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Robust Stabilization of Oblique Wing Aircraft Model Using PID Controller
Proceedings of the 8th IFAC Symposium on Robust Control Design July 8-11, 2015. Bratislava, Republic Proceedings of the 8th IFACSlovak Symposium on Robust Control Design Proceedings of of the the 8th 8th IFAC IFAC Symposium Symposium on on Robust Robust Control Design Design Proceedings AvailableControl online at www.sciencedirect.com July 8-11, 2015. Bratislava, Slovak Republic July 8-11, 8-11, 2015. 2015. Bratislava, Bratislava, Slovak Slovak Republic Republic July
ScienceDirect 48-14 (2015) 265–270 Robust Stabilization ofIFAC-PapersOnLine Oblique Wing Aircraft Model Using PID Controller Robust Stabilization of Oblique Wing Aircraft Robust Stabilization of Oblique Wing Aircraft Model Model Using Using PID PID Controller Controller Radek Matušů*, Roman Prokop**
Radek Radek Matušů*, Matušů*, Roman Roman Prokop** Prokop** Centre for Security, Information and Advanced Technologies (CEBIA – Tech) Centre Security, Information and Technologies Faculty of Applied Informatics, Tomas Bata University(CEBIA in Zlín –– Tech) Centre for for Security, Information and Advanced Advanced Technologies (CEBIA Tech) Faculty of Applied Informatics, TomasZlín, BataCzech University in Zlín nám. T. G. Masaryka 5555, 76001 Republic Faculty of Applied Informatics, Tomas Bata University in Zlín nám. [email protected], G. Masaryka 5555, 76001 Zlín,[email protected] Czech Republic * e-mail: ** e-mail: nám. T. G. Masaryka 5555, 76001 Zlín, Czech Republic * e-mail: [email protected], ** e-mail: [email protected] * e-mail: [email protected], ** e-mail: [email protected] Abstract: The paper is focused on computation of all possible robustly stabilizing Proportional-IntegralAbstract: paper is computation of possible stabilizing Derivative The (PID) controllers for on oblique wing aircraft as interval system. Proportional-IntegralThe main idea of the Abstract: The paper is focused focused on computation of all all modelled possible robustly robustly stabilizing Proportional-IntegralDerivative (PID) controllers forTan’s oblique wing aircraft modelled asofinterval system. The mainPIidea the proposed method is based on technique for calculation (nominally) stabilizing andofPID Derivative (PID) controllers for oblique wing aircraft modelled as interval system. The main idea of the proposed is based on Tan’s techniquebyfor calculation of (nominally) stabilizinglocus PI and PID controllersmethod or robustly stabilizing PI controllers means of plotting the stability boundary in either proposed method is based on Tan’s techniquebyfor calculation of (nominally) stabilizinglocus PI andeither PID controllers or P-I-D robustly stabilizing PI controllers means of plotting the stability boundary P-I plane or space. Refinement of the existing method by consideration of 16 segmentin plants controllers or robustly stabilizing PI controllers by means of plotting the stability boundary locus in either P-I plane space.plants Refinement existing consideration of robustly 16 segment plants instead of or 16 P-I-D Kharitonov providesofanthe elegant andmethod efficientbytool for finding all stabilizing P-I plane or P-I-D space. Refinement of the existing method by consideration of 16 segment plants instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. PID controllers for an interval system. © 2015, IFAC (International Federation of Automatic Control) ElsevierSystems. Ltd. All rights reserved. Keywords: Robust Stabilization, Oblique Wing Aircraft, PIDHosting Control,byInterval Keywords: Robust Stabilization, Oblique Wing Aircraft, PID Control, Interval Systems. Keywords: Robust Stabilization, Oblique Wing Aircraft, PID Control, Interval Systems. combination with the sixteen plant theorem (Barmish, 1994), 1. INTRODUCTION combination the sixteen plant theorem (Barmish et with al., 1992). Nevertheless, this (Barmish, extension 1994), works 1. INTRODUCTION combination with the sixteen plant theorem (Barmish, 1994), 1. INTRODUCTION (Barmish et al., 1992). Nevertheless, this extension works only for PI but not for PID controllers. The Proportional-Integral-Derivative (PID) control (Barmish et al., 1992). Nevertheless, this extension works for PI but not for PID controllers. The Proportional-Integral-Derivative (PID) control only algorithms and their simplifications (P, I, PD and especially for PI PIDpaper controllers. The Proportional-Integral-Derivative (PID) control only The main but aimnot offor this is to present a method for algorithms andthe their simplifications I, PD and especially PI) comprise great majority of (P, contemporary industrial The main aim of this paper is to present a method for algorithms andthe their simplifications (P, I, PD and especially computation ofof all possible robustly The main aim this paper to presentstabilizing a method PID for PI) comprise great majority of contemporary industrial control applications. It has been reported that they represent computation of all possible is robustly stabilizing PID PI) comprise the great majority of contemporary industrial controllers for interval plants and to demonstrate its computation of all possible robustly stabilizing PID control applications. It has been reported that they represent over 95% of all practically applied controllers (Åström and for interval plants and to demonstrate its control applications. It has been reported that they represent controllers serviceability by robust stabilization of an oblique wing plants and to demonstrate its over 95% of1995), all practically applied controllers Hägglund, (O‘Dwyer, 2003). Thus, (Åström despite and the controllers forby interval robust stabilization over 95% of all practically applied controllers (Åström and serviceability aircraft model. More specifically, the of goalanis oblique to refinewing the Hägglund, 1995), (O‘Dwyer, 2003). Thus, despite the serviceability by robust stabilization of an oblique wing existence of many more sophisticated control design methods Hägglund, 1995), (O‘Dwyer, 2003). Thus, despite the aircraft model. More Tan’s specifically, goaland is Kaya, to refine the elegant and effective methodthe(Tan 2003), existence of more methods model. More Tan’s specifically, the(Tan goal is Kaya, to refine the and modern approaches, the effectivecontrol tuning design of PI and PID aircraft existence of many many more sophisticated sophisticated control design methods elegant and effective 2003), (Tan et al., 2006) by the ideasmethod from (Ho et and al., 2001), (Ho et and modern approaches, the effective tuning of PI and PID elegant and effective Tan’s method (Tan and Kaya, 2003), controllers is still very topical because it can bring significant and modern approaches, the effective tuning of PI and PID (Tan et al., and 2006)tobymake the ideas from (Ho for et al., 2001), (Ho of et al., 1998) it applicable computation controllers is very because it (Tan et al., 2006) by the ideas from (Ho et al., 2001), (Ho et saving on energy welltopical as expenses. the significant systematic al., controllers is still stillas very topical becauseEvidently, it can can bring bring significant 1998) and computation of robustly PIDit controllers.for al., 1998) stabilizing and to to make make it applicable applicable for Previously, computation the of saving onon energy as wellof asthe expenses. Evidently, the systematic research application PI(D) controllers under various robustly controllers. stabilizing PID Previously, the saving on energy as well as expenses. Evidently, the systematic computation of all (nominally) stabilizing PI or PID robustly stabilizing PID controllers. Previously, the research on application of the PI(D) controllers under various conditions of uncertainty contributes to this mosaic. of (nominally) stabilizing or research onofapplication the PI(D) controllers under various computation controllers, PI controllers computationrobustly of all allstabilizing (nominally) stabilizingandPI PIconsequent or PID PID conditions uncertaintyofcontributes to this mosaic. controllers, robustly stabilizing PI controllers and consequent conditions of uncertainty contributes to this mosaic. choice of the specific controller with desired performance on Obviously, the stability is the first and most critical controllers, robustly stabilizing PI controllers and consequent choice of the specific controller with desired performance on the basis of the desired model method (formerly known as Obviously, the stability is the first and most critical requirement of all control applications. However, the real-life choice of the specific controller with desired performance on Obviously, the stability is the first and most critical the basis of the desired model method (formerly known as dynamics inversion method) (Vítečková, 2000) is shown in requirement of all control applications. However, the real-life control circumstances differ from the ideal nominal and the basis of the desired model method (formerly known as requirement of all