No Overshoot Step Response of an Integral Controller with a Simple Robust Lead Inverse Compensator t

International Journal of Automation and Control Engineering (IJACE) Volume 5, 2016 doi: 10.14355/ijace.2016.05.001

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No Overshoot Step Response of an Integral Controller with a Simple Robust Lead Inverse Compensator to Improve Saftey of Vehicles Makoto Katoh*1 Department of Mechanical Engineering, Osaka Institute of Technology/Faculty of Engineering, 5-16-1 Ohmiya, Asahi-ku, Osaka, 535-8585, Japan makoto.kato@oit.ac.jp; 1katoh@med.oit.ac.jp

*1

Abstract This paper mainly presents a proposal for a control system intended to provide a stable response with no-overshoot for a firstorder proper system in a loop configuration that includes also an integral controller. An extended theoretical analysis is provided to justify the approach and also an application is presented at the end of paper. The results from the approach proposed in this paper are simpler, mathematically, than those from the conventional approach. Moreover, 7 examples of arrangement of zeros and poles for stability with no-overshoot property on a second order proper controlled object with an integrator were shown though they were not solved easily and analytically.

Keywords No-overshoot; Robust Lead Compensator; Proper System

Introduction Many control systems have been designed that do not use integral control (eg., PD cascade controllers for position control of a mobile robot with integrator to convert to position from velocity etc., but it is not robust for change of laden weight, then the change in the load causes untrodden and excesses even if there is no overshoot.) to improve safety and reduce overshooting or excesses for references, and others have used an integral term (such as an iterative linear quadratic regulator, I-LQR; or a linear quadratic Gaussian control system with integral action, ILQG, which is an expanded version of integral-proportional-derivative, I-PD, control) in hard servo mechanisms which was reported to be effective for the vibration control, too [1, 2, 3].

However, though lead/lag compensators had been studied in frequency domain in order to improve the performance of PID controllers [4,5], it is necessary to have no-overshooting by the integrator. We have studied overshoot reduction in step responses [6].

Liu et al. proposed series double compensators, which combine a proportional-integral (PI) compensator with a first-order delay compensator for a series double first-order delay, and this results in step responses without overshoots [7]. Other robust compensators and controllers have been studied in various areas [1, 2, 8-10]. These are important in order to ensure the safety of control systems. But, they were not for proper systems.

This paper proposes an approach to have no-overshoots for stable minimum-phase first-order proper controlled objects by integral controller; we do this by adding a simple robust (Siro) lead inverse compensator (LEICO). We call the pair of integral and LEICO to KILEICO by adding an integral gain name Ki. First, we will examine the necessary condition to prevent overshoots of a single integral controller in a time-constant parameter plane.

Stability and Safety of Integral Control Condition for No Overshoot and Root Locus Stability An informal committee of the Japanese government is defined safety and peace of mind as follows [10].

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(1) Safety is the objective judgment that a system will not incur damage.

(2) Peace of mind depends on the subjective judgment of an individual. Peace of mind occurs when one believes that the future will not differ greatly from expectations based on knowledge and experience.

The following definition and lemma, along with a conceptual proof, were proposed here.

Definition of Root Locus Stability A closed-loop system is said to have root locus stability if the root locus of the closed-loop system is not in an unstable area for an arbitrary loop gain change.

There are three types of root locus stability in case of the controlled objects with real poles and zeros, which we will label +, - , and ○ as their root locus figures look like them.

Lemma 1 (Nyquist Diagram Condition for Root Locus Stability of Integral Control) If the Nyquist diagram of an open-loop transfer function is limited to the first (ω < 0) and fourth (ω > 0) zones, then the Nyquist diagram of the integral control is limited to the first and second zones (ω < 0) or to the third and fourth zones (ω > 0). Since it does not intersect the negative real axis as the gain margin becomes ∞, the root locus does not become unstable even if the loop gain is increased. (proof) The Nyquist diagram for integral control begins at ±∞ for ω = 0, and it converges to 0 as ω approaches ∞. Because integral control adds a phase lag of 90°, when the phase lag of the original system is at most 90°, the phase lag of the integral control system is at most 180°. Thus, this lemma holds.

The following major property, concept, definition, theorem and lemma were proposed here.

No-Overshoot Property or Safe Stability (SS) The - and small ○ (microbubble) types of root locus stability have the no-overshoot property in step responses.

