An Effective Approach for Reduction of Discrete Time System using Modified Cauer Form

This paper presents an effective approach for lower-order (LO) modeling of an electric vehicle–integrated off-grid microgrid (OMG) system. The seventh-order system (SOS) of the OMG is reduced to a second-order model (SOM) while preserving the original system’s dynamic characteristics and ensuring computational efficiency. The Taylor series (TS) and Laurent series (LS) expansions are employed to simplify the complex system that plays a significant role in the reduction process. The expansion parameters of the higher-order system (HOS) of OMG and its lower-order model (LOM) are exploited to construct the fitness function. The proposed approach constructs three sub-objective functions based on TS and LS. These sub-objective functions are then combined into a single fitness function to obtain an improved LOM by enhancing the transient and steady-state responses with respect to the HOS of OMG. To minimize the error, the resultant fitness function is optimized using the brown bear optimization (BBO) algorithm. The optimization is performed under two key constraints: (i) ensuring zero steady-state error, and (ii) satisfying the Hurwitz stability criterion. To demonstrate the efficacy of the proposed LOM, it is compared with other LOMs obtained from different approximation techniques. The proposed LOM and other LOMs are graphically validated through step, impulse, Bode, Nichols, and Nyquist response comparisons with the HOS. Additionally, the performance error criteria (PEC), time-domain specifications (TDSs) and frequency domain specifications (FDSs) of the proposed LOM are compared with other LOMs using the HOS to establish the validation and applicability of the proposed method.

Interpolation and extrapolation with the CALPHAD method

Problem statement: In this study, a Model Order Reduction (MOR) method is proposed for reducing higher order system into lower order system. Speed controller design is carried out to the lower order system by Genetic Algorithm (GA) approach and this controller is used to higher order system. Approach: This study is used to find a solution to a given objective function employing different procedures and computational techniques. The problem area chosen is that of lower order system modelling used in design of speed controller for Permananent Magnet Synchronous Motor (PMSM) drive. Results: Genetic Algorithm obtains a better controller values that reflects the characteristics of the original higher order system and the performance evaluated using this method are compared with the existing approximation method. Conclusion: Performance of this Speed controller has been verified through Simulation using MATLAB package.

The authors deliver the comparative study between the directly reduced systems and step wise reduced systems. In this paper reduced order model corresponding to a higher order model, say T, in two steps using HSIA's method has been found. This method is very simple and gave stable reduced order system corresponding to the stable original system. This method is used twice to find the final reduced order system. First an intermediate lower order system, is found and then the final lower order system, say system T1, is obtained in a two step manner. Another reduced order model is obtained directly without any intermediate steps from higher order system to desired reduced order, say T2. A comparison is presented between the responses of reduced models T1, T2 and the response of the original system, T. The comparison of the responses is illustrated with an example which vindicates that the response corresponding to, T1 is more closer to the response of the original system, T than that corresponding to T2.

This article is concerned with the problem of the Hankel-norm model reduction for stochastic discrete-time nonlinear systems in interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy framework. The IT2 T-S fuzzy model is an efficient model for describing uncertain nonlinear systems, and the model reduction is to simplify the high-order complex systems by reducing the order of the original system. The aim of this article is to reduce the order of the original stochastic discrete-time IT2 fuzzy system into lower order system without ignoring the influence of IT2 membership functions. First, the Hankel-norm performance of the stochastic discrete-time IT2 fuzzy model is analyzed. Then, based on the projection theorem and cone complementary linearization approach, a convex Hankel-norm-based model reduction approach subject to conditions in the form of linear matrix inequalities (LMIs) is obtained. A membership-functions-dependent (MFD) technique is applied to capture the information of IT2 membership functions and further reduce the conservativeness. A numerical example is presented to illustrate the effectiveness of the proposed results.

Switch-off/Switch-on Algorithms for Robust Adaptive Control

Despite promising results, Curvature wavefront sensing is usually not considered as an option for future AO systems such as AO systems for Extremely Large Telescopes (ELTs) or high order systems for the current generation of 8 to 10m telescopes. CWFS is generally thought to be useful only for low order systems, both for technical reasons (detector and DM technology) and fundamental reasons (noise propagation for high order curvature systems). I show in this paper that these worries are unjustified, and that, thanks to newly developed techniques and algorithms, CWFS is in fact much superior to more traditional Shack-Hartman wavefront sensing: (1) CWFS can be made extremely efficient, even for a high order system, thanks to a new "multi-stroke" curvature wavefront sensing mode (2) CWFS-based systems can efficiently utilize both piezo-stack type deformable mirrors and square pixel detector array, and there is therefore no reason to think that technological considerations limit CWFS-based systems to low-order correction (3) non-linear Fourier-based CWFS control algorithms can dramatically increase the performance of existing and future CWFS-based systems.

