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The synthesis of externally positive filters that can be used to modulate control signals of output-feedback sliding mode controllers is presented. These filters are useful to estimate the norm of unmeasured signals in systems with parametric uncertainties. These norm observers are a robust alternative to state observers, that may be difficult to design for uncertain systems. The first step of the synthesis is the search of the envelope function, which is an upper bound for the norm of each admissible impulse response of the uncertain system. After this step, the approximation of a positive filter with an impulse response that is an upper bound for the envelope function is carried out. The search of the envelope function and the synthesis of the approximation filters are formulated as non-convex optimisation problems that can be solved by genetic algorithms. It is important to design norm observers that are not too conservative, because the modulation signals should be kept small in order to reduce the amplitude of the control signals, power losses, and undesirable chattering effects that may occur in sliding-mode control. The proposed methods are illustrated by numerical examples.

This study presents an output-feedback control algorithm based on unit vector sliding mode for a class of multivariable systems. The control objective is to force each output signal to track a desired reference trajectory, while retaining good performance despite parameter uncertainties, unmatched disturbances and actuators faults that eventually may occur in the plant. Owing to the new approach proposed to tackle this problem, which involves a linear matrix inequality to be satisfied by the control distribution matrix, no upper bound on this matrix is required. Moreover, a remarkable result is that the proposed fault-tolerant controller can even handle variations in the order of the system dynamics. The analysis shows that the resulting closed-loop system is globally exponentially stable. A simulation example with a chain of trailers driven by redundant actuators illustrates the design and effectiveness of the proposed strategy.

An output-feedback sliding-mode controller is proposed for multivariable uncertain systems with unknown time-varying state delays and unmatched non-linear terms. The non-linearities are allowed to depend not only on the system output, but also on its unmeasurable state, and can be polynomial. This scheme is based on model-reference unit vector control under a mild Hurwitz condition for the high-frequency gain matrix. To achieve global stability properties, the amplitude of the control signal is given by a norm observer for the unmeasured state. For the first time, the design of norm observers is applied to a class of time-delay systems. This approach guarantees finite-time convergence of the output-tracking error to zero. Simulation results illustrate the application of the developed control scheme.