Approximate pole placement with dominance for continuous delay systems by PID controllers.

Wang, Qing-Guo ; Liu, Min ; Hang, Chang Chieh 等

INTRODUCTION

In a typical control textbook, the standard second-order system is discussed in great detail and used to guide practical control system design even if the underlying system is not of second-order. The assumption to make such a design to hold is that there is a dominant second-order dynamics. The desired closed-loop poles are calculated from certain control specifications such as percentage overshoot and settling time. However, continuous-time delay control systems are infinite-dimensional (Astrom and Wittenmark, 1997). They have infinite spectrum and it is not possible to assign such infinite spectrum with a finite-dimensional controller (Michiels et al., 2002). Instead, one naturally wishes to assign a pair of poles that dominate all other poles. This idea was first introduced by Persson and Astrom (1993) and further explained in Astrom and Hagglund (1995). In Coelho (1998), this idea is developed for the tuning of lead-lag controllers. Both methods are based on a simplified model of processes and thus cannot guarantee the chosen poles to be indeed dominant in reality. In the case of high-order systems or systems with time delay, these conventional dominant pole designs, if not well handled, could result in sluggish response or even instability of the closed-loop. To the best knowledge of the authors, no method is available in the literature to guarantee dominance of the assigned poles in general.

In this paper, an analytical PID design method is proposed for continuous-time delay systems to achieve approximate placement of two desired poles with dominance. A continuous delay process is converted to a low-order rational discrete model. A discrete PID controller is designed to ensure dominant pole placement in discrete domain. This is a finite-dimensional problem and the solution for pole placement is readily available. The designed discrete PID controller is finally converted back to the continuous one. The poles in continuous domain are generally not precisely the same as originally set. It is argued that exact pole placement is not necessary as practical design specifications are commonly set as ranges instead of precise values, and approximate ones should be sufficient as long as they do not deviate too much from the ideal ones. The dominance and error of the assigned poles are measured and checked for the design. It is shown by simulation that the proposed method works well with great dominance and negligible error of approximately assigned desired poles for a large range of normalized dead time up to at least 4. It should also be pointed out that discretization of a continuous process and discrete PID calculations are purely employed as a design intermediate and can be viewed as a fictitious process to get a workable continuous PID controller. No sampling is applied. Performance of our design should be judged from that of the so-obtained continuous PID controller, rather from discretization errors involved.

[FIGURE 1 OMITTED]

Continuous controller design is always carried out in continuous domain, and this causes an infinite spectrum assignment problem for a delay process under a PID control, a hard and open problem. While the proposed method to transform into and out of a discrete model is the first of its kind and brings the infinite spectrum assignment problem to an approximate finite spectrum assignment problem by a special selection of sampling time, a simple solution is obtained. No method is available in the literature to guarantee dominance of the assigned poles for PID control of a continuous delay process while the proposed method can do so.

The paper is organized as follows. In the next section, the problem under consideration is formulated. In the third section, the design method is presented for monotonic processes, with simulation examples given in the fourth section. In section five, the proposed method is applied to a thermal control system. Positive PID setting is discussed in the sixth section. In section seven the design method is presented for oscillatory processes, with conclusions drawn in the final section.

PROBLEM STATEMENT

A block diagram of a PID control system is shown in Figure 1, where G(s) is a continuous-time delay process and C(s) is the PID controller. Suppose that control system design specifications are represented by the overshoot and settling time on its closedloop step response. The overshoot is usually achieved by setting a suitable damping ratio, [xi]. A reasonable value of the damping ratio is typically in the range of 0:4 to 1. The settling time, [T.sub.s], cannot be taken arbitrarily but largely limited by the process characteristics and available magnitude of the manipulated variable. If Ts is too large, the response is very slow, which is bad performance and should be avoided. On the other hand, if Ts is too small, this may cause a very large control signal and less robust control system. From the point of view of dominant pole placement, pole dominance is also difficult to realize (Astrom and Hagglund, 1995; Zhang et al., 2002) if [T.sub.s] is very small. In this paper, through extensive simulation, we adopt the following empirical equation to choose [T.sub.s] for the process with a monotonic step response:

