# Automatic Regulation Time Series for Industry Processes

Authors: Tain-Sou Tsay
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 seriesG(jTS) is proposed for analyses and designs of industry processes. They are sampled-data feedback control systems. TS represents the sampling interval. The ratios of G((k+1)TS) to G(kTS) of the series are formulated as a function of the reference input command and the output of the plant. The value of G(kTS) is the control input of the plant at time intervals between (k-1)TS and kTS. Thus, the considered system is closed as a feedback control system by use of G(jTS). It will be seen that the output of the plant tracks the reference input command exactly after ratios G((k+1)TS)/G(kTS) of the series being converted to unities. It implies that G(kTS) 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)TS)/G(kTS). This function can be called as “Regulation Function.” It will be proven that the considered system using G(kTS) 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 ofG((k+1)TS)/G(kTS). Furthermore, an adaptive limitation for G(kTS) 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 intervalTS[1–5] can be written as in the form of(2.1)G(jTS),j=1,2,3,…,n,n+1,…,
where G(jTS) represents a constant value between time interval from (j-1)TS and jTS. For simplicity, the representation of G(jTS) 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 of(2.2)F(j)=G(j+1)G(j),j=1,2,3,…,n,n+1,….
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,(2.3)G(n+1)=[∑i=0mai(R(n)YS(n))i]G(n),whereR(n) represents the reference input command and YS(n) represents the nonzero sampled output of the plant at the sampling interval nTS. 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) becomes(2.4)G(n+1)=∑i=0maiG(n).
For steady-state condition, G(n+1) approaches to be a constant value, and it gives(2.5)∑i=0mai=1.
Rearranging (2.3) and taking the derivative of it with respect to YS(n)/R(n), one has(2.6)F(n)=∑i=0mai(YS(n)R(n))-i,(2.7)∂F(n)∂(YS(n)/R(n))=-∑i=0miai(YS(n)R(n))-1-i.
The sufficient but not necessary condition for (2.7) less than zero is ai>0 for YS(n)/R(n)≅1 and (2.6) can be rewritten as in the form of(2.8)F(n)=∑i=0mai|YS(n)R(n)|-i.ai>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, YS(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 of(2.9)G(n+1)=[β|R(n)YS(n)|+1-β]G(n),
where β satisfies constrains stated above and becomes an adjustable parameter. Thus, the ratios F(n) become(2.10)F(n)=β|R(n)YS(n)|+1-β.F(n) can be called as “Regulation Function” also. Similarly, the third-order representation of F(n) is in the form of(2.11)F(n)=α|YS(n)R(n)|-3+γ|YS(n)R(n)|-1+1-α-γ,
where 0<α and 0<γ.Taking the derivative of (2.10) with respect to YS(n)=R(n), one has(2.12)∂F(n)∂(YS(n)/R(n))=-β(YS(n)R(n))-2.
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)/YS(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.Figure1 shows that the value of F(n) is less than one for that of YS(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 YS(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 (YS(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 YS(n)/R(n) is negative. Similar to the derivation of (2.12), (2.11) gives(2.13)∂F(n)∂(YS(n)/R(n))=-{γ(YS(n)R(n))-2+3α(YS(n)R(n))-4},
where 0<α and 0<γ.The constraint of non-zeroYS(n) can be removed by R(n)/YS(n) of (2.9) replaced by (R(n)+Yo)/(YS(n)+Yo). Yo is a positive value and represents the negative maximal control swing. The modified equation of (2.9) becomes(2.14)G(n+1)={β|R(n)+YoYS(n)+Yo|+1-β}G(n).
Equation (2.14) implies ratios G(n+1)/G(n) are in the form of(2.15)F(n)=[β|R(n)+YoYS(n)+Yo|+1-β],n=1,2,3,…,j,j+1,….
Control inputs of the plant are in the form of(2.16)u(n+1)=G(n+1)-YoP(0)
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)=YS(n) is corresponding to R(n)+Yo=YS(n)+Yo. Similar to the evaluation of (2.12), the derivative of (2.15) becomes(2.17)∂F(n)∂((YS(n)+Yo)/(R(n)+Yo))=-β(YS(n)+YoR(n)+Yo)-2.
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 YS(n) and Yo (or R and Yo) is greater than zero with specified values of Yo. All positive values will give the better continuity and regulating characteristic of the time series. Naturally, absolute value of (R(n)+Yo)/(YS(n)+Yo) 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 ofYo can be selected at |R(n)| for the system is well controlled. Then (2.14) and (2.16) can be rewritten as(2.18)G(n+1)={β|R(n)+|R(n)|YS(n)+|R(n)||+1-β}G(n),(2.19)u(n+1)=G(n+1)-|R(n)|P(0),
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 YS(n)+|R(n)|=0. It is easy to replace YS(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 YS(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 Figure2 using G(n) limitations and singularity avoidance of (2.18).