We wish to ensure that step responses of closed-loop systems will have the no-overshoot property, because for many processes and traffic control areas, root locus stability does not ensure sufficient safety or peace of mind.

Then, we call the stability with no-overshoot property to safe stability (SS).

Definition of Real Roots Root Locus (3RL) Stability A closed-loop system is said to have real roots root locus(3RL) stability if the root locus of the closed-loop system has the only stable real roots for an arbitrary loop gain change.

Integral control has been used for various purposes in process control and in hard servo areas. In this study, we focus on integral control. We obtain the following theorem for integral control that will not overshoot.

Theorem 1 The sufficient condition for preventing overshooting of the integral control C(s):

C ( s) 

Ki s

(1)

for a stable minimum-phase first-order proper controlled object G(s): G( s) 

Tn1 s  1 Td 1 s  1

(2)

is that Tn1 > Td1 > 0 in the time-constant parameter plane. There may be double repeated roots and bifurcations of the root locus when 0 < Tn1 < Td1. (Here, suffix n is the numerator and suffix d is the denominator.) (proof)

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W (s)  CG(s) / (1  CG(s))  Tn1  2Td 1

if

then then

Td 1  Tn1

if

D0

double Ki 

if

Td 1  Tn1 then

if

Tn1  Td 1

then

Ki (Tn1s  1) Ki  0

for

 K i  R,

(4) (5)

(Tn1  2Td 1 )  2 (Td 1  Tn1 )Td1

0

Tn12

D0

 Ki  R,sin gle

for

D0

(3)

Td1s 2  (1  KiTn1 )s  Ki

for

D0

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(6)

Ki 

1 0 Tn1

Ki  R

for

(7) (8)

There are no repeated poles on the real axis in this area (Tn1 > Td1 > 0). Thus, the root locus does not bifurcate, and there is no overshoot of the step response in this area. ■ The critical points Sp for integral control for a stable minimum-phase proper first-order system are on the stable real axis only if 0 < Tn1 ≦ Td1. (proof) K T s 1 (9) C ( s)G ( s )  i n1 s Td 1s  1 dC ( s)G ( s) (10) 0  ds T  (Td 1  Tn1 )Td 1 (11) s p  d1 Tn1Td 1 (12) D  (Td 1  Tn1 )Td 1  0  Re( s p )  0 when Tn1  Td 1 ■ Lemma 2 Examination of Condition for No Overshoot From above equations, we can obtain the following robust no overshoot half area (Fig. 1), if you note that there is no repeated root in gain Ki of the real number becoming a repeated root when Ki is an imaginary root.

Tn1

Root locus stability In any Ki 根軌跡 3

根軌跡

0.2

0.15

2

虚軸

0.05

0

-0.05

-0.1

-0.15

-0.2 -3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

実軸

0 stable repeated root with Stable real axis no overshoot In any Ki with no overshoot

1

虚軸

0.1

-2

-3 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

4

5

6

実軸

ステップ応答

1

0

-1

ステップ応答

1.4

0.9

1.2

0.8

1 0.7

Stable repeated roots in double Ki

0.4 0.3 0.2

0.8 振幅

振幅

0.6 0.5

0.6

0.4

0.2

0.1 0

0 0

2

4

6

8

10

12

0

1

2

3 時間 (sec)

時間 (sec)

Stable micro bubble area with almost overshootless

1.2

1

0.8 振幅

Stable large Bubble area

Td1

ステップ応答

1.4

0.6

0.4

0.2

0

0

2

4

6

8

10

12

時間 (sec)

FIG. 1 VARIOUS AREA OF REAL PARAMETER PLANE (TD1, TN1) IN INTEGRAL CONTROL FOR 1ST ORDER PROPER SYSTEM

Siro Lead Inverse Compensator Approach Let us consider the following configuration shown in Fig. 2 for a unity feedback system.

Reference value

+ + -

C(s)

IN(s)

G(s)

Controlled value

FIG. 2 SYSTEM CONFIGURATION OF CING

The LEICO IN(s), defined in Eq. (13), is added to the integral controller C(s) for the fault in the area where Tn1 < Td1.

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IN ( s) 

(Td 1 /  )s 1 (Tn1 /  )s 1

0    1,   1

(13)

where α is a positive parameter that is sufficiently large that IN(s) does not become unstable, and β is a positive parameter that tunes a zero position for compensation of the system.