[T.sub.s] = T(4.5 + 4.5 L/T)(0.35/[xi] + 0.5) (10)

where T and L are the equivalent time constant and dead time of the process. The natural frequency, [[omega].sub.0], is calculated with [[omega].sub.0] = 4/[xi][T.sub.s]. Then, the specifications can be transferred to the corresponding desired second-order dynamics:

[s.sup.2] +2[xi][[omega].sub.0]s+[[omega].sup.2.sub.0] = 0.

[FIGURE 2 OMITTED]

Its two roots are denoted by [p.sub.s,1] with a positive imaginary part and [p.sub.s,2], which are the desired closed-loop poles to be achieved and be dominant by our controller design.

The actual closed-loop system has its characteristic equation:

1 + G(s)C(s) = 0

Let its roots or closed-loop poles be [p.sub.s,i], i = 1,2, ... They are ordered such that [p.sub.s,i] meets Re([p.sub.s,i]) [greater than or equal to] Re([p.sub.s,i+1]) and if Re[p.sub.s,i] = Re([p.sub.s,i+1]), Im([p.sub.s,i+1]) > Im [p.sub.s,i+1] where Re([p.sub.s]) and Im([p.sub.s]) are the real and imaginary parts of [p.sub.s,i], respectively. Note that the actual poles, [p.sub.s,i]i = 1,2, may not be the same as the desired ones: [p.sub.s,1] and [p.sub.s,2], and [p.sub.s,i].i = 1,2, may not be dominant enough with respect to other poles. Thus, we introduce two measures to reflect them: the relative pole assignment error,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

and the relative dominance,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Our problem of approximate pole placement with dominance is to determine a continuous controller C(s) so as to produce reasonably small relative pole assignment error and large relative dominance, say, [E.sub.p] [greater than or equal to] 20% and [E.sub.D] [greater than or equal to] 3, which are used as defaults.

The difficulty of the above problem lies in the existence of an infinite number of closed-loop poles for a continuous delay process under PID control. It is impossible to assign all the closed-loop poles. However, a continuous-time delay process may be converted to a low-dimensional discrete system with some special sampling time selection. In this paper, discrete design is used as a bridge to approximate pole placement in continuous PID control systems, but no sampling is done in the real control system of Figure 1.

[FIGURE 3 OMITTED]

THE PROPOSED METHOD

Let a continuous-time delay process G(s) have a monotonic step response and be represented by a first-order time delay model:

G(s) = K/Ts + 1 [e.sup.-Ls] (4)

In this paper, we choose the sampling time h as h = L to make the discretized process, G(z), have the lowest order. The process has a pole at 1/T. This pole is mapped via z = [e.sup.hs] (adopted in pole-zero matching method in Franklin et al. (1990)), to the pole of its discrete equivalent at T = [e.sup.-L/T], so that K/Ts + 1 is converted to K/z - T, where K is selected to match the static gain, K/Ts + 1[parallel] s = 0 = K/z-T[parallel]z = 1, and thus K = K(1-[e.sup.-L/T]). Note that the discrete equivalent of [e.sup.-Ls] is 1/z under h = L. Overall, the process in form of (4) is converted to

G(z) = K/z(z - T) (5)

The continuous PID controller in form of

C(s) = [K.sub.p] (1 + 1/[T.sub.i]s + [T.sub.d]s)

is also converted to the discrete-time model,

C(z) = [k.sub.1][z.sub.2]+[[k.sub.2]z + [k.sub.3]/z - 1 (7)

where [k.sub.1], [k.sub.2] and [k.sub.3] are the functions of [k.sub.p], [T.sub.i] and [T.sub.d]. The characteristic polynomial of the discrete closed-loop system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