control applications. However, theones real-life dynamics inversion method)the(Vítečková, 2000) shown in (Matušů, 2011). Then, application of isKronecker control circumstances differ from the ideal nominal ones and so the uncertainty of the mathematical models has to be dynamics inversion method) (Vítečková, 2000) is shown in control circumstances differ from the ideal nominal ones and (Matušů, application of stabilization Kronecker summation 2011). method Then, (Fang etthe al., 2009) to robust so the be frequently taken intoof The models attentionhas of to many (Matušů, 2011). Then, the application of Kronecker so the the uncertainty uncertainty ofconsiderations. the mathematical mathematical models has to be summation method (Fang et al., 2009) to robust stabilization of a chemical reactor or robust of astabilization third order frequently The many researchers has into been considerations. focused on investigation summation method (Fang et al., stabilization 2009) to robust frequently taken taken into considerations. The attention attentionofof of robust many of aa chemical reactor or robust stabilization of aaetthird order electronic model is given in (Matušů al., 2011) researchers beenwith focused on investigation robust stability for has systems parametric uncertainty –ofsee e.g. nonlinear of chemical reactor or robust stabilization of third order researchers has beenwith focused on investigation ofsee robust nonlinear electronic model is given in (Matušů et al., 2011) or (Matušů et al., 2010a), respectively. The robust stability for systems parametric uncertainty – e.g. (Barmish, 1994), (Bhattacharyya et al., 1995), (Matušů and nonlinear electronic model is given in (Matušů et al., 2011) stability for1994), systems with parametric – see e.g. or The robust same2010a), nonlinearrespectively. electronic plant the (Barmish, (Bhattacharyya al.,uncertainty 1995), (Matušů Prokop, 2011), (Aguirre and Suárez,et2006). Typical problemand of stabilization or (Matušů (Matušů ofet etthe al., al., 2010a), respectively. Theusing robust (Barmish, 1994), (Bhattacharyya et al., 1995), (Matušů and stabilization of the same nonlinear electronic plant using the Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is Prokop, 2011), (Aguirre and Suárez, 2006). Typical problem of practical PI(D) controller design is to ensure, that the calculated stabilization of the same nonlinear electronic plant using the Prokop, 2011), (Aguirre and Suárez, 2006). Typical problem of Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is presented e.g. in (Matušů et al., 2010b). practical PI(D) controller design is to ensure, that the calculated controllerPI(D) will controller guaranteedesign stability only that for the onecalculated assumed Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is practical is tonot ensure, presented e.g. in (Matušů et al., 2010b). controller will stability not one assumed nominal system, but also the for whole of presented e.g. in (Matušů et al., 2010b). controllercontrolled will guarantee guarantee stability notforonly only for onefamily assumed 2. NOMINAL STABILIZATION nominal controlled system, but the family systems a model with for parametric uncertainty. nominal described controlled by system, but also also for the whole whole family of of 2. NOMINAL STABILIZATION systems described by a model with parametric uncertainty. Such closed-loop control system is called as “robustly stable” 2. NOMINAL STABILIZATION systems described by a model with parametric uncertainty. Such closed-loop is called as “robustly and the controller control itself is system then robustly stabilizing one.stable” 2.1 PI Control Such closed-loop control system is called as “robustly stable” and the controller itself is then robustly stabilizing one. 2.1 and controller itself is then stabilizing one. 2.1 PI PI Control Control An the array of techniques forrobustly calculation of (nominally) First, the fundamentals relating to computation of stability An array of techniques for calculation of (nominally) stabilizing PI and PID controllers have been already An array of techniques for calculation of (nominally) First, the fundamentals of regions for PI controllers relating are goingto be summarized. First, the fundamentals relating totocomputation computation of stability stability stabilizing PI and PID controllers have been already published, such as rules presented in (Söylemez et al., 2003), stabilizing PI and PID controllers have been already regions for PI controllers are going to be summarized. regions for PI controllers are going to be summarized. published, such as rules presented in (Söylemez et al., 2003), the Tan’s method in (Taninand Kaya, 2003), et Suppose the classical closed-loop control system with a published, such asdescribed rules presented (Söylemez et al.,(Tan 2003), the described in Kaya, (Tan al., 2006) method or the Kronecker from (Fang closed-loop system with controller C(s)classical and a controlled plantcontrol G(s). The controller the Tan’s Tan’s method described summation in (Tan (Tan and andmethod Kaya, 2003), 2003), (Tan et et Suppose Suppose the the classical closed-loop control system with isaa al., the summation method from (Fang et 2009).or methods have been also controllerinC(s) andform: a controlled plant G(s). The controller is the PI al., 2006) 2006) or Furthermore, the Kronecker Kronecker these summation method from (Fang et assumed controller C(s) and a controlled plant G(s). The controller is al., these methods also extended for Furthermore, robust stabilization intervalhave plantsbeen by their in the PI form: al., 2009). 2009). Furthermore, these of methods have been also assumed assumed in the PI form: extended for robust stabilization of interval plants by their extended for robust stabilization of interval plants by their
Copyright 265 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 ©©2015 2015,IFAC IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 265 Control. Peer review under responsibility of International Federation of Automatic Copyright © © 2015 2015 IFAC IFAC 265 Copyright 265 10.1016/j.ifacol.2015.09.468
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic 266
C (s) = kP +
Radek Matušů et al. / IFAC-PapersOnLine 48-14 (2015) 265–270
kI kP s + kI = s s
Im [G ( s ) ] = 0
(1)
The obtained intervals could be helpful for the proper frequency scaling.
The principal task is to determine its parameters kP, kI which guarantee stabilization of the controlled plant: G ( s) =
B( s) A( s )
2.2 PID Control
(2)
Now, the issue of feedback stabilization will be elaborated again, but for the case of ideal PID controller given by:
Several effective methods for computation of stabilizing PI controllers have been already published – e.g. (Söylemez et al., 2003), (Tan and Kaya, 2003), (Tan et al., 2006), (Fang et al., 2009). Here, the Tan’s method from (Tan and Kaya, 2003), (Tan et al., 2006) will be revisited and extended. This graphical approach is based on plotting the stability boundary locus. The substitution of s for jω in the plant transfer function (2) and subsequent decomposition of the numerator and denominator into their even and odd parts result in:
G ( jω ) =
BE (−ω 2 ) + jω BO (−ω 2 ) AE (−ω 2 ) + jω AO (−ω 2 )
C (s) = kP +
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k I (ω ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
kI k s + kI + kD s 2 + kD s = P s s
(7)
The principal idea for obtaining the relevant stability regions is to fix one controller parameter to a certain value and calculate the stability boundary locus using two remaining parameters analogous to the procedure presented in the previous subsection 2.1. The expression for the stability boundary locus in the ( k P , k I ) plane for a fixed value of kD leads to a bit modified
(3)
equations for proportional and integral gains:
Further, expressing the closed-loop characteristic polynomial and equating both real and imaginary parts to zero lead to the relations for the proportional and integral gains kP, kI: P (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) k P (ω ) = 5 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
(6)
k P (ω , k D ) =
P5 (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k I (ω , k D ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
(4)
(8)
where P1 (ω ) = −ω 2 BO (−ω 2 )
where
P2 (ω ) = BE (−ω 2 )
P1 (ω ) = −ω 2 BO (−ω 2 )
P3 (ω ) = ω BE (−ω 2 )
P2 (ω ) = BE (−ω 2 ) P3 (ω ) = ω BE (−ω 2 ) P4 (ω ) = ω BO (−ω ) 2
P4 (ω ) = ω BO (−ω 2 ) P5 (ω ) = ω 2 AO (−ω 2 ) + ω 2 BE (−ω 2 )k D
(5)
P6 (ω ) = −ω AE (−ω 2 ) + ω 3 BO (−ω 2 )k D
P5 (ω ) = ω AO (−ω ) 2
(9)
2
Note that the last two terms in (9) depend on derivative constant kD. From the viewpoint of practical computation, kD is considered to be chosen and corresponding set of boundary parameters kP, kI is consequently calculated while this process is repeated for several selected values of kD. Thus, the final stability regions are successively plotted through the “ ( k P , k I ) sections” in the ( k P , k I , k D ) space.
