Identification of Linear Time-Varying Parameters of Nonstationary Systems

2019 ◽  
Vol 20 (5) ◽  
pp. 269-265
Author(s):  
V. T. Le ◽  
M. M. Korotina ◽  
A. A. Bobtsov ◽  
S. V. Aranovskiy ◽  
Q. D. Vo
Keyword(s):  
Linear Time ◽  
Control Object ◽  
Time Interval ◽  
Identification Algorithm ◽  
Time Varying ◽  
State Variables ◽  
Unknown Parameters ◽  
Technical Problems ◽  
True Values ◽  
Stationary Control

The paper considers the identification algorithm for unknown parameters of linear non-stationary control objects. It is assumed that only the object output variable and the control signal are measured (but not their derivatives or state variables) and unknown parameters are linear functions or their derivatives are piecewise constant signals. The derivatives of non-stationary parameters are supposed to be unknown constant numbers on some time interval. This assumption for unknown parameters is not mathematical abstraction because in most electromechanical systems parameters are changing during the operation. For example, the resistance of the rotor is linearly changing, because the resistance of the rotor depends on the temperature changes of the electric motor in operation mode. This paper proposes an iterative algorithm for parameterization of the linear non-stationary control object using stable LTI filters. The algorithm leads to a linear regression model, which includes time-varying and constant (at a certain time interval) unknown parameters. For this model, the dynamic regressor extension and mixing (DREM) procedure is applied. If the persistent excitation condition holds, then, in the case the derivative of each parameter is constant on the whole time interval, DREM provides the convergence of the estimates of configurable parameters to their true values. In the case of a finite time interval, the estimates convergence in a certain region. Unlike well-known gradient approaches, using the method of dynamic regressor extension and mixing allows to improve the convergence speed and accuracy of the estimates to their true values by increasing the coefficients of the algorithm. Additionally, the method of dynamic regressor extension and mixing ensures the monotony of the processes, and this can be useful for many technical problems.

Output Adaptive Observers Design for Linear Non-Stationary Systems with Polynomial Parameters

2021 ◽  
Vol 22 (8) ◽  
pp. 404-410
Author(s):  
K. B. Dang ◽  
A. A. Pyrkin ◽  
A. A. Bobtsov ◽  
A. A. Vedyakov ◽  
S. I. Nizovtsev
Keyword(s):  
Linear Time ◽  
State Observer ◽  
Observer Design ◽  
Model Parameters ◽  
Time Varying ◽  
State Variables ◽  
Unknown Constant ◽  
Established Method ◽  
Time Varying Parameters ◽  
True Values

The article deals with the problem of state observer design for a linear time-varying plant. To solve this problem, a number of realistic assumptions are considered, assuming that the model parameters are polynomial functions of time with unknown coefficients. The problem of observer design is solved in the class of identification approaches, which provide transformation of the original mathematical model of the plant to a static linear regression equation, in which, instead of unknown constant parameters, there are state variables of generators that model non-stationary parameters. To recover the unknown functions of the regression model, we use the recently well-established method of dynamic regressor extension and mixing (DREM), which allows to obtain monotone estimates, as well as to accelerate the convergence of estimates to the true values. Despite the fact that the article deals with the problem of state observer design, it is worth noting the possibility of using the proposed approach to solve an independent and actual estimation problem of unknown time-varying parameters.

Linear Approximations for Swing Leg Motion During Gait

1984 ◽  
Vol 106 (2) ◽  
pp. 137-143 ◽  
Author(s):  
W. H. Lee ◽  
J. M. Mansour
Keyword(s):  
Linear Systems ◽  
System Analysis ◽  
Linear Time ◽  
Time Varying ◽  
Two Dimensional ◽  
State Variables ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Leg Motion ◽  
Time Invariant

The applicability of a linear systems analysis of two-dimensional swing leg motion was investigated. Two different linear systems were developed. A linear time-varying system was developed by linearizing the nonlinear equations describing swing leg motion about a set of nominal system and control trajectories. Linear time invariant systems were developed by linearizing about three different fixed limb positions. Simulations of swing leg motion were performed with each of these linear systems. These simulations were compared to previously performed nonlinear simulations of two-dimensional swing leg motion and the actual subject motion. Additionally, a linear system analysis was used to gain some insight into the interdependency of the state variables and controls. It was shown that the linear time varying approximation yielded an accurate representation of limb motion for the thigh and shank but with diminished accuracy for the foot. In contrast, all the linear time invariant systems, if used to simulate more than a quarter of the swing phase, yielded generally inaccurate results for thigh shank and foot motion.