[FIGURE 4 OMITTED]

On the other hand, the given [p.sub.s,1] and [p.sub.s,2] have the desirable discrete characteristic polynomial as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

where [p.sub.z,1] = [e.sup.Lps,1], [p.sub.z,2] = [e.sub.Lps,2], and [p.sub.z,3] is a user-defined parameter and set at [e.sup.10LRe(ps,1)] in this paper. Equalizing [A.sub.cl](z) with [A.sub.de](z) yields

[k.sub.1] = [p.sub.1] + 1 + T/K

[k.sub.2] = [p.sub.2] - T

[k.sub.3] = [p.sub.3]/K

Once [k.sub.1], [k.sub.2] and k3 are known, the two zeros of C(z) can be calculated as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Using the pole-zero matching method gives the continuous controller as

C(s) = [K.sub.c] (s - log([z.sub.1]/L)(s - log([z.sub.2])/L)

with Kc selected to match the gain of C(s) at 0.1m/L, where m is the smallest integer and meets [e.sup.0.1m][not equal to] 1, [z.sub.1] and [z.sub.2]. Finally, C(s) can be then rearranged into the form in (6) with its settings given as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 5 OMITTED]

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To apply the above method to a non-first-order process G(j[omega]) with monotonic step response, we have to obtain its first-order approximate model G(s) in form of (4). The simplest technique is to match the model frequency response with the process one at two frequency points, [omega] = 0 and [omega] = [[omega].sub.p], the phase cross-over frequency. The formulas are well known (Wang et al., 2003):

K = G(0) (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

L = [pi] + [tan.sup.-1]([-[omega].sub.p]T)/[[omega].sub.p] (12)

Gain and phase margins are basic measure of the system's robustness. In this paper, we apply these specifications to judge robustness of the design results. Tuning [xi] will give suitable robust stability of the closed-loop system against the parameter uncertainties.

SIMULATION STUDY

Example 1

Consider an exact first-order process with G(s) = 1/s+1 [e.sup.-Ls] and study our design with several typical values of L. Let L = 0.5 first. Suppose that the desired damping ratio is [xi] = 0.7. Ts is calculated from (1) as 8.25. We have [p.sub.s,1] = -0.4848 + 0.4946i, [p.sub.s,2] = -0.4848 - 0.4946i. The third pole is then [p.sub.s,3] = 10Re([p.sub.s,1]) = -4.848. The proposed method with these specifications leads to the discrete PID:

C(z) = -0.009146[z.sup.2] + 0.366z - 0.1386/z - 1

and via the pole-zero matching method, the continuous PID:

C(s) = -0.0.321[s.sup.2] + 0.1726s - 0.4505/s

which is rearranged in form of (6) as

C(s) = 0.1726(1 + 1/0.3832s - 0.1859s)

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The closed-loop poles are calculated from the roots of 1 + G(s)C(s) = 0 with a fortieth-order Pade approximate to the time delay as [p.sub.s,1] = -0.5135 + 0.4837i, [p.sub.s,2] = -0.5135 - 0.4837i, [p.sub.s,3] = -5.6623, [p.sub.s,4] = -6.4016 + 13.1493i, [p.sub.s,5] = -6.4016 -13.1493i, .... It follows that [E.sub.P] = 4.43% and [E.sub.D] = 11.03. The gain margin and phase margin are 6.64 and 63.92o, respectively. The step response and the manipulated variable are shown in Figure 2. The settling time of the control system is 8.5 and the overshoot is 3.71% with the corresponding damping ratio of 0.72. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system, 2.326/[s.sup.3] + 5.818[s.sup.2] + 5.181s + 2.326 with its poles at the desired -0.4848 [+ or -] 0.4946i and one extra at -4.848, are given in Figure 3 for comparisons, from which one sees that the designed continuous system is quite close to them. Consider L = 2. Suppose that the desired damping ratio is [xi] = 0.7. [T.sub.s] is calculated from (1) as 19.5. We have [p.sub.s,1] = -0.2051 + 0.2093i and [p.sub.s,2] = -0.2051 + 0.2093i. The third pole is at -2.051. The proposed method with these specifications leads to the discrete PID:

