LTR design of discrete-time proportional-integral observers

Abstract

A Monte Carlo simulation with 500 experiments was executed, and the results are presented in Fig. 3. In each experiment, the measurement model in effect at each point of time was randomly chosen according to (65). In Fig. 3(a), the root-mean-square-error (RMSE) in the state estimate versus time is presented. Averaging the RMSE's over the time interval gives an average error of 10.75 for the IMM filter, 9.32 for the smoother of Method 1, and 9.42 for Method 2. Fig. 3(b) presents the probability of error in the system-stmcture detection versus time (i.e., the probability of choosing the incorrect measurement model at each point of time). Averaging the probabilities over the time interval gives an average probability of error of 0.19 for the IMM filter, 0.15 for Method 1, and 0.16 for Method 2.The two smoothers provided noticeably better performances than the IMM filter, while the smoother of Method 1 provided slightly better performance than Method 2. The smoother of Method 1 provided the best overall performance because it considered the most hypotheses.A simulation example comparing the performances of these algorithms in reconstructing the trajectory of a maneuvering target has also been performed. Detailed results are not presented because of space limitations. Both smoothers provided significantly better performance than the IMM filter in estimating the system state. However, unlike the system-structure simulation results above, the Method 1 smoother provided significantly better mode estimates than the Method 2 smoother. The mode estimates from the Method 2 smoother and the IMM filter were comparable. V. SUMMARYSuboptimal approaches to the one-step fixed-lag smoothing problem for Markovian switching systems were examined in this paper. Two algorithms for generating one-step fixed-lag smoothed estimates were presented. In the first algorithm, the models over the two most recent sampling periods were considered. For n models, there are n2 possible ways of conditioning on models in two sampling periods, and this algorithm evaluated the n2 hypotheses using n2 parallel one-step smoothers. In the second algorithm, only the models in the most recent sampling period were considered, and it evaluated 72 hypotheses using n parallel one-step smoothers. A simulation of a system-structure detection problem was used to compare the performances of the two smoothers and an IMM filter. The results show that the smoother of Method 1 provided the best overall performance. Variants of these one-step smoothing algorithms have been used in conjunction with IMM filtering algorithms to develop techniques for the alignment of asynchronous sensors [ 6 ] . Finally, approaches similar to the ones presented in this paper have been applied to the fixed-interval smoothing problem for Markovian switching systems ~71. Abstract-This paper applies the proportional-integral (PI) observer in connection with loop-transfer recovery (LTR) design for discrete-time systems. Both the prediction and the filtering versions of the PI observer are consid...

Robust proportional-integral Kalman filter design using a convex optimization method

“…Here the value of the objective function will be greater than the estimation error variance defined in Eq. (7). However, by minimizing the objective function, the optimal solution in Eq.…”

“…Several design methods for these PI observers have been introduced: pole-placement methods [6,7], eigenstructure assignments [12], H-infinity norm minimization problems [9,11], or minimum estimation error variance approaches [13,14]. For instance, Duan et al [12] introduced a parametric eigenstructure assignment method that provides complete degrees of freedom in designing PI observers, which leads to lower eigenvalue sensitivities.…”

“…This type of the observer scheme was first introduced by Wojciechowski [3] for single-input single-output systems and then extended to multivariable systems by Kaczorek [4] and Shafai and Carroll [5], addressing a more flexible way for performing design objectives. For example, this loop can be used to identify unknown inputs or nonlinearities [6], to increase the stability margin in the LTR design [7,8], to attenuate or decouple disturbances [9,10], or to detect sensor or actuator faults [9,11].…”

Proportional-Integral Observer in Robust Control, Fault Detection, and Decentralized Control of Dynamic Systems

“…, r, one can compute t 2j ; for all j from (41), which specifies T 2 and the remaining observer parameter matrices G, H, and N can be obtained from (36), (35), and (38). It should be pointed out that for a properly chosen set of eigenvalues and depending on the specified functional, it is possible to have other solutions for (42) leading to functional observers of lower order r < ν − 1. It is also clear that for estimating a vector function of the states, where W ∈ R l×1 , one can design multi-functional observer of order l(ν − 1) using the above technique.…”

“…When the system is not minimum phase, then exact recovery is not possible and one can only achieve recovery asymptotically. A comprehensive treatment of LTR can be found in [38][39][40][41][42].…”