P6 (ω ) = −ω AE (−ω 2 )
Simultaneous calculations of the equations (4) for suitable range of ω and plotting the obtained values into the ( k P , k I ) plane determine the stability boundary locus. The obtained curve together with the line k I = 0 split the ( k P , k I ) plane into the stable and unstable regions. The decision if the respective region represents stabilizing or unstabilizing area can be done simply using a test point within each region. Nonetheless, the appropriate frequency gridding could represent a potential problem. Thus, the Nyquist plot based technique from (Söylemez et al., 2003) can be used for improvement of the method. In this improvement, the frequency ω can be separated into several intervals within which the stability or instability can not change. The borders of such intervals are defined by the real values of ω which fulfill the equation:
Alternatively, the stability boundary locus in the
( kP , kD )
plane for a fixed value of kI can be computed. This scenario would change the equations (8) and (9) to, respectively: k P (ω , k I ) =
P5 (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k D (ω , k I ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
where 266
(10)
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267
P1 (ω ) = −ω 2 BO (−ω 2 )
where i, j ∈ {1, 2,3, 4} ; and B1 ( s) to B4 ( s ) and A1 ( s ) to
P2 (ω ) = −ω BE (−ω )
A4 ( s ) are the Kharitonov polynomials for the numerator and denominator of the interval plant (13).
2
2
P3 (ω ) = ω BE (−ω ) 2
(11)
P4 (ω ) = −ω 3 BO (−ω 2 )
Recall that the construction of Kharitonov polynomials e.g. for the numerator interval polynomial:
P5 (ω ) = ω AO (−ω ) − BE (−ω )k I 2
2
2
P6 (ω ) = −ω AE (−ω 2 ) − ω BO (−ω 2 )k I
m
B ( s, b) = ∑ ⎡⎣bi− ; bi+ ⎤⎦ s i
Obviously, the final stability regions are given by the “ ( k P , k D ) sections” in the ( k P , k I , k D ) space.
is based on use of the lower and upper bounds of interval parameters in compliance with the rule (Kharitonov, 1978):
Nonetheless, the third option of obtaining the stability boundary, which consists in fixing kP and calculating the curves in ( k I , k D ) plane, is not so straightforward as the
B1 ( s ) = b0− + b1− s + b2+ s 2 + b3+ s 3 + " B2 ( s ) = b0+ + b1+ s + b2− s 2 + b3− s 3 + "
previous two alternatives, because for this case it holds true: P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω ) = 0
B3 ( s ) = b0+ + b1− s + b2− s 2 + b3+ s 3 + "
Consequently, the robust stabilization of an interval plant directly follows from the simultaneous stabilization of all 16 fixed Kharitonov plants. Hence, the final area of stability for original interval plant is given by the intersection of all 16 related partial areas obtained individually using the techniques from the sub-section 2.1.
kP can be acquired using the stability region in the ( k P , k I )
plane and ( k P , k D ) plane together as it has been presented in (Tan et al., 2006). In accordance with a linear programming based approach from (Ho et al., 1997), the stability region in the ( k I , k D ) plane under fixed kP is a convex polygon.
3.2 PID Control
3. ROBUST STABILIZATION
Unfortunately, the sixteen plant theorem is not applicable for robust stabilization of interval systems by PID controllers as it is not valid anymore (Pujara and Roy, 2001). However, the suitable method based on the generalized Kharitonov theorem and linear programming techniques has been presented e.g. in (Ho et al., 2001), (Ho et al., 1998). This paper adopts the idea of Kharitonov segments used in the generalized Kharitonov theorem (Chapellat and Bhattacharyya, 1989) and similar thirty-two edge theorem (Barmish, 1994), (Chapellat and Bhattacharyya, 1989) and combines it with the stability boundary locus technique (Tan and Kaya, 2003), (Tan et al., 2006).
3.1 PI Control So far, the previous part was focused only on nominal stabilization of controlled plants with fixed parameters. The following section will deal with robust stabilization. It means that the plant whose coefficients can vary within given intervals (interval system) is considered to represent a controlled object and the aim is to find all controllers which assure stabilization of all possible members of the interval system family. The works (Tan and Kaya, 2003), (Tan et al., 2006), (Fang et al., 2009) have improved a feedback stabilization technique using PI controllers also for interval plants simply by using its combination with the sixteen plant theorem (Barmish, 1994), (Barmish et al., 1992), (Ho et al., 1998). The sixteen plant theorem itself says that a first order controller robustly stabilizes an interval plant:
G ( s, b, a ) =
B ( s, b) = A( s, a )
∑ ⎡⎣b i =0
− i
n −1
i =0
Consider an interval plant: m
B ( s, b) = G ( s, b, a ) = A( s, a )
i =0 n
− i
, bi+ ⎤⎦ s i
∑ ⎡⎣ai− , ai+ ⎤⎦ si
; m
(17)
, b ⎤⎦ s
i
− i
+ i
i
; m
and a PID controller (7).
(13)
The family of interval systems (17) is stabilized by a fixed PID controller if and only if each of sixteen segment plants related to the interval family is stabilized by the same PID controller (Ho et al., 2001).
where bi− , bi+ , ai− , ai+ represent lower and upper bounds for parameters of numerator and denominator, if and only if it stabilizes its 16 Kharitonov plants, defined as: Gi , j ( s ) =
∑ ⎡⎣b i =0
+ i
s + ∑ ⎡⎣ a , a ⎤⎦ s n
(16)
B4 ( s ) = b0− + b1+ s + b2+ s 2 + b3− s 3 + "
(12)
However, the stability region in the ( k I , k D ) plane for a fixed
m
(15)
i =0
Bi ( s ) Aj ( s)
The mentioned 16 segment plants are defined as: Gi , j ( s, λ ) =
(14)
267
BSi ( s, λ ) Aj ( s )
(18)
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where i, j ∈ {1, 2,3, 4} ; A1 ( s ) to A4 ( s ) are the Kharitonov
The stability boundary locus for the segment plant (22) is given by intersection of the stability areas for several sampled values of λ ∈ 0,1 . The Fig. 1 shows 11 curves for the range
polynomials for the denominator of the interval plant (17); and BS1 ( s, λ ) to BS 4 ( s, λ ) are four Kharitonov segments (Chapellat and Bhattacharyya, 1989), (Ho et al., 2001), (Barmish, 1994) which can be written as:
λ = 0 : 0.1:1 intersection.
BS 1 ( s, λ ) = [ B1 ( s ), B3 ( s ) ] = (1 − λ ) B1 ( s ) + λ B3 ( s )
BS 2 ( s, λ ) = [ B1 ( s ), B4 ( s ) ] = (1 − λ ) B1 ( s ) + λ B4 ( s )
BS 3 ( s, λ ) = [ B2 ( s ), B3 ( s ) ] = (1 − λ ) B2 ( s ) + λ B3 ( s )
80 70
(19)
60
BS 4 ( s, λ ) = [ B2 ( s ), B4 ( s ) ] = (1 − λ ) B2 ( s ) + λ B4 ( s )
50 I
and B1 ( s ) to B4 ( s ) are the Kharitonov
k
where λ ∈ 0,1
polynomials for the numerator of the interval plant (17).
( kP , kI )
10 0 0
plane gives the stability
3
4
k
5
6
7
8
9
The same process can be analogous repeated for the remaining 15 segment plants and then the intersection of all 16 partial intersections would lead to the stability boundary locus for the original interval family (20) under assumption of k D = 1 . The Fig. 2 presents the curves for all 16 segment plants and sampled λ = 0 : 0.1:1 in a single plot. The zoomed and highlighted intersection representing the robust stability region for closed loop containing PID controller with k D = 1 and original interval system (20) is depicted in Fig. 3.
Now, the theoretical results outlined in the previous parts are going to be practically utilized to robust stabilization of uncertain mathematical model adopted from (Barmish, 1994), (Dorf, 1974):
80 70 60
[54, 74] s + [90, 166] (20) s 4 + [ 2.8, 4.6] s 3 + [50.4, 80.8] s 2 + [30.1, 33.9 ] s + [ −0.1, 0.1]
50 k
I
which describes the experimental oblique wing aircraft.
40 30
Initially, the derivative constant is chosen and fixed as k D = 1 . The first of the segment plants (18) is constructed using:
20 10
(21)
0 -1
0
1
2
3
4
k
5
6
7
8
9
P
where B1 ( s ) , B3 ( s ) and A1 ( s ) are relevant Kharitonov
Fig. 2. Stability regions for PID controller ( k D = 1 ) and all 16 segment plants
polynomials and λ ∈ 0,1 . More specifically: G1,1 ( s, λ ) =
2
Fig. 1. Stability region for PID controller ( k D = 1 ) and the first segment plant (22)
4. ROBUST STABILIZATION OF OBLIQUE WING AIRCRAFT
BS 1 ( s, λ ) (1 − λ ) B1 ( s ) + λ B3 ( s ) = A1 ( s ) A1 ( s )
1
P
can be plotted in one figure and intersection can be found at a time. Anyway, the whole process should be repeated for the other selected values of kD and the very final robust stability region can be visualized by the simultaneous plotting of the “ ( k P , k I ) sections” into one graph in ( k P , k I , k D ) space.