C(z) = -0.1083[z.sup.2] + 0.3758z - 0.008416/z - 1

and via the pole-zero matching method, the continuous PID:

C(s) = -0.1179[s.sup.2] - 0.1506s - 0.1384/s

which is rearranged in form of (6) as follows

C(s) = 0.1506(1 - 1/1.0883s - 0.7829s)

The closed-loop poles are [p.sub.s,1] = -0.1913 + 0.2284i, [p.sub.s,2] = -0.1913 - 0.2284i, [p.sub.s,3] = -1.0131 + 3.0847i, [p.sub.s,4] = -1.0131 - 3.0847i, ... It follows that EP = 8.04% and ED = 5.30. The gain margin and phase margin are 2.59 and 57.25[degrees], respectively. The step response and the manipulated variable are shown in Figure 4. The settling time is 22.95 and the overshoot is 7.49% with the corresponding damping ratio of 0.64. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z) and the prototype continuous system, 0.1761/[s.sup.3] + 2.462[s.sup.2] + 0.9274s + 0.1761, with its poles at the desired -0.2051 [+ or -] 0.2093i and -2.051, are given in Figure 5 for comparison.

[FIGURE 9 OMITTED]

Consider L = 4. Suppose that the desired damping ratio is [xi] = 0.7. [T.sub.s] is calculated from (1) as 34.5. We have [p.sub.s,1] = -0.1159 + 0.1183i and [p.sub.s,2] = -0.1159 - 0.1183i. The third pole is at -1.159. The proposed method with these specifications leads to the discrete PID:

C(z) = -0.207[s.sup.2] - 0.1743s + 0.07457/z - 1

and via the pole-zero matching method, the continuous PID:

C(s) = -0.207[s.sup.2] - 0.1743s + 0.07457/s

which is rearranged in form of (6) as follows

C(s) = -0.1743 (1 - 1/2.3366s + 1.1880s)

The closed-loop poles are [p.sub.s,1] =-0.1184 + 0.1289i, [p.sub.s,2] = -0.1184 - 0.1289i, [p.sub.s,3] = -0.3704 + 1.5947i, [p.sub.s,4] = -0.3704 - 1.5947i, .... It follows that [E.sub.P] = 6.56% and [E.sub.D] = 3.12. The gain margin and phase margin are 2.48 and 58.02[degrees], respectively. The step response and the manipulated variable are shown in Figure 6. The settling time is 39.73 and the overshoot is 5.94% with the corresponding damping ratio of 0.67. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system,

0.03181/[s.sup.3] + 1.391[s.sup.2] + 0.2963s + 0.03181, with its poles at the desired -0.1159 [+ or -] 0.1183i and -1.159, are given in Figure 7 for comparison.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

Example 2

Consider a high-order process, G(s) = (2s + 1)/(s+1)(4s + 1) [e.sup.s]. By Formulas (10), (11) and (12), we obtain its first-order approximate as

G(s) = 1/3.743s + 1[e.sup.-1.49s]

Suppose that the desired damping ratio is [xi]= 0.7. [T.sub.s] is calculated from (1) as 28. We have [p.sub.s,1] = -0.1427 + 0.1456i and [p.sub.s,2] = -0.1427 - 0.1456i. The third pole is at -1.427. The proposed method with these specifications leads to the discrete PID:

C(z) = -0.07994[z.sup.2] + 0.5168z - 0.2366/z - 1

and via the pole-zero matching method, the continuous PID:

C(s) = -0.2485[s.sup.2] + 0.1808s - 0.14/s

which is rearranged in form of (6) as

C(s) = 0.1808(1 + 1/1.2910s - 1.3747s)

