Automatic Regulation Time Series for Industry Processes

Abstract

A nonlinear digital control scheme is proposed for analyses and designs of stable industry processes. It is derived from the converging characteristic of a specified numerical time series. The ratios of neighbourhoods of the series are formulated as a function of the output of the plant and the reference input command and will be converted to be unities after the output has tracked the reference input command. Lead compensations are also found by another numerical time series to speed up the system responses on the online adjusting manner. A servosystem, a time-delay system, a high-order system, a very-high-order system, and a 2 × 2 multivariable aircraft gas turbine engine are used to illustrate effectiveness of the proposed nonlinear digital controller. Comparisons with other conventional methods are also made.

1. Introduction

For unit feedback discrete-time control systems, the control sequences are usually functions of the difference between the sampled reference input and output of the plant [1–5]. The discrete-time control sequence can be generated by Finite Impulse Response (FIR) filter or Infinite Impulse Response (IIR) filter. The input of FIR or IIR filter is the difference between the sampled reference input and output of the plant. The output of FIR or IIR will be the input of the plant. In general, they are linear controllers.

In this literature, a nonlinear discrete-time control sequence described by periodic numerical series 𝐺(𝑗𝑇_𝑆) is proposed for analyses and designs of industry processes. They are sampled-data feedback control systems. 𝑇_𝑆 represents the sampling interval. The ratios of 𝐺((𝑘+1)𝑇_𝑆) to 𝐺(𝑘𝑇_𝑆) of the series are formulated as a function of the reference input command and the output of the plant. The value of 𝐺(𝑘𝑇_𝑆) is the control input of the plant at time intervals between (𝑘−1)𝑇_𝑆 and 𝑘𝑇_𝑆. Thus, the considered system is closed as a feedback control system by use of 𝐺(𝑗𝑇_𝑆). It will be seen that the output of the plant tracks the reference input command exactly after ratios 𝐺((𝑘+1)𝑇_𝑆)/𝐺(𝑘𝑇_𝑆) of the series being converted to unities. It implies that 𝐺(𝑘𝑇_𝑆) will be converted to a steady-state value for a constant reference input applied. The stability of the closed-loop system is guaranteed by selecting the proper function of ratios 𝐺((𝑘+1)𝑇_𝑆)/𝐺(𝑘𝑇_𝑆). This function can be called as “Regulation Function.” It will be proven that the considered system using 𝐺(𝑘𝑇_𝑆) becomes a negative feedback control system for a stable plant [4].

Note that it needs not integration to get zero tracking error, and performance of controlled systems are dependent on selected functions of 𝐺((𝑘+1)𝑇_𝑆)/𝐺(𝑘𝑇_𝑆). Furthermore, an adaptive limitation for 𝐺(𝑘𝑇_𝑆) can be applied also to minimize the control effort and get better performance. Controlled results will be compared with conventional famous PI and PID controllers [6–15]. In this work, measurement noises of plant outputs are not considered. It is worthwhile to include recent developments of fractional-order systems and controls [16, 17] in the proposed nonlinear automatic regulation time series. They have been applied to signal processing [18], Cyber-physical networking system [19, 20], PMSM position servo system [21], and optimal control [22].

In following sections, basic concepts of the proposed nonlinear discrete-time control sequence are discussed first, and then a servo system, a time-delay system, a high-order system, a very-high-order system, and a 2 × 2 multivariable aircraft gas turbine examples are used to illustrate their tracking behaviour and performance. Simulating results will show that the proposed nonlinear digital controller gives another possible way for analyses and designs of industry processes. Design results of the fourth example give the proposed method can also be applied to multivariable feedback control systems.