Output Control by Linear Time-Varying Systems using Parametric Identification Methods

2020 ◽  
Vol 21 (7) ◽  
pp. 387-393
Author(s):  
V. Q. Dat ◽  
A. A. Bobtsov
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Control Object ◽  
The State ◽  
Time Varying ◽  
State Variables ◽  
Matrix Differential Equation ◽  
Matrix Differential ◽  
Uniform Observability ◽  
Riccati Matrix

In this paper the problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. This procedure involves the synthesis of linear static feedbacks on state variables in the case of known parameters of the plant. If state variables are not measured, then for the observer design using the matrix Riccati differential equation, using the dual scheme, which provides for the transposition of the state matrix and the replacement of the input matrix by the output matrix. It is well known that an observer of state variables built on the basis of a solution of the Riccati matrix differential equation ensures the exponential stability of a closed loop system in the case of uniform observability. Despite the fact that this type of observer can be classified as universal, its have a number of significant drawbacks. The main problem of such observers is the need for accurate knowledge of the parameters and the requirement for uniform observability, which in practice cannot always be realized. Thus, the problem of the new methods design for constructing observers of state variables of linear non-stationary systems is still relevant. Some time ago, a number of methods for the adaptive observers design of state variables for nonlinear systems were proposed. The main idea of the synthesis of observers was based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. This approach in the English language literature is called PEBO. This paper, based on the PEBO method, proposes a new approach for the observers design of state for non-stationary systems. This approach provides the possibility of obtaining monotonic convergence estimates with transient time tuning.

Iterative learning identification and control for point-to-point tracking of linear time-varying systems with unknown parameters and stochastic noise

2017 ◽  
Vol 40 (13) ◽  
pp. 3834-3845 ◽  
Author(s):  
Yan Geng ◽  
Xiaoe Ruan
Keyword(s):  
Linear Time ◽  
Tracking Error ◽  
Iterative Learning ◽  
Time Varying ◽  
Unknown Parameters ◽  
Point Tracking ◽  
Time Varying Systems ◽  
System Output ◽  
Point To Point ◽  
And Control

In this paper, an interactive iterative learning identification and control (ILIC) scheme is developed for a class of discrete-time linear time-varying systems with unknown parameters and stochastic noise to implement point-to-point tracking. The identification is to iteratively estimate the unknown system parameter matrix by adopting the gradient-type technique for minimizing the distance of the system output from the estimated system output, whilst the control law is to iteratively upgrade the current control input with the current point-to-point tracking error scaled by the estimated system parameter matrix. Thus, the iterative learning identification and the iterative learning control are scheduled in an interactive mode. By means of norm theory, the boundedness of the discrepancy between the system matrix estimation and the real one is derived, whilst, by the manner of the statistical technique, it is conducted that the mathematical expectation of the tracking error monotonically converges to nullity and the variance of the tracking error is bounded. Numerical simulations exhibit the validity and effectiveness of the proposed ILIC scheme.

A Simple Iterative Procedure for the Identification of the Unknown Parameters of a Linear Time Varying Discrete System

1963 ◽  
Vol 85 (2) ◽  
pp. 227-235 ◽  
Author(s):  
Harold J. Kushner
Keyword(s):  
Impulse Response ◽  
Discrete System ◽  
Linear Time ◽  
Time Varying ◽  
Unknown Parameters ◽  
Random Drift ◽  
Process Dynamics ◽  
Linear Discrete System ◽  
Single Input ◽  
Time Variations

A new “steepest descent” approach to the “adaptive control system” problem of the determination of the process dynamics of a time varying system is analyzed in considerable detail. The unknown parameters are the parameters of the impulse response of a linear discrete system. The identification procedure is a first-order iterative process and is designed to operate with the natural inputs of the system to be identified. After each new (single) input, new estimates of all the unknown parameters are computed. The method is computationally simple and, in its analysis, the effects of additive noise in the observations (of both input and output), random drift with time, or neglected parameters of the impulse response are handled with relative ease and become transparent. Time variations are taken directly into account, thus eliminating the necessity of the assumption of stationarity over a period of time.

scholarly journals Circuits with a mem-element: invariant manifolds control via pulse programmed sources

2021 ◽  
Author(s):  
Mauro Di Marco ◽  
Giacomo Innocenti ◽  
Alberto Tesi ◽  
Mauro Forti
Keyword(s):  
Invariant Manifolds ◽  
Linear Time ◽  
Operational Amplifiers ◽  
Time Interval ◽  
State Variables ◽  
Finite Time Interval ◽  
Independent Source ◽  
Linear Time Invariant ◽  
Terminal Element ◽  
Time Invariant

AbstractThe paper considers the problem of controlling multistability in a general class of circuits composed of a linear time-invariant two-terminal (one port) element, containing linear R, L, C components and ideal operational amplifiers, coupled with one of the mem-elements (memory elements) introduced by Prof. L.O. Chua, i.e., memristors, memcapacitors, and meminductors. First, explicit expressions of the invariant manifolds of the circuit are directly given in terms of the state variables of the two-terminal element and the mem-element. Then, the problem of steering the circuit dynamics from an initial invariant manifold to a final one, and hence to potentially switch among different attractors of the circuit, is addressed by designing pulse shaped control inputs. The control inputs ensure that the transition between the initial and final manifolds is accomplished within a given finite time interval. Moreover, it is shown how the designed control inputs can be implemented by introducing independent voltage and current sources in the two-terminal element. Notably, it turns out that it is always possible to solve the considered control problem by using a unique independent source. Several examples are provided to illustrate how the proposed approach can be applied to different circuits with mem-elements and to highlight the influence of the features of the designed sources on the behavior of the controlled dynamics.

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