Design of optimal switching surfaces for discretely controlled systems

depend on the activated mode of operation q(t) ∈ Q and are influenced by disturbances d(t). The discrete controller implements a state-dependent switching law to trigger mode transitions in the plant, such that specifications r(t) imposed on the continuous state x(t) are met. A central control task of DCCS concerns the stabilization of stationary periodic operations, which map into p-periodic limit cycles Γ in the state space (see Fig. 2). Each limit cycle Γ can be characterized by its periodic mode sequence QΓ = (q̄?0 . . . q̄ ? p−1) and the associated switch points XΓ = {x̄? ( q̄?0 ) , . . . , x̄ ( q̄?p−1 ) }. In general, orbital stability must be achieved by timing mode transitions properly, as the mode order is fixed through Γ.