2. The Basic Approach

2.1. Automatic Regulation Time Series

A numerical series with time interval 𝑇_𝑆[1–5] can be written as in the form of𝐺𝑗𝑇_𝑆,𝑗=1,2,3,…,𝑛,𝑛+1,…, (2.1) where 𝐺(𝑗𝑇_𝑆) represents a constant value between time interval from (𝑗−1)𝑇_𝑆 and 𝑗𝑇_𝑆. For simplicity, the representation of 𝐺(𝑗𝑇_𝑆) will be replaced by 𝐺(𝑗) in the following evaluations. The ratios 𝐺(𝑗+1)/𝐺(𝑗) of the series are defined as in the form of𝐹𝑗=𝐺𝑗+1𝐺𝑗,𝑗=1,2,3,…,𝑛,𝑛+1,…. (2.2) Equation (2.2) gives the value of 𝐺(𝑛+1) approaches to be a constant value when the value of 𝐹(𝑛) approaches to be unity. Now, the problem for closing the considered system is to find the formula of 𝐹(𝑗) which is the function of the reference input command 𝑅 and the output of the plant 𝑌. 𝐺(𝑛+1) is used as the input of the considered system. Considering a series given below,𝐺(𝑛+1)=^𝑚_𝑖=0𝑎_𝑖𝑅(𝑛)𝑌_𝑆(𝑛)^𝑖𝐺(𝑛), (2.3)

where 𝑅(𝑛) represents the reference input command and 𝑌_𝑆(𝑛) represents the nonzero sampled output of the plant at the sampling interval 𝑛𝑇_𝑆. Note that this non-zero constraint will be removed later by level shifting. Equation (2.3) is a possible way to close the considered system as a sampled-data feedback control system. Assuming the reference input command has been tracked by applying control effort 𝐺(𝑗), (2.3) becomes𝐺(𝑛+1)=^𝑚_𝑖=0𝑎_𝑖𝐺(𝑛). (2.4) For steady-state condition, 𝐺(𝑛+1) approaches to be a constant value, and it gives^𝑚_𝑖=0𝑎_𝑖=1. (2.5) Rearranging (2.3) and taking the derivative of it with respect to 𝑌_𝑆(𝑛)/𝑅(𝑛), one has𝐹(𝑛)=^𝑚_𝑖=0𝑎_𝑖𝑌_𝑆(𝑛)𝑅(𝑛)^−𝑖,𝜕𝐹(𝑛)𝜕𝑌_𝑆(𝑛)/𝑅(𝑛)=−^𝑚_𝑖=0𝑖𝑎_𝑖𝑌_𝑆(𝑛)𝑅(𝑛)^−1−𝑖. (2.6)(2.7) The sufficient but not necessary condition for (2.7) less than zero is 𝑎_𝑖>0 for 𝑌_𝑆(𝑛)/𝑅(𝑛)≅1 and (2.6) can be rewritten as in the form of𝐹(𝑛)=^𝑚_𝑖=0𝑎_𝑖𝑌_𝑆(𝑛)𝑅(𝑛)^−𝑖. (2.8)𝑎_𝑖>0 will be used in the following evaluations. Negative value of (2.7) represents the closed-loop system using (2.3) activated as a negative feedback system around the equilibrium condition; that is, 𝑌_𝑆(𝑛)=𝑅(𝑛). This statement will be illustrated and discussed by a graph in the next paragraph. The first-order polynomial described in (2.3) can be written as in the form of𝐺()=𝛽𝑅(𝑛)𝑛+1𝑌_𝑆(𝑛)+1−𝛽𝐺(𝑛), (2.9) where 𝛽 satisfies constrains stated above and becomes an adjustable parameter. Thus, the ratios 𝐹(𝑛) become𝐹(𝑛)𝑅(𝑛)=𝛽𝑌_𝑆(𝑛)+1−𝛽. (2.10)𝐹(𝑛) can be called as “Regulation Function” also. Similarly, the third-order representation of 𝐹(𝑛) is in the form of𝐹(𝑛)𝑌=𝛼^𝑆(𝑛)𝑅(𝑛)⁻³𝑌+𝛾^𝑆(𝑛)𝑅(𝑛)⁻¹+1−𝛼−𝛾, (2.11) where 0<𝛼 and 0<𝛾.

