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We study the stability of a certain class of switched systems where discontinuous jumps (resets) on some of the state components are allowed, at the switching instants. It is known that, if all components of the state are available for reset, the system can be stabilisable by an adequate choice of resets. However, this question may have negative answer if there are forbidden state components for reset. We give a sufficient condition for the stabilisability of a switched system, under arbitrary switching, by partial state reset in terms of a block simultaneous triangularisability condition. Based on this sufficient condition, we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset. We also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems. [ABSTRACT FROM PUBLISHER]