G1,1 ( s, λ ) =
Stability Region
20
boundary locus for this specific segment plant. The calculations are repeated for all the remaining segment plants and the robust stability region for the original interval plant and chosen value of kD is determined by intersection of areas for all 16 segment plants. From the practical viewpoint, the curves for all sampled λ ∈ 0,1 and all 16 segment plants
G(s) =
40 30
The computation of robustly stabilizing PID controllers can be performed as follows: First, a certain value of controller parameter kD is chosen and fixed. Then, the stability boundary for one of segment plants (18) is calculated for several sampled values of λ ∈ 0,1 using the equations (8), (9). The intersection of the obtained areas in
and the highlighted area represents the
BS1 ( s, λ ) (1 − λ )( 54s + 90 ) + λ ( 54 s + 166 ) (22) = 4 A1 ( s ) s + 4.6s 3 + 80.8s 2 + 30.1s − 0.1 268
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Moreover, the red curve represents the output variable for the system with average values of uncertain parameters from (20):
10 9 8
GA ( s ) =
7 6
64 s + 128 s + 3.7 s 3 + 65.6 s 2 + 32s
(24)
4
As can be seen, the control loop is really robustly stable.
I
5 k
269
4 3
3
Stability Region
2 1
2.5
0 0
0.2
0.4
0.6
0.8
1
k
1.2
1.4
1.6
2
1.8
Output Signals
-1 -0.2
P
Fig. 3. Robust stability region for PID controller with k D = 1 and interval system (20)
1.5
1
0.5
The whole previous process was repeated for the other values of derivative constant kD, more specifically for k D = 0 : 0.5 : 5 . The very final robust stability region
0 0
visualized by means of corresponding eleven “ ( k P , k I )
20
40
60 Time
80
100
120
Fig. 5. Output signals of “representative” plants for robustly stabilizing controller (23)
sections” in ( k P , k I , k D ) space is shown in Fig. 4.
The intentional choice of a controller outside the robust stability region from Fig. 4 obviously leads to robustly unstable closed control loop. The representative of such robustly non-stabilizing controller is e.g.:
5 4 3 k
D
C2 ( s ) = k P +
2
10 5 k
0 I
0
1
0.5
1.5
2
25 k
P
20
Fig. 4. Final robust stability region for PID controller and interval system (20)
15
Output Signals
10
The example of robustly stabilizing PID controller, which obviously lies inside the robust stability region from Fig. 4, can be chosen as: C1 ( s ) = k P +
(25)
Then, the corresponding set of control responses obtained under the same conditions as in the previous case are depicted in Fig. 6.
1 0 15
kI 0.5 + kD s = 2 + + 0.5s s s
kI 0.5 + kD s = 1 + + 0.5s s s
5 0 -5 -10
(23)
-15 -20 0
The Fig. 5 shows the control responses of the loop with the controller C1 (23) and 729 “representative” systems from the interval family (20). Each interval parameter has been divided into 2 subintervals and thus these 3 values and 6 parameters result in 36 = 729 systems for simulation.
5
10 Time
15
20
Fig. 6. Output signals of “representative” plants for robustly non-stabilizing controller (25)
269
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Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (1997). A linear programming characterization of all stabilizing PID controllers, In: Proceedings of the American Control Conference, Albuquerque, New Mexico, USA. Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (1998). Design of P, PI and PID Controllers for Interval Plants, In: Proceedings of the American Control Conference, Philadelphia, Pennsylvania, USA. Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (2001). Robust and non-fragile PID controller design. International Journal of Robust and Nonlinear Control, Vol. 11, No. 7, pp. 681-708. Kharitonov, V.L. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differentsial'nye Uravneniya, Vol. 14, pp. 2086-2088. Matušů, R. (2011). Calculation of all stabilizing PI and PID controllers. International Journal of Mathematics and Computers in Simulation, Vol. 5, No. 3, pp. 224-231. Matušů, R. and Prokop, R. (2011). Graphical analysis of robust stability for systems with parametric uncertainty: an overview. Transactions of the Institute of Measurement and Control, Vol. 33, No. 2, pp. 274-290. Matušů, R., Prokop, R., Matejičková, K. and Bakošová, M. (2010a). Robust Stabilization of Interval Plants using Kronecker Summation Method. WSEAS Transactions on Systems, Vol. 9, No. 9, pp. 917-926. Matušů, R., Vaneková, K., Prokop, R. and Bakošová, M. (2010b). Design of Robust PI Controllers and their Application to a Nonlinear Electronic System. Journal of Electrical Engineering, Vol. 61, No. 1, pp. 44-51. Matušů, R., Závacká, J., Prokop, R. and Bakošová, M. (2011). The Kronecker Summation Method for Robust Stabilization Applied to a Chemical Reactor. Journal of Control Science and Engineering, Vol. 2011, 7 p. O‘Dwyer, A. (2003). Handbook of PI and PID Controller Tuning Rules, Imperial College Press, London, UK. Pujara, L.R. and Roy, A. (2001). On Computing Stabilizing Controllers for SISO Interval Plants, In: Proceedings of the American Control Conference, Arlington, Virginia, USA. Söylemez, M.T., Munro, N. and Baki, H. (2003). Fast calculation of stabilizing PID controllers. Automatica, Vol. 39, No. 1, pp. 121–126. Tan, N. and Kaya, I. (2003). Computation of stabilizing PI controllers for interval systems, In: Proceedings of the 11th Mediterranean Conference on Control and Automation, Rhodes, Greece. Tan, N., Kaya, I., Yeroglu, C., and Atherton, D.P. (2006). Computation of stabilizing PI and PID controllers using the stability boundary locus. Energy Conversion and Management, Vol. 47, No. 18-19, pp. 3045-3058. Vítečková, M. (2000). Tuning of controllers by dynamics inversion method (Seřízení regulátorů metodou inverze dynamiky), VSB – Technical University of Ostrava, Ostrava, Czech Republic, (In Czech).
5. CONCLUSIONS The key aim of the contribution has been to present the improved method for computation of stabilizing controllers with conventional structure on the basis of plotting the stability boundary locus in either P-I plane or P-I-D space. Now, thanks to the combination of the original method with stabilization of so-called segment plants, the refined technique can be conveniently used for determination of all possible robustly stabilizing PID controllers for interval plants. In illustrative example, the model of an experimental oblique wing aircraft is considered as a controlled object. The final robust stability region has been computed and visualized and selected representatives from stable or intentionally unstable areas have been chosen and used for supporting control simulations. ACKNOWLEDGMENTS The work was performed with financial support of research project NPU I No. MSMT-7778/2014 by the Ministry of Education of the Czech Republic and also by the European Regional Development Fund under the Project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089. This assistance is very gratefully acknowledged. REFERENCES Aguirre, B. and Suárez, R. (2006). Algebraic test for the Hurwitz stability of a given segment of polynomials. Boletín de la Sociedad Matemática Mexicana, Vol. 12, No. 3, pp. 261-275. Åström, K.J. and Hägglund, T. (1995). PID Controllers: Theory, Design and Tuning, 2nd ed., Instrument Society of America, Research Triangle Park, North Carolina, USA. Barmish, B.R. (1994). New Tools for Robustness of Linear Systems. Macmillan. New York, USA. Barmish, B.R., Hollot, C.V., Kraus, F.J. and Tempo, R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators. IEEE Transactions on Automatic Control, Vol. 37, No. 6, pp. 707-714. Bhattacharyya, S.P., Chapellat, H. and Keel, L.H. (1995). Robust control: The parametric approach, Prentice Hall, Englewood Cliffs, New Jersey, USA. Chapellat, H. and Bhattacharyya, S.P. (1989). A generalization of Kharitonov’s theorem: Robust stability of interval plants. IEEE Transactions on Automatic Control, Vol. 34, No. 3, pp. 306-311. Dorf, R.C. (1974). Modern Control Systems, AddisonWesley Publishing, Reading, Massachusetts, USA. Fang, J., Zheng, D. and Ren, Z. (2009). Computation of stabilizing PI and PID controllers by using Kronecker summation method. Energy Conversion and Management, Vol. 50, No. 7, pp. 1821-1827.