The closed-loop poles are [p.sub.s,1] = -0.1530 + 0.1369i, [p.sub.s,2] = -0.1530 - 0.1369i, [p.sub.s,3] = -0.7307 + 0.3366i, [p.sub.s,4] = -0.7307 - 0.3366i,.... It follows that EP = 6.62% and ED = 4.77. The gain margin and phase margin are 5.47 and 63.81[degrees], respectively. The step response and the manipulated variable is shown in Figure 8. The settling time of the control system is 28.28 and the overshoot is 3.41% with the corresponding damping ratio of 0.73. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system, 0.05931/[s.sup.3] + 1.713[s.sup.2] + 0.4489s + 0.05931 with its poles at -0.1427 [+ or -] 0.1456i and -1.427, are also given in Figure 9 for comparison.

[FIGURE 12 OMITTED]

In practice, the measurement noise and unmodelled dynamics, such as disturbances, are generally present. For the same example, the measurement noise is simulated by adding a white noise to the output and a disturbance with the magnitude of -0.3 is added to the output at t = 30. The response, y(t), the measured output, [y.sub.n](t), and the manipulated variable, u(t), are shown in Figure 10. The effectiveness of our method is shown.

REAL TIME TESTING

In this section, the proposed PID tuning method is also applied to a temperature chamber system, which is made by National Instruments Corp. and shown in Figure 11. The experiment setup consists of a thermal chamber and a personal computer with data acquisition cards and labView software. The system input, u, is the adjustable power supply to 20W Halogen bulb. The system output, y, is the temperature of the temperature chamber. The model of the process is

G(s) = 29.49[e.sup.-0.106s]/0.6853s+1

The proposed method with ??= 0.8 leads to the PID controller as

C(s) = 0.0047(1 + 1/0.1535 - 1.3408s)

This ideal PID is not physically realizable and is thus replaced by

C(s) = 0.0047(1 + 1/0.1535s - 1.3408s/(1.3408/N) s + 1)

where N = 4, in the real time testing. Before the test is applied, the control system is at a steady state. At t = 0, the reference input is changed from 29 to 27. The process input and output are given in Figure 12. The step response of the prototype continuous system, 20.8/[s.sup.3] + 13.2[s.sup.2] + 26.09s + 20.8 are also given in Figure 12 for comparison. The designed system has satisfying performance.

POSITIVE PID SETTINGS

It is noted from the simulation results in the preceding section that some of the PID parameters are not positive. In many applications, it is not permissible. To avoid this problem, we choose the controller in the form of

C(s) = [K.sub.p](1 + 1/[T.sub.i]s)(s + [beta]/s + [alpha]) (13)

which corresponds to the practical form (no pure D) of PID controller in the cascaded structure (Astrom and Hagglund, 1995; Ang et al., 2005). We choose [T.sub.i] = T to cancel the pole of G(s). The open-loop transfer function, G(s)C(s), is converted by the pole-zero matching method to its discrete equivalent,

G(z)C(z) = K/z [k.sub.1]z+[k.sub.2]/z(z-1)(z + [k.sub.3]) (14)

where K = K/T, and [k.sub.1], [k.sub.2], [k.sub.3] are the functions of [K.sub.p], [beta] and [alpha]. The discrete closed-loop characteristic polynomial is

[A.sub.cl](z) = [z.sup.3] + ([k.sub.3] - 1)[z.sup.2] + (K[k.sub.1] - [k.sub.3])z + K[k.sub.2]

By making [A.sub.cl](z) = [A.sub.de](z), we can solve for [k.sub.1], [k.sub.2] and [k.sub.3] as

[k.sub.1] = [p.sub.1] + [p.sub.2] + 1 (15)

[k.sub.2] = [p.sub.3]/K (16)

[k.sub.3] = [p.sub.1] + 1 (17)

Once [k.sub.1], [k.sub.2] and [k.sub.3] are known, we obtain the controller parameters in continuous domain as

[beta] = -log (-[k.sub.2]/[k.sub.1]/L (18)

[alpha] = -log(-[k.sub.3])/L (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where m is the smallest integer, which meets [e.SUP.0.1m] [not equal to] 0, -[k.sub.3], [k.sub.2]/[k.sub.1].

