**6.4 Simplification of the tuning rules for on-line tuning**

The tuning rules presented in the previous sections can significantly be simplified, in the case where *τD=τS*. In this case, the loop transfer function is given by (17), and the solutions of the algorithms presented in Subsections 6.1.1 and 6.2.1-6.2.3, can easily be approximated with satisfactory accuracy for all systems with *0<d<0.9*. In particular, the solutions for *wmin* and *wmax*, can be approximated by relations (21)-(23). Note that, here, ,min ˆ ( ) *<sup>I</sup> d* is an accurate

approximation of the smallest value of the integral term *τI*, for which (8) has a solution, when *τD=τS*, and when the atan function takes values in the range *(-π/2, π/2)*. Table 6 summarizes useful approximations of some other parameters involved in the aforementioned algorithms. Note that the maximum normalized errors for the parameters *KC,min* and *KC,max*, when their estimates are obtained by (20), using min *w*ˆ and max *w*ˆ as given by (21), never exceed 2.2% for *d≤0.9* and *τI>1.2* ,min ˆ *I* .

In Table 7, numerical applications of the PM, GM and PGM tuning methods are presented for three processes with normalized dead time 0.1, 0.5 and 0.9. The controller parameters obtained from the application of these tuning methods are presented in the left section of Table 7 for both the exact *(KC, τI)* and the approximated controller parameters ( ˆ *KC* , ˆ *I* ). In

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 67

1.6 -1.4 -1.2 -1 -0.8

PGM

PGM

*dx dt v* / , 2 2 *dv dt g c M i x x* / / /( ) (31)

Real Axis

PM

Real Axis

GM

**o o**

**o o**

PM

GM



0




Imaginary Axis




Imaginary Axis


0

Specifications Controller parameters Nyquist Plots

*I I* 

*I I* 

*I I* 

*I I* 

*I I* 

*I I* 

 

**7. Application to an experimental magnetic levitation system** 

coil of the electromagnet has an inductance *L* and a total resistance *R*.

 

 

 

6.5667 ˆ 6.5160

5.5286 ˆ 5.7115

6.6907 ˆ 6.9608

777.17 ˆ 744.56

511.24 ˆ 590.60

971.4 ˆ 930.70

Table 7. Some characteristic numerical examples of the proposed tuning methods reported

In this section the tuning methods presented above will be applied to the experimental magnetic levitation system shown in Figure 5. This experimental system is a popular gravity-biased one degree of freedom magnetic levitation system in which an electromagnet exerts attractive force to levitate a steel ball. The dynamics of the MagLev system can be described by the following simplified state space model (Yang & Tateishi,

where *x*, *v* and *M* are the air gap (vertical position), the velocity and the mass of the steel ball respectively, *g* is the gravity acceleration, *i* is the coil current, *c* and *x∞* are constants that are determined by the magnetic properties of the electromagnet and the steel ball. Moreover the

*d*=0.5 , *τD= τS*

*d*=0.9 , *τD= τS*

*PM*=0.018, *GMinc*=1.07, *GMdec* =1.07

in Section 6.

2001)

*PM*=0.15, *GMinc*=1.3, *GMdec*=1.5

PM

GM

PGM

PM

GM

PGM Method

Method

Method

Method

Method

Method

1.5690 ˆ 1.5688

1.7581 ˆ 1.7505

1.6933 ˆ 1.6916

1.0602 ˆ 1.0602

1.0811 ˆ 1.0795

1.0756 ˆ 1.0759

*C C*

*C C*

*C C*

*C C*

*C C*

*C C*  

*K K*  

*K K*  

*K K*  

*K K*  

*K K*  

*K K*

the right section of Table 7, the polar plots of the resulting closed-loop systems are presented. Solid and dashed lines are used for the exact and the approximate controller, respectively. The gain margin specifications are indicated by the symbol '**o**' and the point on the unit circle which determines the phase margin specification is indicated by the symbol ''. From all these polar plots, it becomes obvious that the approximate solution is very accurate and in most cases cannot be distinguished from the exact solution. Note that, since the proposed tuning methods provide a controller that satisfies the required stability robustness specifications with significant accuracy, it is possible to design a closed loop system with any desired design specifications. The most robust (but slow) closed loop system possible (when *τD=τS*) can be obtained when *PMdes→PMmax* or when *GMprod,des→ GMpred,max* (i.e. *τI→∞*), while it is possible to design a faster but less robust system with less conservative stability margins specifications.


Table 6. Approximations of parameters involved in the PM, GM and PGM algorithms, when *τD=τS*.


the right section of Table 7, the polar plots of the resulting closed-loop systems are presented. Solid and dashed lines are used for the exact and the approximate controller, respectively. The gain margin specifications are indicated by the symbol '**o**' and the point on the unit circle which determines the phase margin specification is indicated by the symbol ''. From all these polar plots, it becomes obvious that the approximate solution is very accurate and in most cases cannot be distinguished from the exact solution. Note that, since the proposed tuning methods provide a controller that satisfies the required stability robustness specifications with significant accuracy, it is possible to design a closed loop system with any desired design specifications. The most robust (but slow) closed loop system possible (when *τD=τS*) can be obtained when *PMdes→PMmax* or when *GMprod,des→ GMpred,max* (i.e. *τI→∞*), while it is possible to design a faster but less robust system with less

Function Approximation MNE Valid Range

*des*

1

*d*

*g(d)*=10-2[-0.18+5 *d* -32*d*+75*d*2-51*d*3+(-2.3*d*2+3*d*4)/(1-*d*)3] Table 6. Approximations of parameters involved in the PM, GM and PGM algorithms, when

*<sup>A</sup> <sup>d</sup> g d <sup>A</sup>*

max

*A*

PM

/ () <sup>ˆ</sup> ()1 () 1 / ()

,min <sup>ˆ</sup> ( ) 1 0.65 ( ) <sup>1</sup>

*fPM(d)=*(-0.0153+0.436 *<sup>d</sup>* +0.632*d*)/*d* , ,

Specifications Controller parameters Nyquist Plots

*I I* 

*I I* 

*I I* 

 

 

0.3010 ˆ 0.2980

0.3184 ˆ 0.3216

0.3598 ˆ 0.3597

 

*PM PM d d fd PM PM d*

*d<0.9* and

*GM >1+0.2× prod des* , *(GMprod,max-1)*

> -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

Imaginary Axis

5% *PMdes>0.2PMmax*

3%

3%

1 () 1

,max

Real Axis

**o o**

*prod*

*GM d* 

*GM*


PGM

GM

*prod des*

conservative stability margins specifications.

(, ) *des*

, (, ) *<sup>I</sup> prod des*

*I* 

*d GM*

*τD=τS*.

*d*=0.1 , *τD= τS*

*PM*=0.3, *GMinc*=4, *GMdec*=2

PM

GM

PGM

Method

Method

Method

*C C*

*C C*

*C C*  

*K K*  

*K K*  

*K K*

,max( ) *GM d prod* 2 1 0.4085 /(1 0.2864 ) *dd d*

*d PM* max ,min

*I*

5.2293 ˆ 5.2170

3.0225 ˆ 3.0400

3.1333 ˆ 3.1411

*I PM des*


Table 7. Some characteristic numerical examples of the proposed tuning methods reported in Section 6.