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ScienceDirect 48-14 (2015) 265–270 Robust Stabilization ofIFAC-PapersOnLine Oblique Wing Aircraft Model Using PID Controller Robust Stabilization of Oblique Wing Aircraft Robust Stabilization of Oblique Wing Aircraft Model Model Using Using PID PID Controller Controller Radek Matušů*, Roman Prokop**
Radek Radek Matušů*, Matušů*, Roman Roman Prokop** Prokop** Centre for Security, Information and Advanced Technologies (CEBIA – Tech) Centre Security, Information and Technologies Faculty of Applied Informatics, Tomas Bata University(CEBIA in Zlín –– Tech) Centre for for Security, Information and Advanced Advanced Technologies (CEBIA Tech) Faculty of Applied Informatics, TomasZlín, BataCzech University in Zlín nám. T. G. Masaryka 5555, 76001 Republic Faculty of Applied Informatics, Tomas Bata University in Zlín nám. [email protected], G. Masaryka 5555, 76001 Zlín,[email protected] Czech Republic * e-mail: ** e-mail: nám. T. G. Masaryka 5555, 76001 Zlín, Czech Republic * e-mail: [email protected], ** e-mail: [email protected] * e-mail: [email protected], ** e-mail: [email protected] Abstract: The paper is focused on computation of all possible robustly stabilizing Proportional-IntegralAbstract: paper is computation of possible stabilizing Derivative The (PID) controllers for on oblique wing aircraft as interval system. Proportional-IntegralThe main idea of the Abstract: The paper is focused focused on computation of all all modelled possible robustly robustly stabilizing Proportional-IntegralDerivative (PID) controllers forTan’s oblique wing aircraft modelled asofinterval system. The mainPIidea the proposed method is based on technique for calculation (nominally) stabilizing andofPID Derivative (PID) controllers for oblique wing aircraft modelled as interval system. The main idea of the proposed is based on Tan’s techniquebyfor calculation of (nominally) stabilizinglocus PI and PID controllersmethod or robustly stabilizing PI controllers means of plotting the stability boundary in either proposed method is based on Tan’s techniquebyfor calculation of (nominally) stabilizinglocus PI andeither PID controllers or P-I-D robustly stabilizing PI controllers means of plotting the stability boundary P-I plane or space. Refinement of the existing method by consideration of 16 segmentin plants controllers or robustly stabilizing PI controllers by means of plotting the stability boundary locus in either P-I plane space.plants Refinement existing consideration of robustly 16 segment plants instead of or 16 P-I-D Kharitonov providesofanthe elegant andmethod efficientbytool for finding all stabilizing P-I plane or P-I-D space. Refinement of the existing method by consideration of 16 segment plants instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. PID controllers for an interval system. © 2015, IFAC (International Federation of Automatic Control) ElsevierSystems. Ltd. All rights reserved. Keywords: Robust Stabilization, Oblique Wing Aircraft, PIDHosting Control,byInterval Keywords: Robust Stabilization, Oblique Wing Aircraft, PID Control, Interval Systems. Keywords: Robust Stabilization, Oblique Wing Aircraft, PID Control, Interval Systems. combination with the sixteen plant theorem (Barmish, 1994), 1. INTRODUCTION combination the sixteen plant theorem (Barmish et with al., 1992). Nevertheless, this (Barmish, extension 1994), works 1. INTRODUCTION combination with the sixteen plant theorem (Barmish, 1994), 1. INTRODUCTION (Barmish et al., 1992). Nevertheless, this extension works only for PI but not for PID controllers. The Proportional-Integral-Derivative (PID) control (Barmish et al., 1992). Nevertheless, this extension works for PI but not for PID controllers. The Proportional-Integral-Derivative (PID) control only algorithms and their simplifications (P, I, PD and especially for PI PIDpaper controllers. The Proportional-Integral-Derivative (PID) control only The main but aimnot offor this is to present a method for algorithms andthe their simplifications I, PD and especially PI) comprise great majority of (P, contemporary industrial The main aim of this paper is to present a method for algorithms andthe their simplifications (P, I, PD and especially computation ofof all possible robustly The main aim this paper to presentstabilizing a method PID for PI) comprise great majority of contemporary industrial control applications. It has been reported that they represent computation of all possible is robustly stabilizing PID PI) comprise the great majority of contemporary industrial controllers for interval plants and to demonstrate its computation of all possible robustly stabilizing PID control applications. It has been reported that they represent over 95% of all practically applied controllers (Åström and for interval plants and to demonstrate its control applications. It has been reported that they represent controllers serviceability by robust stabilization of an oblique wing plants and to demonstrate its over 95% of1995), all practically applied controllers Hägglund, (O‘Dwyer, 2003). Thus, (Åström despite and the controllers forby interval robust stabilization over 95% of all practically applied controllers (Åström and serviceability aircraft model. More specifically, the of goalanis oblique to refinewing the Hägglund, 1995), (O‘Dwyer, 2003). Thus, despite the serviceability by robust stabilization of an oblique wing existence of many more sophisticated control design methods Hägglund, 1995), (O‘Dwyer, 2003). Thus, despite the aircraft model. More Tan’s specifically, goaland is Kaya, to refine the elegant and effective methodthe(Tan 2003), existence of more methods model. More Tan’s specifically, the(Tan goal is Kaya, to refine the and modern approaches, the effectivecontrol tuning design of PI and PID aircraft existence of many many more sophisticated sophisticated control design methods elegant and effective 2003), (Tan et al., 2006) by the ideasmethod from (Ho et and al., 2001), (Ho et and modern approaches, the effective tuning of PI and PID elegant and effective Tan’s method (Tan and Kaya, 2003), controllers is still very topical because it can bring significant and modern approaches, the effective tuning of PI and PID (Tan et al., and 2006)tobymake the ideas from (Ho for et al., 2001), (Ho of et al., 1998) it applicable computation controllers is very because it (Tan et al., 2006) by the ideas from (Ho et al., 2001), (Ho et saving on energy welltopical as expenses. the significant systematic al., controllers is still stillas very topical becauseEvidently, it can can bring bring significant 1998) and computation of robustly PIDit controllers.for al., 1998) stabilizing and to to make make it applicable applicable for Previously, computation the of saving onon energy as wellof asthe expenses. Evidently, the systematic research application PI(D) controllers under various robustly controllers. stabilizing PID Previously, the saving on energy as well as expenses. Evidently, the systematic computation of all (nominally) stabilizing PI or PID robustly stabilizing PID controllers. Previously, the research on application of the PI(D) controllers under various conditions of uncertainty contributes to this mosaic. of (nominally) stabilizing or research onofapplication the PI(D) controllers under various computation controllers, PI controllers computationrobustly of all allstabilizing (nominally) stabilizingandPI PIconsequent or PID PID conditions uncertaintyofcontributes to this mosaic. controllers, robustly stabilizing PI controllers and consequent conditions of uncertainty contributes to this mosaic. choice of the specific controller with desired performance on Obviously, the stability is the first and most critical controllers, robustly stabilizing PI controllers and consequent choice of the specific controller with desired performance on the basis of the desired model method (formerly known as Obviously, the stability is the first and most critical requirement of all control applications. However, the real-life choice of the specific controller with desired performance on Obviously, the stability is the first and most critical the basis of the desired model method (formerly known as dynamics inversion method) (Vítečková, 2000) is shown in requirement of all control applications. However, the real-life control circumstances differ from the ideal nominal and the basis of the desired model method (formerly known as requirement of all control applications. However, theones real-life dynamics inversion method)the(Vítečková, 2000) shown in (Matušů, 2011). Then, application of isKronecker control circumstances differ from the ideal nominal ones and so the uncertainty of the mathematical models has to be dynamics inversion method) (Vítečková, 2000) is shown in control circumstances differ from the ideal nominal ones and (Matušů, application of stabilization Kronecker summation 2011). method Then, (Fang etthe al., 2009) to robust so the be frequently taken intoof The models attentionhas of to many (Matušů, 2011). Then, the application of Kronecker so the the uncertainty uncertainty ofconsiderations. the mathematical mathematical models has to be summation method (Fang et al., 2009) to robust stabilization of a chemical reactor or robust of astabilization third order frequently The many researchers has into been considerations. focused on investigation summation method (Fang et al., stabilization 2009) to robust frequently taken taken into considerations. The attention