A physical system is of higher-order and it is hectic for researchers to understand these systems in higher mathematical form. So, there is a requirement for systematic conversion of higher-order into a lower order. The lower order approximately gives the same result as that of the higher-order by preserving the important properties of higher-order. But the lower order retains some approximation error. Objective: The objective is to optimise the reduced-order by minimizing the integral square error between the higherorder system (HOS) and the lower-order system (LOS). Methodology: For the optimization process the novel harris hawk hunting behaviour is optimized. It is applied to find the unknown numerator by applying the novel algorithm. The denominator parameter is obtained by the Routh Hurwitz Array technique. Finding: The proposed technique is applied on a linear time-invariant single input single output system of higher-order which is randomly selected from the literature. To justify the proposed technique, the result obtained is compared with the result available in the literature. The comparison is based on the step response characteristics of the diminished order with original and result accessed from literature. The response indices such as integral square, integral absolute, integral time absolute errors are also compared. The error gets minimized and results improved as associated with the result presented in the literature. Keywords: Harris Hawk Optimization; Routh Array Technique; Integral square error; step response characteristics; reduced order

The Model Order Reduction has indeed gained significant attention due to the advancements in automation and control system engineering. The need of more accurate reduced-order models arises for achieving effective control of the process or the plant. The objective of MOR is to reduce the complexity of higher order model into lower order model while preserving the system's critical characteristics. The two commonly used methods i.e. Factor Division and Dominant Pole Retention are employed for reducing higher-order models. Dominant pole retention selects the dominant poles of the higher-order complex system and transfers them to the desired lower-order system to obtain the characteristic equation of the desired system. Unknown parameters of numerator are computed using factor division algorithm. The effectiveness of the proposed method is verified by comparing it with other recently published works through examples. Further, PID (Proportional-Integral-Derivative) controller design is a well-established technique for controlling dynamic systems. Once a reduced-order model is obtained, a PID controller is designed based on the model's characteristics. Particle Swarm Optimization (PSO) approach has been used to find the unknown controller parameters, resulting in an optimized controller that can provide superior performance to the system. The time domain characteristics and the response of the original and the reduced systems have been examined to test the efficacy of the method proposed in this article. It was observed that the use of PID controller improves overall time-domain performance of the system with negligible steady-state error.

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

A two-time-scale system can be reduced to a lower order system, slow mode subsystem, by the technique of singular perturbation, in which the quasi- steady state approximation and the L-K transformation are used. However, as pointed out in [4], the quasi-steady state approximation is good only for the sufficiently singularly perturbed systems and the L-K transformation always loses some steady state component of the full order system. A new technique, modified L-K transformation, is proposed here to transform A system into weak and strong subsystems representing different modes of the full order system respectively. The singular value decomposition is employed in minimizing the weakness indices. The reduced system is obtained by keeping the strong subsystem. By using the formulas of constructing L and K matrices [4], this technique is very easy to be applied.

This research article suggests a novel procedure for representing the higher order system with its reduced order models of the desired order. The modelling of the systems by reducing the order of the system has been the requirement of the researchers for diminishing the computational effort and the execution time in critical missions of defence and space technologies. Many of the available methods in international literature suffer from the limitation of retention of the stability of the higher order system in its inheritance models. Few other methods have the disadvantage of framing the interval tables of full order and even the significant methods need the formation of two different tables at length for deriving the numerator and denominator polynomials of the models. Most of the physical applications of the modelling techniques require the matching of time and frequency responses of both the higher order and lower order systems, often fail to attain, but the suggested method guarantees the matching of the responses in addition to retaining the performance indicators such as Time-moments and Markov parameters. The impact of the suggested methodology is highlighted by retaining the Integral Square Error coefficient of the higher order system in its models, unlike other prominent methods available as the interval coefficients of the model become the subset of those of the higher order systems.

High-order systems are commonly encountered in production processes. Due to the uncertainty and load disturbances, the performance of the systems which are designed with reduced approximated model are often not satisfactory. In this paper, a reduced-order model plus dead time is used to approximate high-order systems and to compensate the inertial lag. Based on the Popov hyperstability theorem, an adaptive mechanism is designed to minimize the broad sense predictive output error. Simulation results show that the adaptive reduced-order model has good ability to auto-approximate high-order systems and to compensate the inertial lag. Based on the adaptive dead-time compensation, an H/sub /spl infin// robust controller is designed for high-order systems. Control simulation results prove that the H/sub /spl infin// robust control system, which is described by the adaptive low-order model with dead time, has good robust stability and dynamical performance.

Part I: Synthesis of Immittances with Two Poles and Two Zeros. Cauer's continued-fraction technique is generalized for use in realizing RLC network immittances. Using this method, element values are determined by simple processes of forward and reverse division. Immittances F(p) with two poles and zeros may be grouped in three classes according to the frequency j?m at which Re F(j?) is a minimum. For Class I, wm= 0; for Class II, ?m = ?; and for Class III, ?m is finite. Class I and II F(p) can be realized immediately by making a continued-fraction expansion. If both F(p) and 1/F(p) are Class III, to obtain a realization without unity-coupled coils the function must be split into two terms which may then be expanded in continued fractions. Simple formulas are presented which enable one to easily classify F(p) and determine an appropriate realization method. Part II: A Constant-Resistance Ladder for Transfer Function Synthesis. The physical factors that determine the poles and zeros of a transfer function are examined. By use of physical insight, a design procedure for a constant-resistance ladder network is arrived at. This ladder network is found to have the same realm of application as the conventional, RLC, constant-resistance bridged-T. However, the ladder has the advantages of: fewer elements, more flexibility, and requiring less flat loss if the bridged-T requires flat loss. Complicated transfer functions may be realized in a chain of ladder sections having arm immittances with only two poles and two zeros.

Singular perturbation techniques for order reduction are based on decomposing the high-order system into two parts. Approximation of the high-order system dynamics by the low-order model is mainly dependent on this partition. This paper introduces a measure for the degree of dependence of the state variables on the high-order dominant eigenvalues. By means of this measure it is not only possible to find an appropriate decomposition yielding a good low-order model, but also to get an indication of the order of the reduced model. Two illustrative examples demonstrate the main results.