### 2.3. Phase Lead Compensation

A conventional digital filterC(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 as(2.20)C(z)=Tns+1Tns/ρ+1|s=(2/TS)((z-1)/(z+1))
for ρ>1. The parameter Tn can be found by another numerical time series. It is(2.21)W(n+1)=[η(TcTcs)j+1-η]W(n),Tn=W(n+1),
where Tc is the time constant of the closed loop system and Tcs is the wanted time constant.Considering a illustrating example [6] is shown in Figure 3, in which P(s) is in the form of(2.22)P(s)=30s2+10s+30.
DC gain of P(s) is unity. The sampling period TS 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,Yo=|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 forβ=0.50 and TS=0.1 sec.Figure4 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.Figure5 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., TS=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 of(2.23)C(z)=0.15923s+10.03185s+1|s=(2/TS)((z-1)/(z+1)).
It can speed up the time responses while keeping system performance.Figure 5
Time responses of the illustrating example forβ=0.50 and sampling frequencyequating to 100, 40, 20, 10, and 5 Hz, respectively.Figure 6
Time responses of the illustrating example with/withoutC(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.


## 2.1. Automatic Regulation Time Series

A numerical series with time intervalTS[1–5] can be written as in the form of(2.1)G(jTS),j=1,2,3,…,n,n+1,…,
where G(jTS) represents a constant value between time interval from (j-1)TS and jTS. For simplicity, the representation of G(jTS) 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 of(2.2)F(j)=G(j+1)G(j),j=1,2,3,…,n,n+1,….
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,(2.3)G(n+1)=[∑i=0mai(R(n)YS(n))i]G(n),whereR(n) represents the reference input command and YS(n) represents the nonzero sampled output of the plant at the sampling interval nTS. 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) becomes(2.4)G(n+1)=∑i=0maiG(n).
For steady-state condition, G(n+1) approaches to be a constant value, and it gives(2.5)∑i=0mai=1.
Rearranging (2.3) and taking the derivative of it with respect to YS(n)/R(n), one has(2.6)F(n)=∑i=0mai(YS(n)R(n))-i,(2.7)∂F(n)∂(YS(n)/R(n))=-∑i=0miai(YS(n)R(n))-1-i.
The sufficient but not necessary condition for (2.7) less than zero is ai>0 for YS(n)/R(n)≅1 and (2.6) can be rewritten as in the form of(2.8)F(n)=∑i=0mai|YS(n)R(n)|-i.ai>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, YS(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 of(2.9)G(n+1)=[β|R(n)YS(n)|+1-β]G(n),
where β satisfies constrains stated above and becomes an adjustable parameter. Thus, the ratios F(n) become(2.10)F(n)=β|R(n)YS(n)|+1-β.F(n) can be called as “Regulation Function” also. Similarly, the third-order representation of F(n) is in the form of(2.11)F(n)=α|YS(n)R(n)|-3+γ|YS(n)R(n)|-1+1-α-γ,
where 0<α and 0<γ.Taking the derivative of (2.10) with respect to YS(n)=R(n), one has(2.12)∂F(n)∂(YS(n)/R(n))=-β(YS(n)R(n))-2.
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)/YS(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.Figure1 shows that the value of F(n) is less than one for that of YS(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 YS(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 (YS(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 YS(n)/R(n) is negative. Similar to the derivation of (2.12), (2.11) gives(2.13)∂F(n)∂(YS(n)/R(n))=-{γ(YS(n)R(n))-2+3α(YS(n)R(n))-4},
where 0<α and 0<γ.The constraint of non-zeroYS(n) can be removed by R(n)/YS(n) of (2.9) replaced by (R(n)+Yo)/(YS(n)+Yo). Yo is a positive value and represents the negative maximal control swing. The modified equation of (2.9) becomes(2.14)G(n+1)={β|R(n)+YoYS(n)+Yo|+1-β}G(n).
Equation (2.14) implies ratios G(n+1)/G(n) are in the form of(2.15)F(n)=[β|R(n)+YoYS(n)+Yo|+1-β],n=1,2,3,…,j,j+1,….
Control inputs of the plant are in the form of(2.16)u(n+1)=G(n+1)-YoP(0)
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)=YS(n) is corresponding to R(n)+Yo=YS(n)+Yo. Similar to the evaluation of (2.12), the derivative of (2.15) becomes(2.17)∂F(n)∂((YS(n)+Yo)/(R(n)+Yo))=-β(YS(n)+YoR(n)+Yo)-2.
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 YS(n) and Yo (or R and Yo) is greater than zero with specified values of Yo. All positive values will give the better continuity and regulating characteristic of the time series. Naturally, absolute value of (R(n)+Yo)/(YS(n)+Yo) 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 ofYo can be selected at |R(n)| for the system is well controlled. Then (2.14) and (2.16) can be rewritten as(2.18)G(n+1)={β|R(n)+|R(n)|YS(n)+|R(n)||+1-β}G(n),(2.19)u(n+1)=G(n+1)-|R(n)|P(0),
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 YS(n)+|R(n)|=0. It is easy to replace YS(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 YS(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 Figure2 using G(n) limitations and singularity avoidance of (2.18).