Taking the derivative of (2.10) with respect to 𝑌_𝑆(𝑛)=𝑅(𝑛), one has𝜕𝐹(𝑛)𝜕𝑌_𝑆(𝑛)(𝑛)𝑌/𝑅=−𝛽^𝑆(𝑛)𝑅(𝑛)⁻². (2.12) For negative value of (2.12), the value of 𝛽 must be greater than zero. This implies the range of 𝛽 is 0<𝛽<1. The suitability of the proposed nonlinear adaptive digital controller is based on this negative regulation characteristic. Figure 1 shows ratios 𝐹(𝑛) versus 𝑅(𝑛)/𝑌_𝑆(𝑛) represented by (2.9) for 𝛽 = 0.9, 0.7, 0.5, 0.3 and 0.1, respectively.

Figure 1 

𝐺(𝑛+1)/𝐺(𝑛) Versus 𝑅/𝑌𝑆 for 𝛽 = 0.9, 0.7, 0.5, 0.3, and 0.1.

Figure 1 shows that the value of 𝐹(𝑛) is less than one for that of 𝑌_𝑆(𝑛) greater than that of 𝑅(𝑛), then the value of 𝐺(𝑛+1) will be decreased; the value of 𝐹(𝑛) is greater than one for that of 𝑌_𝑆(𝑛) less than that of 𝑅(𝑛), and the value of 𝐺(𝑛+1) will be increased. This implies that the controlled system connected using (2.9) will be regulated to the equilibrium point (𝑌_𝑆(𝑛)/𝑅(𝑛)=1) and gives a negative feedback control system for deviation from the equilibrium point. From Figure 1, it can be seen that one can adjust 𝛽 to get desired regulating slope; that is, regulating characteristic. Certainly, other tracking functions can be formulated and proposed also for the considered system, if its derivative with respect to 𝑌_𝑆(𝑛)/𝑅(𝑛) is negative. Similar to the derivation of (2.12), (2.11) gives𝜕𝐹(𝑛)𝜕𝑌_𝑆(𝑛)(𝑛)𝛾𝑌/𝑅=−^𝑆(𝑛)𝑅(𝑛)⁻²𝑌+3𝛼^𝑆(𝑛)𝑅(𝑛)⁻⁴, (2.13) where 0<𝛼 and 0<𝛾.

The constraint of non-zero 𝑌_𝑆(𝑛) can be removed by 𝑅(𝑛)/𝑌_𝑆(𝑛) of (2.9) replaced by (𝑅(𝑛)+𝑌_𝑜)/(𝑌_𝑆(𝑛)+𝑌_𝑜). 𝑌_𝑜 is a positive value and represents the negative maximal control swing. The modified equation of (2.9) becomes𝐺()=𝛽𝑅(𝑛)𝑛+1+𝑌_𝑜𝑌_𝑆(𝑛)+𝑌_𝑜+1−𝛽𝐺(𝑛). (2.14) Equation (2.14) implies ratios 𝐺(𝑛+1)/𝐺(𝑛) are in the form of𝐹(𝑛)=𝛽𝑅(𝑛)+𝑌_𝑜𝑌_𝑆(𝑛)+𝑌_𝑜+1−𝛽,𝑛=1,2,3,…,𝑗,𝑗+1,…. (2.15) Control inputs of the plant are in the form of𝑢()()−𝑌𝑛+1=𝐺𝑛+1^𝑜𝑃(0) (2.16) for the negative swing control using positive values of 𝛽, 𝐺(𝑛), and 𝐹(𝑛). Equation (2.14) gives negative regulation characteristics also for𝑅(𝑛)=𝑌_𝑆(𝑛) is corresponding to 𝑅(𝑛)+𝑌_𝑜=𝑌_𝑆(𝑛)+𝑌_𝑜. Similar to the evaluation of (2.12), the derivative of (2.15) becomes𝜕𝐹(𝑛)𝜕𝑌_𝑆(𝑛)+𝑌_𝑜/𝑅(𝑛)+𝑌_𝑜𝑌=−𝛽^𝑆(𝑛)+𝑌_𝑜𝑅(𝑛)+𝑌_𝑜⁻². (2.17) Figure 2 shows the connected system configuration using (2.14) and (2.16) in which 𝑈 is the sampled with hold output of the controller. The values of 𝐺(𝑛) and 𝐹(𝑛) will be all positive for the summation of 𝑌_𝑆(𝑛) and 𝑌_𝑜 (or 𝑅 and 𝑌_𝑜) is greater than zero with specified values of 𝑌_𝑜. All positive values will give the better continuity and regulating characteristic of the time series. Naturally, absolute value of (𝑅(𝑛)+𝑌_𝑜)/(𝑌_𝑆(𝑛)+𝑌_𝑜) can be used in (2.14) to guarantee positive of 𝐺(𝑛) and 𝐹(𝑛) for negative of 𝑅(𝑛).

