System Approximation via Restructured Hankel Matrix

Abstract

This paper presents a modified minimal realization technique to reduce single input single output (SISO) systems from higher-order SISO systems. The reduction process is based on restructuring the Hankel matrix, which consists of two major elements, i.e., Time Moments and Markov parameters. The system transformation is executed to reduce the order of the system by maintaining the desired system properties. The modified Hankel Matrix is proposed to obtain an expected reduce order model, i.e., kth order reduced model by selecting $\left[ {k \times k} \right]$ order square matrix and using Silverman’s algorithm. This paper presents a simple solution of model order reduction with the advantages of minimizing the steady-state error, fast convergence of steady-state behavior, better approximation in terms of rise time, peak time, and settling time at higher frequencies. Three different cases have been taken from the literature to validate the proposed technique with the comparisons of performance in terms of a quality check through performance indices and response matching between original and reduced-order models.

Appendices

Appendix 1

Numerical Case 1: Let the state space representation of a 4th order system taken from Mittal [10].

$$ \dot{X} = \left[ {\begin{array}{*{20}c} { - 10} & { - 4.375} & { - 3.125} & { - 1.5} \\ 8 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right]X + \left[ {\begin{array}{*{20}c} 2 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right]U $$

(36)

$$ Y = \left[ {\begin{array}{*{20}c} {0.5} & {0.4375} & {0.75} & {0.75} \\ \end{array} } \right]X $$

(37)

From the Eq. (5), the time moments and Markov parameters are obtained as:

$$ \begin{aligned} & T_{1} = - 1, \, T_{2} = 1.0833, \, T_{3} = - 1.0903, \, T_{4} = 1.066, \, T_{5} = - 1.0417, \, T_{6} = 1.0240 \, \ldots \\ & M_{1} = 1, \, M_{2} = - 3, \, M_{3} = 19, \, M_{4} = - 119, \, M_{5} = 571 \ldots \\ \end{aligned} $$

To obtain 2nd order reduced-order model, modified Hankel matrix $A_{44}$ and modified next Hankel matrix ${\mathop{A}\limits^{ \otimes }}_{44}$ are obtained from (8) and (9) as:

$$ H_{44}^{{\prime }} = \left[ {\begin{array}{*{20}c} 1 & { - 1} & {1.0833} & { - 1.0903} \\ { - 1} & 1 & { - 1.0903} & {1.066} \\ {1.0833} & { - 1.0903} & {1.066} & { - 1.0417} \\ { - 1.0903} & {1.066} & { - 1.0417} & {1.0240} \\ \end{array} } \right] $$

(38)

$$ {\mathop{H}\limits^{ \otimes }}_{44}^{{\prime }} = \left[ {\begin{array}{*{20}c} { - 1} & {1.0833} & { - 1.0903} & {1.066} \\ {1.0833} & 1 & {1.066} & { - 1.0417} \\ { - 1.0903} & {1.066} & { - 1.0417} & {1.0240} \\ {1.066} & { - 1.0417} & {1.0240} & {1.0131} \\ \end{array} } \right] $$

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where The matrix $H_{22}^{{\prime }}$ and ${\mathop{H}\limits^{ \otimes }}_{44}^{{\prime }}$ are obtained from $H_{44}^{{\prime }}$ and ${\mathop{H}\limits^{ \otimes }}_{44}^{{\prime }}$, respectively, and $H_{12}^{{\prime }}$ is obtained from first row of $H_{44}^{{\prime }}$ by selecting only two elements given as:

$$ H_{22}^{{\prime }} = \left[ {\begin{array}{*{20}c} {1.0000} & { - 1.0000} \\ { - 1.0000} & {1.0000} \\ \end{array} } \right] $$

(40)

$$ {\mathop{H}\limits^{ \otimes }}_{22}^{\prime } = \left[ {\begin{array}{*{20}c} { - 1.0000} & {1.0833} \\ {1.0833} & {1.0000} \\ \end{array} } \right] $$

(41)

$$ H_{12}^{{\prime }} = \left[ {\begin{array}{*{20}c} {1.0000} & { - 1.0000} \\ \end{array} } \right] $$

(42)

From (10), the triple matrices of the 2nd order reduced system are obtained as

$$ A_{2} = H^{\prime}_{22} /{\mathop{H}\limits^{ \otimes}}{^{\prime}_{22}} = \left[ {\begin{array}{*{20}c} { - 0.9585} & {0.0383} \\ {0.9585} & { - 0.0383} \\ \end{array} } \right] $$

(43)

$$ B_{2} = (H^{\prime}_{12} )^{T} = \left[ {\begin{array}{*{20}c} {1.0000} \\ { - 1.0000} \\ \end{array} } \right] $$

(44)

$$ C_{2} = (H_{12}^{{\prime }} )*({\mathop{H}\limits^{ \otimes }}_{22}^{\prime } ) = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right] $$

(45)

$$ {\text{And}}\quad D_{4} = 0 $$

(46)

Thus, the 2nd order system is obtained as:

$$ \dot{X}_{2} = \left[ {\begin{array}{*{20}c} { - 0.9585} & {0.0383} \\ {0.9585} & { - 0.0383} \\ \end{array} } \right]X_{2} + \left[ {\begin{array}{*{20}c} {1.0000} \\ { - 1.0000} \\ \end{array} } \right] \, U $$

(47)

$$ Y_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right]X_{2} $$

(48)

Appendix 2

The parameters used in thermal, hydro, and gas systems demonstrated in Fig. 3 are given as follows:

2.1 Thermal Power Plant

$\Delta X_{g11}$ = Governor output of thermal power plant in pu MW.

$\Delta X_{R11}$ = Reservoir output power of thermal power plant in pu MW.

$\Delta P_{g11}$ = Turbine output power of thermal power plant in pu MW.

$T_{g} = 0.08$ Sec = Generator time constant.

$T_{r}$ = Reheat time constant = 10 s.

$K_{r}$ = Coefficient of reheat steam turbine = 0.3

$T_{t}$ = Turbine time constant = 0.3 s.

$R_{1}$ = Speed governor regulation = 2.4 Hz/pu MW.

2.2 Hydro Power Plant

$\Delta X_{g12}$ = Governor output of hydro power plant in pu MW.

$\Delta X_{R12}$ = Reservoir output power of hydro power plant in pu MW.

$\Delta P_{g12}$ = Turbine output power hydro power plant in pu MW.

$T_{H}$ = Main servo time constant = 0.2 s.

$T_{R}$ = Speed governor rest time = 0.5 s.

$T_{w}$ = Water time constant = 1.0 s.

$R_{2}$ = Speed governor regulation = 2.4 Hz/pu MW.

2.3 Gas Power Plant

$\Delta X_{g13}$ = Valve position of gas power plant in pu MW.

$\Delta X_{R13}$ = Gas governor output power of gas power plant in pu MW.

$\Delta P_{g13}$ = Fuel system output power of gas power plant in pu MW.

$c$ = Valve position constant = 1.

$b$ = Valve position constant = 0.05 s.

$X$ = Speed governor lead time constant = 0.6 s.

$Y$ = Speed governor lag time constant = 1.0 s.

$T_{F}$ = Fuel time constant = 0.23 s.

$T_{CR}$ = Combustion reaction time delay = 0.01 s.

$T_{CD}$ = Compressor discharge volume time constant = 0.2 s.

$R_{3}$ = Speed governor regulation = 2.4 Hz/pu MW.

2.4 Power system

$K_{p}$ = Power system gain = 120 Hz/pu MW.

$T_{p}$ = Power system time constant = 20 s.