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

Example 1 (continued)

Consider Example 1 again with L = 0.5. Suppose that the desired damping ratio is [xi]= 0.7 and Ts = 8.25 as before. The controller in form of (13) is obtained as

C(s) = 0.2195(1 + 1/s)(s + 1.8901/s + 0.9878)

The closed-loop poles are calculated as [p.sub.s,1] = -0.5382 + 0.4020i, [p.sub.s,2] = -0.5382 - 0.4020i, [p.sub.s,3] = -7.32,.... For this example, [E.sub.P] = 15.43% and [E.sub.D] = 13.6. The closed-loop pole at -1 is concealed by the closed-loop zero at -1. The gain margin and phase margin are 10.31 and 68.53[degrees], respectively. The step response and the manipulated variable are shown in Figure 13. The settling time of the resultant control system is 5.45 and the overshoot is 1.63% with the corresponding damping ratio of 0.79. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system, 2.326/[s.sup.3] + 5.818[s.sup.2] + 5.181s + 2.326, are given in Figure 14 for comparison.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

OSCILLATION PROCESSES

Some practical processes such as temperature loops exhibit oscillatory or essentially second-order behaviour in its step response. The first-order modelling is not adequate for them. Instead, one has to use the following model:

G(s) = K/[s.sup.2] + as + b [e.sup.-Ls] (21)

Define [p.sub.g,i], i = 1, 2 as the roots of [s.sub.2] + as + b = 0. The equivalent time constant of the oscillation process is defined as T = -1/Re([p.sub.g,i]). To set a desired second-order dynamic properly, the damping ratio is chosen as before, while the following new formula,

[T.sub.s] = T(1 + 15 L/T) (0.35/[xi] + 0.5) (22)

is used for determining [T.sub.s]. For this kind of processes, we exploit the controller in the form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

and choose [T.sub.i] and [T.sub.d] to cancel the poles of G(s):

[T.sub.d] = 1/a

[T.sub.i] = a/b

Then, the resulting open-loop G(s)C(s) and its discrete equivalent are the same as those in the previous section with K = K/a. The procedure there applies to obtain [K.sub.p], [beta] and [alpha] from (15), (16) and (17). With [k.sub.1], [k.sub.2] and [k.sub.3], we calculate [K.sub.p], [beta] and [alpha] according to (18), (19) and (20).

Example 3

Consider a oscillation process, G(s) = 1/[s.sup.2] + 1.2s + 1 [e.sup.-0.7s]. The equivalent time constant of the process is T = 1.667. Suppose that the desired damping ratio is [xi]= 0.7. [T.sub.s] is calculated from (22) as 13. We have [p.sub.s,1] = -0.3288 + 0.3354i and [p.sub.s,2] = -0.3288 - 0.3354i. The third pole is at -3.288. The proposed method in this section with these specifications leads to the continuous controller:

C(s) = 0.1953 (1 + 1/1.2s + 0.8333s)(s + 1.1410/s + 0.6256)

The closed-loop poles are calculated as [p.sub.s,1] = -0.3585 + 0.2755i, [p.sub.s,2] = -0.3585 - 0.2755i, [p.sub.s,3] = -5.0949, [p.sub.s,4] = -6.1839 + 10.3973i, [p.sub.s,5] = -6.1839 - 10.3973i,.... It follows [E.sub.P] = 14.24% and [E.sub.D] = 14.21. The closed-loop pole at -0.6000 [+ or -] 0.8000i are concealed by the closed-loop zeros. The gain margin and phase margin are 10.72 and 68.51[degrees], respectively. The step response and the manipulated variable are shown in Figure 15. The settling time of the control system is 11 and the overshoot is 2.04% with the corresponding damping ratio of 0.77. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system, 0.7252/[s.sup.3] + 3.945[s.sup.2]+ 2.382 + 0.7252 are given in Figure 16 for comparison.