Figure 2 

A nonlinear digital controller using automatic regulation time series.

2.2. Control Effort Limitation

An adaptive value of 𝑌_𝑜 can be selected at |𝑅(𝑛)| for the system is well controlled. Then (2.14) and (2.16) can be rewritten as𝐺()=𝛽𝑅(𝑛)+|𝑅(𝑛𝑛+1)|𝑌_𝑆(𝑛)+|𝑅(𝑛𝐺(𝑛),𝑢()()−|𝑅(𝑛)|+1−𝛽𝑛+1=𝐺𝑛+1)|𝑃(0), (2.18)(2.19) respectively. The maximal value of 𝐺(𝑛) can be limited by an adaptive constraint |𝑅(𝑛)+|𝑅(𝑛)|| to minimize the control effort. The control input 𝑈 of the plant is now described by (2.19).

Note that the singularity of (2.18) must be avoided when 𝑌_𝑆(𝑛)+|𝑅(𝑛)|=0. It is easy to replace 𝑌_𝑆(𝑛)+|𝑅(𝑛)|=0 by a small value. A small value of 𝐺(𝑛) is selected also to avoid null time series. Figure 3 shows an equivalent block diagram of Figure 2 using constraint of 𝐺(𝑛) and singularity avoidance of 𝑌_𝑆(𝑛)+|𝑅(𝑛)|=0. The constrain of 𝐺(𝑛) cannot be only for minimizing the control effort but also for improving system performance.

Figure 3 

Equivalent block diagram of Figure 2 using 𝐺(𝑛) limitations and singularity avoidance of (2.18).

2.3. Phase Lead Compensation

A conventional digital filter 𝐶(𝑧) in Figure 3 can be applied for filtering 𝐺(𝑛), if it is necessary. In general, phase lead is used for speeding up the time response. The first-order phase lead can be expressed as𝐶(𝑧)=𝑇_𝑛𝑠+1𝑇_𝑛𝑠/𝜌+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))} (2.20) for 𝜌>1. The parameter 𝑇_𝑛 can be found by another numerical time series. It is𝑊()=𝜂𝑇𝑛+1^𝑐𝑇_𝑐𝑠^𝑗𝑊(𝑛),𝑇+1−𝜂_𝑛=𝑊(𝑛+1), (2.21) where 𝑇_𝑐 is the time constant of the closed loop system and 𝑇_𝑐𝑠 is the wanted time constant.