We now show the root locus stability of CING in lemma 3.

Lemma 3 When C(s)G(s) are as defined in Eqs. (1)(2), and IN(s) is as defined in Eq. (13), then the closed-loop of CING has root locus stability. (proof) CING ( s) 

Ki (Td 1 /  ) s  1 Tn1s  1 s (Tn1 /  ) s  1 Td 1s  1

characterestic

equation

of

(14) closed

loop

 ( s)  (Tn1 /  )Td 1s  {(Tn1 /  )  Td 1  Tn1 (Td 1 /  ) Ki }s 2  {1  (Tn1  Td 1 /  ) Ki }s  Ki  0 if 0    1 then H1  0, H 2  0, H 3  0 for where 3

H2 

{(Tn1 /  )  Td 1  Tn1 (Td 1 /  ) Ki }

Ki

(Tn1 /  )Td 1

{1  (Tn1  Td 1 /  ) Ki }

Ki  0

(15) (16)

(17) ■ Remark 1: It is difficult to directly obtain a sufficient condition for real roots root locus (3RL) stability because it is necessary to rewrite the inequality and the equation condition concerning the distinction type of the equation from the third equation for an arbitrary Ki > 0.

Corollary 1 Necessary conditions for the real roots root locus (3RL) stability of CING are 0 < α and Tn1  2Td 1 when β = ∞ in LEICO, as defined in Eq. (13). (proof) β =0∞ K Tn1s  1 1 CING  i 1 (Tn1 /  ) s  1 Td 1s  1 if

(18)

Ki (Tn1s  1) CING  1  CING (Td 1s  1){(Tn1 /  ) s  1}  Ki (Tn1s  1)

(19)

 ( s)  Td 1 (Tn1 /  )s 2  {Td 1  (Tn1 /  )  Tn1Ki }s Ki  1  0

(20)

D  {Td 1  (Tn1 /  )  Tn1Ki }  4Td 1 (Tn1 /  )( Ki  1)  0 2

 Tn12 Ki 2  [2Tn1{Td 1  (Tn1 /  )}  4Td 1 (Tn1 /  )]Ki  {Td 1  (Tn1 /  )}2  0 if   0, Tn1  2Td 1 then

D  0 for

Ki  0

(21) (22)

This also holds if there is no repeated root for Ki < 0.

D  Tn12 Ki 2  2{Tn1Td 1  (Tn1  2Td 1 )(Tn1 /  )}K i  {Td 1  (Tn1 /  )}2  0

(23)

{Tn1Td 1  (Tn1  2Td 1 )(Tn1 /  )} Ki 

 {Tn1Td 1  (Tn1  2Td 1 )(Tn1 /  )}2  Tn12{Td 1  (Tn1 /  )}2

if {Tn1Td 1  (Tn1  2Td 1 )(Tn1 /  )}2  Tn12{Td 1  (Tn1 /  )}2 thenKi  R ifTn1  Td 1thenKi  R

4

(24)

Tn12

(25)

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if   0, Tn1  2Td1thenKi  0, D  0

(26)

Union set of above three conditions (22) (25) and (26) is 0 < α and Tn1  2Td 1

Real roots root locus (3RL) stability

Remark 2: Setting β = 0 to cancel the pole at the origin is meaningless because the area of 3RL stability becomes very narrow.

Remark 3: The case of letting α = ∞ in order to delete the pole of LEICO by adding a PD compensator to C is omitted here because it is complex.

Remark 4: The case of letting α = 1 and β = 1 is shown in discussions and considerations later.

Remark 5: The hole appeared by easy analysis is not seen in the numerical calculations as shown in Discussions A when using MATLAB and SimulinkTM (MathWorks Co., Ltd.), since it was removed to facilitate analysis. It is left to our future work to determine a sufficient condition for real roots root locus (3RL) stability in integral control with LEICO for the lower half of the parameter plane of a stable minimum-phase first-order proper system.

Smith Compensator Figure 3 shows CINOGG with Smith’s local feedback.

Reference value +

+ -

C(s)

IN(s)

G1(s)

G2(s)

Controlled value

O(s) FIG. 3 SYSTEM CONFIGURATION OF CINOGG

The partial controlled object model of G2 can be removed from the characteristic equation, and the equivalent control loop will be the same, as follows.

