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Register a new userAn arbitrary-order predefined-time exact differentiator for signals with exponential growth bound
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Added by
David G\\'omez-Guti\\'errez
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Constructing differentiation algorithms with a fixed-time convergence and a predefined Upper Bound on their Settling Time (textit{UBST}), i.e., predefined-time differentiators, is attracting attention for solving estimation and control problems under time constraints. However, existing methods are limited to signals having an $n$-th Lipschitz derivative. Here, we introduce a general methodology to design $n$-th order predefined-time differentiators for a broader class of signals: for signals, whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic derivative, i.e., whose $(n+1)$-th derivative grows at most exponentially. Our approach is based on a class of time-varying gains known as Time-Base Generators (textit{TBG}). The only assumption to construct the differentiator is that the class of signals to be differentiated $n$-times have a $(n+1)$-th derivative bounded by a known function with a known bound for its $(n+1)$-th logarithmic derivative. We show how our methodology achieves an textit{UBST} equal to the predefined time, better transient responses with smaller error peaks than autonomous predefined-time differentiators, and a textit{TBG} gain that is bounded at the settling time instant.
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Algorithms having uniform convergence with respect to their initial condition (i.e., with fixed-time stability) are receiving increasing attention for solving control and observer design problems under time constraints. However, we still lack a general methodology to design these algorithms for high-order perturbed systems when we additionally need to impose a user-defined upper-bound on their settling time, especially for systems with perturbations. Here, we fill this gap by introducing a methodology to redesign a class of asymptotically, finite- and fixed-time stable systems into non-autonomous fixed-time stable systems with a user-defined upper-bound on their settling time. Our methodology redesigns a system by adding time-varying gains. However, contrary to existing methods where the time-varying gains tend to infinity as the origin is reached, we provide sufficient conditions to maintain bounded gains. We illustrate our methodology by building fixed-time online differentiators with user-defined upper-bound on their settling time and bounded gains.
There is an increasing interest in designing differentiators, which converge exactly before a prespecified time regardless of the initial conditions, i.e., which are fixed-time convergent with a predefined Upper Bound of their Settling Time (UBST), due to their ability to solve estimation and control problems with time constraints. However, for the class of signals with a known bound of their $(n+1)$-th time derivative, the existing design methodologies are either only available for first-order differentiators, yielding a very conservative UBST, or result in gains that tend to infinity at the convergence time. Here, we introduce a new methodology based on time-varying gains to design arbitrary-order exact differentiators with a predefined UBST. This UBST is a priori set as one parameter of the algorithm. Our approach guarantees that the UBST can be set arbitrarily tight, and we also provide sufficient conditions to obtain exact convergence while maintaining bounded time-varying gains. Additionally, we provide necessary and sufficient conditions such that our approach yields error dynamics with a uniformly Lyapunov stable equilibrium. Our results show how time-varying gains offer a general and flexible methodology to design algorithms with a predefined UBST.
Differentiation is an important task in control, observation and fault detection. Levants differentiator is unique, since it is able to estimate exactly and robustly the derivatives of a signal with a bounded high-order derivative. However, the convergence time, although finite, grows unboundedly with the norm of the initial differentiation error, making it uncertain when the estimated derivative is exact. In this paper we propose an extension of Levants differentiator so that the worst case convergence time can be arbitrarily assigned independently of the initial condition, i.e. the estimation converges in emph{Fixed-Time}. We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.
This paper aims to provide a methodology for generating autonomous and non-autonomous systems with a fixed-time stable equilibrium point where an Upper Bound of the Settling Time (UBST) is set a priori as a parameter of the system. In addition, some conditions for such an upper bound to be the least one are provided. This construction procedure is a relevant contribution when compared with traditional methodologies for generating fixed-time algorithms satisfying time constraints since current estimates of an UBST may be too conservative. The proposed methodology is based on time-scale transformations and Lyapunov analysis. It allows the presentation of a broad class of fixed-time stable systems with predefined UBST, placing them under a common framework with existing methods using time-varying gains. To illustrate the effectiveness of our approach, we generate novel, autonomous and non-autonomous, fixed-time stable algorithms with predefined least UBST.
This paper presents an efficient suboptimal model predictive control (MPC) algorithm for nonlinear switched systems subject to minimum dwell time constraints (MTC). While MTC are required for most physical systems due to stability, power and mechanical restrictions, MPC optimization problems with MTC are challenging to solve. To efficiently solve such problems, the on-line MPC optimization problem is decomposed into a sequence of simpler problems, which include two nonlinear programs (NLP) and a rounding step, as typically done in mixed-integer optimal control (MIOC). Unlike the classical approach that embeds MTC in a mixed-integer linear program (MILP) with combinatorial constraints in the rounding step, our proposal is to embed the MTC in one of the NLPs using move blocking. Such a formulation can speedup on-line computations by employing recent move blocking algorithms for NLP problems and by using a simple sum-up-rounding (SUR) method for the rounding step. An explicit upper bound of the integer approximation error for the rounding step is given. In addition, a combined shrinking and receding horizon strategy is developed to satisfy closed-loop MTC. Recursive feasibility is proven using a $l$-step control invariant ($l$-CI) set, where $l$ is the minimum dwell time step length. An algorithm to compute $l$-CI sets for switched linear systems off-line is also presented. Numerical studies demonstrate the efficiency and effectiveness of the proposed MPC algorithm for switched nonlinear systems with MTC.
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