Considering a illustrating example [6] is shown in Figure 3, in which 𝑃(𝑠) is in the form of𝑃(𝑠)=30𝑠²+10𝑠+30. (2.22) DC gain of 𝑃(𝑠) is unity. The sampling period 𝑇_𝑆 is selected to be equal to 0.1 second for illustrating variations of 𝐺(𝑛) and 𝐹(𝑛). Time responses of the overall system using the nonlinear digital controller for 𝛽=0.5,𝑌_𝑜=|𝑅(𝑛)| and 𝐶(𝑧)=1 are shown in Figure 4. Magnitudes of reference inputs between 0 and 5 seconds are equal to 1, between 5 and 10 seconds are equal to −0.7, between 10 and 14 seconds are equal to 0.5, and between 14 and 17 seconds are equal to −0.3, in which gives reference input 𝑅(𝑛) (dash line), output 𝑌 (solid line), time series 𝐺(𝑛) (dotted line), and ratios 𝐹(𝑛) (dash-dotted line) of 𝐺(𝑛). Figure 4 shows that all values of 𝐺(𝑛) and 𝐹(𝑛) are positive while the value of output 𝑌 is tracking the negative value of the reference input 𝑅(𝑛). The value of 𝑅(𝑛) can be positive or negative.

Figure 4 

Time responses of the illustrating example for 𝛽=0.50 and 𝑇𝑆=0.1 sec.

Figure 4 shows also that ratios 𝐹(𝑛) are converted to be unities quickly; that is, the controlled output tracks the reference input quickly. The proposed method gives a good performance and zero steady-state error without integration. Note that maximal values of 𝐺(𝑛) are set to be |𝑅(𝑛)+|𝑅(𝑛)|| for better performance and minimal the control effort. Equation (2.14) gives that𝐹(𝑛) will be converted to 0.5 for zero input (𝑅(𝑛)=0) and 𝛽=0.5. Equation (2.14) and Figure 1 give that the less the value of 𝛽 is, the larger the regulation slope will be. 𝛽=0.5 is the optimal value for the considered system.

Figure 5 shows time responses for 𝛽=0.50 and sampling frequency equal to 100, 40, 20, 10, and 5 Hz, respectively. It shows that 40 Hz (i.e., 𝑇_𝑆=25 ms) is fast enough for the considered system. Figure 6 shows comparisons with a phase-lead compensator 𝐶(𝑧) which is included in the control loop. The phase-lead compensator 𝐶(𝑧) is in the form of𝐶(𝑧)=0.15923𝑠+10.03185𝑠+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))}. (2.23) It can speed up the time responses while keeping system performance.

Figure 5 

Time responses of the illustrating example for 𝛽=0.50 and sampling frequency equating to 100, 40, 20, 10, and 5 Hz, respectively.

Figure 6 

Time responses of the illustrating example with/without 𝐶(𝑧) for 𝛽=0.5, 𝑇𝑆=25 ms.

The proposed control scheme using numerical time series will be applied to three numerical SISO (single-input single-output) examples in next section on online adjusting manner. Equation (2.21) will be used for finding phase-lead compensators 𝐶(𝑧) to meet design specifications.

3. Numerical Examples

Example 3.1. Consider a stable plant that has the transfer function [7, 8]𝑃₁(𝑠)=𝑒^−𝑠(𝑠+1)². (3.1) It has pure time delay of 1 second. The specification for time constant 𝑇_𝑐𝑠=1.85sec is selected. Parameters of (2.18) and (2.21) are 𝛽=0.7, 𝑇_𝑆=25 ms,𝜌=50, 𝜂=0.9,and 𝑗=1. Figure 7 shows online adjusting processes for finding 𝐶(𝑧). The initial guess of 𝑇_𝑛 is equal to 1.00 and converted to 0.5195 after third adjusting processes. The found lead compensator is 𝐶(𝑧)=0.5195𝑠+10.5195𝑠/𝜌+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))}. (3.2) Time constants of each step are 1.4107, 1.8488, 1.8498, and 1.8500. Figure 7 shows the proposed method provides an automatic regulation procedure to get wanted design specifications. It gives good performance and zero steady-state error.
Simulation results of the proposed method and four other methods are presented for comparisons. They are Ziegler-Nichols method [9–12] for finding PI and PID compensators, Tan et al. [13, 14] for finding PID compensator, and Majhi [7, 8] for finding PI compensator. The controller is in the form of𝑢(𝑡)=𝐾_𝑝𝑒(𝑡)+𝐾_𝑖𝑒(𝑡)𝑑𝑡+𝐾_𝑑𝑑𝑒(𝑡)𝑑𝑡. (3.3) Parameters of four found compensators are given below:(1)ZN(PI): 𝐾_𝑝=1.240 and𝐾_𝑖=0.251;(2) ZN(PID): 𝐾_𝑝=1.6367, 𝐾_𝑖=0.4187, and𝐾_𝑑=0.5972;(3) Tan’s(PID): 𝐾_𝑝=0.620, 𝐾_𝑖=0.5636, and𝐾_𝑑=0.1705;(4) Majhi’s(PI): 𝐾_𝑝=0.864 and 𝐾_𝑖=0.3653.Integral of the Square Error (ISE) and Integral of the Absolute Error (IAE) are given in Table 1. Time responses are shown in Figure 8. From Table 1 and Figure 8, one can see that the proposed method gives faster and better performance than those of other methods presented.

