimental verification and more examples can be found in a companion paper.