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This paper considers an output-based event-triggered control approach for discrete-time systems and proposes three new types of performance measures under unknown disturbances. These measures are motivated by the fact that signals in practical systems are often associated with bounded energy or bounded magnitude, and they should be described in the $ \ell_{2} $ and $ \ell_\infty $ spaces, respectively. More precisely, three performance measures from $ \ell_{q} $ to $ \ell_{p} $, denoted by the $ \ell_{p/q} $ performances with $ (p, q) = (2, 2), \ (\infty, 2) $ and $ (\infty, \infty) $, are considered for event-triggered systems (ETSs) in which the corresponding event-trigger mechanism is defined as a function from the measured output of the plant to the input of the dynamic output-feedback controller with the triggering parameter $ \sigma (>0) $. Such a selection of the pair $ (p, q) $ represents the $ \ell_{p/q} $ performances to be bounded and well-defined, and the three measures are natural extensions of those in the conventional feedback control, such as the $ H_\infty $, generalized $ H_2 $ and $ \ell_1 $ norms. We first derive the corresponding closed-form representation with respect to the relevant ETSs in terms of a piecewise linear difference equation. The asymptotic stability condition for the ETSs is then derived through the linear matrix inequality approach by developing an adequate piecewise quadratic Lyapunov function. This stability criterion is further extended to compute the $ \ell_{p/q} $ performances. Finally, a numerical example is given to verify the effectiveness of the overall arguments in both the theoretical and practical aspects, especially for the trade-off relation between the communication costs and $ \ell_{p/q} $ performances. |