For comparison with first-order design method, by (10), (11) and (12), we obtain its first-order model as

G(s) = 1/1.235s + 1 [e.sup.-1.44s]

Suppose the desired damping ratio is [xi] = 0.7. [T.sub.s] = 16.4 is calculated from (1) with T = 1.235 and L = 1.44. The proposed method in the third section with these specification leads to the continuous PID:

C(s) = -0.0457(1 - 1/0.2481s + 1.6713s)

The closed-loop poles are calculated as [p.sub.s,1] = -0.3437, [p.sub.s,2] = -0.4066 + 0.5870i, [p.sub.s,3] = -0.4066 - 0.5870i, [p.sub.s,4] = -6.69 + 5.47i, [p.sub.s,5] = -6.69 - 5.47i, ... The resulting dominant poles are -0.3437 and -0.4066 [+ or -] 0.5870i, which are far from the desired ones.

Example 4

Consider a high-order oscillation process, G(s) = 1/(0.8s + 1)([s.sup.2] + 1.1s + 1) [e.sup.-2s]. Applying the identification method proposed by Liu et al. (2007), we obtain one of its estimations as

G(s) = 0.702/[s.sup.2] + 0.9708s + 0.7114 [e.eup.-2.33s]

with the equivalent time constant of T = 2.06. Suppose that the desired damping ratio is [xi] = 0.7. Ts is calculated from (22) as 37. We have [p.sub.s,1] = -0.1081 + 0.1103i and [p.sub.s,2] = -0.1081 -0.1103i. The third pole is at -1.081. The proposed method in this section with these specifications leads to the continuous controller:

C(s) = 0.0637 (1 + 1/1.3646s + 0.0301s)(s + 0.4567/s + 0.2306)

[FIGURE 18 OMITTED]

The closed-loop poles are calculated as [p.sub.s,1] = -0.1178 + 0.0941i, [p.sub.s,2] = -0.1178 - 0.0941i, [p.sub.s,3] = -0.5221 + 0.8374i, [p.sub.s,4] = -0.5221 -0.8374i, ... It follows [E.sub.P] = 12.24% and [E.sub.D] = 4.43. The gain margin and phase margin are 10.02 and 67.34[degrees], respectively. The step response and the manipulated variable are shown in Figure 17. The settling time of the resultant control system is 35.36 and the overshoot is 2.1% with the corresponding damping ratio of 0.77. The step responses of the discrete system, G(z)C(z)/1 + G(z)C(z), and the prototype continuous system, 0.02576/[s.sup.3] + 1.297[s.sup.2] + 0.2575s + 0.02576 are given in Figure 18 for comparison.

CONCLUSION

In this paper, an analytical PID design method has been presented for continuous-time delay systems to achieve approximate pole placement with dominance. It greatly simplifies the continuous infinite spectrum assignment problem with a delay process to a third-order pole placement problem in discrete domain for which the closed-form solution exists and is converted back to its continuous PID controller. The method works well for both monotonic and oscillatory processes of low or high order.

ACKNOWLEDGEMENT

This work was sponsored by the Ministry of Education's AcRF Tier 1 funding, R-263-000-306-112, Singapore.

Manuscript received January 3, 2007; revised manuscript received March 4, 2007; accepted for publication May 8, 2007.

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Qing-Guo Wang *, Min Liu and Chang Chieh Hang

Department of Electrical and Computer Engineering, National University of Singapore,

10 Kent Ridge Crescent, Singapore 119260

* Author to whom all correspondence may be addressed. E-mail address: [email protected]