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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 710690, 16 pages
Automatic Regulation Time Series for Industry Processes
Department of Aeronautical Engineering, National Formosa University, Huwei, Yunlin 63208, Taiwan
Received 28 January 2012; Accepted 20 March 2012
Academic Editor: Ming Li
Copyright © 2012 Tain-Sou Tsay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 is proposed for analyses and designs of industry processes. They are sampled-data feedback control systems. represents the sampling interval. The ratios of to of the series are formulated as a function of the reference input command and the output of the plant. The value of is the control input of the plant at time intervals between and . Thus, the considered system is closed as a feedback control system by use of . It will be seen that the output of the plant tracks the reference input command exactly after ratios of the series being converted to unities. It implies that 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 . This function can be called as “Regulation Function.” It will be proven that the considered system using becomes a negative feedback control system for a stable plant .
Note that it needs not integration to get zero tracking error, and performance of controlled systems are dependent on selected functions of . Furthermore, an adaptive limitation for 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 , Cyber-physical networking system [19, 20], PMSM position servo system , and optimal control .
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 [1–5] can be written as in the form of where represents a constant value between time interval from and . For simplicity, the representation of will be replaced by in the following evaluations. The ratios of the series are defined as in the form of Equation (2.2) gives the value of approaches to be a constant value when the value of approaches to be unity. Now, the problem for closing the considered system is to find the formula of which is the function of the reference input command and the output of the plant . is used as the input of the considered system. Considering a series given below,
where represents the reference input command and represents the nonzero sampled output of the plant at the sampling interval . 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 , (2.3) becomes For steady-state condition, approaches to be a constant value, and it gives Rearranging (2.3) and taking the derivative of it with respect to , one has The sufficient but not necessary condition for (2.7) less than zero is for and (2.6) can be rewritten as in the form of 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, . 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 where satisfies constrains stated above and becomes an adjustable parameter. Thus, the ratios become can be called as “Regulation Function” also. Similarly, the third-order representation of is in the form of where and .
Taking the derivative of (2.10) with respect to , one has For negative value of (2.12), the value of must be greater than zero. This implies the range of is . The suitability of the proposed nonlinear adaptive digital controller is based on this negative regulation characteristic. Figure 1 shows ratios versus represented by (2.9) for = 0.9, 0.7, 0.5, 0.3 and 0.1, respectively.
Figure 1 shows that the value of is less than one for that of greater than that of , then the value of will be decreased; the value of is greater than one for that of less than that of , and the value of will be increased. This implies that the controlled system connected using (2.9) will be regulated to the equilibrium point () and gives a negative feedback control system for deviation from the equilibrium point. From Figure 1, it can be seen that one can adjust 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 is negative. Similar to the derivation of (2.12), (2.11) gives where and .
The constraint of non-zero can be removed by of (2.9) replaced by . is a positive value and represents the negative maximal control swing. The modified equation of (2.9) becomes Equation (2.14) implies ratios are in the form of Control inputs of the plant are in the form of for the negative swing control using positive values of , , and . Equation (2.14) gives negative regulation characteristics also for is corresponding to . Similar to the evaluation of (2.12), the derivative of (2.15) becomes Figure 2 shows the connected system configuration using (2.14) and (2.16) in which is the sampled with hold output of the controller. The values of and will be all positive for the summation of and (or and ) is greater than zero with specified values of . All positive values will give the better continuity and regulating characteristic of the time series. Naturally, absolute value of can be used in (2.14) to guarantee positive of and for negative of .
2.2. Control Effort Limitation
An adaptive value of can be selected at for the system is well controlled. Then (2.14) and (2.16) can be rewritten as respectively. The maximal value of can be limited by an adaptive constraint to minimize the control effort. The control input of the plant is now described by (2.19).
Note that the singularity of (2.18) must be avoided when . It is easy to replace by a small value. A small value of is selected also to avoid null time series. Figure 3 shows an equivalent block diagram of Figure 2 using constraint of and singularity avoidance of . The constrain of cannot be only for minimizing the control effort but also for improving system performance.
2.3. Phase Lead Compensation
A conventional digital filter in Figure 3 can be applied for filtering , 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 for . The parameter can be found by another numerical time series. It is where is the time constant of the closed loop system and is the wanted time constant.
Considering a illustrating example  is shown in Figure 3, in which is in the form of DC gain of is unity. The sampling period is selected to be equal to 0.1 second for illustrating variations of and . Time responses of the overall system using the nonlinear digital controller for , and 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 (dash line), output (solid line), time series (dotted line), and ratios (dash-dotted line) of . Figure 4 shows that all values of and are positive while the value of output is tracking the negative value of the reference input . The value of can be positive or negative.
Figure 4 shows also that ratios 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 are set to be for better performance and minimal the control effort. Equation (2.14) gives that will be converted to 0.5 for zero input () and . Equation (2.14) and Figure 1 give that the less the value of is, the larger the regulation slope will be. is the optimal value for the considered system.
Figure 5 shows time responses for and sampling frequency equal to 100, 40, 20, 10, and 5 Hz, respectively. It shows that 40 Hz (i.e., ms) is fast enough for the considered system. Figure 6 shows comparisons with a phase-lead compensator which is included in the control loop. The phase-lead compensator is in the form of It can speed up the time responses while keeping system performance.
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 to meet design specifications.
3. Numerical Examples
Example 3.1. Consider a stable plant that has the transfer function [7, 8]
It has pure time delay of 1 second. The specification for time constant is selected. Parameters of (2.18) and (2.21) are , ms,, and . Figure 7 shows online adjusting processes for finding . The initial guess of is equal to 1.00 and converted to 0.5195 after third adjusting processes. The found lead compensator is
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 Parameters of four found compensators are given below:(1)ZN(PI): and;(2) ZN(PID): , , and;(3) Tan’s(PID): , , and;(4) Majhi’s(PI): and .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.
Example 3.2. Consider a sixth order plant [7, 8] The specification of time constant sec is selected. Parameters of (2.18) and (2.21) are , ms,,, and . The initial guess of is equal to 1.0 and converted to 0.7759 after second adjusting process. The found lead compensator is Figure 9 shows on-line adjusting processes for finding to meet sec. 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.  for finding PID compensator, and Majhi [7, 8] for finding PI compensator. Parameters of five found compensators are given below:(1)ZN(PI): and ;(2) ZN(PID): , and;(3) Majhi’s(PI): , and ;(4) Ho’s(PID): .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.
Example 3.3. Consider the very-high-order plant [7, 8]:
The design specification for time constant sec is selected. Parameters of (2.18) and (2.21) are , ms,,and . The initial guess of is equal to 1.00 and converted to 0.9586 after the fourth adjusting process. The found lead compensator is
Figure 11 shows time response of the controlled system, which gives reference input (dash line), output (solid line), time series (dotted line), and ratios (dash-dotted line) of . 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., ) 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  for finding PI compensator, and Majhi [7, 8] for finding PI compensator. Parameters of four found compensators are given below:
(1) ZN(PI): and;(2) ZN(PID): ,nd;(3) Majhi’s(PI): and;(4) Zhuang’s(PI): and.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.
Example 3.4. Consider a gas turbine engine with plant transfer function matrix [24–26]: where . It is a 2 × 2 multivariable plant. The steady-state gain of open loop is in the form of A pre-compensating matrix 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 where ms is the sampling period. Figure 13 shows time responses of this controlled system for . It shows that the proposed control scheme can be applied to the multivariable feedback control system also.
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 to get better performance.
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