O G1 (1  G2 )

(27)

CIN CINO  1  CING1 (1  G2 )

(28)

W ( s) 

CINOG1G2 CING1G2 CING1G2   1  CINOG1G2 1  CING1 (1  G2 )  CING1G2 1  CING1

 ( s)  1  CING1  0

(29) (30)

When G1 is a stable first-order proper system and G2 is a lag-time system, then the simulated step-up and step-down responses become lag-time shift responses.

When the system has a Smith compensator, there is no overshoot on the lag time, even if the loop gain and the lag time are both large, e.g., Ki = 10 and L=1 s. However, when the system does not have a Smith compensator, there is overshoot on the lag time, even if the loop gain is small and the lag time is very small, e.g., Ki = 1 and L = 0.1 s.

The stability analysis must consider the equivalent minimum loop without the lag-time transfer function G2; this will be shown Discussion and Considerations later.

There are many applications of lag time recently in measurement and communication using PSDs or Cameras among vehicular like presenting in the application section.

Pole Movement by Local Feedback Approach Because there is a zero in the proper controlled object G, therefore a pole cannot be arbitrarily assigned (Tn1 > Td1) by the proportional local output feedback (we call it to GO), then the zero is invalid and the pole moves as follows.

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G(s) 

Tn1s  1 Td 1s  1

Tn1  Td 1

when

(31)

R( s)  K pR G(s) 

Tn1s  1 K (Tn1s  1)  (Td 1  Tn1 K pR ) s  K pR  1 Td 1s  1

K

1

Td 1 

K pR  1

(Td 1  Tn1 K pR ) K pR  1

(32)

 Td 1 ~ Td 1  Tn1  Tn1

(33)

The pole will become double, although this is controllable by differentiation feedback.

G( s) 

Tn1s  1 Td 1s  1

when

Tn1  Td 1

(34)

R( s)  K dR s G ( s) 

Tn1s  1 K dRTn1s  (Td 1  K dR ) s  1

(35)

2

(36)

(Td 1  K dR )  (Td 1  K dR )  4 K dRTn1 2

sp 

2 K dRTn1



1 Tn1

(37)

(Td 1  K dR ) 2  4 K dRTn1  (Td 1  K dR ) Td 1  Tn1 , Td 1  K dR

if

then

(38)

K dR ( Td 1  Tn1 )  0

(39)

Td 1  Tn1 , Td 2  Tn1

(40)

K dR  0  Td 1  Tn1 , Td 2  Tn1

Remark 6: We note that CGO (where O means local feedback around G(s)) control cannot be used in this object, although it is useful for objects with strictly proper control, as discussed above.

The CINGO output local feedback control (see Fig. 4), which follows after the Siro LEICO, can move poles in order to allow gentle responses, and it can create a safe space on the real axis for LEICO to assign an additional zero near the origin, which allows rapid responses.

Reference value

+ + -

C(s)

IN(s)

+ -

G(s)

Controlled value

O(s)

FIG. 4 SYSTEM CONFIGURATION OF CINGO

Safety of Integral Control for A First-Order Proper System We present the following results for the safety of integral control of a first-order proper system.

1) If time constants of the denominator and numerator are constant in time, and the integration gain is positive, the root locus will not become unstable for any loop gain change.

2) The system has real roots root locus (3RL) stability and it will not overshoot for an arbitrary integration gain Ki. 3) There are two repeated roots, which result in repeated stability; there is critical damping with no overshoot and a rapid increase when Tn1 < Td1.

The system does not overshoot or become unstable near Tn1 < Td1 if the Siro LEICO is added with the integral controller.

Application System Configuration of NOR CINONOME We examined the dual agent convoying simulation in before study [12]. It is called also the leader and follower

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system [13], which is a virtual vehicle system with possibility of expansion to formation control. No overshoot property of the control systems mentioned above section may be effective for the convoying (leader and follower) simulation, especially, the following system configuration used the series CINOGG systems as shown in Figure 3 will be useful even if the physical model became more complex. We call it NOR (not or) CINONOME which means both Eyes of CINO in Japanese which named the upper view of Fig. 5.

Reference value

Controlled value

Central Control Room FIG. 5 SYSTEM CONFIGURATION OF NOR CINONOME

Here, the lag time is measurement or communication lag. And the role of nose is to give the reference value of distance from left of leader to right of follower. Then, the role of an additional big mouth is a central control room which may be realized in the larger vehicles for construction etc..

