This article is concerned with tackling the L1$$ {L}_1 $$ performance analysis problem of continuous and piecewise continuous nonlinear systems with non‐unique solutions by using the involved arguments of set‐invariance principles. More precisely, this article derives a sufficient condition for the L1$$ {L}_1 $$ performance of continuous nonlinear systems in terms of the invariant set. With respect to the case such that solving a nonlinear differential equation is difficult and thus an employment of the invariant set‐based sufficient condition is a non‐trivial task, we also derive another sufficient condition through the extended invariance domain approach. Because this extended approach characterizes set‐invariance properties in terms of the corresponding vector field and an extended version of contingent cones, the L1$$ {L}_1 $$ performance analysis problem could be solved without considering both the explicit solutions for the differential equation and the relevant solution uniqueness. These arguments associated with the L1$$ {L}_1 $$ performance of continuous systems are further extended to the involved case of piecewise continuous nonlinear systems, and we establish parallel results relevant to the set‐invariance principles obtained for the continuous nonlinear systems. Finally, numerical examples are provided to demonstrate the effectiveness as well as the applicability of the overall results derived in this article.
This article provides a synthesis method of the controller for piecewise continuous nonlinear systems in terms of set invariance principles. Following the arguments of Filippov's differential equations, solutions of piecewise continuous differential equations are first characterized by set‐valued functions. Based on such characteristics of solutions equipped with set‐valued functions, the performance of piecewise continuous nonlinear systems, that is, the gain from the associated disturbance to regulated output, is formulated. To clarify the existence of a controller such that the resulting closed‐loop system satisfies the performance in the presence of piecewise continuity, the so‐called generalized controlled invariance domain is proposed. Taking this advanced argument allows us to establish a synthesis procedure for the controller by showing a lower‐semicontinuity of the set‐valued functions involved in the generalized controlled invariance domain. Finally, some numerical examples are provided to evaluate the validity of the existence condition for the controller.
This paper aims at computing two types of system gains for discrete‐time observer‐based event‐triggered systems (ETSs) via the linear matrix inequality (LMI) approach. More precisely, an event‐trigger mechanism (ETM) is considered for the discrete‐time control systems consisting of a static controller and a Luenberger observer to determine whether or not the input signal for the controller is updated to the estimated value from the observer. For a tractable treatment of the input/output behavior of such ETMs, we derive their closed‐form representation through a piecewise linear difference equation. Based on this representation, we establish LMI‐based computation approaches to the gains for the ETSs from ℓp$$ {\ell}_p $$ to ℓ∞$$ {\ell}_{\infty } $$ (denoted by ℓ∞false/p$$ {\ell}_{\infty /p} $$) with p=2,∞$$ p=2,\infty $$. Finally, the effectiveness of the overall arguments is verified through a numerical example.
This paper is concerend with tackling the L 1 performance analysis problem of continuous and piecewise continuous nonlinear systems with non-unique solutions by using the involved arguments of set-invariance principles. More precisely, this paper derives a sufficient condition for the L 1 performance of continuous nonlinear systems in terms of the invariant set. However, because this sufficient condition intrinsically involves analytical representations of solutions of the differential equations corresponding to the nonlinear systems, this paper also establishes another sufficient condition for the L 1 performance by introducing the so-called extended invariance domain, in which it is not required to directly solving the nonlinear differential equations. These arguments associated with the L 1 performance analysis is further extended to the case of piecewise continuous nonlinear systems, and we obtain parallel results based on the set-invariance principles used for the continuous nonolinear systems. Finally, numerical examples are provided to demonstrate the effectiveness as well as the applicability of the overall results derived in this paper.
<abstract><p>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.</p></abstract>
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