Annotation of the results obtained in 2023
1. For 2023 year, existing papers devoted to interval observer design for different classes of systems were considered; the conclusion that the main shortcoming of these papers is that they design interval observers estimating full state vector of the original system has been made. At the same time, one may need to obtain estimates only for the prescribed function of the state vector. As a result, the corresponding interval observer will be simpler than the observer estimating the full state vector, the interval width is reduced that corresponds to more precise estimate, and class of systems for which such observer can be designed will be extended. The suggested approach is based on the reduced order model of the original system. The methods and algorithms to design such a model of minimal dimension insensitive to the external disturbances and estimating the prescribed function of the system state vector are developed. Depending on class of systems (continuous-time or discrete-time) two canonical forms of the matrix describing the system dynamics are suggested: Jordan canonical form for continuous-time and identification canonical form for discrete-time systems. Advantage of these forms is that they satisfy requirements for designing interval observers: Jordan canonical form with negative eigenvalues is stable and Metzler (it has nonnegative non-diagonal elements) and identification canonical form is stable and nonnegative as well. In known papers, to transform the system into the form with Metzler matrix, some restrictions should be satisfied. The suggested approach is free of such restrictions. Based on the obtained model and assuming that the external disturbances and measurement noise are bounded, interval observer is designed. Unlike the model, such an observer contains not unknown measurement noise but its upper and lower bounds which form corresponding bounds for prescribed function of the system state vector. The obtained observer differs from that designed by known methods: it has minimal dimension and is free of the disturbance. The first fact guarantees fewer calculation complexity of program for computer; in particular, this is important for computer in underwater vehicles; the second fact reduces the interval width and improves the estimation accuracy. The advantage of internal observers is that it gives interval estimates which describe a quality (accuracy) of estimation. It was shown that traditional restriction for initial conditions of the system is not essential and can be removed. If this restriction is absent, then after some time where the estimated function can be out of interval, it will be within the interval. In some cases, the model insensitive to the external disturbances and estimating the prescribed function of the system state vector cannot be designed. To overcome this difficulty, the methods and algorithms based on singular value decomposition of matrices describing the systems and disturbances are developed. They allow designing the model of minimal dimension with minimal sensitivity to the disturbances. The observer based on such a model forms wider interval but its width is fewer that that forming by observers estimating the full state vector. 2. Based on the methods described above, methods and algorithms to design interval observers for continuous-time or discrete-time systems with parametric uncertainties were developed. It is assumed that such uncertainties are bounded in the forms of matrix inequalities. Two approaches are developed to solve the problem: the first one under some restrictions imposed on the original system produces simple solution, the second one is free of such restriction but the observer is more complex. The suggested solution is based on the reduced order model of the original system estimation the prescribed function containing the transformed parametric uncertainties. In some cases, such a model can be free of such uncertainties, and the observer without parametric uncertainties can be designed that reduces the interval width essentially. The appropriate algorithm to design such a model is developed. When parametric uncertainties are unknown but constant, the algorithm to design simpler interval observer is developed. 3. It was shown how based on interval observer estimating the prescribed function of the system state vector, a bank of interval observers estimating the full state vector and forming more precise interval estimates can be designed. On the first step of the appropriate algorithm, interval estimation for those components of the state vector which are available due to system physical sensors is performed. Note that the external disturbances are not presented in this procedure therefore estimates are of maximal accuracy. On the second step of this algorithm, the suggested above algorithms form interval estimates for other components of the state vector. They will be free of the disturbance or have minimal sensitivity to them (dependently on whether or not the model insensitive to the disturbances exists). Estimates obtaining on these steps are collected and form the full state vector estimate. Since estimates obtaining on the first step are formed based on simple algebraic operations, general complexity of such a bank is fewer than that of interval observer estimating the system full state vector. 4. In addition to interval observer, methods and algorithms to design so-called virtual sensors are developed. Such sensors can supplement existing physical sensors. Virtual sensors are designed based on the reduced order model of the original system insensitive or having minimal sensitivity to the disturbances and estimating those components of the system state vector which are not measured by the existing physical sensors. Virtual sensors can be used for replacement the faulty physical sensors as well. 5. All developed algorithms and designed interval observers and virtual sensors are computer simulated based on the mathematical package Matlab. Real technical systems (robot servoactuator and well known three-tank system) have been used for simulation. The external disturbances and measurement noise are simulated by random series. Based on simulation results, some algorithms have been corrected. Simulation shows an effectiveness of the designed interval observers and virtual sensors. In particular, it was shown by simulation that the traditional restriction for initial conditions of the system is not essential and can be removed.

 

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