The position of follower which keeps distance to leader must have no overshoot property for avoiding collision.

Figure 6 shows an example of simple simulation result using a qualitative model for theoretical examination.

Both no overshoot property of the leader and the follower were confirmed in case of safe PTP (Point to Point) operation by repeated step response as it was easy to assure theoretically.

Blue broken line denotes the reference of a leader and blue solid line does the response of a leader in the left of Fig.6. Magenta broken line denotes the reference of a follower and magenta solid line does the response of a follower. Both do not knock against because they do not overshoot though they high-speed follow to the targeted value change.

4

4 r1 y1 r2 y2

3.5

3

3

Leader

2.5

r1,r2,y1.y2

r1,r2,y1,y2

2.5

2

follower

1.5

0.5

0.5

5

10

15 time(s)

20

Leader

1.5

1

0

follower

2

1

0

r1 y1 r2 y2

3.5

25

30

0

0

time(s)

5

10

15 time(s)

20

25

30

time(s)

FIG. 6 POINT-TO-POINT CONVOYING SIMULATION RESULTS (L = 0.1, TN1 = 2.2, TD1 = 2.5, Α = 5, Β = 0.5,KI1 = 10, RATE-LIMIT = 0.25)

Discussions and Considerations Necessary Conditions for Real Roots Root Locus (3RL) Stability Here, we will show the necessary conditions only for real roots root locus (3RL) stability for the case in which α = 1 and β = 1 omitted β = ∞, of which case was described in main text, because only partial range was shown as a necessary condition in both case.

Corollary 2 A necessary condition for real roots root locus (3RL) stability of CING is 0 < β < 0.853 when α = 1 in LEICO, as defined in Eq. (13). (proof)

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CING 

Ki (Td 1 /  ) s  1 1 s 1 Td 1s  1

(41)

 ( s)  Td 1s 2  {1  (Td 1 /  ) Ki }s  Ki  0

(42) (43)

D  (Td 1 /  ) K  2Td 1{(1/  )  2}Ki  1  0 2 i

0    0.5

if

D0

then

for

K i  0

(44)

This also holds if there is no repeated root for Ki < 0.

(Td 1 /  ) 2 Ki2  2Td 1{(1/  )  2}K i  1  0 Ki 

(45) (46)

Td 1{(1/  )  2}  2 Td21{(1/  )  1} (Td 1 /  ) 2

0    1; 0    0.5 & (1/  )  8(1/  )  8  0  0    0.146 2

0    0.5 & (1/  ) 2  8(1/  )  8  0  0.5    0.853

(47) (48) (49)

The union of the conditions (44), (48), and (49) is 0 < β < 0.853.

(50)

Corollary 3 A necessary condition for real roots root locus (3RL) stability of CING is 1 < α <1.17 or 2 < α when β = 1 in LEICO, as defined in Eq. (13). (proof)

if

 1

(51)

K Tn1s  1 1 CING  i s (Tn1 /  ) s  1 1

(52)

K i (Tn1s  1) CING  1  CING s{(Tn1 /  ) s  1}  K i (Tn1s  1)

 ( s )  (Tn1 /  ) s 2  (1  Tn1 K i ) s  K i  0 D  (1  Tn1 K i ) 2  4(Tn1 /  ) K i  0  Tn12 K i 2  2Tn1 (1  2 /  ) K i  1  0 if   2 then This also holds if there is no repeated root for Ki < 0.

D  0 for

K i  0

D  Tn21 K i2  2Tn1 (1  2 /  ) K i  1  0 (1  2 /  )  2 (1/  )(1  1/  ) Ki  0 Tn1

  1& (1  2 /  ) 2  4(1/  )(1  1/  ) 2 2 2 2 8A  8A 1  0  A  ,A 4 4 2

(53) (54) (55) (56)

(57)

1    1.17,   4.66

(58) (59) (60)

The union of conditions (56) and (60) is 1 < α <1.17, 2 < α

real roots root locus (3RL) stability.

Puncturing by all means like these results is regrettable analyzing the necessary condition though it seems not to be puncturing in the simulation of the following section C. Stability Analysis of CINONOME We must compare stability analysis for an equivalent minimum loop without lag time shown before and Pade approximation of lag time for original loop. Then, each loop stability analysis like Fig. 7 is important for the multi loop matter like NOR CINONOME of Figure 5.

