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Drought and water supply
Published in Stephen A. Thompson, Hydrology for Water Management, 2017
The author combined runs analysis, Markov chain analysis, and the PDSI in a study of drought in the central United States from 1895 to 1988 (Thompson 1990). The transition probability matrix for Missouri climate division 4 (southwest Missouri) is given as Table 12.9. The number of PDSI states was reduced from eleven to nine by combining the incipient wet and dry categories with the normal category. The main diagonal of the state transition matrix (i = j) is a measure of the persistence of any state. In southwestern Missouri extreme drought is the most persistent state with a state transition probability of PED,ED = 0.825. Once extreme drought becomes established it is very difficult to break. On the other hand extreme wet spells are only about half as persistent (0.438). The persistence for drought in general, that is going from a drought state to any other drought state, was found to be 0.254. Once drought becomes established there is about a one-in-four chance of it continuing in some form into the next month. Not all elements of the matrix are filled, which means it is impossible to go from certain states to other states, such as extreme wetness to extreme drought, in one month. The equilibrium matrix for southwestern Missouri is shown in Table 12.10. % %
Hidden Markov models for sequential damage detection of bridges
Published in Hiroshi Yokota, Dan M. Frangopol, Bridge Maintenance, Safety, Management, Life-Cycle Sustainability and Innovations, 2021
O. Bahrami, W. Wang, J.P. Lynch
The schematic of an HMM is shown in. An HMM is described by three properties. First is the state transition probabilities. Suppose there are a total of N hidden states (for simplicity, denote each unique state with numbers 1 to N). At each time instance, the hidden state can be only one of the N hidden states. The dynamics of the model is governed by a matrix whose terms are transition probabilities: A(i,j)=p(zt+1=j|zt=i) for i,j=1,2,…,N. This matrix is referred to as the state transition matrix. The second important property of the HMM, is the distribution of the observations given the states, also known as emission probabilities. In this work, we assume that the emission probabilities are Gaussian:p(xt|zt=i)=N(xt|μi,Σi)
Pseudo predictor feedback stabilisation of linear systems with both state and input delays
Published in International Journal of Control, 2021
Zhe Zhang, Bin Zhou, Wim Michiels
Inspired by the fundamental matrix-based approach proposed in Kharitonov (2015), in this paper, we extend the PPF approach presented in Liu and Zhou (2019), Zhou (2014b) and Zhou et al. (2019) for linear systems with only input delays to the case of linear systems with both state and input delays. By the PPF controller developed in this paper, arbitrarily large yet bounded input delays can be compensated properly. It is shown that the stability of the closed-loop system controlled by the PPF control can be recast into the stability of the target closed-loop system (without input delay) and the stability of a certain integral delay system. Differently from traditional predictor feedback, the distributed terms in the proposed PPF control scheme can be safely implemented without using input filters. We also show how the aforementioned eigenvalue-based approach can be used to synthesise stabilising controller gains. The main difference of the problem considered in this paper and the one considered in Zhou et al. (2019) is that the input delay in the present paper is larger than the state delay, while in Zhou et al. (2019) the input delay equals the state delay. As a result, the fundamental matrix has to be employed to predict the future states, instead of the closed-form state transition matrix used in Zhou et al. (2019). However, we will show that in case of commensurate delay the control law can be reformulated in terms of the controller gains and system matrices of the uncontrolled system, thereby avoiding the explicit use of the fundamental matrix.
Optimal H 2 output-feedback control of sampled-data systems
Published in International Journal of Control, 2020
Matheus F. Amorim, Alim P. C. Gonçalves, André R. Fioravanti, Matheus Souza
As in any linear system, it suffices to find a state transition matrix associated with (2) to completely describe its dynamic behaviour. As stated in Ichicawa and Katayama (2001), is a state transition matrix associated with (2) if, and only if, φ satisfies In this case, it follows that any state trajectory of satisfies for all . Moreover, other properties such as and , for all , , can also be verified. In the particular but important case in which the jump instants sequence is such that for all , the state transition matrix also satisfies , for all ; see Ichicawa and Katayama (2001).
Stability of switched positive linear delay systems with mixed impulses
Published in International Journal of Systems Science, 2019
Consider the following discrete-time switched linear impulsive systems with time-varying delays: The solution of (24) can be written as where is the state transition matrix of linear system and given by where denotes the number of in , denotes the number of in such that , .