Table 1 

IAE and ISE errors of Example 3.1 using different control methods.

Figure 7 

Time responses of Example 3.1 for finding 𝐶(𝑧).

Figure 8 

Time responses of Example 3.1 using different control methods.

Example 3.2. Consider a sixth order plant [7, 8] 𝑃₂(𝑠)=1(𝑠+1)⁶. (3.4) The specification of time constant 𝑇_𝑐𝑠=4.8 sec is selected. Parameters of (2.18) and (2.21) are 𝛽=0.5, 𝑇_𝑆=25ms,𝜌=50,𝜂=0.9, and 𝑗=3. The initial guess of 𝑇_𝑛 is equal to 1.0 and converted to 0.7759 after second adjusting process. The found lead compensator is 𝐶(𝑧)=0.7759𝑠+10.7759𝑠/𝜌+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))}. (3.5) Figure 9 shows on-line adjusting processes for finding 𝐶(𝑧) to meet 𝑇_𝑐𝑠=4.8sec. Simulation results of the proposed and four other methods are presented for comparisons. They are Ziegler-Nichols rule [9–12] for finding PI and PID compensators, Ho et al. [15] for finding PID compensator, and Majhi [7, 8] for finding PI compensator. Parameters of five found compensators are given below:(1)ZN(PI): 𝐾_𝑝=1.079 and 𝐾_𝑖=0.110;(2) ZN(PID): 𝐾_𝑝=1.4248, 𝐾_𝑖=0.1838,and𝐾_𝐷=1.360;(3) Majhi’s(PI): 𝐾_𝑝=0.7736, and 𝐾_𝑖=0.1547;(4) Ho’s(PID): 𝐾(𝑠)=1.3(1+0.189/𝑠+1.3𝑠/(0.13𝑠+1)).Integral of the Square Error (ISE) and Integral of the Absolute Error (IAE) are given in Table 2. Time responses are shown in Figure 10. From Table 2 and Figure 10, one can see that the proposed method gives faster and better performance than those of other methods.

Table 2 

IAE and ISE errors of Example 3.2 using different control methods.

Figure 9 

Time responses of Example 3.2 for finding 𝐶(𝑧).

Figure 10 

Time responses of Example 3.2 using different control methods.