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It was surprising that no overshoot property occurred not only by a real axis stable system (B), but also an equivalent approximated conversion lag time system (A) even if the loop gain increase larger. There will be some reasons. 根軌跡

250

根軌跡 0.4

200 0.3

150 0.2

100 0.1

虚軸

50 虚軸

0

0

-50 -0.1

-100 -0.2

-150 -0.3

-200 -250 -300

-200

-100

0

100

200

300

400

-0.4 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

実軸

Real実軸Axis

Real Axis

(A) PADE APPROXIMATION OF LAG TIME FOR ORIGINAL LOOP1

(B) MINIMUM LOOP1 WITHOUT LAG TIME

FIG. 7 ROOT LOCUS ANALYSIS (KI1 = 1000)

Experience of Pole-Zero Displacement Though the first order proper systems only are presented in main text, 7 examples of arrangement of poles and zeros which were confirmed SS by Matlab & SimulinkTM (Mathworks Co. Ltd) for second order proper controlled object with an integral controller and LEICO in Table 1 are shown in this section.

Here, × describes original pole, ○ describes original zero, ✖ describes additional pole and ◎describes additional zero by a lead compensator for integral (KILEICO). And the most left pole × means an origin pole by an integral, the others are poles or zeros ordered on real axes. The double ×× in the same box denotes a repeated pole and the double ○○ in the same box does a repeated zero. Lead 2 denotes the lead compensator with double poles and zeros.

The SS was confirmed by numerical approach for an example with the parameters (61).

Parameters

1 1 1 1 1 1 1  1,  0.6,  0.5,  0.45,  0.4,  0.3,  0.25 Td 1 Td 2 Td 3 Tn1 Td 4 Tn 2 Tn3

(61)

TABLE 1 ARRANGEMENT OF POLES AND ZEROS FOR SS ON SECOND ORDER PROPER CONTROLLED OBJECT WITH INTEGRAL IN A SINGLE LOOP

No. (1)

Arrangement of poles and zeros for SS compensator ×

(2)

×

(3) (4)

(5)

○

×

○

IN(s)

× Without compe. ------------

SS OK

○ ✖ ○

× ◎ × Lead 1

OK

✖ ○

×

○

× ◎ × Lead1

OK

×

×

○

○

OK

○ ✖ ○

×

× ◎ × Lead1

OK

(6)

✖ ○ ✖ ○

× ◎ × Lead2 × ◎

OK

(7)

× ×

○ ○

OK

× Without compe. ------------

× Without compe. ----------

Double lead is not necessary except for double lag type controlled object when the controlled object is double.

This clear results were obtained because the controlled object was only proper real poles and zeros system. If the controlled object is a strictly proper system, such simple results for SS could not be obtained. If non-dominant poles and bifurcation of root locus by LEICO and the original system in the most left plane can be neglected in the system, the dominant portion may be approximated to the results of above table by considering that all zeros are made by LEICO. It must be noticed that the distance relation of dominant pole in the origin and additional dominant zero by original or LEICO is more influential to robustness and responsibility of the step response than non-dominant poles and zeros.

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If the cascade or the included multi loop may be allowed also double integral (control and object etc.) system.

Conclusions After examining the conditions necessary to avoid overshooting by the integral control for a stable minimum-phase first-order proper controlled object, we proposed a system that avoids overshooting by the use of a general lead compensator, and that increases the phase margin to increase the stability and speed of a closed-loop system.

We determined the necessary and sufficient conditions for real roots root locus (3RL) stability in the simple robust lead inverse compensator that is used to recover from a fault in integral control.

The Smith compensator approach was applied to CINOGG by removing the lag time from the characteristic equation, and it was then applied to a convoy simulation.

The local feedback compensator approach was shown for a double first-order proper system; this was done to assign any eigenvalues in the real axis, and this allowed for movement of the pole in order to improve the response and create a safe space on the real axis.

An application for a leader and follower system was examined, and it was shown to have real roots root locus (3RL) stability and no problems with overshooting.

The results from the approach proposed in this paper are simpler, mathematically, than those from the conventional approach.

Therefore, we hope to extend the proposed approach to filter model matching control with integral and without integral in future.

