I am also thankful to all other faculty members of Department of Electronics and Electrical Engineering, IIT Guwahati for all seminars/talks/conferences etc. I would like to thank all past and present faculty of control system laboratory, Dr.
Nonlinear Systems
For an insightful and further in-depth analysis of nonlinear dynamics, reading Strogatz's book [4] is highly recommended. A nonlinear system in its simplest form can be represented using a set of nonlinear differential equations given as.
Block Oriented Nonlinear Models
However, it is not impossible that such a unifying approach will not exist in the future. Another important advantage of these models is that they allow the use of standard linear controller design methods which is possible because the static non-linearity in the process can be canceled by inserting an inverse at the appropriate position in the loop.
System Identification
An extensive literature has already been established for the identification of BONL models, including functional series approaches, correlation analysis, neural network models, fuzzy models, differential equation models, regression models, etc.
Identification using Relay
Relay feedback is such a method that produces a deterministic output signal, ie. persistent swing output with no input. The state of oscillation in relay feedback BONL systems requires in-depth analysis and has been proposed as future work.
Motivation
Thesis Organization
Similar to the previous case of the Wiener model, the identification method also applies to unstable cases. Hanjali'c and Juri'c [39] used a frequency domain approach to identify the Wiener model where the relay is used in the feedback path instead of the conventional feed forward path.
Problem Formulation
To simplify the calculation, r is taken as 0, since the operating point of the system is considered as the starting point. The static nonlinearity is assumed to be passive and monotonic, which is estimated as a polynomial of the form.
Wiener Model of Second-Order Systems
Linear Subsystem Identification
Now equations and (2.27) will be treated separately for various kinds of roots in the denominator of the linear subsystem. Therefore, the value of T1 can be obtained by solving the non-linear equations and (2.27) considering a very high initial value of T2 compared to T1.
Nonlinearity Estimation
The unknowns required to be calculated for a Wiener model with an integrator are, the delay which can be calculated using the method proposed by Majhi [43]. The equation of v(t) for this case can be derived using the modified values of T1 and T2 from (2.28) to (2.43).
Linear Structure Identification
An integrator can be seen as a special case of the integrating SOPDT system where the time constant T1=0 in Equation (2.39). If it is a linear system, the coefficient of the first degree term i.e. the slope gives the static gain of the system. 44] derived three equations to be solved simultaneously for the two time constants and the gain.
Mitigation of Measurement Noise
However, in the proposed work, two equations are simultaneously solved to find the time constants and the gain is derived from the slope of the graph of v(t)andy(t).
Simulation Study
An unstable linearFOPDT model studied by Majhi and Atherton [50] is considered which is given as. An example simulation for repeated roots is considered for a linear system which is given as. Now, the linear subsystem also has a static gain in G(s) which is 15 =0.2 and can be identified along with the static nonlinearity.
Discussion
Online Identification and Complexity
Therefore, the complexity in identification depends on the method used to solve the nonlinear equations in Section 3. While relay feedback for Wiener system modeling seems easy, it should be noted that for identification of systems whose parameters vary continuously, it is not very efficient. Having said that, it should also be noted that in system modeling where relay itself is a controller, relay feedback identification can be more efficient.
Existence of Solution and Convergence
If the smallest peak is below the line F(a) =0, it will cross it again at some value aw, which is a unique solution of (2.41). If the smallest vertex touches the origin, then a=0 is the only solution that defines a system with a very high time constant as T1= 1a. The degree of convergence, however, will depend on the procedure used to solve the nonlinear equations to find the time constants.
A discussion on the static nonlinearity
Similarly, the existence and uniqueness of all equations to find T1 and T2 can be proved using geometric analysis. It can be seen that it is difficult to find the peak of v(t) by looking at y(t) since the nonlinearity is unknown. The monotonicity constraint can be relaxed in the case of the Wiener model if the vertex can be found by some other means.
Summary
In the Hammerstein model, however, identification is not a necessity, which will be studied in the next chapter. Wiener model was developed from the Volterra series, similarly the Hammerstein model was derived from a class of operator called the Urysohn operator. A summary of their work is presented here to describe the derivation of the Hammerstein model.
Problem Formulation
The transfer function of the linear subsystem is considered to be the same as the case of the Wiener model. The DC gain in the transfer function G(s) is assumed to be unity because its effect can be considered together with the static nonlinearity. However, the static nonlinearity must be continuous since discontinuous nonlinearities will not produce limit cycle oscillation and absence of limit cycle oscillation does not mean a discontinuity in the nonlinearity.
Hammerstein Model of Second-Order Systems
Linear Subsystem Identification
Now, equations (3.17) and will be treated separately for different types of roots in the denominator of the linear subsystem. However, the gain term in the numerator, (a2+b2) can be considered together with the nonlinearity, therefore. For the linear subsystem with critical damping, the values of the matrices in the state space equations (3.5) and (3.6) are modified as follows.
Nonlinearity Estimation
When the roots are complex conjugates, note that in the modified expression for G(s) in (3.25), (a2+b2) is multiplied, which although does not affect the equations to find and b while converting from real roots to complex conjugates. will however affect v(t). This does not affect the equations to find T1 and T2 since the denominator is unchanged, but it will affect the expressions for v1 and v2. Although this did not affect the calculation of the time constant (which is always 1 in this case), it will affect the static nonlinearity.