## 2.3. Phase Lead Compensation

A conventional digital filterC(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 as(2.20)C(z)=Tns+1Tns/ρ+1|s=(2/TS)((z-1)/(z+1))
for ρ>1. The parameter Tn can be found by another numerical time series. It is(2.21)W(n+1)=[η(TcTcs)j+1-η]W(n),Tn=W(n+1),
where Tc is the time constant of the closed loop system and Tcs is the wanted time constant.Considering a illustrating example [6] is shown in Figure 3, in which P(s) is in the form of(2.22)P(s)=30s2+10s+30.
DC gain of P(s) is unity. The sampling period TS 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,Yo=|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 forβ=0.50 and TS=0.1 sec.Figure4 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.Figure5 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., TS=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 of(2.23)C(z)=0.15923s+10.03185s+1|s=(2/TS)((z-1)/(z+1)).
It can speed up the time responses while keeping system performance.Figure 5
Time responses of the illustrating example forβ=0.50 and sampling frequencyequating to 100, 40, 20, 10, and 5 Hz, respectively.Figure 6
Time responses of the illustrating example with/withoutC(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](3.1)P1(s)=e-s(s+1)2.
It has pure time delay of 1 second. The specification for time constant Tcs=1.85sec is selected. Parameters of (2.18) and (2.21) are β=0.7, TS=25 ms,ρ=50, η=0.9,and j=1. Figure 7 shows online adjusting processes for finding C(z). The initial guess of Tn is equal to 1.00 and converted to 0.5195 after third adjusting processes. The found lead compensator is
(3.2)C(z)=0.5195s+10.5195s/ρ+1|s=(2/TS)((z-1)/(z+1)).
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 of(3.3)u(t)=Kpe(t)+Ki∫e(t)dt+Kdddte(t).
Parameters of four found compensators are given below:(1)
ZN(PI):Kp=1.240 andKi=0.251;(2)
ZN(PID):Kp=1.6367, Ki=0.4187, andKd=0.5972;(3)
Tan’s(PID):Kp=0.620, Ki=0.5636, andKd=0.1705;(4)
Majhi’s(PI):Kp=0.864 and Ki=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 Example3.1 using different control methods.
MethodsProposedZN(PI)ZN(PID)Tan’sMajhi’sISE1.46102.26751.76942.24712.4654IAE1.87264.01072.87573.07254.0659Figure 7
Time responses of Example3.1 for finding C(z).Figure 8
Time responses of Example3.1 using different control methods.Example 3.2.
Consider a sixth order plant [7, 8]
(3.4)P2(s)=1(s+1)6.
The specification of time constant Tcs=4.8 sec is selected. Parameters of (2.18) and (2.21) are β=0.5, TS=25ms,ρ=50,η=0.9, and j=3. The initial guess of Tn is equal to 1.0 and converted to 0.7759 after second adjusting process. The found lead compensator is
(3.5)C(z)=0.7759s+10.7759s/ρ+1|s=(2/TS)((z-1)/(z+1)).
Figure 9 shows on-line adjusting processes for finding C(z) to meet Tcs=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):Kp=1.079 and Ki=0.110;(2)
ZN(PID):Kp=1.4248, Ki=0.1838,andKD=1.360;(3)
Majhi’s(PI):Kp=0.7736, and Ki=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 Example3.2 using different control methods.
MethodsProposedZN(PI)ZN(PID)Ho’sMajhi’sISE3.38955.3354.0235.2153.746IAE4.47989.2796.4927.2195.425Figure 9
Time responses of Example3.2 for finding C(z).Figure 10
Time responses of Example3.2 using different control methods.Example 3.3.
Consider the very-high-order plant [7, 8]:(3.6)P3(s)=1(s+1)20.
The design specification for time constant Tcs=19.0sec is selected. Parameters of (2.18) and (2.21) are β=0.5, TS=25ms,ρ=50,η=0.9,and j=3. The initial guess of Tn is equal to 1.00 and converted to 0.9586 after the fourth adjusting process. The found lead compensator is
(3.7)C(z)=0.9586s+10.9586s/ρ+1|s=(2/TS)((z-1)/(z+1)).
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):Kp=0.585 andKi=0.0305;(2)
ZN(PID):Kp=0.77256,Ki=0.05088,andKd=4.9135;(3)
Majhi’s(PI):Kp=0.5097andKi=0.0443;(4)
Zhuang’s(PI):Kp=0.473 andKi=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 Example3.3 using different control methods.
MethodsProposedZN(PI)ZN(PID)Majji’sZhuang’sISE15.731321.227116.216020.190821.8142IAE18.089232.708422.970726.829532.9125Figure 11
Time responses of Example3.3 with C(z) for β=0.5; TS=25 ms.Figure 12
Time responses of Example3.3 using different control methods.Example 3.4.
Consider a gas turbine engine with plant transfer function matrix [24–26]:
(3.8)P4(s)=1Δ(s)[2533+1515.33s+14.9s21805947+1132094.7s+95150s212268.8+8642.68s+85.2s22525880+1492588s+124000s2],
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
(3.9)P4(0)=[1.00316715.22654.858931000.3485].
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
(3.10)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)),
where TS=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 Example3.4 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 regulationF(n) to get better performance.



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*Source: 710690-2012-05-20.xml*