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Register a new userDiscrete-time MPC for switched systems with applications to biomedical problems
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Switched systems in which the manipulated control action is the time-depending switching signal describe many engineering problems, mainly related to biomedical applications. In such a context, to control the system means to select an autonomous system - at each time step - among a given finite family. Even when this selection can be done by solving a Dynamic Programming (DP) problem, such a solution is often difficult to apply, and state/control constraints cannot be explicitly considered. In this work a new set-based Model Predictive Control (MPC) strategy is proposed to handle switched systems in a tractable form. The optimization problem at the core of the MPC formulation consists in an easy-to-solve mixed-integer optimization problem, whose solution is applied in a receding horizon way. Two biomedical applications are simulated to test the controller: (i) the drug schedule to attenuate the effect of viral mutation and drugs resistance on the viral load, and (ii) the drug schedule for Triple Negative breast cancer treatment. The numerical results suggest that the proposed strategy outperform the schedule for available treatments.
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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.
This paper deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$-switching-cycle. These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. Furthermore, it has been proven that the proposed L-switching-cycle can eventually achieve the nonconservativeness in stability analysis as long as a sufficiently long L-switching-cycle is adopted. A numerical example is provided to illustrate our theoretical results.
We propose an extension of the theory of control sets to the case of inputs satisfying a dwell-time constraint. Although the class of such inputs is not closed under concatenation, we propose a suitably modified definition of control sets that allows to recover some important properties known in the concatenable case. In particular we apply the control set construction to dwell-time linear switched systems, characterizing their maximal Lyapunov exponent looking only at trajectories whose angular component is periodic. We also use such a construction to characterize supports of invariant measures for random switched systems with dwell-time constraints.
This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibriums. We show that an admissible pair $(hat{u},C)$ of control-stopping policy is equilibrium if and only if the axillary function associated to it solves the extended HJB system. We provide almost equivalent conditions to the boundary term of this extended HJB system, which is related to the celebrated smooth fitting principles. As applications of our theoretical results, we develop an investment-withdrawal decision model for time-inconsistent decision makers in infinite time horizon. We provide two concrete examples, one of which includes constant proportion investment with one side threshold withdrawal strategy as equilibrium; in another example, all strategies with constant proportion investment are proved to be irrational, no matter what the withdrawal strategy is.
In this paper, we develop a compositional scheme for the construction of continuous approximations for interconnections of infinitely many discrete-time switched systems. An approximation (also known as abstraction) is itself a continuous-space system, which can be used as a replacement of the original (also known as concrete) system in a controller design process. Having designed a controller for the abstract system, it is refined to a more detailed one for the concrete system. We use the notion of so-called simulation functions to quantify the mismatch between the original system and its approximation. In particular, each subsystem in the concrete network and its corresponding one in the abstract network are related through a notion of local simulation functions. We show that if the local simulation functions satisfy certain small-gain type conditions developed for a network containing infinitely many subsystems, then the aggregation of the individual simulation functions provides an overall simulation function quantifying the error between the overall abstraction network and the concrete one. In addition, we show that our methodology results in a scale-free compositional approach for any finite-but-arbitrarily large networks obtained from truncation of an infinite network. We provide a systematic approach to construct local abstractions and simulation functions for networks of linear switched systems. The required conditions are expressed in terms of linear matrix inequalities that can be efficiently computed. We illustrate the effectiveness of our approach through an application to AC islanded microgirds.
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