Linear Structure Identification
Therefore, to derive the equations of the intermediate signal v(t), the assumed linear static gainT1 is divided into Equations (3.54) and (3.55).
Hammerstein Model of nth Order Systems
Linear Subsystem Identification
The identification of the linear subsystem involves the identification of G(s), where the only two unknowns are θ and the time constant T1. It should be noted that the order of the system is assumed to be known. However, it also applies to cases where the order is unknown but very high and the estimation is performed using smaller (say first/second/third) orders.
Nonlinearity Estimation
Mitigation of Measurement Noise
Simulation Study
This example illustrates that the procedure does not require knowing in advance whether the system is stable or unstable, the equations used for parameter estimation automatically give the sign of the time constant and win accordingly. A Hammerstein model with two real and distinct roots of the linear subsystem is considered for analysis, which was previously investigated by Juric et al. The non-linearity is estimated for the interval (−2,2) with 7 Chebyshev nodes, i.e. three relay tests were performed since the known point 0 is also one of the Chebyshev nodes.
Discussion
Online Identification and Complexity
Existence of Solution and Convergence
It is evident that a=0 is a solution of (3.85), so the curve F(a) will pass through the origin. Therefore, the second derivative in equation (3.87) will be positive for all values, therefore the critical point a∗ is a global minimum. 2.5.2, there are only two possible solutions for F(a) =0: first, when the minimum peak is below F(a) =0, the curve will intercept it somewhere other than the origin which will be the solution, second , when the minimum peak touches the origin, then a=0 is the only solution that defines a system with a very high time constant.
A discussion on the static nonlinearity
However, the rate of convergence will depend on the procedure used to solve the nonlinear equations to find the time constants. A similar geometric analysis can be done to prove the existence and uniqueness of the equations to find time constants for all other cases as well.
Summary
Some important works on nonlinear modeling of the buck converter are presented by Tymerski et al. 68] is the first to use relay feedback as well as Hammerstein modeling for the identification of a buck converter. Although there are some works on the identification of buck converter using relay feedback (Stefanutti et al. [79], Zhao and Prodi [80], Ramana et al. [81] ), it is only limited to small-signal modeling and linearization.
Buck Converter Operation
68] on non-linear identification of buck converter using relay feedback, to the author's knowledge, no work has been done in this area. From equation (4.2), the output voltage is directly proportional to the control input, but this is not the case with a non-ideal buck converter. A non-ideal buck converter circuit is shown in Figure 4.3, which accounts for voltage drop across various circuit components.
Identification Setup and Model
This transfer function describes how the average output voltage changes with a change in the control input. For the detailed derivation of the above equation, Chapter 8 of the book by Erickson and Maksimovic [83] is recommended. Since the main objective of this research is to study the nonlinear static characteristic and not the transient behavior, this transfer function assumption suffices the purpose without any significant deviation from the actual system.
Proposed Identification Procedure
The unknown parameters of the Hammerstein model are the poles a±ib, the time delay θ and the coefficients of the polynomial in (4.4). Along with equation (4.8), equations are chosen based on the position of the peak of the output relative to the relay circuit. In a buck converter, the output voltage increases with the increase of the control input i.e. the duty ratiod.
Simulation Results
However, by extending the linear region of the buck converter towards the x-axis, it can be observed that it has a negative slope, which means that it is a line of the type vo=a1d−a0, which defines an affine function, whereas1anda0er positive real constants. Although any linear model would provide a reasonable approximation of the buck converter characteristics, the steady-state error will be high at lower/higher (depending on the slope) values of the duty ratio. The ramp and step response (for d =0.3) characteristics of the identified models are shown in Figure 4.9 and Figure 4.10, which show smaller steady-state errors compared to the average model.
![Figure 4.6: Simulink set-up used for buck converter identification Ramana et al. [84] is considered whose specification is given in the Table 4.1.](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10343410.0/122.892.116.738.139.662/figure-simulink-converter-identification-ramana-considered-specification-table.webp)
Summary
It can be observed that while the Hammerstein model gives the best result, both the identified models in this work give better results compared to the average model. In this chapter the contribution of the thesis is summarized and a list of probable future works is given.
Thesis Contribution
Thus, the broad contributions of this thesis addressing the limitations of literature relay feedback in identifying BONL models can be summarized as. Requires relatively less prior knowledge: given the structure of the system as Wiener or Hammerstein, identification can be performed without knowing the sign of the time constants or the nature of the static gain (i.e. linear or non-linear again). Considers relatively broad cases of linear subsystems: In most literature, Wiener and Hammerstein modeling using relay feedback only takes into account real and distinct time constants of the linear subsystem.
Scope for Future Work
Greblicki, "Nonparametric Identification of Wiener Systems by Orthogonal Series," IEEE Transactions on Automatic Control, vol. Toivonen, "Support Vector Method for Identification of Wiener Models," Journal of Process Control, vol. Gallman, "An Iterative Method for the identification of nonlinear systems using a Hammerstein model,” IEEE Transactions on Automatic Control , vol.