ACKNOWLEDGMENT

The author would like to give special thanks to the Mr. Shikichi who have interesting in AI and deep learning.

REFERENCES

[1]

M. Katoh. Basic theory of systems control. Corona publisher, Tokyo, 2014 (in Japanese).

[2]

T. Hosokawa and M. Katoh, “Non-linearity and Vibration Isolation Property of Repilsion Type Magnetic Levitation System Using Two Permanent Magnet,” This paper presented at JSME Annual Conference 2008, 2008, 129-130. (in Japanese)

[3]

N. S. Nise, “Control systems engineering third edition,” Jon Willey and Sons, Inc. 2000.

[4]

Y. Chen, “Replacing a PID controller by a lag-lead compensator for a robot-a frequency response robot,” IEEE Trans. on robotics and automation, vol.5, No,2, 1989, 174-182.

[5]

F. Blanchini, A. Lepschy, S. Miani and U. Viaro, “Characterizing of PID and lead/lag compensators satisfying given H∞ specifications,” IEEE Trans. on Automatic Control, vol.49, No.5, 2004, 736-740

[6]

E.Meyer, Z. Zhang and Y-F Liu, “Controlled current source circuit (CCSC) for reduction of output voltage overshoot in buck converter,” IEEE PESC’08, 2008 , 815-820

[7]

L. Liu et al., “Double closed loop control and analysis for a static synchronous series compensator,” IEEE 36th PESC’05, 2005, 354-360

[8]

C.V. Hollot, Y. Fan, “Robust stabilizing of interval plants using lead or lag compensators,” 28th IEEE Conference on Decision and Control, vol.1, 1989, 43-45

[9]

M. Katoh, T. Eguchi, “A robust proper compensator design and an application for magnetic levitation,” Proceedings of SICE2013 , 2013, 666-670

[10] A. Bartoszewicz, “Robust control theory and applications,” INTECK, 2011 [11] “Informal social gathering concerning science and technology policy to contribute to construction of society of safety and peace of mind,” Report of Informal social gathering concerning science and technology policy to contribute to construction of society of safety and peace of mind, http://www.mext.go.jp/a_menu/kagaku/anzen/houkoku/04042302.htm , 2004 (in Japanese)

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[12] M. Katoh and N. Imura, “Development of a case-rule based reasoning engine for the double agent convoying simulation,” IEEE International Conference on SMC2009, 2009, 1148-1152 [13] Z. Yi, Z.Li, Y.Li and Q.Liu, “Multi-robot formation control using leader-follower for MANET,” IEEE International conference on Robotics and Biomimetics (ROBIO), 2009, 337-342 Makoto Katoh was born in Kobe, Japan, on 27/11/1953. He had earned the degree of Bachelor, Master and Doctor of Engineering at Osaka, Osaka-Fu, University of Osaka, Japan on 1976, 1978 and 1981. His major field of study is Control Engineeing, Intelligent System and Mechatronics.

He had joined to Osaka University as an ASSISTANT PROFESSOR from 1981 tp 1984, Mitsubishi Heavy Industries Co. Ltd. as an ENGINEER from 1984 to 1996, Toin Yokohama University as an ASSOCIATE PROFESSOR from 1996 to 2000, Osaka Institute of Technology as a PROFESSOR from 2000 to current time at Osaka. His major books were Basic theory of systems control (Tokyo, Tokyo-to, Corona publisher, 2014), Robust control theory and applications, INTECK publisher, 2011), Standard automatic control (Tokyo, Tokyo-to, Zikkyo publisher, 1999). His recent published English articles were “Trial of Discrete Values Control by a Tracked Model for Wing Sail Eco-ship” (JMMA, 2013), “Safe Dynamics and Control of a Rotor Vehcle on a Track”(JMMA, 2013), “A Plan for Fuzzy Passage Decision of Elavators using Equivalent Person Number”(IJACE, 2012). His current research interests is TBS with zero, Hard Servo with disturbance and Process Control with non-minimum phase system although previous research interests were General Control Theory and Applications, or Intelligent Systems.

Prof. Katoh is a member in IEEE, SICE and JSME. He had received Enhanced Award for Study from SICE on 1980, Works Award from Nagasaki Shipyard&Machinary Works in MHI on 1986, Head Quarter Award from MHI on 1992.

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