attentionofof of robust many of aa chemical reactor or robust stabilization of aaetthird order electronic model is given in (Matušů al., 2011) researchers beenwith focused on investigation robust stability for has systems parametric uncertainty –ofsee e.g. nonlinear of chemical reactor or robust stabilization of third order researchers has beenwith focused on investigation ofsee robust nonlinear electronic model is given in (Matušů et al., 2011) or (Matušů et al., 2010a), respectively. The robust stability for systems parametric uncertainty – e.g. (Barmish, 1994), (Bhattacharyya et al., 1995), (Matušů and nonlinear electronic model is given in (Matušů et al., 2011) stability for1994), systems with parametric – see e.g. or The robust same2010a), nonlinearrespectively. electronic plant the (Barmish, (Bhattacharyya al.,uncertainty 1995), (Matušů Prokop, 2011), (Aguirre and Suárez,et2006). Typical problemand of stabilization or (Matušů (Matušů ofet etthe al., al., 2010a), respectively. Theusing robust (Barmish, 1994), (Bhattacharyya et al., 1995), (Matušů and stabilization of the same nonlinear electronic plant using the Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is Prokop, 2011), (Aguirre and Suárez, 2006). Typical problem of practical PI(D) controller design is to ensure, that the calculated stabilization of the same nonlinear electronic plant using the Prokop, 2011), (Aguirre and Suárez, 2006). Typical problem of Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is presented e.g. in (Matušů et al., 2010b). practical PI(D) controller design is to ensure, that the calculated controllerPI(D) will controller guaranteedesign stability only that for the onecalculated assumed Tan’s method (Tan and Kaya, 2003), (Tan et al., 2006) is practical is tonot ensure, presented e.g. in (Matušů et al., 2010b). controller will stability not one assumed nominal system, but also the for whole of presented e.g. in (Matušů et al., 2010b). controllercontrolled will guarantee guarantee stability notforonly only for onefamily assumed 2. NOMINAL STABILIZATION nominal controlled system, but the family systems a model with for parametric uncertainty. nominal described controlled by system, but also also for the whole whole family of of 2. NOMINAL STABILIZATION systems described by a model with parametric uncertainty. Such closed-loop control system is called as “robustly stable” 2. NOMINAL STABILIZATION systems described by a model with parametric uncertainty. Such closed-loop is called as “robustly and the controller control itself is system then robustly stabilizing one.stable” 2.1 PI Control Such closed-loop control system is called as “robustly stable” and the controller itself is then robustly stabilizing one. 2.1 and controller itself is then stabilizing one. 2.1 PI PI Control Control An the array of techniques forrobustly calculation of (nominally) First, the fundamentals relating to computation of stability An array of techniques for calculation of (nominally) stabilizing PI and PID controllers have been already An array of techniques for calculation of (nominally) First, the fundamentals of regions for PI controllers relating are goingto be summarized. First, the fundamentals relating totocomputation computation of stability stability stabilizing PI and PID controllers have been already published, such as rules presented in (Söylemez et al., 2003), stabilizing PI and PID controllers have been already regions for PI controllers are going to be summarized. regions for PI controllers are going to be summarized. published, such as rules presented in (Söylemez et al., 2003), the Tan’s method in (Taninand Kaya, 2003), et Suppose the classical closed-loop control system with a published, such asdescribed rules presented (Söylemez et al.,(Tan 2003), the described in Kaya, (Tan al., 2006) method or the Kronecker from (Fang closed-loop system with controller C(s)classical and a controlled plantcontrol G(s). The controller the Tan’s Tan’s method described summation in (Tan (Tan and andmethod Kaya, 2003), 2003), (Tan et et Suppose Suppose the the classical closed-loop control system with isaa al., the summation method from (Fang et 2009).or methods have been also controllerinC(s) andform: a controlled plant G(s). The controller is the PI al., 2006) 2006) or Furthermore, the Kronecker Kronecker these summation method from (Fang et assumed controller C(s) and a controlled plant G(s). The controller is al., these methods also extended for Furthermore, robust stabilization intervalhave plantsbeen by their in the PI form: al., 2009). 2009). Furthermore, these of methods have been also assumed assumed in the PI form: extended for robust stabilization of interval plants by their extended for robust stabilization of interval plants by their
Copyright 265 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 ©©2015 2015,IFAC IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 265 Control. Peer review under responsibility of International Federation of Automatic Copyright © © 2015 2015 IFAC IFAC 265 Copyright 265 10.1016/j.ifacol.2015.09.468
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic 266
C (s) = kP +
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kI kP s + kI = s s
Im [G ( s ) ] = 0
(1)
The obtained intervals could be helpful for the proper frequency scaling.
The principal task is to determine its parameters kP, kI which guarantee stabilization of the controlled plant: G ( s) =
B( s) A( s )
2.2 PID Control
(2)
Now, the issue of feedback stabilization will be elaborated again, but for the case of ideal PID controller given by:
Several effective methods for computation of stabilizing PI controllers have been already published – e.g. (Söylemez et al., 2003), (Tan and Kaya, 2003), (Tan et al., 2006), (Fang et al., 2009). Here, the Tan’s method from (Tan and Kaya, 2003), (Tan et al., 2006) will be revisited and extended. This graphical approach is based on plotting the stability boundary locus. The substitution of s for jω in the plant transfer function (2) and subsequent decomposition of the numerator and denominator into their even and odd parts result in:
G ( jω ) =
BE (−ω 2 ) + jω BO (−ω 2 ) AE (−ω 2 ) + jω AO (−ω 2 )
C (s) = kP +
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k I (ω ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
kI k s + kI + kD s 2 + kD s = P s s
(7)
The principal idea for obtaining the relevant stability regions is to fix one controller parameter to a certain value and calculate the stability boundary locus using two remaining parameters analogous to the procedure presented in the previous subsection 2.1. The expression for the stability boundary locus in the ( k P , k I ) plane for a fixed value of kD leads to a bit modified
(3)
equations for proportional and integral gains:
Further, expressing the closed-loop characteristic polynomial and equating both real and imaginary parts to zero lead to the relations for the proportional and integral gains kP, kI: P (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) k P (ω ) = 5 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
(6)
k P (ω , k D ) =
P5 (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k I (ω , k D ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
(4)
(8)
where P1 (ω ) = −ω 2 BO (−ω 2 )
where
P2 (ω ) = BE (−ω 2 )
P1 (ω ) = −ω 2 BO (−ω 2 )
P3 (ω ) = ω BE (−ω 2 )
P2 (ω ) = BE (−ω 2 ) P3 (ω ) = ω BE (−ω 2 ) P4 (ω ) = ω BO (−ω ) 2
P4 (ω ) = ω BO (−ω 2 ) P5 (ω ) = ω 2 AO (−ω 2 ) + ω 2 BE (−ω 2 )k D
(5)
P6 (ω ) = −ω AE (−ω 2 ) + ω 3 BO (−ω 2 )k D
P5 (ω ) = ω AO (−ω ) 2
(9)
2
Note that the last two terms in (9) depend on derivative constant kD. From the viewpoint of practical computation, kD is considered to be chosen and corresponding set of boundary parameters kP, kI is consequently calculated while this process is repeated for several selected values of kD. Thus, the final stability regions are successively plotted through the “ ( k P , k I ) sections” in the ( k P , k I , k D ) space.
P6 (ω ) = −ω AE (−ω 2 )
Simultaneous calculations of the equations (4) for suitable range of ω and plotting the obtained values into the ( k P , k I ) plane determine the stability boundary locus. The obtained curve together with the line k I = 0 split the ( k P , k I ) plane into the stable and unstable regions. The decision if the respective region represents stabilizing or unstabilizing area can be done simply using a test point within each region. Nonetheless, the appropriate frequency gridding could represent a potential problem. Thus, the Nyquist plot based technique from (Söylemez et al., 2003) can be used for improvement of the method. In this improvement, the frequency ω can be separated into several intervals within which the stability or instability can not change. The borders of such intervals are defined by the real values of ω which fulfill the equation:
Alternatively, the stability boundary locus in the
( kP , kD )
plane for a fixed value of kI can be computed. This scenario would change the equations (8) and (9) to, respectively: k P (ω , k I ) =
P5 (ω ) P4 (ω ) − P6 (ω ) P2 (ω ) P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
P (ω ) P1 (ω ) − P5 (ω ) P3 (ω ) k D (ω , k I ) = 6 P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω )
where 266
(10)
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267
P1 (ω ) = −ω 2 BO (−ω 2 )
where i, j ∈ {1, 2,3, 4} ; and B1 ( s) to B4 ( s ) and A1 ( s ) to
P2 (ω ) = −ω BE (−ω )
A4 ( s ) are the Kharitonov polynomials for the numerator and denominator of the interval plant (13).