Example 3.3. Consider the very-high-order plant [7, 8]:𝑃₃(𝑠)=1(𝑠+1)²⁰. (3.6) The design specification for time constant 𝑇_𝑐𝑠=19.0sec is selected. Parameters of (2.18) and (2.21) are 𝛽=0.5, 𝑇_𝑆=25ms,𝜌=50,𝜂=0.9,and 𝑗=3. The initial guess of 𝑇_𝑛 is equal to 1.00 and converted to 0.9586 after the fourth adjusting process. The found lead compensator is 𝐶(𝑧)=0.9586𝑠+10.9586𝑠/𝜌+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))}. (3.7) Figure 11 shows time response of the controlled system, which gives reference input 𝑅(𝑛) (dash line), output 𝑌 (solid line), time series 𝐺(𝑛)(dotted line), and ratios 𝐹(𝑛)(dash-dotted line) of 𝐺(𝑛). It gives good performance and zero steady-state errors. Figure 11 shows the considered plant is a large time-lag system. The high order system model is usually used to describe the industry process for replacing pure time delay (e.g., 𝑒^{−𝑇𝑑𝑠}) such that conventional analysis and design techniques can be applied [7, 8]. Figure 11 shows the proposed method can be applied to a large time-delayed system.
Final results and four other methods are presented for comparison and show the merit of the proposed method. They are Ziegler-Nichols method [9–12] for finding PI and PID compensators, Zhuang and Atherton [23] for finding PI compensator, and Majhi [7, 8] for finding PI compensator. Parameters of four found compensators are given below:
(1) ZN(PI): 𝐾_𝑝=0.585 and𝐾_𝑖=0.0305;(2) ZN(PID): 𝐾_𝑝=0.77256,𝐾_𝑖=0.05088,𝑎nd𝐾_𝑑=4.9135;(3) Majhi’s(PI): 𝐾_𝑝=0.5097and𝐾_𝑖=0.0443;(4) Zhuang’s(PI): 𝐾_𝑝=0.473 and𝐾_𝑖=0.058.Time responses are shown in Figure 12. Table 3 gives integration of absolute error (IAE) and integration of square error (ISE) of them. From Table 3 and Figure 12, one can see that the proposed method gives better performance than those of other methods.

Table 3 

IAE and ISE errors of Example 3.3 using different control methods.

Figure 11 

Time responses of Example 3.3 with 𝐶(𝑧) for 𝛽=0.5; 𝑇𝑆=25 ms.

Figure 12 

Time responses of Example 3.3 using different control methods.

Example 3.4. Consider a gas turbine engine with plant transfer function matrix [24–26]: 𝑃₄(𝑠)=1Δ(𝑠)2533+1515.33𝑠+14.9𝑠²1805947+1132094.7𝑠+95150𝑠²12268.8+8642.68𝑠+85.2𝑠²2525880+1492588𝑠+124000𝑠², (3.8) where Δ(𝑠)=2525+3502.7𝑠+1357.3𝑠²+113.22𝑠³+𝑠⁴. It is a 2 × 2 multivariable plant. The steady-state gain of open loop 𝑃₄(𝑠) is in the form of 𝑃₄(0)=1.00316715.22654.858931000.3485. (3.9) A pre-compensating matrix 𝑃₄⁻¹(0) is first applied to decouple the plant in low-frequency band. Then, two digital filters are used in the diagonal to filter outputs of two time series for speeding up transient responses. They are in the form of 𝐶₁(𝑧)=0.75𝑠+10.15𝑠+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))},𝐶₂(𝑧)=0.60𝑠+10.25𝑠+1_{𝑠=(2/𝑇𝑆)((𝑧−1)/(𝑧+1))}, (3.10) where 𝑇_𝑆=25ms is the sampling period. Figure 13 shows time responses of this controlled system for 𝛽=0.5. It shows that the proposed control scheme can be applied to the multivariable feedback control system also.

Figure 13 

Time responses of Example 3.4 for 𝛽=0.5 and 𝑇𝑆=25ms.

4. Conclusions

In this literature, a new nonlinear digital controller has been proposed for analyses and designs of industry processes. They are sampled-data feedback control systems. It was applied to four simple and complicated numerical examples to get good performance and zero steady-state errors. No integrations of tracking errors are needed to get zero steady-state errors. Lead compensations are also found by another numerical time series to speed up the system responses on the on-line adjusting manner. From simulation and comparison results with other famous control methods, it can be seen that the proposed method provides another possible control scheme for sampled-data feedback control systems, and it is worthwhile to find other regulation 𝐹(𝑛) to get better performance.