2
2
P3 (ω ) = ω BE (−ω ) 2
(11)
P4 (ω ) = −ω 3 BO (−ω 2 )
Recall that the construction of Kharitonov polynomials e.g. for the numerator interval polynomial:
P5 (ω ) = ω AO (−ω ) − BE (−ω )k I 2
2
2
P6 (ω ) = −ω AE (−ω 2 ) − ω BO (−ω 2 )k I
m
B ( s, b) = ∑ ⎡⎣bi− ; bi+ ⎤⎦ s i
Obviously, the final stability regions are given by the “ ( k P , k D ) sections” in the ( k P , k I , k D ) space.
is based on use of the lower and upper bounds of interval parameters in compliance with the rule (Kharitonov, 1978):
Nonetheless, the third option of obtaining the stability boundary, which consists in fixing kP and calculating the curves in ( k I , k D ) plane, is not so straightforward as the
B1 ( s ) = b0− + b1− s + b2+ s 2 + b3+ s 3 + " B2 ( s ) = b0+ + b1+ s + b2− s 2 + b3− s 3 + "
previous two alternatives, because for this case it holds true: P1 (ω ) P4 (ω ) − P2 (ω ) P3 (ω ) = 0
B3 ( s ) = b0+ + b1− s + b2− s 2 + b3+ s 3 + "
Consequently, the robust stabilization of an interval plant directly follows from the simultaneous stabilization of all 16 fixed Kharitonov plants. Hence, the final area of stability for original interval plant is given by the intersection of all 16 related partial areas obtained individually using the techniques from the sub-section 2.1.
kP can be acquired using the stability region in the ( k P , k I )
plane and ( k P , k D ) plane together as it has been presented in (Tan et al., 2006). In accordance with a linear programming based approach from (Ho et al., 1997), the stability region in the ( k I , k D ) plane under fixed kP is a convex polygon.
3.2 PID Control
3. ROBUST STABILIZATION
Unfortunately, the sixteen plant theorem is not applicable for robust stabilization of interval systems by PID controllers as it is not valid anymore (Pujara and Roy, 2001). However, the suitable method based on the generalized Kharitonov theorem and linear programming techniques has been presented e.g. in (Ho et al., 2001), (Ho et al., 1998). This paper adopts the idea of Kharitonov segments used in the generalized Kharitonov theorem (Chapellat and Bhattacharyya, 1989) and similar thirty-two edge theorem (Barmish, 1994), (Chapellat and Bhattacharyya, 1989) and combines it with the stability boundary locus technique (Tan and Kaya, 2003), (Tan et al., 2006).
3.1 PI Control So far, the previous part was focused only on nominal stabilization of controlled plants with fixed parameters. The following section will deal with robust stabilization. It means that the plant whose coefficients can vary within given intervals (interval system) is considered to represent a controlled object and the aim is to find all controllers which assure stabilization of all possible members of the interval system family. The works (Tan and Kaya, 2003), (Tan et al., 2006), (Fang et al., 2009) have improved a feedback stabilization technique using PI controllers also for interval plants simply by using its combination with the sixteen plant theorem (Barmish, 1994), (Barmish et al., 1992), (Ho et al., 1998). The sixteen plant theorem itself says that a first order controller robustly stabilizes an interval plant:
G ( s, b, a ) =
B ( s, b) = A( s, a )
∑ ⎡⎣b i =0
− i
n −1
i =0
Consider an interval plant: m
B ( s, b) = G ( s, b, a ) = A( s, a )
i =0 n
− i
, bi+ ⎤⎦ s i
∑ ⎡⎣ai− , ai+ ⎤⎦ si
; m
(17)
, b ⎤⎦ s
i
− i
+ i
i
; m
and a PID controller (7).
(13)
The family of interval systems (17) is stabilized by a fixed PID controller if and only if each of sixteen segment plants related to the interval family is stabilized by the same PID controller (Ho et al., 2001).
where bi− , bi+ , ai− , ai+ represent lower and upper bounds for parameters of numerator and denominator, if and only if it stabilizes its 16 Kharitonov plants, defined as: Gi , j ( s ) =
∑ ⎡⎣b i =0
+ i
s + ∑ ⎡⎣ a , a ⎤⎦ s n
(16)
B4 ( s ) = b0− + b1+ s + b2+ s 2 + b3− s 3 + "
(12)
However, the stability region in the ( k I , k D ) plane for a fixed
m
(15)
i =0
Bi ( s ) Aj ( s)
The mentioned 16 segment plants are defined as: Gi , j ( s, λ ) =
(14)
267
BSi ( s, λ ) Aj ( s )
(18)
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where i, j ∈ {1, 2,3, 4} ; A1 ( s ) to A4 ( s ) are the Kharitonov
The stability boundary locus for the segment plant (22) is given by intersection of the stability areas for several sampled values of λ ∈ 0,1 . The Fig. 1 shows 11 curves for the range
polynomials for the denominator of the interval plant (17); and BS1 ( s, λ ) to BS 4 ( s, λ ) are four Kharitonov segments (Chapellat and Bhattacharyya, 1989), (Ho et al., 2001), (Barmish, 1994) which can be written as:
λ = 0 : 0.1:1 intersection.
BS 1 ( s, λ ) = [ B1 ( s ), B3 ( s ) ] = (1 − λ ) B1 ( s ) + λ B3 ( s )
BS 2 ( s, λ ) = [ B1 ( s ), B4 ( s ) ] = (1 − λ ) B1 ( s ) + λ B4 ( s )
BS 3 ( s, λ ) = [ B2 ( s ), B3 ( s ) ] = (1 − λ ) B2 ( s ) + λ B3 ( s )
80 70
(19)
60
BS 4 ( s, λ ) = [ B2 ( s ), B4 ( s ) ] = (1 − λ ) B2 ( s ) + λ B4 ( s )
50 I
and B1 ( s ) to B4 ( s ) are the Kharitonov
k
where λ ∈ 0,1
polynomials for the numerator of the interval plant (17).
( kP , kI )
10 0 0
plane gives the stability
3
4
k
5
6
7
8
9
The same process can be analogous repeated for the remaining 15 segment plants and then the intersection of all 16 partial intersections would lead to the stability boundary locus for the original interval family (20) under assumption of k D = 1 . The Fig. 2 presents the curves for all 16 segment plants and sampled λ = 0 : 0.1:1 in a single plot. The zoomed and highlighted intersection representing the robust stability region for closed loop containing PID controller with k D = 1 and original interval system (20) is depicted in Fig. 3.
Now, the theoretical results outlined in the previous parts are going to be practically utilized to robust stabilization of uncertain mathematical model adopted from (Barmish, 1994), (Dorf, 1974):
80 70 60
[54, 74] s + [90, 166] (20) s 4 + [ 2.8, 4.6] s 3 + [50.4, 80.8] s 2 + [30.1, 33.9 ] s + [ −0.1, 0.1]
50 k
I
which describes the experimental oblique wing aircraft.
40 30
Initially, the derivative constant is chosen and fixed as k D = 1 . The first of the segment plants (18) is constructed using:
20 10
(21)
0 -1
0
1
2
3
4
k
5
6
7
8
9
P
where B1 ( s ) , B3 ( s ) and A1 ( s ) are relevant Kharitonov
Fig. 2. Stability regions for PID controller ( k D = 1 ) and all 16 segment plants
polynomials and λ ∈ 0,1 . More specifically: G1,1 ( s, λ ) =
2
Fig. 1. Stability region for PID controller ( k D = 1 ) and the first segment plant (22)
4. ROBUST STABILIZATION OF OBLIQUE WING AIRCRAFT
BS 1 ( s, λ ) (1 − λ ) B1 ( s ) + λ B3 ( s ) = A1 ( s ) A1 ( s )
1
P
can be plotted in one figure and intersection can be found at a time. Anyway, the whole process should be repeated for the other selected values of kD and the very final robust stability region can be visualized by the simultaneous plotting of the “ ( k P , k I ) sections” into one graph in ( k P , k I , k D ) space.
G1,1 ( s, λ ) =
Stability Region
20
boundary locus for this specific segment plant. The calculations are repeated for all the remaining segment plants and the robust stability region for the original interval plant and chosen value of kD is determined by intersection of areas for all 16 segment plants. From the practical viewpoint, the curves for all sampled λ ∈ 0,1 and all 16 segment plants
G(s) =
40 30
The computation of robustly stabilizing PID controllers can be performed as follows: First, a certain value of controller parameter kD is chosen and fixed. Then, the stability boundary for one of segment plants (18) is calculated for several sampled values of λ ∈ 0,1 using the equations (8), (9). The intersection of the obtained areas in
and the highlighted area represents the
BS1 ( s, λ ) (1 − λ )( 54s + 90 ) + λ ( 54 s + 166 ) (22) = 4 A1 ( s ) s + 4.6s 3 + 80.8s 2 + 30.1s − 0.1 268
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Moreover, the red curve represents the output variable for the system with average values of uncertain parameters from (20):
10 9 8
GA ( s ) =
7 6
64 s + 128 s + 3.7 s 3 + 65.6 s 2 + 32s
(24)
4
As can be seen, the control loop is really robustly stable.
I
5 k
269
4 3
3
Stability Region
2 1
2.5
0 0
0.2
0.4
0.6
0.8
1
k
1.2
1.4
1.6
2
1.8
Output Signals
-1 -0.2
P
Fig. 3. Robust stability region for PID controller with k D = 1 and interval system (20)
1.5
1
0.5
The whole previous process was repeated for the other values of derivative constant kD, more specifically for k D = 0 : 0.5 : 5 . The very final robust stability region
0 0
visualized by means of corresponding eleven “ ( k P , k I )
20
40
60 Time
80
100
120
Fig. 5. Output signals of “representative” plants for robustly stabilizing controller (23)
sections” in ( k P , k I , k D ) space is shown in Fig. 4.
The intentional choice of a controller outside the robust stability region from Fig. 4 obviously leads to robustly unstable closed control loop. The representative of such robustly non-stabilizing controller is e.g.:
5 4 3 k
D
C2 ( s ) = k P +
2
10 5 k
0 I
0
1
0.5
1.5
2
25 k
P
20
Fig. 4. Final robust stability region for PID controller and interval system (20)
15
Output Signals
10
The example of robustly stabilizing PID controller, which obviously lies inside the robust stability region from Fig. 4, can be chosen as: C1 ( s ) = k P +
(25)
Then, the corresponding set of control responses obtained under the same conditions as in the previous case are depicted in Fig. 6.
1 0 15
kI 0.5 + kD s = 2 + + 0.5s s s
kI 0.5 + kD s = 1 + + 0.5s s s
5 0 -5 -10
(23)
-15 -20 0
The Fig. 5 shows the control responses of the loop with the controller C1 (23) and 729 “representative” systems from the interval family (20). Each interval parameter has been divided into 2 subintervals and thus these 3 values and 6 parameters result in 36 = 729 systems for simulation.
5
10 Time
15
20
Fig. 6. Output signals of “representative” plants for robustly non-stabilizing controller (25)
269
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Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (1997). A linear programming characterization of all stabilizing PID controllers, In: Proceedings of the American Control Conference, Albuquerque, New Mexico, USA. Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (1998). Design of P, PI and PID Controllers for Interval Plants, In: Proceedings of the American Control Conference, Philadelphia, Pennsylvania, USA. Ho, M.-T., Datta, A. and Bhattacharyya, S.P. (2001). Robust and non-fragile PID controller design. International Journal of Robust and Nonlinear Control, Vol. 11, No. 7, pp. 681-708. Kharitonov, V.L. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differentsial'nye Uravneniya, Vol. 14, pp. 2086-2088. Matušů, R. (2011). Calculation of all stabilizing PI and PID controllers. International Journal of Mathematics and Computers in Simulation, Vol. 5, No. 3, pp. 224-231. Matušů, R. and Prokop, R. (2011). Graphical analysis of robust stability for systems with parametric uncertainty: an overview. Transactions of the Institute of Measurement and Control, Vol. 33, No. 2, pp. 274-290. Matušů, R., Prokop, R., Matejičková, K. and Bakošová, M. (2010a). Robust Stabilization of Interval Plants using Kronecker Summation Method. WSEAS Transactions on Systems, Vol. 9, No. 9, pp. 917-926. Matušů, R., Vaneková, K., Prokop, R. and Bakošová, M. (2010b). Design of Robust PI Controllers and their Application to a Nonlinear Electronic System. Journal of Electrical Engineering, Vol. 61, No. 1, pp. 44-51. Matušů, R., Závacká, J., Prokop, R. and Bakošová, M. (2011). The Kronecker Summation Method for Robust Stabilization Applied to a Chemical Reactor. Journal of Control Science and Engineering, Vol. 2011, 7 p. O‘Dwyer, A. (2003). Handbook of PI and PID Controller Tuning Rules, Imperial College Press, London, UK. Pujara, L.R. and Roy, A. (2001). On Computing Stabilizing Controllers for SISO Interval Plants, In: Proceedings of the American Control Conference, Arlington, Virginia, USA. Söylemez, M.T., Munro, N. and Baki, H. (2003). Fast calculation of stabilizing PID controllers. Automatica, Vol. 39, No. 1, pp. 121–126. Tan, N. and Kaya, I. (2003). Computation of stabilizing PI controllers for interval systems, In: Proceedings of the 11th Mediterranean Conference on Control and Automation, Rhodes, Greece. Tan, N., Kaya, I., Yeroglu, C., and Atherton, D.P. (2006). Computation of stabilizing PI and PID controllers using the stability boundary locus. Energy Conversion and Management, Vol. 47, No. 18-19, pp. 3045-3058. Vítečková, M. (2000). Tuning of controllers by dynamics inversion method (Seřízení regulátorů metodou inverze dynamiky), VSB – Technical University of Ostrava, Ostrava, Czech Republic, (In Czech).
5. CONCLUSIONS The key aim of the contribution has been to present the improved method for computation of stabilizing controllers with conventional structure on the basis of plotting the stability boundary locus in either P-I plane or P-I-D space. Now, thanks to the combination of the original method with stabilization of so-called segment plants, the refined technique can be conveniently used for determination of all possible robustly stabilizing PID controllers for interval plants. In illustrative example, the model of an experimental oblique wing aircraft is considered as a controlled object. The final robust stability region has been computed and visualized and selected representatives from stable or intentionally unstable areas have been chosen and used for supporting control simulations. ACKNOWLEDGMENTS The work was performed with financial support of research project NPU I No. MSMT-7778/2014 by the Ministry of Education of the Czech Republic and also by the European Regional Development Fund under the Project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089. This assistance is very gratefully acknowledged. REFERENCES Aguirre, B. and Suárez, R. (2006). Algebraic test for the Hurwitz stability of a given segment of polynomials. Boletín de la Sociedad Matemática Mexicana, Vol. 12, No. 3, pp. 261-275. Åström, K.J. and Hägglund, T. (1995). PID Controllers: Theory, Design and Tuning, 2nd ed., Instrument Society of America, Research Triangle Park, North Carolina, USA. Barmish, B.R. (1994). New Tools for Robustness of Linear Systems. Macmillan. New York, USA. Barmish, B.R., Hollot, C.V., Kraus, F.J. and Tempo, R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators. IEEE Transactions on Automatic Control, Vol. 37, No. 6, pp. 707-714. Bhattacharyya, S.P., Chapellat, H. and Keel, L.H. (1995). Robust control: The parametric approach, Prentice Hall, Englewood Cliffs, New Jersey, USA. Chapellat, H. and Bhattacharyya, S.P. (1989). A generalization of Kharitonov’s theorem: Robust stability of interval plants. IEEE Transactions on Automatic Control, Vol. 34, No. 3, pp. 306-311. Dorf, R.C. (1974). Modern Control Systems, AddisonWesley Publishing, Reading, Massachusetts, USA. Fang, J., Zheng, D. and Ren, Z. (2009). Computation of stabilizing PI and PID controllers by using Kronecker summation method. Energy Conversion and Management, Vol. 50, No. 7, pp. 1821-1827.
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